Something terrible just happened. My wife, Elise Cawley, was recovering from heart surgery and had just attended virtually a celebration for one of our children when she had a freak, vast cardiovascular event—and died instantly. We had been together for 36 years. The picture above was taken just hours before she died. In all the […]
“My Program Did the Wrong Thing!” Bugs are a ubiquitous phenomenon in the software world. And—essentially by definition—each one of them is somehow unique and unexpected. But—particularly given their ubiquity—one might wonder whether there could perhaps be some kind of general “scientific” theory that could be developed about them. My goal here is to explore […]
June 23, 1988 is when we launched Version 1.0 of Mathematica. Today—almost 38 years later—we’re launching Version 15 of what—in recognition of how far it’s expanded beyond “math”—we now call Wolfram Language. It’s an impressive release, with a lot of new core functionality. It might perhaps seem surprising that after 38 years there’d still be more to add. But it’s like the typical arc of intellectual history: the more one’s figured out, the further one can see, and the more one becomes able to do. And for all of us working on it, it’s been a very satisfying process: year after year building an ever taller tower of ideas and technology, with which we can reach ever further—today to all the functionality of Version 15.
The Basic Setup Whether one’s dealing with biology, economics, politics or a host of other fields, it’s common to encounter situations that can be modeled as involving two agents that repeatedly compete with each other. One imagines that at each step each agent can take one of a certain set of actions, and that then—in […]
Foundation Models Need a Foundation Tool LLMs don’t—and can’t—do everything. What they do is very impressive—and useful. It’s broad. And in many ways it’s human-like. But it’s not precise. And in the end it’s not about deep computation. So how can we supplement LLM foundation models? We need a foundation tool: a tool that’s broad […]
The Wolfram Institute recently received a grant from the Templeton World Charity Foundation for “Computational Metaphysics”. I wrote this piece in part as a launching point for discussions with experts in traditional philosophy. Moving Metaphysics from Philosophy to Science “What ultimately is there?” has always been seen as a fundamental—if thorny—question for philosophy, or perhaps […]
Empirical Theoretical Computer Science “Could there be a faster program for that?” It’s a fundamental type of question in theoretical computer science. But except in special cases, such a question has proved fiendishly difficult to answer. And, for example, in half a century, almost no progress has been made even on the rather coarse (though […]
Ruliology is taking off! And more and more people are talking about it. But what is ruliology? Since I invented the term, I decided I should write something to explain it. But then I realized: I actually already wrote something back in 2021 when I first invented the term. What I wrote back then was […]
To immediately enable Wolfram Compute Services in Version 14.3 Wolfram Desktop systems, run RemoteBatchSubmissionEnvironment["WolframBatch"]. (The functionality is automatically available in the Wolfram Cloud.) Scaling Up Your Computations Let’s say you’ve done a computation in Wolfram Language. And now you want to scale it up. Maybe 1000x or more. Well, today we’ve released an extremely streamlined […]
Towards a Theory of Bulk Orchestration It’s a key feature of living systems, perhaps even in some ways the key feature: that even right down to a molecular scale, things are orchestrated. Molecules (or at least large ones) don’t just move around randomly, like in a liquid or a gel. Instead, what molecular biology has […]
Click any diagram to get Wolfram Language code to reproduce it. What Are Lambdas? It’s a story of pure, abstract computation. In fact, historically, one of the very first. But even though it’s something I for one have used in practice for nearly half a century, it’s not something that in all my years of […]
Theories of the World Several often arrive in a single day. Sometimes they’re marked “urgent”. Sometimes they’re long. Sometimes they’re short. Sometimes they’re humble. Sometimes they’re conspiratorial. And sometimes, these days, they’re written “in collaboration with” an AI. But there’s a common theme: they’re all emails that present some kind of fundamental theory invented by […]
June 1, 1984—forty years ago today—is when it would be fair to say I made my all-time favorite science discovery. Like with basically all significant science discoveries (despite the way histories often present them) it didn’t happen without several long years of buildup. But June 1, 1984, was when I finally had my “aha” moment—even though in retrospect the discovery had actually been hiding in plain sight for more than two years.
My diary from 1984 has a cryptic note that shows what happened on June 1, 1984:
There’s a part that says “BA 9 pm → LDN”, recording the fact that at 9pm that day I took a (British Airways) flight to London (from New York; I lived in Princeton at that time). “Sent vega monitor → SUN” indicates that I had sent the broken display of a computer I called “vega” to Sun Microsystems. But what’s important for our purposes here is the little “side” note:
Take C10 pict.
R30
R110
What did that mean? C10, R30 and R110 were my shorthand designations for particular, very simple programs of types I’d been studying: “code 10”, “rule 30” and “rule 110”. And my note reminded me that I wanted to take pictures of those programs with me that evening, making them on the laser printer I’d just got (laser printers were rare and expensive devices at the time).
I’d actually made (and even published) pictures of all these programs before, but at least for rule 30 and rule 110 those pictures were very low resolution:
But on June 1, 1984, my picture was much better:
For several years I’d been studying the question of “where complexity comes from”, for example in nature. I’d realized there was something very computational about it (and that had even led me to the concept of computational irreducibility—a term I coined just a few days before June 1, 1984). But somehow I had imagined that “true complexity” must come from something already complex or at least random. Yet here in this picture, plain as anything, complexity was just being “created”, basically from nothing. And all it took was following a very simple rule, starting from a single black cell.
Our usual intuition that to make something complex required “complex effort” was, I realized, simply wrong. In the computational universe one needed a new intuition. And the picture of rule 30 I generated that day was what finally made me understand that. Still, although I hadn’t internalized it before, several years of work had prepared me for this. And just days later I was at a conference already talking confidently about the implications of what I’d seen in rule 30.
Over the years that followed, rule 30 became basically the face of the phenomenon I had discovered. By 1985 I devoted a whole paper to it, in A New Kind of Science it was my initial and quintessential example, for the past quarter century a picture of rule 30 has adorned my personal business cards, and in 2019 we launched the Rule 30 Prizes to promote the rich basic science of rule 30:
But what about “C10”—the first item in my cryptic note? What was that? And what became of it?
Well, C10 was “code 10”, or, more fully, “k = 2, r = 2 totalistic code 10 cellular automaton”. (I used the term “code” as a way to indicate a totalistic, rather than general, “rule”.) And, actually, I had looked at code 10 several times before, never really paying much attention to it.
The first explicit mention I find in my archives is from February 1983 (apparently reporting on something I’d done in January of that year). I had been doing all sorts of computer experiments on cellular automata, recording the results in a lab notebook. One page has observations about what I then called “summational rules” (I would soon rename these “totalistic”). And there’s code 10:
Mostly I had been studying the behavior starting from random initial conditions, but for code 10 I noted: “very irregular, even from simple initial state”. Within a couple of months I had even made (on an electrostatic printer) a high-resolution picture of code 10 starting from a single black cell—and here it is, prepared for publication, Scotch tape and all:
It appeared in a paper I wrote in May 1983. But the paper (entitled “Universality and Complexity in Cellular Automata”) was mostly about other things (for example, introducing my four general classes of cellular automaton behavior and talking quite a lot about code 20 as an example of class 4 rule), and it contained only a passing comment about code 10:
Code 10 is a range-2 rule, which means that the patterns it generates can grow by 2 cells on each side at each step. And the result is that the patterns quickly get quite wide, so that if one cuts them off when they “hit the edge of the page” (as my early programs “conveniently” tended to do) they don’t go very far, and one doesn’t get to see much of code 10’s behavior.
And it was this piece of “ergonomics” that caused me to basically ignore code 10—and not to recognize the “rule 30 phenomenon” until I happened to produce that high-resolution image of rule 30 on June 1, 1984.
I didn’t entirely forget code 10, for example mentioning it in a note to “Why These Discoveries Were Not Made Before” in my 2002 book A New Kind of Science:
But now that forty years have passed since I made—and basically ignored—that “C10” picture, I thought it would be nice to go back and see what I missed, and to use our modern Wolfram Language tools to spend a few hours checking out the story of code 10.
It’s an exercise in what I now call “ruliology”—the basic science of studying what simple rules do. And whenever one does ruliology there are certain standard things one can look at—that I showed many examples of in A New Kind of Science. But in a quintessential reflection of computational irreducibility there are also always “surprises”, and special phenomena one did not expect. And so it is with code 10.
Note: Click any diagram to get Wolfram Language code to reproduce it.
Code 10 is a cellular automaton operating on a line of black and white cells, at each step adding up the values of the 5 cells up to distance 2 from any given cell (black is 1, white is 0). If the total is 1 or 3, the cell is black on the next step; otherwise it’s white (in base 2 the number 10 is 001010):
And, yes, in many ways this rule is even simpler to describe—at least in words—than rule 30. And if one thinks of it in terms of Boolean expressions, it can also be written in a very simple form:
(By the way, as a general k = 2, r = 2 rule, code 10 is rule 376007062.)
So what does code 10 do? Here are a few steps of its evolution starting from a single black cell:
And here are 2000 steps:
And, yes, even though it’s a simple rule, its behavior looks highly complex, and in many ways quite random. One immediate observation is that—unlike rule 30—code 10 is symmetric, so the pattern it generates is left-right symmetric. The center column isn’t interesting: after having black cells for 2 steps, it’s white thereafter. (And by substituting values yx0xy into the Boolean expression above, it’s easy to prove this.)
Filling the white region around the center column with red we get:
There doesn’t seem to be any long-range regularity to the way the width of this region changes:
And indeed the (even) widths seem at least close to exponentially distributed:
What if one goes one column to the left or right of the center? Here’s the beginning of the sequence one gets:
And, yes, every other cell is white. Picking only “even-numbered positions” we get:
Looking at the accumulated mean for 100,000 steps suggests that this sequence isn’t “uniformly random”, and that slightly fewer than 50% of the cells end up being black:
Going away from the center line, every other column has white cells every two steps. Sampling the pattern only at “odd positions” in both “space and time” we get a pattern that looks similar—though not identical—to our original one:
Looking at every cell, the overall density of the pattern seems to approach about 0.361. Looking only at “odd positions” the overall density seems to be about 0.49. And, yes, the fact that it doesn’t seem to become exactly 1/2 is one of those typical “not-quite-as-expected” things that one routinely finds in doing ruliology.
There are some aspects of the code 10 pattern, though, that inevitably work in particular ways. For example, if we “rotate” the pattern so that its boundary is vertical, we can see that close to the boundary the pattern is periodic:
The period progressively doubles at depths separated by 1, 1, 4, 6, 8, 14, 124, …—yielding what may perhaps ultimately be a logarithmic growth of period with depth:
We’ve looked at what happens with an initial condition consisting of a single black cell. But what about other initial conditions? Here are a few examples:
We might have thought that the “strength of randomness” would be large enough that we’d get patterns that look basically the same in all cases. But so what’s going on in the 
case? Running twice and five times as long reveals it’s actually nothing special; there just happen to be a few large triangles near the top:
So will nothing else notable happen with larger initial conditions?
What about 
? Let’s run that a little longer:
And OMG! It’s not random and unpredictable at all. It’s a nested pattern!
Even in the midst of all that randomness and computational irreducibility, here is a dash of computational reducibility—and a reminder that there are always pockets of reducibility to be found in any computationally irreducible system, though there’s no guarantee how difficult they will be to find in any given case.
The particular nested pattern we get here is a bit like the one from the additive elementary rule 150, that simply computes the total mod 2 of the three cells in each neighborhood:
And it turns out to be almost exactly the r = 2 analog of this—the additive rule (code 42) that takes the total mod 2 of the five cells in the r = 2 neighborhood:
The limiting fractal dimension of this pattern is:
Is 
unique, or does this same phenomenon happen with other “seeds”? It turns out to happen again for:
So what’s going on here? Comparing the detailed pattern in the code 10 
case with the additive rule 
case, there’s no immediate obvious correspondence:
But if we look at the rules for code 10 and code 42 respectively:
We notice that there’s really only one difference. In code 10, 
gives 
while in code 42, it gives 
. In other words, if code 10 avoids ever generating any 
block, it will inevitably behave just like code 42—and shows nested before. And that’s what happens for the initial conditions above; they can for example lead to 
blocks, but never 
.
Another notable and at first unexpected phenomenon concerns the overall density of black cells in patterns from different initial conditions:
And what we find is that for even-length initial blocks the density is about 0.47, while for odd ones it’s about 0.36. At first it might seem very strange that something as global as overall density could be affected by the initial conditions. But once again, it’s a story of what blocks can occur: in the odd-length case, there’s a checkerboard of guaranteed-white cells, which just doesn’t exist in the even-length case.
We’ve been looking at what code 10 does with specific, simple initial conditions. What about with random initial conditions? Well, it’s not terribly exciting. It basically just looks random all the way through—which, by the way, is part of the reason I didn’t pay much attention to code 10 back in 1983:
But even though this looks quite random, it’s for example not the case that every single possible block of values can occur. Though it’s very close. Let’s say we start from all possible sequences of 0s and 1s in the initial conditions. Then—using methods I developed in 1984 based on finite automata—it’s possible to determine that even after 1 step there are some blocks of values that can’t occur. But it turns out that one has to go all the way to blocks of length 36 before one finds an example:
Although the patterns generated by code 10 generally look quite random, if we look closely we can see at least patches that are fairly regular. The most obvious examples are white triangles. But there are other examples, most notably associated with regions consisting of repetitions of blocks with periodic behavior:
Complementary to this is the question of what code 10 does in regions of limited size—say with cyclic boundary conditions, starting from a single black cell. The result is quite different for regions of different sizes:
For a region of size n, a symmetric rule like code 10 must repeat with a period of at most 2n/2. Here are the actual repetition periods as a function of size, shown on a log plot:
These are results specifically for the single-cell initial condition. We can also generate state transition diagrams for all 2n possible states in a size n code 10 system:
And mostly what we see is highly contractive behavior, with many different initial states evolving to the same final state—even though “eventually” we should start seeing larger cycles of the kind we picked up above when we looked at evolution from a single-cell initial condition.
And, yes, I could go on, for example repeating analyses I’ve done in the past for rule 30. A lot of what we’d see would be at least qualitatively much the same as for rule 30—and essentially the result of the appearance of computational irreducibility in both cases. But it’s a feature of the computational universe—and indeed one of the many consequences of computational irreducibility—that different computational systems will inevitably have different “idiosyncrasies”. And so it is for rule 30 and code 10. Rule 30 has an “Xor on one side” which gives it special surjectivity properties. Code 10 on the other hand has its block emulations, which lead, for example, to the surprise of nesting.
I’ve now spent many years studying the ruliology of simple programs, and if there’s one thing that still amazes me after all that time it’s that there are always surprises. Even with very simple underlying rules one can never be sure what will happen; there’s no choice but to just do the experiments and see. And, in my experience, pretty much whenever one thinks one’s “got to the end” and “seen everything there is to see”, something completely unexpected will pop out—a reminder that, as the Principle of Computational Equivalence tells us, these simple computational systems are in some sense a microcosm of everything that’s possible.
Ruliology is in many ways the ultimate foundational science—a science concerned with pure abstract rules not set up with any particular reference either to nature or to human choice. In a sense ruliology is our best path to ultimate pure abstraction—and unfettered exploration of the ruliad. And at least for me, it’s also something very satisfying to do. These days, with modern Wolfram Language, it’s all very streamlined and fast. Sitting at one’s computer, one can immediately start visiting vast uncharted areas of the computational universe, seeing things—and often very beautiful things—that have never been seen before, and discovering new but everlasting things anchored in the bedrock of computation and of simple programs.
It’s been fun spending a few hours studying the ruliology of code 10. Essentially everything I’ve done here I could have done (though not nearly as efficiently) back in 1983 when I first came up with code 10. But as it was, code 10 in a sense had to “wait patiently” for someone to come and look at it. The form of the rule 30 pattern is in some ways more “human-scaled” than code 10. But, as we’ve seen here, code 10 still manifests the same core phenomenon as rule 30. And now, forty years after printing that “C10” picture, I’m happy to be able to say that I think I’ve finally gotten at least a passing acquaintance with another remarkable “computational world” out there in the computational universe: the world of code 10.
Why does biological evolution work? And, for that matter, why does machine learning work? Both are examples of adaptive processes that surprise us with what they manage to achieve. So what’s the essence of what’s going on? I’m going to concentrate here on biological evolution, though much of what I’ll discuss is also relevant to machine learning—but I’ll plan to explore that in more detail elsewhere.
OK, so what is an appropriate minimal model for biology? My core idea here is to think of biological organisms as computational systems that develop by following simple underlying rules. These underlying rules in effect correspond to the genotype of the organism; the result of running them is in effect its phenotype. Cellular automata provide a convenient example of this kind of setup. Here’s an example involving cells with 3 possible colors; the rules are shown on the left, and the behavior they generate is shown on the right:
Note: Click any diagram to get Wolfram Language code to reproduce it.
We’re starting from a single (
But let’s say we start from the trivial (“null”) rule that makes any pattern die out immediately. Can we end up “adaptively evolving” to the rule above? Imagine making a sequence of randomly chosen “point mutations”—each changing just one outcome in the rule, as in:
Then suppose that at each step—in a minimal analog of natural selection—we “accept” any mutation that makes the lifetime longer (though not infinite), or at least the same as before, and we reject any mutation that makes the lifetime shorter, or infinite. It turns out that with this procedure we can indeed “adaptively evolve” to the rule above (where here we’re showing only “waypoints” of progressively greater lifetime):
Different sequences of random mutations give different sequences of rules. But the remarkable fact is that in almost all cases it’s possible to “make progress”—and routinely reach rules that give long-lived patterns (here with lifetimes 107, 162 and 723) with elaborate morphological structure:
Is it “obvious” that our simple process of adaptive evolution will be able to successfully “wrangle” things to achieve this? No. But the fact that it can seems to be at the heart of why biological evolution manages to work.
Looking at the sequences of pictures above we see that there are often in effect “different mechanisms” for producing long lifetimes that emerge in different sequences of rules. Typically we first see the mechanism in simple form, then as the adaptive process continues, the mechanism gets progressively more developed, elaborated and built on—not unlike what we often appear to see in the fossil record of biological evolution.
But let’s drill down and look in a little more detail at what’s happening in the simple model we’re using. In the 3-color nearest-neighbor (k = 3, r = 1) cellular automata we’re considering, there are 26 (= 33 – 1) relevant cases in the rule (there’d be 27 if we didn’t insist that
). “Point mutations” affect a single case, changing it to one of two (= 3 – 1) possible alternative outcomes—so that there are altogether 52 (= 26 × 2) possible distinct “point mutations” that can be made to a given rule.
For example, starting from the rule
the results of possible single point mutations are:
And even with such point mutations there’s usually considerable diversity in the behavior they generate:
In quite a few cases the pattern generated is exactly the same as the one for the original rule. In other cases it dies out more quickly—or it doesn’t die out at all (either becoming periodic, or growing forever). And in this particular example, in just one case it achieves “higher fitness”, surviving longer.
If we make a sequence of random mutations, many will produce shorter-lived or infinite lifetime (“tumor”) patterns, and these we’ll reject (or, in biological terms, we can imagine they’re “selected out”):
But still there can be many “neutral mutations” that don’t change the final pattern (or at least give a pattern of the same length). And at first we might think that these don’t achieve anything. But actually they’re critical in allowing single point mutations to build up to larger mutations that can eventually give longer-lived patterns:
Tracing our whole adaptive evolution process above, the total number of point mutations involved in getting from one (increasingly long-lived) “fitness waypoint” to another is:
Here are the underlying rules associated with these fitness waypoints (where the numbers count cumulative “accepted mutations”, ignoring ones that go “back and forth”):
One way to get a sense of what’s going on is to take the whole sequence of (“accepted”) rules in the adaptive evolution process, and plot them in a dimension-reduced rendering of the (27-dimensional) rule space:
There are periods when there’s a lot of “wandering around” going on, with many mutations needed to “make progress”. And there are other periods when things go much faster, and fewer mutations are needed.
As another way to see what’s going on, we can plot the maximum lifetime achieved so far against the total number of mutation steps made:
We see plateaus (including an extremely long one) in which “no progress” is made, punctuated by sometimes-quite-large, sudden changes, often brought on by just a single mutation.
If we include “rejected mutations” we see that there’s a lot of activity going on even in the plateaus; it just doesn’t manage to make progress (one can think of each red dot that lies below a plateau as being like a mutation—or an organism—that “doesn’t make it”, and is selected out):
It’s worth noting that there can be multiple different (“phenotype”) patterns that occur across a plateau. Here’s what one sees in the particular example we’re considering:
But even between these “phenotypically different” cases, there can be many “genotypically different” rules. And in a sense this isn’t surprising, because usually only parts of the underlying rule are “coding”; other parts are “noncoding”, in the sense that they’re not sampled during the generation of the pattern from that rule.
And for example this highlights for each “fitness waypoint rule” which cells make use of a “fresh” case in the rule that hasn’t so far been sampled during the generation of the pattern:
And we see that even in the last rule shown here, only 18 of the 26 relevant cases in the rule are actually ever sampled during the generation of the pattern (from the particular, single-red-cell initial condition used). So this means that 8 cases in the rule are “undetermined” from the phenotype, implying that there are 38 = 6561 possible genotypes (i.e. rules) that will give the same result.
So far we’ve mostly been talking about one particular random sequence of mutations. But what happens if we look at many possible such sequences? Here’s how the longest lifetime (or, in effect, “fitness”) increases for 100 different sequences of random mutations:
And what’s perhaps most notable here is that it seems as if these adaptive processes indeed don’t “get stuck”. It may take a while (with the result that there are long plateaus) but these pictures suggest that eventually “adaptive evolution will find a way”, and one will get to rules that show longer lifetimes—as the progressive development of the distribution of lifetimes reflects:
In what we’ve done so far we’ve always been discussing particular paths of adaptive evolution, determined by particular sequences of random mutations. But a powerful way to get a more global view of the process of adaptive evolution is to look—in the spirit of our Physics Project, the ruliad, etc.—not just at individual paths of adaptive evolution, but instead at the multiway graph of all possible paths. (And in making a correspondence with biology, multiway graphs give us a way to talk about adaptive evolution not just of individual sequences of organisms, but also populations.)
To start our discussion, let’s consider not the 3-color cellular automata of the previous section, but instead (nearest-neighbor) 2-color cellular automata—for which there are just 128 possible relevant rules. How are these rules related by point mutations? We can construct a graph of every possible way that one rule from this set can be transformed to another by a single point mutation:
If we imagine 5-bit rather than 7-bit rules, there are only 16 relevant ones, and we can readily see that the graph of possible mutations has the form of a Boolean hypercube:
Let’s say we start from the “null rule” 
. Then we enumerate the rules obtained by a single point mutation (and therefore directly connected to the null rule in the graph above)—then we see what behavior they produce, say from the initial condition …
…:
Some of these rules we can view as “making progress”, in the sense that they yield patterns with longer lifetimes (not impressively longer, just 2 rather than 1). But other rules “make no progress” or generate patterns that “live forever”. Keeping only mutations that don’t lead to shorter or infinite lifetimes, we can construct a multiway graph that shows all possible mutation paths:
Although this is a very small graph (with just 15 rules appearing), we can already see hints of some important phenomena. There are “fitness-neutral” mutations that can “go both ways”. But there are also plenty of mutations that only “go one way”—because the other way they would decrease fitness. And a notable feature of the graph is that once one’s “committed” to a particular part of the graph, one often can’t reach a different one—suggesting an analogy to the existence of distinct branches in the tree of life.
Moving beyond 2-color, nearest-neighbor (k = 2, r = 1) cellular automata, we can consider k = 2, r = 
ones. A typical such cellular automaton is:
For k = 2, r = 1 there were a total of 128 (= 223 – 1) relevant rules. For k = 2, r = , there are a total of 32,768 (= 224 – 1). Starting with the null rule, and again using initial condition …,
And here is the beginning of the multiway graph for k = 2, r = 
rules—showing rules reached by up to two mutations starting from the null rule:
This graph contains many examples of “fitness-neutral sets”—rules that have the same fitness and that can be transformed into each other by mutations. A few examples of such fitness-neutral sets:
In the first case here, the “morphology of the phenotypic patterns” is the same for all “genotypic rules” in the fitness-neutral set. But in the other cases there are multiple morphologies within a single fitness-neutral set.
If we included all individual rules we’d get a complete k = 2, r = 
multiway graph with a total of 1884 nodes. But if we just include one representative from every fitness-neutral set, we get a more manageable multiway graph, with a total of 86 nodes:
Keeping only one representative from pairs of patterns that are related by left-right symmetry, we get a still-simpler graph, now with a total of 49 nodes:
There’s quite a lot of structure in this graph, with both divergence and convergence of possible paths. But overall, there’s a certain sense that different sections of the graph separate into distinct branches in which adaptive evolution in effect “pursues different ideas” about how to increase fitness (i.e. lifetime of patterns).
We can think of fitness-neutral sets as representing a certain kind of equivalence class of rules. There’s quite a range of possible structures to these sets—from ones with a single element to ones with many elements but few distinct morphologies, to ones with different morphologies for every element:
What about larger spaces of rules? For k = 2, r = 2 there are altogether about 2 billion (225 – 1) relevant rules. But if we choose to look only at left-right symmetric ones, this number is reduced to 524,288 (= 219). Here are some examples of sequences of rules produced by adaptive evolution in this case, starting from the null rule, and allowing only mutations that preserve symmetry (and now using a single black cell as the initial condition):
Once again we can identify fitness-neutral sets—though this time, in the vast majority of cases, the patterns generated by all members of a given set are the same:
Reducing out fitness-neutral sets, we can then compute the complete (transitively reduced) multiway graph for symmetric k = 2, r = 2 rules (containing a total of 60 nodes):
By reducing out fitness-neutral sets, we’re creating a multiway graph in which every edge represents a mutation that “makes progress” in increasing fitness. But actual paths of adaptive evolution based on random sequences of mutations can do any amount of “rattling around” within fitness-neutral sets—not to mention “trying” mutations that decrease fitness—before reaching mutations that “make progress”. So this means that even though the reduced multiway graph we’ve drawn suggests that the maximum number of steps (i.e. mutations) needed to adaptively evolve from the null rule to any other is 9, it can actually take any number of steps because of the “rattling around” within fitness-neutral sets.
Here’s an example of a sequence of accepted mutations in a particular adaptive evolution process—with the mutations that “make progress” highlighted, and numbers indicating rejected mutations:
We can see “rattling around” in a fitness-neutral set, with a cycle of morphologies being generated. But while this represents one way to reach the final pattern, there are also plenty of others, potentially involving many fewer mutations. And indeed one can determine from the multiway graph that an absolutely shortest path is:
This involves the sequence of rules:
We’re starting from the null rule, and at each step making a single point mutation (though because of symmetry two bits can sometimes be changed). The first few mutations don’t end up changing the “phenotypic behavior”. But after a while, enough mutations (here 6) have built up that we get morphologically different behavior. And after just 3 more mutations, we end up with our final pattern.
Our original random sequence of mutations gets to the same result, but in a much more tortuous way, doing a total of 169 mutations which often cancel each other out:
In drawing a multiway graph, we’re defining what evolutionary paths are possible. But what about probabilities? If we assume that every point mutation is equally likely, we can in effect “analyze the flow” in the multiway graph, and determine the ultimate probability that each rule will be reached (with higher probabilities here shown redder):
Multiway graphs give a very global view of adaptive evolution. But in understanding the process of adaptive evolution, it’s also often useful to think somewhat more locally. We can imagine that all the possible rules are laid out in a certain space, and that adaptive evolution is trying to find appropriate paths in this space. Potentially we can suppose that there’s a “fitness landscape” defined in the space, and that adaptive evolution is trying to follow a path that progressively ascends to higher peaks of fitness.
Let’s consider again the very first example we gave above—of adaptive evolution in the space of 3-color cellular automata. At each step in this adaptive evolution, there are 52 possible point mutations that can be made to the rule. And one can think of each of these mutations as corresponding to making an “elementary move” in a different direction in the (26-dimensional) space of rules.
Here’s a visual representation of what’s going on, based on the particular path of adaptive evolution from our very first example above:
What we’re showing here is in effect the sequence of “decisions” that are being made to get us from one “fitness waypoint” to another. Different possible mutations are represented by different radial directions, with the length of each line being proportional to the fitness achieved by doing that mutation. At each step the gray disk represents the previous fitness. And what we see is that many possible mutations lead to lower fitness outcomes, shown “within the disk”. But there are at least some mutations that have higher fitness, and “escape the disk”.
In the multiway graph, we’d trace every mutation that leads to higher fitness. But for a particular path of adaptive evolution as we’ve discussed it so far, we imagine we always just pick at random one mutation from this set—as indicated here by a red line. (Later we’ll discuss different strategies.)
Our radial icons can be thought of as giving a representation of the “local derivative” at each point in the space of rules, with longer lines corresponding to directions with larger slopes “up the fitness landscape”.
But what happens if we want to “knit together” these local derivatives to form a picture of the whole space? Needless to say, it’s complicated. And as a first example, consider k = 2, r = 1 cellular automaton rules.
There are a total of 128 relevant such rules, that (as we discussed above) can be thought of as connected by point mutations to form a 7-dimensional Boolean hypercube. As also discussed above, of all 128 relevant rules, only 15 appear in adaptive evolution processes (the others are in effect never selected because they represent lower fitness). But now we can ask where these rules lie on the whole hypercube:
Each node here represents a rule, with the size of the highlighted nodes indicating their corresponding fitness (computed from lifetime with initial condition ……). The node shown in green corresponds to the null rule.
Rendering this in 3D, with fitness shown as height, we get what we can consider a “fitness landscape”:
And now we can think of our adaptive evolution as proceeding along paths that never go to nodes with lower height on this landscape.
We get a more filled-in “fitness landscape” when we look at k = 2, r = rules (here with initial condition ……):
Adaptive evolution must trace out a “never-go-down” path on this landscape:
Along this path, we can make “derivative” pictures like the ones above to represent “local topography” around each point—indicating which of the possible upwards-on-the-landscape directions is taken:
The rule space over which our “fitness landscape” is defined is ultimately discrete and effectively very high-dimensional (15-dimensional for k = 2, r = 
rules)—and it’s quite challenging to produce an interpretable visualization of it in 3D. We’d like it if we could lay out our rendering of the rule space so that rules which differ just by one mutation are a fixed (“elementary”) 2D distance apart. In general this won’t be possible, but we’re trying to at least approximate this by finding a good layout for the underlying “mutation graph”.
Using this layout we can in principle make a “fitness landscape surface” by interpolating between discrete points. It’s not clear how meaningful this is, but it’s perhaps useful in engaging our spatial intuition:
We can try machine learning and dimension reduction, operating on the set of “rule vectors” (i.e. outcome lists) that won’t be rejected in our adaptive evolution process—and the results are perhaps slightly better:
By the way, if we use this dimension reduction for rule space, here’s how the behavior of rules lays out:
And here, for comparison, is a feature space plot based on the visual appearance of these patterns:
In adaptive evolution, we start, say, from the null rule and then make random mutations to try to reach rules with progressively larger fitness. But what about just exhaustively searching the complete space of possible rules? The number of rules rapidly becomes unmanageably big—but some cases are definitely accessible:
For example, there are just 524,288 symmetric k = 2, r = 2 rules—of which 77,624 generate patterns with finite lifetimes. Ultimately, though, there are just 77 distinct phenotypic patterns that appear—with varying lifetimes and varying multiplicity (where at least in this case the multiplicity is always associated with “unused bits” in the rule):
How do these exhaustive results compare with what’s generated in the multiway graph of adaptive evolutions? They’re almost the same, but for the addition of the two extra cases
which are generated by rules of the form (where the gray entries don’t matter):
Why don’t such rules ever appear in our adaptive evolution? The reason is that there isn’t a chain of point mutations starting from the null rule that can reach these rules without going through rules that would be rejected by our adaptive evolution process. If we draw a multiway graph that includes every possible “acceptable” rule, then we’ll see a separate part in the graph, with its own root, that contains rules that can’t be reached by our adaptive evolution from the null rule:
So now if we look at all (symmetric k = 2, r = 2) rules, here’s the distribution of lifetimes we get:
The maximum, as seen above, is 65. The overall distribution roughly follows a power law, with an exponent around –3:
As we saw above, not all rules make use of all their bits (i.e. outcomes) in generating phenotypic patterns. But what we see is that the larger the lifetime achieved, the more bits tend to be needed:
And in a sense this isn’t surprising: as we’ll discuss later, we can expect to need “more bits in the program” to specify more elaborate behavior—or, in particular, behavior that embodies a larger number for lifetime.
So what about general (i.e. not necessarily symmetric) k = 2, r = 2 rules? There are
Again it’s roughly a power law, but now with exponent around –3.5:
Here are the actual 100 patterns produced that have the longest lifetimes (in all asymmetric cases there are also rules giving left-right flipped patterns):
It’s interesting to see here a variety of “qualitatively different ideas” being used by different rules. Some (like the one with lifetime 151) we might somehow imagine could have been constructed specifically “for the purpose” of having their particular, long lifetime. But others (like the one with lifetime 308) somehow seem more “coincidental”—behaving in an apparently random way, and then just “happening to die out” after a certain number of steps.
Since we found these rules by exhaustive search, we know they’re the only possible ones with such long lifetimes (at least with k = 2, r = 2). So then we can infer that the ornate structures we see are in some sense necessary to achieve the objective of, say, having a finite lifetime of more than 100 steps. So that means that if we go through a process of adaptive evolution and achieve a lifetime above 100 steps, we see a complex pattern of behavior not because of “complicated choices in our process of adaptive evolution”, but rather because to achieve such a lifetime one has no choice but to use such a complex pattern. Or, in other words, the complexity we see is a reflection of “computational necessity”, not historical accidents of adaptive evolution.
Note also that (as we’ll discuss in more detail below) there are certain behaviors we can get, and others that we cannot. So, for example, there is a rule that gives lifetime 308, but none that gives lifetime 300. (Though, yes, if we used more complicated initial conditions or a more complicated family of rules we could find such a rule.)
Much as we saw in the symmetric k = 2, r = 2 case above, almost any long lifetimes require using all the available bits in the rule:
But, needless to say, there’s an exception—a pair of rules with lifetime 84 where the outcome for the 
case doesn’t matter:
But, OK, so can these long-lifetime rules be reached by single-mutation adaptive evolution from the null rule? Rather than trying to construct the whole multiway graph for the general k = 2,
And what we see is that at least in this case such a procedure never reaches the null rule. The “furthest” it gets is to lifetime-2 rules, and among these rules the closest to the null rule are:
But it turns out that there’s no way to reach these 2-bit rules by a single point mutation from any of the 26 1-bit rules that aren’t rejected by our adaptive evolution process. And in fact this isn’t just an issue for this particular long-lifetime rule; it’s something quite general among k = 2 rules. And indeed, constructing the “forward” multiway graph starting from the null rule, we find we can only ever reach lifetime-1 rules.
Ultimately this is a particular feature of rules with just 2 colors—and it’s specific to starting with something like the null rule that has lifetime 1—but it’s an illustration of the fact that there can even be large swaths of rule space that can’t be reached by adaptive evolution with point mutations.
What about symmetric k = 2, r = 2 rules? Well, to maintain symmetry we have to deal with mutations that change not just one but two bits. And this turns out to mean that (except in the cases we discovered above) the inverse multiway system starting from long-lifetime rules always successfully reaches the null rule:
There’s something else to notice here, however. Looking at this graph, we see that there’s a way to get with just one 2-bit mutation from a lifetime-1 to a lifetime-65 rule:
We didn’t see this in our multiway graph above because we had applied transitive reduction to it. But if we don’t do that, we find that a few large lifetime jumps are possible—as we can see on this plot of possible lifetimes before and after a single point mutation:
Going beyond k = 2, r = 2 rules, we can consider symmetric k = 3, r = 1 rules, of which there are 317, or about 129 million. The distribution of lifetimes in this case is
which again roughly fits a power law, again with exponent around –3.5:
But now the maximum lifetime found is not just 308, but 2194:
One again, there are some different “ideas” on display, with a few curious examples of convergence—such as the rules we see with lifetimes 989 and 990 (as well as 1068 and 1069) which give essentially the same patterns after just exchanging colors, and adding one “prefatory” step.
What about general k = 3, r = 1 rules? There are too many to easily search exhaustively. But directed random sampling reveals plenty of long-lifetime examples, such as:
And now the tail of very long lifetimes extends further, for example with:
It’s a little easier to see what the lifetime-10863 rule does if one visualizes it in sections (and adjusts colors to get more contrast):
Sampling 100 steps out every 2000 (as well as at the very end), we see elaborate alternation between periodic and seemingly random behavior—but none of it gives any obvious clue of the remarkable fact that after 10863 steps the whole pattern will die out:
As our example criterion for the “fitness” of cellular automaton rules, we’ve used the lifetimes of the patterns they generate—always assuming that if the patterns don’t terminate at all they should be considered to have fitness zero.
But how can we tell if a pattern is going to terminate? In the previous section, for example, we saw patterns that live a very long time—but do eventually terminate.
Here are some examples of the first 100 steps of patterns generated by a few k = 3, r = 1 symmetric rules:
What will happen with these patterns? We know from what we see here that none of them have lifetimes less than 100 steps. But what would allow us to say more? In a few cases we can see that the patterns are periodic, or have obvious repeating structures, which means they’ll never terminate. But in the other cases there’s no obvious way to predict what will happen. Explicitly running the rules for another 100 steps we discover some more outcomes:
Going to 500 steps there are some surprises. Rule (a) becomes periodic after 388 steps; rules (o) and (v) terminate after 265 and 377 steps, respectively:
But is there a way to systematically say what will happen “in the end” with all the remaining rules? The answer is that in general there is not; it’s something that must be considered undecidable by any finite computation.
Given how comparatively simple the cellular automaton rules we’re considering are, we might have assumed that with all our sophisticated mathematical and computational methods we’d always be able to “jump ahead of them”—and figure out their outcome without the computational effort of explicitly running each step.
But the Principle of Computational Equivalence suggests that pretty much whenever the behavior of these rules isn’t obviously simple, it will in effect be of equal computational sophistication to any other system, and in particular to any methods that we might use to predict it. And the result is the phenomenon of computational irreducibility that implies that in many systems—presumably including most of the cellular automata here—there isn’t any way to figure out their outcome much more efficiently than by explicitly tracing each of their steps. So this means that to know what will happen “in the end”—after an infinite number of steps—might take an unlimited amount of computational effort. Or, in other words, it must be considered effectively undecidable by any finite computation.
As a practical matter we might look at the observed distribution of lifetimes for a particular type of cellular automaton, and become pretty confident that there won’t be longer finite lifetimes for that type of cellular automaton. But for the k = 3, r = 1 rules from the previous section, we might have been fairly confident that a few thousand steps was the longest lifetime that would ever occur—until we discovered the 10,863-step example.
So let’s say we run a particular rule for 10,000 steps and it hasn’t died out. How can we tell if it never will? Well, we have to construct a proof of some kind. And that’s easy to do if we can see that the pattern becomes, say, completely periodic. But in general, computational irreducibility implies we won’t be able to do it. Might there, though, still be special cases where we can? In effect, those would have to correspond to “pockets of computational reducibility” where we manage to find a compressed description of the cellular automaton behavior.
There are cases like this where there isn’t strict periodicity, but where in the end there’s basically repetitive behavior (here with period 480):
And there are cases of nested behavior, which is never periodic, but is nevertheless simple enough to be predictable:
But there are always surprises. Like this example—which eventually resolves to have period 6, but only after 7129 steps:
So what does all this mean for our adaptive evolution process? It implies that in principle we could miss a very long finite lifetime for a particular rule, assuming it to be infinite. In a biological analogy we might have a genome that seems to lead to unbounded perhaps-tumor-like growth—but where actually the growth in the end “unexpectedly” stops.
What we’re asking about the dying out of patterns in cellular automata is directly analogous to the classic halting problem for Turing machines, or the termination problem for term rewriting, Post tag systems, etc. And in looking for cellular automata that have the longest-lived patterns, we’re studying a cellular automaton analog of the so-called busy beaver problem for Turing machines.
We can summarize the results we’ve found so far (all for single-cell initial conditions):
The profiles (i.e. widths of nonzero cells) for the patterns generated by these rules are
and the “integrals” of these curves are what give the “areas” in the table above.
For the reasons described in the previous section, we can only be certain that we’ve found lower bounds on the actual maximum lifetime—though except in the last few cases listed it seems very likely that we do in fact have the maximum lifetime.
It’s somewhat sobering, though, to compare with known results for maximum (“busy beaver”) lifetimes for Turing machines (where now s is the number of Turing machine states, the Turing machines are started from blank tapes, and they are taken to “halt” when they reach a particular halt state):
Sufficiently small Turing machines can have only modest lifetimes. But even slightly bigger Turing machines can have vastly larger lifetimes. And in fact it’s a consequence of the undecidability of the halting problem for Turing machines that the maximum lifetime grows with the size of the Turing machine faster than any computable function (i.e. any function that can be computed in finite time by a Turing machine, or whose value can be proved by a finite proof in a finite axiom system).
But, OK, the maximum lifetime increases with the “size of the rule” for a Turing machine, or a cellular automaton. But what defines the “size of a rule”? Presumably it should be roughly the number of independent bits needed to specify the rule (which we can also think of as an approximate measure of its “information content”)—or something like log2 of the number of possible rules of its type.
At the outset, we might imagine that all 232 k = 2, r = 2 rules would need 32 bits to specify them. But as we discussed above, in some cases some of the bits in the rule don’t matter when it comes to determining the patterns they produce. And what we see is that the more bits that matter (and so have to be specified), the longer the lifetimes that are possible:
So far we’ve only been discussing cellular automata with single-cell initial conditions. But if we use more complicated initial conditions what we’re effectively doing is adding more information content into the system—with the result that maximum lifetimes can potentially get larger. And as an example, here are possible lifetimes for k = 2, r = 
rules with a sequence of possible initial conditions:
Cellular automata are at their core deterministic systems: given a particular cellular automaton rule and a particular initial condition, every aspect of the behavior that is generated is completely determined. But is there any way that we can approximate this behavior by some probabilistic model? Or might we at least usefully be able to use such a model if we look at the aggregate properties of large numbers of different rules?
One hint along these lines comes from the power-law distributions we found above for the frequencies of different possible lifetimes for cellular automata of given types. And we might wonder whether such distributions—and perhaps even their exponents—could be found from some probabilistic model.
One possible approach is to approximate a cellular automaton by a probabilistic process—say one in which a cell becomes black with probability p if it or either of its neighbors were black on the step before. Here are some examples of what can happen with this (“directed percolation”) setup:
The behavior varies greatly with p; for small p everything dies out, while for large p it fills in:
And indeed the final density—starting from random initial conditions—has a sharp (phase) transition at around p = 0.54 as one varies p:
If instead one starts from a single initial black cell one sees a slightly different transition:
One can also plot the probabilities for different “survival times” or “lifetimes” for the pattern:
And right around the transition the distribution of lifetimes follows a power law—that’s roughly τ–1 (which happens to be what one gets from a mean field theory estimate).
So how does this relate to cellular automata? Let’s say we have a k = 2 rule, and we suppose that the colors of cells can be approximated as somehow random. Then we might suppose that the patterns we get could be like in our probabilistic model. And a potential source for the value of p to use would be the fraction of cases in the rule that give a black cell as output.
Plotting the lifetimes for k = 2, r = 2 rules against these fractions, we see that the longest lifetimes do occur when a little under half the outcomes are black (though notice this is also where the binomial distribution implies the largest number of rules are concentrated):
If we don’t try thinking about the details of cellular automaton evolution, but instead just consider the boundaries of finite-lifetime patterns we generate, we can imagine approximating these (say for symmetric rules) just by random walks—that when they collide correspond to the pattern dying out:
The standard theory of random walks then tells us that the probability to survive τ steps is proportional to τ–3/2 for large τ—a power law, though not immediately one of the same ones that we’ve observed for our cellular automaton lifetimes.
In what we’ve done so far, we’ve always taken each step of our process of adaptive evolution to pick an outcome of equal or greater fitness. But what if we adopt a “more impatient” procedure in which at each step we insist on an outcome that has strictly greater fitness?
For k = 2 it’s simply not possible with this procedure (at least with a null initial condition) to “escape” the null rule; everything that can be reached with 1 mutation still has lifetime 1. With
But we’re assuming here that we have to reach greater fitness with just one mutation. What if we allow two mutations at a time? Well, then we can “make progress”. And here’s the multiway graph in this case for symmetric k = 2, r = 2 rules:
We don’t reach as many phenotypic patterns as by using single mutations and allowing “fitness-neutral moves”, but where we do get, we get much quicker, without any “back and forth” in fitness-neutral spaces.
If we allow up to 3 mutations, we get still further:
And indeed we seem to get a pretty good representative sampling of “what’s out there” in this rule space, even though we reach only 37 rules, compared to the 77,624 (albeit with many duplicated phenotypic patterns) from our standard approach allowing neutral moves.
For k = 3, r = 1 symmetric rules single mutations can get 2 steps:
But now if we allow up to 2 mutations, we can go much further—and the fact that we now don’t have to deal with neutral moves means we can explicitly construct at least the first few steps of the multiway graph in this case:
We can go further if at each step we just pick a random higher-fitness rule reached with two or fewer mutations:
The adaptive evolution histories we just showed can be generated in effect by randomly trying a series of possibilities at each step, then picking the first one that exhibits increased fitness. Another approach is to use what amounts to “local exhaustive search”: at each step, look at results from all possible mutations, and pick one that gives the largest fitness. At least in smaller rule spaces, it’s common that there will be several results with the same fitness, and as an example we’ll just pick among these at random:
One might think that this approach would in effect always be an optimization of the adaptive evolution process. But in practice its systematic character can end up making it get stuck, in some sense repeatedly “trying to do the same thing” even if it “isn’t working”.
Something of an opposite approach involves loosening our criteria for which paths can be chosen—and for example allowing paths that temporarily reduce fitness, say by one step of lifetime:
In effect here we’re allowing less-than-maximally-fit organisms to survive. And we can represent the overall structure of what’s happening by a multiway graph—which now includes “backtracking” to lower fitnesses:
But although the details are different, in the end it doesn’t seem as if allowing this kind of backtracking has any dramatic effect. Somehow the basic phenomena around the process of adaptive evolution are strong enough that most of the details of how the adaptive evolution is done don’t ultimately matter much.
In everything we’ve done so far, we’ve been making mutations only to individual rules. But there’s another mechanism that exists in many biological organisms: sexual reproduction, in which in effect a pair of rules (i.e. genomes) get mixed to produce a new rule. As a simple model of the crossover that typically happens with actual genomes, we can take two rules, and splice together the beginning of one with the end of the other:
In general there will be many ways to combine pairs of rules like this. In a direct analogy to our Physics Project, we can represent such “recombinations” as “events” that take two rules and produce one:
The analog of our multiway graph for all possible paths of adaptive evolution by mutations is now what we call in our Physics Project a token-event graph:
In dealing just with mutations we were able to take a single rule and progressively modify it. Now we always have to work with a “population” of rules, combining them two at a time to generate new rules. We can represent conceivable combinations among one set of rules as follows:
There are at this point many different choices we could make about how to set up our model. The particular approach we’ll use selects just n of the 
= n (n – 1)/2 possible combinations:
Then for each of these selected combinations we attempt a crossover, keeping those “children” (drawn here between their parents) that are not rejected as a result of having lower fitness:
Finally, to “maintain our gene pool”, we carry forward parents selected at random, so that we still end up with n rules. (And, yes, even though we’ve attempted to make this whole procedure as clean as possible, it’s still a mess—which seems to be inevitable, and which has, as we’ll discuss below, bedeviled computational studies of evolution in the past.)
OK, so what happens when we apply this procedure, say to k = 3, r = 1 rules? We’ll pick 4 rules at random as our initial population (and, yes, two happen to produce the same pattern):
Then in a sequence of steps we’ll successively pick various combinations:
And here are the distinct “phenotype patterns” produced in this process (note that even though there can be multiple copies of the same phenotype pattern, the underlying genotype rules are always distinct):
As a final form of summarization we can just plot the successive fitnesses of the patterns we generate (with the size of each dot reflecting the number of times a particular fitness occurs):
In this case we reach a steady state after 9 steps. The larger the population the longer the adaptive evolution will typically keep going. Here are a couple of examples with population 10, showing all the patterns obtained:
Showing in each case only the longest-lifetime rule found so far we get:
The results aren’t obviously different from what we were finding with mutation alone—even though now we’ve got a much more complicated model, with a whole population of rules rather than a single rule. (One obvious difference, though, is that here we can end up with overall cycles of populations of rules, whereas in the pure-mutation case that can only happen among fitness-neutral rules.)
Here are some additional examples—now obtained after 500 steps with population 25
and with population 50:
And so far as one can tell, even here there are no substantial differences from what we saw with mutation. Certainly there are detailed features introduced by sexual reproduction and crossover, but for our purposes in understanding the big picture of what’s happening in adaptive evolution it seems sufficient to do as we have done so far, and consider only mutation.
By investigating adaptive evolution in cellular automata we’re already making dramatic simplifications relative, say, to actual biology. But in the effort to understand the essence of phenomena we see, it’s helpful to go even further—and instead of thinking about computational rules and their behavior, just think about vertices on a “mutation graph”, each assigned a certain fitness.
As an example, we can set up a 2D grid, assigning each point a certain random fitness:
And then, starting from a minimum-fitness point, we can follow the same kind of adaptive evolution procedure as above, at each step going to a neighboring point with an equal or greater fitness:
Typically we don’t manage to go far before we get stuck, though with the uniform distribution of fitness values used here, we still usually end on a fairly large fitness value.
We can summarize the possible paths we can take by the multiway graph:
In our cellular automaton rule space—and, for that matter, in biology—neighboring points don’t just have independent random fitnesses; instead, the fitnesses are determined by a definite computational procedure. So as a simple approximation, we can just take the fitness of each point to be a particular function of its graph coordinates. If the function forms something like a “uniform hill”, then the adaptive evolution procedure will just climb it:
But as soon as the function has “systematic bumpiness” there’s a tremendous tendency to quickly get stuck:
And if there’s some “unexpected spot of high fitness”, adaptive evolution typically won’t find it (and it certainly won’t if it’s surrounded by a lower-fitness “moat”):
So what happens if we increase the dimensionality of the “mutation space” in which we’re operating? Basically it becomes easier to find a path that increases fitness:
And we can see this, for example, if we look at Boolean hypercubes in increasing numbers of dimensions:
But ultimately this relies on the fact that in the neighborhood reachable by mutations from a given point, there’ll be a “sufficiently random” collection of fitness values that it’ll (likely) be possible to find a “direction” that’s “going up” in fitness. Yet this alone won’t in general be enough, because we also need it to be the case that there’s enough regularity in the fitness landscape that we can systematically navigate it to find its maximum—and that the maximum is not somehow “unexpected and isolated”.
We’ve seen that adaptive evolution can be surprisingly successful at finding cellular automata that produce patterns with long but finite lifetimes. But what about other types of “traits”? What can (and cannot) adaptive evolution ultimately manage to do?
For example, what if we’re trying to find cellular automata whose patterns don’t just live “as long as possible” but instead die after a specific number of steps? It’s clear that within any finite set of rules (say with particular k and r) there’ll only be a limited collection of possible lifetimes. For symmetric k = 2, r = 2 rules, for example, the possible lifetimes are:
But as soon as we’re dealing even with k = 3, r = 1 symmetric rules it’s already in principle possible to get every lifetime up to 100. But what about adaptive evolution? How well does it do at reaching rules with all those lifetimes? Let’s say we do single point mutation as before, but now we “accept” a mutation if it leads not specifically to a larger finite lifetime, but to a lifetime that is closer in absolute magnitude to some desired lifetime. (Strictly, and importantly, in both cases we also allow “fitness-neutral” mutations that leave the lifetime the same.)
Here are examples of what happens if we try to adaptively evolve to get lifetime exactly 50 in
It gets close—and sometimes it overshoots—but, at least in these particular examples, it never quite makes it. Here’s what we see if we look at the lifetimes achieved with 100 different random sequences of mutations:
Basically they mostly get stuck at lifetimes close to 50, but not exactly 50. It’s not that k = 3,
It’s just that our adaptive evolution process usually gets stuck before it reaches rules like these. Even though there’s usually enough “room to maneuver” in k = 3, r = 1 rule space to get to generally longer lifetimes, there’s not enough to specifically get to lifetime 50.
But what about k = 4, r = 1 rule space? There are now not 1012 but about 1038 possible rules. And in this rule space it becomes quite routine to be able to reach lifetime 50 through adaptive evolution:
It can sometimes take a while, but most of the time in this rule space it’s possible to get exactly to lifetime 50:
What happens with other “lifetime goals”? Even symmetric k = 3, r = 1 rules can achieve many lifetime values:
Indeed, the first “missing” values are 129, 132, 139, etc. And, for example, many multiples of 50 can be achieved:
But it becomes increasingly difficult for adaptive evolution to reach these specific goals. Increasing the size of the rule space always seems to help; so for example with k = 4, r = 1, if one’s aiming for lifetime 100, the actual distribution of lifetimes reached is:
In general the distribution gets broader as the lifetime sought gets larger:
We saw above that across the whole space of, say, k = 4, r = 1 rules, the frequency of progressively larger lifetimes falls roughly according to a power law. So this means that the fractional region in rule space that achieves a given lifetime gets progressively smaller—with the result that typically the paths followed by adaptive evolution are progressively more likely to get stuck before they reach it.
OK, so what about other kinds of objectives? Say ones more related to the morphologies of patterns? As a simple example, let’s consider the objective of maximizing the “widths” of finite-lifetime patterns. We can try to achieve this by adaptive evolution in which we reject any mutations that lead to decreased width (where “width” is defined as the maximum horizontal extent of the pattern). And once again this process manages to “discover” all sorts of “mechanisms” for achieving larger widths (here each pattern is labeled by its height—i.e. lifetime—and width):
There are certain structural constraints here. For example, the width can’t be too large relative to the height—because if it’s too large, patterns tend to grow forever.
But what if we specifically try to select for maximal “pattern aspect ratio” (i.e. ratio of width to height)? In essentially every case so far, adaptive evolution has in effect “invented many different mechanisms” to achieve whatever objective we’ve defined. But here it turns out we essentially see “the same idea” being used over and over again—presumably because this is the only way to achieve our objective given the overall structure of how the underlying rules we’re using work:
What if we ask something more specific? Like, say, that the aspect ratio be as close to 3 as possible. Much of the time the “solution” that adaptive evolution finds is the correct if trivial:
But sometimes it finds another solution—and often a surprisingly elaborate and complicated one:
How about if our goal is an aspect ratio of π ≈ 3.14? It turns out adaptive evolution can still do rather well here even just with the symmetric k = 3, r = 1 rules that we’re using:
We can also ask about properties of the “inside” of the pattern. For example, we can ask to maximize the lengths of uniform runs of nonwhite cells in the center column of the pattern. And, once again, adaptive evolution can successfully lead us to rules (like these random examples) where this is large:
We can go on and get still more detailed, say asking about runs of particular lengths, or the presence or number of particular subpatterns. And eventually—just like when we asked for too long a lifetime—we’ll find that the cases we’re looking for are “too sparse”, and adaptive evolution (at least in a given rule space) won’t be able to find them, even if exhaustive search could still identify at least a few examples.
But just what kinds of objectives (or fitness functions) can be handled how well by adaptive evolution, operating for example on the “raw material” of cellular automata? It’s an important question—an analog of which is also central to the investigation of machine learning. But as of now we don’t really have the tools to address it. It’s somehow reminiscent of asking what kinds of functions can be approximated how well by different methods or basis functions. But it’s more complicated. Solving it, though, would tell us a lot about the “reach” of adaptive evolution processes, not only for biology but also for machine learning.
How do biological organisms manage to be the way they are, with all their complex and seemingly clever solutions to such a wide range of challenges? Is it just natural selection that does it, or is there in effect more going on? And if “natural selection does it”, how does it actually manage it?
From the point of view of traditional engineering what we see in biology is often very surprising, and much more complex and “clever” than we’d imagine ever being able to create ourselves. But is the secret of biology in a sense just natural selection? Well, actually, there’s often an analog of natural selection going on even in engineering, as different designs get tried and only some get selected. But at least in traditional engineering a key feature is that one always tries to come up with designs where one can foresee their consequences.
But biology is different. Mutations to genomes just happen, without any notion that their consequences can be foreseen. But still one might assume that—when guided by natural selection—the results wouldn’t be too different to what we’d get in traditional engineering.
But there’s a crucial piece of intuition missing here. And it has to do with how randomly chosen programs behave. We might have assumed (based on our typical experience with programs we explicitly construct for particular purposes) that at least a simple random program wouldn’t ever do anything terribly interesting or complicated.
But the surprising discovery I made in the early 1980s is that this isn’t true. And instead, it’s a ubiquitous phenomenon that in the computational universe of possible programs, one can get immense complexity even from very simple programs. So this means that as mutation operates on a genome, it’s essentially inevitable that it’ll end up sampling programs that show highly complex behavior. At the outset, one might have imagined that such complexity could only be achieved by careful design and would inevitably be at best rare. But the surprising fact is that—because of how things fundamentally work in the computational universe—it’s instead easy to get.
But what does complexity have to do with creating “successful organisms”? To create a “successful organism” that can prosper in a particular environment there fundamentally has to be some way to get to a genome that will “solve the necessary problems”. And this is where natural selection comes in. But the fact that it can work is something that’s not at all obvious.
There are really two issues. The first is whether a program (i.e. genome) even exists that will “solve the necessary problems”. And the second is whether such a program can be found by a “thread” of adaptive evolution that goes only through intermediate states that are “fit enough” to survive. As it turns out, both these issues are related to the same fundamental features of computation—which are also responsible for the ubiquitous appearance of complexity.
Given some underlying framework—like cellular automata, or like the basic apparatus of life—is there some rule that can be implemented in that framework that will achieve some particular (computational) objective? The Principle of Computational Equivalence says that generically the answer will be yes. In effect, given almost any “underlying hardware”, it’ll ultimately be possible to come up with “software” (i.e. a rule) that achieves almost any (“physically possible”) given objective—like growing an organism of at least some kind that can survive in a particular environment. But how can we actually find a rule that achieves this?
In principle we could do exhaustive search. But that will be exponentially difficult—and in all but toy cases will be utterly infeasible in practice. So what about adaptive evolution? Well, that’s the big question. And what we’ve seen here is that—rather surprisingly—simple mutation and selection (i.e. the mechanisms of natural selection) very often provide a dramatic shortcut for finding rules that do what we want.
So why is this? In effect, adaptive evolution is finding a path through rule space that gets to where we want to go. But the surprising part is that it’s managing to do this one step at a time. It’s just trying random variations (i.e. mutations) and as soon as it finds one that’s not a “step down in fitness”, it’ll “take it”, and keep going. At the outset it’s certainly not obvious that this will work. In particular, it could be that at some point there just won’t be any “way forward”. All “directions” will lead only to lower fitness, and in effect the adaptive evolution will get stuck.
But the key observation from the experiments in our simple model here is that this typically doesn’t happen. And there seem to be basically two things going on. The first is that rule space is in effect very high-dimensional. So that means there are “many directions to choose from” in trying to find one that will allow one to “take a step forward”. But on its own this isn’t enough. Because there could be correlations between these directions that would mean that if one’s blocked in one direction one would inevitably be blocked in all others.
So why doesn’t this happen? Well, it seems to be the result of the fundamental computational phenomenon of computational irreducibility. A traditional view based on experience with mathematical science had been that if one knew the underlying rule for a system then this would immediately let one predict what the system would do. But what became clear from my explorations in the 1980s and 1990s is that in the computational universe this generically isn’t true. And instead, that the only way one can systematically find out what most computational systems will do is explicitly to run their rules, step by step, doing in effect the same irreducible amount of computational work that they do.
So if one’s just presented with behavior from the system one won’t be in a position to “decode it” and “see its simple origins”. Unless one’s capable of doing as much computational work as the system itself, one will just have to consider what it’s doing as (more or less) “random”. And indeed this seems to be at the root of many important phenomena, such as the Second Law of thermodynamics. And I also suspect it’s at the root of the effectiveness of adaptive evolution, notably in biology.
Because what computational irreducibility implies is that around every point in rule space there’ll be a certain “effective randomness” to fitnesses one sees. And if there are many dimensions to rule space that means it’s overwhelmingly likely that there’ll be “paths to success” in some directions from that point.
But will the adaptive evolution find them? We’ve assumed that there are a series of mutations to the rule, all happening “at random”. And the point is that if there are n elements in the rule, then after some fraction of n mutations we should find our “success direction”. (If we were doing exhaustive search, we’d instead have to try about kn possible rules.)
At the outset it might seem conceivable that the sequence of mutations could somehow “cleverly probe” the structure of rule space, “knowing” what directions would or would not be successful. But the whole point is that going from a rule (i.e. genotype) to its behavior (i.e. phenotype) is generically a computationally irreducible process. So assuming that mutations are generated in a computationally bounded way it’s inevitable that they can’t “break computational irreducibility” and so will “experience” the fitness landscape in rule space as “effectively random”.
OK, but what about “achieving the characteristics an organism needs”? What seems to be critical is that these characteristics are in a sense computationally simple. We want an organism to live long enough, or be tall enough, or whatever. It’s not that we need the organism to perform some specific computationally irreducible task. Yes, there are all sorts of computationally irreducible processes happening in the actual development and behavior of an organism. But as far as biological evolution is concerned all that matters is ultimately some computationally simple measure of fitness. It’s as if biological evolution is—in the sense of my recent observer theory—a computationally bounded observer of underlying computationally irreducible processes.
And to the observer what emerges is the “simple law” of biological evolution, and the idea that, yes, it is possible just by natural selection to successfully generate all sorts of characteristics.
There are all sorts of consequences of this for thinking about biology. For example, in thinking about where complexity in biology “comes from”. Is it “generated by natural selection”, perhaps reflecting the complicated sequence of historical accidents embodied in the particular collection of mutations that occurred? Or is it from somewhere else?
In the picture we’ve developed here it’s basically from somewhere else—because it’s essentially a reflection of computational irreducibility. Having said that, we should remember that the very possibility of being able to have organisms with such a wide range of different forms and functions is a consequence of the universal computational character of their underlying setup, which in turn is closely tied to computational irreducibility.
And it’s in effect because natural selection is so coarse in its operation that it does not somehow avoid the ubiquitous computational irreducibility that exists in rule space, with the result that when we “look inside” biological systems we tend to see computational irreducibility and the complexity associated with it.
Something that we’ve seen over and over again here is that, yes, adaptive evolution manages to “solve a problem”. But its solution looks very complex to us. There might be some “simple engineering solution”—involving, say, a very regular pattern of behavior. But that’s not what adaptive evolution finds; instead it finds something that to us is typically very surprising—very often an “unexpectedly clever” solution in which lots of pieces fit together just right, in a way that our usual “understand-what’s-going-on” engineering practices would never let us invent.
We might not have expected anything like this to emerge from the simple process of adaptive evolution. But—as the models we’ve studied here highlight—it seems to be an inevitable formal consequence of core features of computational systems. And as soon as we recognize that biological systems can be viewed as computational, then it also becomes something inevitable for them—and something we can view as in a sense formally derivable for them.
At the outset we might not have been able to say “what matters” in the emergence of complexity in biology. But from the models we’ve studied, and the arguments we’ve made, we seem to have quite firmly established that it’s a fundamentally computational phenomenon, that relies only on certain general computational features of biological systems, and doesn’t depend on their particular detailed components and structure.
But in the end, how “generic” is the complexity that comes out of adaptive evolution? In other words, if we were to pick programs, say completely at random, how different would the complexity they produce be from the complexity we see in programs that have been adaptively evolved “for a purpose”? The answer isn’t clear—though to know it would provide important foundational input for theoretical biology.
One has the general impression that computational irreducibility is a strong enough phenomenon that it’s the “dominant force” that determines behavior and produces complexity. But there’s still usually something a bit different about the patterns we see from rules we’ve found by adaptive evolution, compared to rules we pick at random. Often there seems to be a certain additional level of “apparent mechanism”. The details still look complicated and in some ways quite random, but there seems to be a kind of “overall orchestration” to what’s going on.
And whenever we can identify such regularities it’s a sign of some kind of computational reducibility. There’s still plenty of computational irreducibility at work. But “high fitness” rules that we find through adaptive evolution typically seem to exhibit traces of their specialness—that manifests in at least a certain amount of computational reducibility.
Whenever we manage to come up with a “narrative explanation” or a “natural law” for something, it’s a sign that we’ve found a pocket of computational reducibility. If we say that a cellular automaton manages to live long because it generates certain robust geometric patterns—or, for that matter, that an organism lives long because it proofreads its DNA—we’re giving a narrative that’s based on computational reducibility.
And indeed whenever we can successfully identify a “mechanism” in our cellular automaton behavior, we’re in effect seeing computational reducibility. But what can we say about the aggregate of a whole collection of mechanisms?
In a different context I’ve discussed the concept of a “mechanoidal phase”, distinguished, say, from solids and liquids by the presence of a “bulk orchestration” of underlying components. It’s something closely related to class 4 behavior. And it’s interesting to note that if we look, for example, at the rules we found from adaptive evolution at the end of the previous section, their evolution from random initial conditions mostly shows characteristic class 4 behavior:
In other words, adaptive evolution is potentially bringing us to “characteristically special” places in rule space—perhaps suggesting that there’s something “characteristically special” about the kind of structures that are produced in biological systems. And if we could find a way to make general statements about that “characteristic specialness” it would potentially lead us to a framework for constructing a new broad formal theory in biology.
The models we’ve studied here are extremely simple in their basic construction. And at some level it’s remarkable that—without for example including any biophysics or biochemistry—they can get anywhere at all in capturing features of biological systems and biological evolution.
In a sense this is ultimately a reflection of the fundamentally computational character of biology—and the generality of computational phenomena. But it’s very striking that even the patterns of cellular automaton behavior we see look very “lifelike and organic”.
In actual biology even the shortest genomes are vastly longer than the tiny cellular automaton rules we’ve considered. But even by the time we’re looking at the length-27 “genomic sequences” in k = 3, r = 1 cellular automata, there are already 3 trillion possible sequences, which seems to be enough to see many core “combinatorially driven” biology-like phenomena.
The running of a cellular automaton rule might also at first seem far away from the actual processes that create biological organisms—involving as they do things like the construction of proteins and the formation of elaborate functional and spatial structures. But there are more analogies than one might at first imagine. For example, it’s common for only particular cases in the cellular automaton rule to be used in a given region of the pattern that’s formed, much as particular genes are typically turned on in different tissues in biological organisms.
And, for example, the “geometrical restriction” to a simple 1D array of “cells” doesn’t seem to matter much as soon as there’s sophisticated computation going on; we still get lots of structures that are actually surprisingly reminiscent of typical patterns of biological growth.
One of the defining features of biological organisms is their capability for self-reproduction. And indeed if it wasn’t for this kind of “copying” there wouldn’t be anything like adaptive evolution to discuss. Our models don’t attempt to derive self-reproduction; they just introduce it as something built into the models.
And although we’ve considered several variants, we’re basically also just building into our models the idea of mutations. And what we find is that it seems as if single point mutations made one at a time are enough to capture basic features of adaptive evolution.
We’ve also primarily considered what amounts to a single lineage—in which there’s just a single rule (or genome) at a given step. We do mutations, and we mostly “implement natural selection” just by keeping only rules that lead to patterns whose fitness is no less than what we had before.
If we had a whole population of rules it probably wouldn’t be so significant, but in the simple setup we’re using, it turns out to be important that we don’t reject “fitness-neutral mutations”. And indeed we’ve seen many examples where the system wanders around fitness-neutral regions of rule space before finally “discovering” some “innovation” that allows it to increase fitness. The way our models are set up, that “wandering” always involves changes in the “genotype”—but usually at most minor changes in the phenotype. So it’s very typical to see long periods of “apparent equilibrium” in which the phenotype changes rather little, followed by a “jump” to a new fitness level and a rather different phenotype.
And this observation seems quite aligned with the phenomenon of “punctuated equilibrium” often reported in the fossil record of actual biological evolution.
Another key feature of biological organisms and biological evolution is the formation of distinct species, as well as distinct phyla, etc. And indeed we ubiquitously see something that seems to be directly analogous in our multiway graphs of all possible paths of adaptive evolution. Typically we see distinct branches forming, based on what seem like “different mechanisms” for achieving fitness.
No doubt in actual biology there are all sorts of detailed phenomena related to reproductive or spatial isolation. But in our models the core phenomenon that seems to lead to the analog of “branching in the tree of life” is the existence of “distinctly different computational mechanisms” in different parts of rule space. It’s worth noting that at least with our finite rule spaces, branches can die out, with no “successor species” appearing the multiway graph.
And indeed looking at the actual patterns produced by rules in different parts of the multiway graph it’s easy to imagine morphologically based taxonomic classifications—that would be somewhat, though not perfectly, aligned with the phylogenetic tree defined by actual rule mutations. (At a morphological level we quite often see some level of “convergent evolution” in our multiway graphs; in small examples we sometimes also see actual “genomic convergence”—which will typically be astronomically rare in actual biological systems.)
One of the remarkable features of our models is that they allow quite global investigation of the “overall history” of adaptive evolution. In many of the simple cases we’ve discussed, the rule space we’re using is small enough that in a comparatively modest number of mutation steps we get to the “highest fitness we can reach”. But (as the examples we saw in Turing machines suggest) expanding the size of the rules we’re using even just a little can be expected to be sufficient to allow us to get astronomically further.
And the further we go, the more “mechanisms” will be “invented”. It’s an inevitable feature of systems that involve computational irreducibility that there are new and unpredictable things that will go on showing up forever—along with new pockets of computational reducibility. So even after a few billion years—and the trillion generations and 1040 or so organisms that have ever lived—there’s still infinitely further for biological evolution to go, and more and more branches to be initiated in the tree of life, involving more and more “new mechanisms”.
I suppose one might imagine that at some point biological organisms would reach a “maximum fitness”, and go no further. But even in our simple model with fitness measured in terms of pattern lifetime, there’ll be no upper limit on fitness; given any particular lifetime, it’s a feature of the fundamental theory of computation that there’ll always be a program that will yield a larger lifetime. Still, one might think, at some point enough is enough: the giraffe’s neck is long enough, etc. But if nothing else, competition between organisms will always drive things forward: yes, a particular lineage of organisms achieved a certain fitness, but then another lineage can come along and get to that fitness too, forcing the first lineage to go even further so as to not lose out.
Of course, in our simple model we’re not explicitly accounting for interactions with other organisms—or for detailed properties of the environment, as well as countless other effects. And no doubt there are many biological phenomena that depend on these effects. But the key point here is that even without explicitly accounting for any of these effects, our simple model still seems to capture many core features of biological evolution. Biological evolution—and, indeed, adaptive evolution in general—is, it seems, fundamentally a computational phenomenon that robustly emerges quite independent of the details of systems.
In the past few years our Physics Project has given strong evidence that the foundations of physics are fundamentally computational—with the core laws of physics arising as inevitable consequences of the way observers like us “parse” the ruliad of all possible computational processes. And what we’ve seen here now suggests that there’s a remarkable commonality between the foundations of physics and biology. Both are anchored in computational irreducibility. And both sample slices of computational reducibility. Physics because that’s what observers like us do to get descriptions of the world that fit in our finite minds. Biology because that’s what biological evolution does in order to achieve the “coarse objectives” set by natural selection.
The intuition of physics tends to be that there are ultimately simple models for things, whereas in biology there’s a certain sense that everything is always almost infinitely complicated, with a new effect to consider at every turn. But presumably that’s in large part because what we study in biology tends to quickly come face to face with computational irreducibility—whereas in physics we’ve been able to find things to study that avoid this. But now the commonality in foundations between physics and biology suggests that there should also be in biology the kind of structure we have in physics—complete with general laws that allow us to make useful, broad statements. And perhaps the simple model I’ve presented here can help lead us there—and in the end help build up a new paradigm for thinking about biology in a fundamentally theoretical way.
There’s a long—if circuitous—history to the things I’m discussing here. Basic notions of heredity—particularly for humans—were already widely recognized in antiquity. Plant breeding was practiced from the earliest days of agriculture, but it wasn’t until the late 1700s that any kind of systematic selective breeding of animals began to be commonplace. Then in 1859 Charles Darwin described the idea of “natural selection” whereby competition of organisms in their natural environment could act like artificial selection, and, he posited, would over long periods lead to the development of new species. He ended his Origin of Species with the claim that:
… from the war of nature … the production of the higher animals, directly follows. … and whilst this planet has gone cycling on according to the fixed law of gravity, from so simple a beginning endless forms most beautiful and most wonderful have been, and are being, evolved.
What he appears to have thought is that there would somehow follow from natural selection a general law—like the law of gravity—that would lead to the evolution of progressively more complex organisms, culminating in the “higher animals”. But absent the kind of model I’m discussing here, nothing in the later development of traditional evolutionary theory really successfully supported this—or was able to give much analysis of it.
Right around the same time as Darwin’s Origin of Species, Gregor Mendel began to identify simple probabilistic laws of inheritance—and when his work was rediscovered at the beginning of the 1900s it was used to develop mathematical models of the frequencies of genetic traits in populations of organisms, with key contributions to what became the field of population genetics being made in the 1920s and 1930s by J. B. S. Haldane, R. A. Fisher and Sewall Wright, who came up with the concept of a “fitness landscape”.
On a quite separate track there had been efforts ever since antiquity to classify and understand the growth and form of biological organisms, sometimes by analogy to physical or mathematical ideas—and by the 1930s it seemed fairly clear that chemical messengers were somehow involved in the control of growth processes. But the mathematical methods used for example in population genetics basically only handled discrete traits (or simple numerical ones accessible to biometry), and didn’t really have anything to say about something like the development of complexity in the forms of biological organisms.
The 1940s saw the introduction of what amounted to electrical-engineering-inspired approaches to biology, often under the banner of cybernetics. Idealized neural networks were introduced by Warren McCulloch and Walter Pitts in 1943, and soon the idea emerged (notably in the work of Donald Hebb in 1949) that learning in such systems could occur through a kind of adaptive evolution process. And by the time practical electronic computing began to emerge in the 1950s there was widespread belief that ideas from biology—including evolution—would be useful as an inspiration for what could be done. Often what would now just be described as adaptive algorithms were couched in biological evolution terms. And even when iterative methods were used for optimization (say in industrial production or engineering design) they were sometimes presented as being grounded in biological evolution.
Meanwhile, by the 1960s, there began to be what amounted to Monte Carlo simulations of population-genetics-style evolutionary processes. A particularly elaborate example was work by Nils Barricelli on what he called “numeric evolution” in which a fairly complicated numerical-cellular-automaton-like “competition between organisms” program with “randomness” injected from details of data layout in computer memory showed what he claimed to be biological-evolution-like phenomena (such as symbiosis and parasitism).
In a different direction there was an attempt—notably by John von Neumann—to “mathematicize the foundations of biology” leading by the late 1950s to what we’d now call 2D cellular automata “engineered” in complicated ways to show phenomena like self-reproduction. The followup to this was mostly early-theoretical-computer-science work, with no particular connection to biology, and no serious mention of adaptive evolution. When the Game of Life was introduced in 1970 it was widely noted as “doing lifelike things”, but essentially no scientific work was done in this direction. By the 1970s, though, L-systems and fractals had introduced the idea of recursive tree-like or nested structures that could be generated by simple algorithms and rendered by computer graphics—and seemed to give forms close to some seen in biology. My own work on 1D cellular automata (starting in 1981) focused on systematic scientific investigation of simple programs and what they do—with the surprising conclusion that even very simple programs can produce highly complex behavior. But while I saw this as informing the generation of complexity in things like the growth of biological organisms, I didn’t at the time (as I’ll describe below) end up seriously exploring any adaptive evolution angles.
Still another thread of development concerned applying biological-like evolution not just to parameters but to operations in programs. And for example in 1958 Richard Friedberg at IBM tried making random changes to instructions in machine-code programs, but didn’t manage to get this to do much. (Twenty years later, superoptimizers in practical compilers did begin to successfully use such techniques.) Then in the 1960s, John Holland (who had at first studied learning in neural nets, and was then influenced by Arthur Burks who had worked on cellular automata with von Neumann) suggested representing what amounted to programs by simple strings of symbols that could readily be modified like genomic sequences. The typical idea was to interpret the symbols as computational operations—and to assign a “fitness” based on the outcome of those operations. A “genetic algorithm” could then be set up by having a population of strings that was adaptively evolved. Through the 1970s and 1980s occasional practical successes were reported with this approach, particularly in optimization and data classification—with much being made of the importance of sexual-reproduction-inspired crossover operations. (Something that began to be used in the 1980s was the much simpler approach of simulated annealing—that involves randomly changing values rather than programs.)
By the beginning of the 1980s the idea had also emerged of adaptively modifying the structure of mathematical expressions—and of symbolic expressions representing programs. There were notable applications in computer graphics (e.g. by Karl Sims) as well as to things like the 1984 Core War “game” involving competition between programs in a virtual machine. In the 1990s John Koza was instrumental in developing the idea of “genetic programming”, notably as a way to “automatically create inventions”, for example in areas like circuit and antenna design. And indeed to this day scattered applications of these methods continue to pop up, particularly in geometrical and mechanical design.
From the very beginning there’d been controversy around Darwin’s ideas about evolution. First, there was the issue of conflict with religious accounts of creation. But there were also—often vigorous—disagreements within the scientific community about the interpretation of the fossil record and about how large-scale evolution was really supposed to operate. A notable issue—still very active in the 1980s—was the relationship between the “freedom of evolution” and the constraints imposed by the actual dynamics of growth in organisms (and interactions between organisms). And despite much insistence that the only reasonable “scientific” (as opposed to religious) point of view was that “natural selection is all there is”, there were nagging mysteries that suggested there must be other forces at work.
Building on the possibilities of computer experimentation (as well as things like my work on cellular automata) there emerged in the mid-1980s, particularly through the efforts of Chris Langton, a focus on investigating computational models of “artificial life”. This resulted in all sorts of simulations of ecosystems, etc. that did produce a variety of evolution-related phenomena known from field biology—but typically the models were far too complex in their structure for it to be possible to extract fundamental conclusions from them. Still, there continued to be specific, simpler experiments. For example, in 1986, for his book The Blind Watchmaker, Richard Dawkins made pictures of what he called “biomorphs”, produced by adaptively adjusting parameters for a simple tree-growth algorithm based on the overall shapes generated.
In the 1980s, stimulated by my work, there were various isolated studies of “rule evolution” in cellular automata (as well as art and museum exhibits based on this), and in the 1990s there was more systematic work—notably by Jim Crutchfield and Melanie Mitchell—on using genetic algorithms to try to evolve cellular automaton rules to solve tasks like density classification. (Around this time “evolutionary computation” also began to emerge as a general term covering genetic algorithms and other usually-biology-inspired adaptive computational methods.)
Meanwhile, accelerating in the 1990s, there was great progress in understanding actual molecular mechanisms in biology, and in figuring out how genetic and developmental processes work. But even as huge amounts of data accumulated, enthusiasm for large-scale “theories of biology” (that might for example address the production of complexity in biological evolution) seemed to wane. (The discipline of systems biology attempted to develop specific, usually mathematical, models for biological systems—but there never emerged much in the way of overarching theoretical principles, except perhaps, somewhat specifically, in areas like immunology and neuroscience.)
One significant exception in terms of fundamental theory was Greg Chaitin’s concept from around 2010 of “metabiology”: an effort (see below) to use ideas from the theory of computation to understand very general features of the evolution of programs and relate them to biological evolution.
Starting in the 1950s another strand of development (sometimes viewed as a practical branch of artificial intelligence) involved the idea of “machine learning”. Genetic algorithms were one of half a dozen common approaches. Another was based on artificial neural nets. For decades machine learning languished as a somewhat esoteric field, dominated by engineering solutions that would occasionally deliver specific application results. But then in 2011 there was unexpectedly dramatic success in using neural nets for image identification, followed in subsequent years by successes in other areas, and culminating in 2022 with the arrival of large language models and ChatGPT.
What hadn’t been anticipated was that the behavior of neural nets can change a lot if they’re given sufficiently huge amounts of training. But there isn’t any good understanding of just why this is so, and just how successful neural nets can be at what kinds of tasks. Ever since the 1940s it has been recognized that there are relations between biological evolution and learning in neural nets. And having now seen the impressive things neural nets can do, it seems worthwhile to look again at what happens in biological evolution—and to try to understand why it works, not least as a prelude to understanding more about neural nets and machine learning.
It’s strange to say, but most of what I’ve done here I should really have done forty years ago. And I almost did. Except that I didn’t try quite the right experiments. And I didn’t have the intuition to think that it was worth trying more.
Forty years later, I have new intuition, particularly informed by experience with modern machine learning. But even now, what made possible what I’ve done here was a chance experiment done for a somewhat different purpose.
Back in 1981 I had become very interested in the question of how complexity arises in the natural world, and I was trying to come up with models that might capture this. Meanwhile, I had just finished Version 1.0 of SMP, the forerunner to Mathematica and the Wolfram Language—and I was wondering how one might generalize its pattern-matching paradigm to “general AI”.
As it happened, right around that time, neural nets gained some (temporary) popularity. And seeing them as potentially relevant to both my topics I started simulating them and trying to see what kind of general theory I could develop about them. But I found them frustrating to work with. There seemed to be too many parameters and details to get any clear conclusions. And, at a practical level, I couldn’t get them to do anything particularly useful.
I decided that for my science question I needed to come up with something much simpler. And as a kind of minimal merger of spin systems and neural nets I ended up inventing cellular automata (only later did I discover that versions of them had been invented several times before).
As soon as I started doing experiments on them, I discovered that cellular automata were a window into an amazing new scientific world—that I have continued to explore in one way or another ever since. My key methodology, at least at first, was just to enumerate the simplest possible cellular automaton rules, and see what they did. The diversity—and complexity—of their behavior was remarkable. But the simplicity of the rules meant that the details of “successive rules” were usually fairly different—and while there were common themes in their overall behavior, there didn’t seem to be any particular structure to “rule space”. (Occasionally, though, particularly in finding examples for exposition, I would look at slightly more complicated and “multicolored” rules, and I certainly anecdotally noticed that rules with nearby rule numbers often had definite similarities in their behavior.)
It so happened that around the time I started publishing about cellular automata in 1983 there was a fair amount of ambient interest in theoretical biology. And (perhaps in part because of the “cellular” in “cellular automata”) I was often invited to theoretical biology conferences. People would sometimes ask about adaptation in cellular automata, and I would usually just emphasize what individual cellular automata could do, without any adaptation, and what significance it might have for the development of organisms.
But in 1985 I was going to a conference (at Los Alamos) on “Evolution, Games and Learning” and I decided I should take a look at the relation of these topics to cellular automata. But, too quickly, I segued away from investigating adaptation to trying to see what kind of pattern matching and other operations cellular automata might be able to be explicitly set up to do:
Many aspects of this paper still seem quite modern (and in fact should probably be investigated more now!). But—even though I absolutely had the tools to do it—I simply failed at that time to explore what I’ve now explored here.
Back in 1984 Problem 7 in my “Twenty Problems in the Theory of Cellular Automata” was “How is different behavior distributed in the space of cellular automaton rules?” And over the years I’d occasionally think about “cellular automaton rule space”, wondering, for example, what kind of geometry it might have, particularly in the continuum limit of infinitely large rules.
By the latter half of the 1980s “theoretical biology” conferences had segued to “artificial life” ones. And when I went to such conferences I was often frustrated. People would show me simulations that seemed to have far too many parameters to ever be able to conclude much. People would also often claim that natural selection was a “very simple theory”, but as soon as it was “implemented” there’d be all kinds of issues—and choices to be made—about population sizes, fitness cutoffs, interactions between organisms, and so on. And the end result was usually a descent into some kind of very specific simulation without obvious robust implications.
(In the mid-1980s I put a fair amount of effort into developing both the content and the organization of a new direction in science that I called “complex systems research”. My emphasis was on systems—like cellular automata—that had definite simple rules but highly complex behavior. Gradually, though, “complexity” started to be a popular general buzzword, and—I suspect partly to distinguish themselves from my efforts—some people started emphasizing that they weren’t just studying complex systems, they were studying complex adaptive systems. But all too often this seemed mostly to provide an excuse to dilute the clarity of what could be studied—and I was sufficiently put off that I paid very little attention.)
By the mid-1990s, I was in the middle of writing A New Kind of Science, and I wanted to use biology as an example application of my methodology and discoveries in the computational universe. In a section entitled “Fundamental Issues in Biology” I argued (as I have here) that computational irreducibility is a fundamentally stronger force than natural selection, and that when we see complexity in biology it’s most likely of “computational origin” rather than being “sculpted” by natural selection. And as part of that discussion, I included a picture of the “often-somewhat-gradual changes” in behavior that one sees with successive 1-bit changes in a k = 3,
This wasn’t done adaptively; it was basically just looking along a “random straight line” in rule space. And indeed both here and in most of the book, I was concerned with what systems like cellular automata “naturally do”, not what they can be constructed (or adaptively evolved) to do. I did give “constructions” of how cellular automata can perform particular computational tasks (like generating primes), and, somewhat obscurely, in a section on “Intelligence in the Universe” I explored finding k = 3, r = 1 rules that can successfully “double their input” (my reason for discussing these rules was to highlight the difficulty of saying whether one of these cellular automata was “constructed for a purpose” or was just “doing what it does”):
Many years went by. There’d be an occasional project at our Summer School about rule space, and occasionally about adaptation. I maintained an interest in foundational questions in biology, gradually collecting information and sometimes giving talks about the subject. Meanwhile—though I didn’t particularly internalize the connection then—by the mid-2010s, through our practical work on it in the Wolfram Language, I’d gotten quite up to speed with modern machine learning. Around the same time I also heard from my friend Greg Chaitin about his efforts (as he put it) to “prove Darwin” using the kind of computational ideas he’d applied in thinking about the foundations of mathematics.
Then in 2020 came our Physics Project, with its whole formalism around things like multiway graphs. It didn’t take long to realize that, yes, what I was calling “multicomputation” wasn’t just relevant for fundamental physics; it was something quite general that could be applied in many areas, which by 2021 I was trying to catalog:
I did some thinking about each of these. The one I tackled most seriously first was metamathematics, about which I finished a book in 2022. Late that year I was finishing a (50-year-in-gestation) project—informed by our Physics Project—on understanding the Second Law of thermodynamics, and as part of this I made what I thought was some progress on thinking about the fundamental character of biological systems (though not their adaptive evolution).
And then ChatGPT arrived. And in addition to being involved with it technologically, I started to think about the science of it, and particularly about how it could work. Part of it seemed to have to do with unrecognized regularities in human language, but part of it was a reflection of the emerging “meta discovery” that somehow if you “bashed” a machine learning system hard enough, it seemed like it could manage to learn almost anything.
But why did this work? At first I thought it must just be an “obvious” consequence of high dimensionality. But I soon realized there was more to it. And as part of trying to understand the boundaries of what’s possible I ended up a couple of months ago writing a piece exploring “Can AI Solve Science?”:
I talked about different potential objectives for science (making predictions, generating narrative explanations, etc.) And deep inside the piece I had a section entitled “Exploring Spaces of Systems” in which I talked about science problems of the form “Can one find a system that does X?”—and asked whether systems like neural nets could somehow let one “jump ahead” in what would otherwise be huge exhaustive searches. As a sideshow to this I thought it might be interesting to compare with what a non-neural-net adaptive evolution process could do.
Remembering Greg Chaitin’s ideas about connecting the halting problem to biological evolution I wondered if perhaps one could just adaptively evolve cellular automaton rules to find ones that generated a pattern with a particular finite lifetime. I imagined it as a classic machine learning problem, with a “loss function” one needed to minimize.
And so it was that just after 1 am on February 22 I wrote three lines of Wolfram Language code—and tried the experiment:
And it worked! I managed to find cellular automaton rules that would generate patterns living exactly 50 steps:
In retrospect, I was slightly lucky. First, that this ended up being such a simple experiment to try (at least in the Wolfram Language) that I did it even though I didn’t really expect it to work. And second, that for my very first experiment I picked parameters that happened to immediately work (k = 4, lifetime 50, etc.).
But, yes, I could in principle have done the same experiment 40 years ago, though without the Wolfram Language it wouldn’t have been so easy. Still, the computers I had back then were powerful enough that I could in principle have generated the same results then as now. But without my modern experience of machine learning I don’t think I would have tried—and I would certainly have given up too easily. And, yes, it’s a little humbling to realize that I’ve gone so many years assuming adaptive evolution was out of the reach of simple, clean experiments. But it’s satisfying now to be able to check off another mystery I’ve long wondered about. And to think that much more about the foundations of biology—and machine learning—might finally be within reach.
Thanks to Brad Klee, Nik Murzin and Richard Assar for their help.
The specific results and ideas I’ve presented here are mostly very recent, but they build on background conversations I’ve had—some recently, some more than 40 years ago—with many people, including: Sydney Brenner, Greg Chaitin, Richard Dawkins, David Goldberg, Nigel Goldenfeld, Jack Good, Jonathan Gorard, Stephen J. Gould, Hyman Hartman, John Holland, Christian Jacob, Stuart Kauffman, Mark Kotanchek, John Koza, Chris Langton, Katja Della Libera, Aristid Lindenmayer, Pattie Maes, Bill Mydlowec, John Novembre, Pedro de Oliveira, George Oster, Norman Packard, Alan Perelson, Thomas Ray, Philip Rosedale, Robert Rosen, Terry Sejnowski, Brian Silverman, Karl Sims, John Maynard Smith, Catherine Wolfram, Christopher Wolfram and Elizabeth Wolfram.
Updated and expanded from a post for the eclipse of August 21, 2017.
On April 8, 2024, there’s going to be a total eclipse of the Sun visible on a line across the US. But when exactly will the eclipse occur at a given location? Being able to predict astronomical events has historically been one of the great triumphs of exact science. But how well can it actually be done now?
The answer is well enough that even though the edge of totality moves at just over 1000 miles per hour, it’s possible to predict when it will arrive at a given location to within perhaps a second. And as a demonstration of this, for the total eclipse back in 2017 we created a website to let anyone enter their geo location (or address) and then immediately compute when the eclipse would reach them—as well as generate many pages of other information.
These days it’s easy to find out when the next solar eclipse will be; indeed built right into the Wolfram Language there’s just a function that tells you (in this form the output is the “time of greatest eclipse”):
It’s also easy to find out, and plot, where the region of totality will be:
Or to determine that the whole area of totality (including lots of ocean and some of Canada) will be about a third of the area of the US:
But computing eclipses is not exactly a new business. In fact, the Antikythera device from 2000 years ago even tried to do it—using 37 metal gears to approximate the motion of the Sun and Moon (yes, with the Earth at the center). To me there’s something unsettling—and cautionary—about the fact that the Antikythera device stands as such a solitary piece of technology, forgotten but not surpassed for more than 1600 years.
But right there on the bottom of the device there’s an arm that moves around, and when it points to an Η or Σ marking, it indicates a possible Sun or Moon eclipse. The way of setting dates on the device is a bit funky (after all, the modern calendar wouldn’t be invented for another 1500 years), but if one takes the simulation on the Wolfram Demonstrations Project (which was calibrated back in 2012 when the Demonstration was created), and turns the crank to set the device for April 8, 2024, here’s what one gets:
And, yes, all those gears move so as to line the Moon indicator up with the Sun—and to make the arm on the bottom point right at an H—just as it should for a solar eclipse. It’s amazing to see this computation successfully happen on a device designed 2000 years ago.
Of course the results are a lot more accurate today. Though, strangely, despite all the theoretical science that’s been done, the way we actually compute the position of the Sun and Moon is conceptually very much like the gears—and effectively epicycles—of the Antikythera device. It’s just that now we have the digital equivalent of hundreds of thousands of gears.
A total solar eclipse occurs when the Moon gets in front of the Sun from the point of view of a particular location on the Earth. And it so happens that at this point in the Earth’s history the Moon can just block the Sun because it has almost exactly the same angular diameter in the sky as the Sun (about 0.5° or 30 arc-minutes).
So when does the Moon get between the Sun and the Earth? Well, basically every time there’s a new moon (i.e. once every lunar month). But we know there isn’t an eclipse every month. So how come?
Well, actually, in the analogous situation of Ganymede and Jupiter, there is an eclipse every time Ganymede goes around Jupiter (which happens to be about once per week). Like the Earth, Jupiter’s orbit around the Sun lies in a particular plane (the “plane of the ecliptic”). And it turns out that Ganymede’s orbit around Jupiter also lies in essentially the same plane. So every time Ganymede reaches the “new moon” position (or, in official astronomy parlance, when it’s aligned “in syzygy”—pronounced sizz-ee-gee), it’s in the right place to cast its shadow onto Jupiter, and to eclipse the Sun wherever that shadow lands. (From Jupiter, Ganymede appears about 3 times the size of the Sun.)
But our Moon is different. Its orbit doesn’t lie in the plane of the ecliptic. Instead, it’s inclined at about 5°. (How it got that way is unknown, but it’s presumably related to how the Moon was formed.) But that 5° is what makes eclipses so comparatively rare: they can only happen when there’s a “new moon configuration” (syzygy) right at a time when the Moon’s orbit passes through the plane of the ecliptic.
To show what’s going on, let’s draw an exaggerated version of everything. Here’s the Moon going around the Earth, colored red whenever it’s close to the plane of the ecliptic:
Now let’s look at what happens over the course of about a year. We’re showing a dot for where the Moon is each day. And the dot is redder if the Moon is closer to the plane of the ecliptic that day. (Note that if this was drawn to scale, you’d barely be able to see the Moon’s orbit, and it wouldn’t ever seem to go backwards like it does here.)
Now we can start to see how eclipses work. The basic point is that there’s a solar eclipse whenever the Moon is both positioned between the Earth and the Sun, and it’s in the plane of the ecliptic. In the picture, those two conditions correspond to the Moon being as far as possible towards the center, and as red as possible. So far we’re only showing the position of the (exaggerated) Moon once per day. But to make things clearer, let’s show it four times a day—and now prune out cases where the Moon isn’t at least roughly lined up with the Sun:
And now we can see that at least in this particular case, there are two points (indicated by arrows) where the Moon is lined up and in the plane of the ecliptic (so shown in red)—and these points will then correspond to solar eclipses.
In different years, the picture will look slightly different, essentially because the Moon is starting at a different place in its orbit at the beginning of the year. Here are schematic pictures for a few successive years:
It’s not so easy to see exactly when eclipses occur here—and it’s also not possible to tell which are total eclipses where the Moon is exactly lined up, and which are only partial eclipses. But there’s at least an indication, for example, that there are “eclipse seasons” in different parts of the year where eclipses happen.
OK, so what does the real data look like? Here’s a plot for 20 years in the past and 20 years in the future, showing the actual days in each year when total and partial solar eclipses occur (the small dots everywhere indicate new moons):
The reason for the “drift” between successive years is just that the lunar month (29.53 days) doesn’t line up with the year, so the Moon doesn’t go through a whole number of orbits in the course of a year, with the result that at the beginning of a new year, the Moon is in a different phase. But as the picture makes clear, there’s quite a lot of regularity in the general times at which eclipses occur—and for example there are usually 2 eclipses in a given year—though there can be more (and in 0.2% of years there can be as many as 5, as there last were in 1935).
To see more detail about eclipses, let’s plot the time differences (in fractional years) between all successive solar eclipses for 100 years in the past and 100 years in the future:
And now let’s plot the same time differences, but just for total solar eclipses:
There’s obviously a fair amount of overall regularity here, but there are also lots of little fine structure and irregularities. And being able to correctly predict all these details has basically taken science the better part of a few thousand years.
It’s hard not to notice an eclipse, and presumably even from the earliest times people did. But were eclipses just reflections—or omens—associated with random goings-on in the heavens, perhaps in some kind of soap opera among the gods? Or were they things that could somehow be predicted?
A few thousand years ago, it wouldn’t have been clear what people like astrologers could conceivably predict. When will the Moon be at a certain place in the sky? Will it rain tomorrow? What will the price of barley be? Who will win a battle? Even now, we’re not sure how predictable all of these are. But the one clear case where prediction and exact science have triumphed is astronomy.
At least as far as the Western tradition is concerned, it all seems to have started in ancient Babylon—where for many hundreds of years, careful observations were made, and, in keeping with the ways of that civilization, detailed records were kept. And even today we still have thousands of daily official diary entries written in what look like tiny chicken scratches preserved on little clay tablets. “Night of the 14th: Cold north wind. Moon was in front of α Leonis. From 15th to 20th river rose 1/2 cubit. Barley was 1 kur 5 siit. 25th, last part of night, moon was 1 cubit 8 fingers behind ε Leonis. 28th, 74° after sunrise, solar eclipse…”
If one looks at what happens on a particular day, one probably can’t tell much. But by putting observations together over years or even hundreds of years, it’s possible to see all sorts of repetitions and regularities. And back in Babylonian times the idea arose of using these to construct an ephemeris—a systematic table that said where a particular heavenly body such as the Moon was expected to be at any particular time.
(Needless to say, reconstructing Babylonian astronomy is a complicated exercise in decoding what’s by now basically an alien culture. A key figure in this effort was a certain Otto Neugebauer, who happened to work down the hall from me at the Institute for Advanced Study in Princeton in the early 1980s. I would see him almost every day—a quiet white-haired chap, with a twinkle in his eye—and just sometimes I’d glimpse his huge filing system of index cards which I now realize was at the center of understanding Babylonian astronomy.)
One thing the Babylonians did was to measure surprisingly accurately the repetition period for the phases of the Moon—the so-called synodic month (or “lunation period”) of about 29.53 days. And they noticed that 235 synodic months was very close to 19 years—so that about every 19 years, dates and phases of the Moon repeat their alignment, forming a so-called Metonic cycle (named after Meton of Athens, who described it in 432 BC).
It probably helps that the random constellations in the sky form a good pattern against which to measure the precise position of the Moon (it reminds me of the modern fashion of wearing fractals to make motion capture for movies easier). But the Babylonians noticed all sorts of details of the motion of the Moon. They knew about its “anomaly”: its periodic speeding up and slowing down in the sky (now known to be a consequence of its slightly elliptical orbit). And they measured the average period of this—the so-called anomalistic month—to be about 27.55 days. They also noticed that the Moon went above and below the plane of the ecliptic (now known to be because of the inclination of its orbit)—with an average period (the so-called draconic month) that they measured as about 27.21 days.
And by 400 BC they’d noticed that every so-called saros of about 18 years 11 days all these different periods essentially line up (223 synodic months, 239 anomalistic months and 242 draconic months)—with the result that the Moon ends up at about the same position relative to the Sun. And this means that if there was an eclipse at one saros, then one can make the prediction that there’s going to be an eclipse at the next saros too.
When one’s absolutely precise about it, there are all sorts of effects that prevent precise repetition at each saros. But over timescales of more than 1300 years, there are in fact still strings of eclipses separated from each other by one saros. (Over the course of such a saros series, the locations of the eclipses effectively scan across the Earth; the upcoming eclipse is number 30 in a series of 71 that began in 1501 AD with an eclipse near the North Pole and will end in 2763 AD with an eclipse near the South Pole.)
Any given moment in time will be in the middle of quite a few saros series (right now it’s 40)—and successive eclipses will always come from different series. But knowing about the saros cycle is a great first step in predicting eclipses—and it’s for example what the Antikythera device uses. In a sense, it’s a quintessential piece of science: take many observations, then synthesize a theory from them, or at least a scheme for computation.
It’s not clear what the Babylonians thought about abstract, formal systems. But the Greeks were definitely into them. And by 300 BC Euclid had defined his abstract system for geometry. So when someone like Ptolemy did astronomy, they did it a bit like Euclid—effectively taking things like the saros cycle as axioms, and then proving from them often surprisingly elaborate geometrical theorems, such as that there must be at least two solar eclipses in a given year.
Ptolemy’s Almagest from around 150 AD is an impressive piece of work, containing among many other things some quite elaborate procedures—and explicit tables—for predicting eclipses. (Yes, even in the later printed version, numbers are still represented confusingly by letters, as they always were in ancient Greek.)
In Ptolemy’s astronomy, Earth was assumed to be at the center of everything. But in modern terms that just meant he was choosing to use a different coordinate system—which didn’t affect most of the things he wanted to do, like working out the geometry of eclipses. And unlike the mainline Greek philosophers he wasn’t trying to make a fundamental theory of the world; he just wanted whatever epicycles and so on he needed to fit what he observed.
For more than a thousand years Ptolemy’s theory of the Moon defined the state of the art. In the 1300s Ibn al-Shatir revised Ptolemy’s models, achieving somewhat better accuracy. In 1472 Regiomontanus (Johannes Müller), systematizer of trigonometry, published more complete tables as part of his launch of what was essentially the first-ever scientific publishing company. But even in 1543 when Nicolaus Copernicus introduced his Sun-centered model of the solar system, the results he got were basically the same as Ptolemy’s, even though his underlying description of what was going on was quite different.
It’s said that Tycho Brahe got interested in astronomy in 1560 at age 13 when he saw a solar eclipse that had been predicted—and over the next several decades his careful observations uncovered several effects in the motion of the Moon (such as speeding up just before a full moon)—that eventually resulted in perhaps a factor 5 improvement in the prediction of its position. To Tycho eclipses were key tests, and he measured them carefully, and worked hard to be able to predict their timing more accurately than to within a few hours. (He himself never saw a total solar eclipse, only partial ones.)
Armed with Tycho’s observations, Johannes Kepler developed his description of orbits as ellipses—introducing concepts like inclination and eccentric anomaly—and in 1627 finally produced his Rudolphine Tables, which got right a lot of things that had been got wrong before, and included all sorts of detailed tables of lunar positions, as well as vastly better predictions for eclipses.
Using Kepler’s Rudolphine Tables (and a couple of pages of calculations) the first known actual map of a solar eclipse was published in 1654. And while there are some charming inaccuracies in overall geography, the geometry of the eclipse isn’t too bad.
Whether it was Ptolemy’s epicycles or Kepler’s ellipses, there were plenty of calculations to do in determining the motions of heavenly bodies (and indeed the first known mechanical calculator—excepting the Antikythera device—was developed by a friend of Kepler’s, presumably for the purpose). But there wasn’t really a coherent underlying theory; it was more a matter of describing effects in ways that could be used to make predictions.
So it was a big step forward in 1687 when Isaac Newton published his Principia, and claimed that with his laws for motion and gravity it should be possible—essentially from first principles—to calculate everything about the motion of the Moon. (Charmingly, in his “Theory of the World” section he simply asserts as his Proposition XXII “That all the motions of the Moon… follow from the principles which we have laid down.”)
Newton was proud of the fact that he could explain all sorts of known effects on the basis of his new theory. But when it came to actually calculating the detailed motion of the Moon he had a frustrating time. And even after several years he still couldn’t get the right answer—in later editions of the Principia adding the admission that actually “The apse of the Moon is about twice as swift” (i.e. his answer was wrong by a factor of 2).
Still, in 1702 Newton was happy enough with his results that he allowed them to be published, in the form of a 20-page booklet on the “Theory of the Moon”, which proclaimed that “By this Theory, what by all Astronomers was thought most difficult and almost impossible to be done, the Excellent Mr. Newton hath now effected, viz. to determine the Moon’s Place even in her Quadratures, and all other Parts of her Orbit, besides the Syzygys, so accurately by Calculation, that the Difference between that and her true Place in the Heavens shall scarce be two Minutes…”
Newton didn’t explain his methods (and actually it’s still not clear exactly what he did, or how mathematically rigorous it was or wasn’t). But his booklet effectively gave a step-by-step algorithm to compute the position of the Moon. He didn’t claim it worked “at the syzygys” (i.e. when the Sun, Moon and Earth are lined up for an eclipse)—though his advertised error of two arc-minutes was still much smaller than the angular size of the Moon in the sky.
But it wasn’t eclipses that were the focus then; it was a very practical problem of his day: knowing the location of a ship out in the open ocean. It’s possible to determine what latitude you’re at just by measuring how high the Sun gets in the sky. But to determine longitude you have to correct for the rotation of the Earth—and to do that you have to accurately keep track of time. But back in Newton’s day, the clocks that existed simply weren’t accurate enough, especially when they were being tossed around on a ship.
But particularly after various naval accidents, the problem of longitude was deemed important enough that the British government in 1714 established a “Board of Longitude” to offer prizes to help get it solved. One early suggestion was to use the regularity of the moons of Jupiter discovered by Galileo as a way to tell time. But it seemed that a simpler solution (not requiring a powerful telescope) might just be to measure the position of our Moon, say relative to certain fixed stars—and then to back-compute the time from this.
But to do this one had to have an accurate way to predict the motion of the Moon—which is what Newton was trying to provide. In reality, though, it took until the 1760s before tables were produced that were accurate enough to be able to determine time to within a minute (and thus distance to within 15 miles or so). And it so happens that right around the same time a marine chronometer was invented that was directly able to keep good time.
One of Newton’s great achievements in the Principia was to solve the so-called two-body problem, and to show that with an inverse square law of gravity the orbit of one body around another must always be what Kepler had said: an ellipse.
In a first approximation, one can think of the Moon as just orbiting the Earth in a simple elliptical orbit. But what makes everything difficult is that that’s just an approximation, because in reality there’s also a gravitational pull on the Moon from the Sun. And because of this, the Moon’s orbit is no longer a simple fixed ellipse—and in fact it ends up being much more complicated. There are a few definite effects one can describe and reason about. The ellipse gets stretched when the Earth is closer to the Sun in its own orbit. The orientation of the ellipse precesses like a top as a result of the influence of the Sun. But there’s no way in the end to work out the orbit by pure reasoning—so there’s no choice but to go into the mathematics and start solving the equations of the three-body problem.
In many ways this represented a new situation for science. In the past, one hadn’t ever been able to go far without having to figure out new laws of nature. But here the underlying laws were supposedly known, courtesy of Newton. Yet even given these laws, there was difficult mathematics involved in working out the behavior they implied.
Over the course of the 1700s and 1800s the effort to try to solve the three-body problem and determine the orbit of the Moon was at the center of mathematical physics—and attracted a veritable who’s who of mathematicians and physicists.
An early entrant was Leonhard Euler, who developed methods based on trigonometric series (including much of our current notation for such things), and whose works contain many immediately recognizable formulas:
In the mid-1740s there was a brief flap—also involving Euler’s “competitors” Clairaut and d’Alembert—about the possibility that the inverse-square law for gravity might be wrong. But the problem turned out to be with the calculations, and by 1748 Euler was using sums of about 20 trigonometric terms and proudly proclaiming that the tables he’d produced for the three-body problem had predicted the time of a total solar eclipse to within minutes. (Actually, he had said there’d be 5 minutes of totality, whereas in reality there was only 1—but he blamed this error on incorrect coordinates he’d been given for Berlin.)
Mathematical physics moved rapidly over the next few decades, with all sorts of now-famous methods being developed, notably by people like Lagrange. And by the 1770s, for example, Lagrange’s work was looking just like it could have come from a modern calculus book (or from a Wolfram|Alpha step-by-step solution):
Particularly in the hands of Laplace there was increasingly obvious success in deriving the observed phenomena of what he called “celestial mechanics” from mathematics—and in establishing the idea that mathematics alone could indeed generate new results in science.
At a practical level, measurements of things like the position of the Moon had always been much more accurate than calculations. But now they were becoming more comparable—driving advances in both. Meanwhile, there was increasing systematization in the production of ephemeris tables. And in 1767 the annual publication began of what was for many years the standard: the British Nautical Almanac.
The almanac quoted the position of the Moon to the arc-second, and systematically achieved at least arc-minute accuracy. The primary use of the almanac was for navigation (and it was what started the convention of using Greenwich as the “prime meridian” for measuring time). But right at the front of each year’s edition were the predicted times of the eclipses for that year—in 1767 just two solar eclipses:
At a mathematical level, the three-body problem is about solving a system of three ordinary differential equations that give the positions of the three bodies as a function of time. If the positions are represented in standard 3D Cartesian coordinates 
, the equations can be stated in the form:
The {x, y, z} coordinates here aren’t, however, what traditionally show up in astronomy. For example, in describing the position of the Moon one might use longitude and latitude on a sphere around the Earth. Or, given that one knows the Moon has a roughly elliptical orbit, one might instead choose to describe its motions by variables that are based on deviations from such an orbit. In principle it’s just a matter of algebraic manipulation to restate the equations with any given choice of variables. But in practice what comes out is often long and complex—and can lead to formulas that fill many pages.
But, OK, so what are the best kinds of variables to use for the three-body problem? Maybe they should involve relative positions of pairs of bodies. Or relative angles. Or maybe positions in various kinds of rotating coordinate systems. Or maybe quantities that would be constant in a pure two-body problem. Over the course of the 1700s and 1800s many treatises were written exploring different possibilities.
But in essentially all cases the ultimate approach to the three-body problem was the same. Set up the problem with the chosen variables. Identify parameters that, if set to zero, would make the problem collapse to some easy-to-solve form. Then do a series expansion in powers of these parameters, keeping just some number of terms.
The calculations were difficult, and people’s results often didn’t agree. And for example in 1843 Ada Lovelace noted that “In the solution of the famous problem of the Three Bodies, there are, out of about 295 coefficients of lunar perturbations [that had recently been computed]… only 101… agree precisely both in signs and in amount [with previous works]” (going on to say, rather farsightedly, that this was something the Analytical Engine would be able to solve).
By the 1860s Charles Delaunay had, however, spent 20 years developing the most extensive theory of the Moon using series expansions. He’d identified five parameters with respect to which to do his expansions (eccentricities, inclinations, and ratios of orbit sizes)—and in the end he generated about 1800 pages like this (yes, he really needed Mathematica!):
But the sad fact was that despite all this effort, he didn’t get terribly good answers. And eventually it became clear why. The basic problem was that Delaunay wanted to represent his results in terms of functions like sin and cos. But in his computations, he often wanted to do series expansions with respect to the frequencies of those functions. Here’s a minimal case:
And here’s the problem. Take a look even at the second term. Yes, the δ parameter may be small. But how about the 
parameter, standing for time? If you don’t want to make predictions very far out, that’ll stay small. But what if you want to figure out what will happen further in the future?
Well, eventually that term will get big. And higher-order terms will get even bigger. But unless the Moon is going to escape its orbit or something, the final mathematical expressions that represent its position can’t have values that are too big. So in these expressions the so-called secular terms that increase with 
must somehow cancel out.
But the problem is that at any given order in the series expansion, there’s no guarantee that will happen in a numerically useful way. And in Delaunay’s case—even though with immense effort he often went to 7th order or beyond—it didn’t.
One nice feature of Delaunay’s computation was that it was in a sense entirely algebraic: everything was done symbolically, and only at the very end were actual numerical values of parameters substituted in.
But even before Delaunay, Peter Hansen had taken a different approach—substituting numbers as soon as he could, and dropping terms based on their numerical size rather than their symbolic form. His presentations look less pure (notice things like all those 
, where 
is the time in years), and it’s more difficult to tell what’s going on. But as a practical matter, his results were much better, and in fact were used for many national almanacs from about 1862 to 1922, achieving errors as small as 1 or 2 arc-seconds at least over periods of a decade or so. (Over longer periods, the errors could rapidly increase because of the lack of terms that had been dropped as a result of what amounted to numerical accidents.)
Both Delaunay and Hansen tried to represent orbits as series of powers and trigonometric functions (so-called Poisson series). But in the 1870s, George Hill in the US Nautical Almanac Office proposed instead using as a basis numerically computed functions that came from solving an equation for two-body motion with a periodic driving force of roughly the kind the Sun exerts on the Moon’s orbit. A large-scale effort was mounted, and starting in 1892 Ernest W. Brown (who had moved to the US, but had been a student of George Darwin, Charles Darwin’s physicist son) took charge of the project and in 1918 produced what would stand for many years as the definitive “Tables of the Motion of the Moon”.
Brown’s tables consist of hundreds of pages like this—ultimately representing the position of the Moon as a combination of about 1400 terms with very precise coefficients:
He says right at the beginning that the tables aren’t particularly intended for unique events like eclipses, but then goes ahead and does a “worked example” of computing an eclipse from 381 BC, reported by Ptolemy:
It was an impressive indication of how far things had come. But ironically enough the final presentation of Brown’s tables had the same sum-of-trigonometric-functions form that one would get from having lots of epicycles. At some level it’s not surprising, because any function can ultimately be represented by epicycles, just as it can be represented by a Fourier or other series. But it’s a strange quirk of history that such similar forms were used.
It’s all well and good that one can find approximations to the three-body problem, but what about just finding an outright solution—like as a mathematical formula? Even in the 1700s, there’d been some specific solutions found—like Euler’s collinear configuration, and Lagrange’s equilateral triangle. But a century later, no further solutions had been found—and finding a complete solution to the three-body problem was beginning to seem as hopeless as trisecting an angle, solving the quintic, or making a perpetual motion machine. (That sentiment was reflected for example in a letter Charles Babbage wrote Ada Lovelace in 1843 mentioning the “horrible problem [of] the three bodies”—even though this letter was later misinterpreted by Ada’s biographers to be about a romantic triangle, not the three-body problem of celestial mechanics.)
In contrast to the three-body problem, what seemed to make the two-body problem tractable was that its solutions could be completely characterized by “constants of the motion”—quantities that stay constant with time (in this case notably the direction of the axis of the ellipse). So for many years one of the big goals with the three-body problem was to find constants of the motion.
In 1887, though, Heinrich Bruns showed that there couldn’t be any such constants of the motion, at least expressible as algebraic functions of the standard {x, y, z} position and velocity coordinates of the three bodies. Then in the mid-1890s Henri Poincaré showed that actually there couldn’t be any constants of the motion that were expressible as any analytic functions of the positions, velocities and mass ratios.
One reason that was particularly disappointing at the time was that it had been hoped that somehow constants of the motion would be found in n-body problems that would lead to a mathematical proof of the long-term stability of the solar system. And as part of his work, Poincaré also saw something else: that at least in particular cases of the three-body problem, there was arbitrarily sensitive dependence on initial conditions—implying that even tiny errors in measurement could be amplified to arbitrarily large changes in predicted behavior (the classic “chaos theory” phenomenon).
But having discovered that particular solutions to the three-body problem could have this kind of instability, Poincaré took a different approach that would actually be characteristic of much of pure mathematics going forward: he decided to look not at individual solutions, but at the space of all possible solutions. And needless to say, he found that for the three-body problem, this was very complicated—though in his efforts to analyze it he invented the field of topology.
Poincaré’s work all but ended efforts to find complete solutions to the three-body problem. It also seemed to some to explain why the series expansions of Delaunay and others hadn’t worked out—though in 1912 Karl Sundman did show that at least in principle the three-body problem could be solved in terms of an infinite series, albeit one that converges outrageously slowly.
But what does it mean to say that there can’t be a solution to the three-body problem? Galois had shown that there couldn’t be a solution to the generic quintic equation, at least in terms of radicals. But actually it’s still perfectly possible to express the solution in terms of elliptic or hypergeometric functions. So why can’t there be some more sophisticated class of functions that can be used to just “solve the three-body problem”?
Here are some pictures of what can actually happen in the three-body problem, with various initial conditions:
And looking at these immediately gives some indication of why it’s not easy to just “solve the three-body problem”. Yes, there are cases where what happens is fairly simple. But there are also cases where it’s not, and where the trajectories of the three bodies continue to be complicated and tangled for a long time.
So what’s fundamentally going on here? I don’t think traditional mathematics is the place to look. But I think what we’re seeing is actually an example of a general phenomenon I call computational irreducibility that I discovered in the 1980s in studying the computational universe of possible programs.
Many programs, like many instances of the three-body problem, behave in quite simple ways. But if you just start looking at all possible simple programs, it doesn’t take long before you start seeing behavior like this:
How can one tell what’s going to happen? Well, one can just keep explicitly running each program and seeing what it does. But the question is: is there some systematic way to jump ahead, and to predict what will happen without tracing through all the steps?
The answer is that in general there isn’t. And what I call the Principle of Computational Equivalence suggests that pretty much whenever one sees complex behavior, there won’t be.
Here’s the way to think about this. The system one’s studying is effectively doing a computation to work out what its behavior will be. So to jump ahead we’d in a sense have to do a more sophisticated computation. But what the Principle of Computational Equivalence says is that actually we can’t—and that whether we’re using our brains or our mathematics or a Turing machine or anything else, we’re always stuck with computations of the same sophistication.
So what about the three-body problem? Well, I strongly suspect that it’s an example of computational irreducibility: that in effect the computations it’s doing are as sophisticated as any computations that we can do, so there’s no way we can ever expect to systematically jump ahead and solve the problem. (We also can’t expect to just define some new finite class of functions that can just be evaluated to give the solution.)
I’m hoping that one day someone will rigorously prove this. There’s some technical difficulty, because the three-body problem is usually formulated in terms of real numbers that immediately have an infinite number of digits—but to compare with ordinary computation one has to require finite processes to set up initial conditions. (Ultimately one wants to show for example that there’s a “compiler” that can go from any program, say for a Turing machine, and can generate instructions to set up initial conditions for a three-body problem so that the evolution of the three-body problem will give the same results as running that program—implying that the three-body problem is capable of universal computation.)
I have to say that I consider Newton in a sense very lucky. It could have been that it wouldn’t have been possible to work out anything interesting from his theory without encountering the kind of difficulties he had with the motion of the Moon—because one would always be running into computational irreducibility. But in fact, there was enough computational reducibility and enough that could be computed easily that one could see that the theory was useful in predicting features of the world (and not getting wrong answers, like with the apse of the Moon)—even if there were some parts that might take two centuries to work out, or never be possible at all.
Newton himself was certainly aware of the potential issue, saying that at least if one was dealing with gravitational interactions between many planets then “to define these motions by exact laws admitting of easy calculation exceeds, if I am not mistaken, the force of any human mind”.
And even today it’s extremely difficult to know what the long-term evolution of the solar system will be.
It’s not particularly that there’s sensitive dependence on initial conditions: we actually have measurements that should be precise enough to determine what will happen for a long time. The problem is that we just have to do the computation—a bit like computing the digits of π—to work out the behavior of the 
-body problem that is our solar system.
Existing simulations show that for perhaps a few tens of millions of years, nothing too dramatic can happen. But after that we don’t know. Planets could change their order. Maybe they could even collide, or be ejected from the solar system. Computational irreducibility implies that at least after an infinite time it’s actually formally undecidable (in the sense of Gödel’s theorem or the halting problem) what can happen.
One of my children, when they were very young, asked me whether when dinosaurs existed the Earth could have had two moons. For years when I ran into celestial mechanics experts I would ask them that question—and it was notable how difficult they found it. Most now say that at least at the time of the dinosaurs we couldn’t have had an extra moon—though a billion years earlier it’s not clear.
We used to only have one system of planets to study. And the fact that there were (then) 9 of them used to be a classic philosopher’s example of a truth about the world that just happens to be the way it is, and isn’t “necessarily true” (like 2 + 2 = 4). But now of course we know about lots of exoplanets. And it’s beginning to look as if there might be a theory for things like how many planets a solar system is likely to have.
At some level there’s presumably a process like natural selection: some configurations of planets aren’t “fit enough” to be stable—and only those that are survive. In biology it’s traditionally been assumed that natural selection and adaptation is somehow what’s led to the complexity we see. But actually I suspect much of it is instead just a reflection of what generally happens in the computational universe—both in biology and in celestial mechanics. Now in celestial mechanics, we haven’t yet seen in the wild any particularly complex forms (beyond a few complicated gap structures in rings, and tumbling moons and asteroids). But perhaps elsewhere we’ll see things like those obviously tangled solutions to the three-body problem—that come closer to what we’re used to in biology.
It’s remarkable how similar the issues are across so many different fields. For example, the whole idea of using “perturbation theory” and series expansions that has existed since the 1700s in celestial mechanics is now also core to quantum field theory. But just like in celestial mechanics there’s trouble with convergence (maybe one should try renormalization or resummation in celestial mechanics). And in the end one begins to realize that there are phenomena—no doubt like turbulence or the three-body problem—that inevitably involve more sophisticated computations, and that need to be studied not with traditional mathematics of the kind that was so successful for Newton and his followers but with the kind of science that comes from exploring the computational universe.
But let’s get back to the story of the motion of the Moon. Between Brown’s tables, and Poincaré’s theoretical work, by the beginning of the 1900s the general impression was that whatever could reasonably be computed about the motion of the Moon had been computed.
Occasionally there were tests. Like for example in 1925, when there was a total solar eclipse visible in New York City, and the New York Times perhaps overdramatically said that “scientists [were] tense… [wondering] whether they or Moon is wrong as eclipse lags five seconds behind”. The fact is that a prediction accurate to 5 seconds was remarkably good, and we can’t do all that much better even today. (By the way, the actual article talks extensively about “Professor Brown”—as well as about how the eclipse might “disprove Einstein” and corroborate the existence of “coronium”—but doesn’t elaborate on the supposed prediction error.)
As a practical matter, Brown’s tables were not exactly easy to use: to find the position of the Moon from them required lots of mechanical desk calculator work, as well as careful transcription of numbers. And this led Leslie Comrie in 1932 to propose using a punch-card-based IBM Hollerith automatic tabulator—and with the help of Thomas Watson, CEO of IBM, what was probably the first “scientific computing laboratory” was established—to automate computations from Brown’s tables.
(When I was in elementary school in England in the late 1960s—before electronic calculators—I always carried around, along with my slide rule, a little book of “4-figure mathematical tables”. I think I found it odd that such a book would have an author—and perhaps for that reason I still remember the name: “L. J. Comrie”.)
By the 1950s, the calculations in Brown’s tables were slowly being rearranged and improved to make them more suitable for computers. But then with John F. Kennedy’s 1962 “We choose to go to the Moon”, there was suddenly urgent interest in getting the most accurate computations of the Moon’s position. As it turned out, though, it was basically just a tweaked version of Brown’s tables, running on a mainframe computer, that did the computations for the Apollo program.
At first, computers were used in celestial mechanics purely for numerical computation. But by the mid-1960s there were also experiments in using them for algebraic computation, and particularly to automate the generation of series expansions. Wallace Eckert at IBM started using FORMAC to redo Brown’s tables, while in Cambridge David Barton and Steve Bourne (later the creator of the “Bourne shell” (sh) in Unix) built their own CAMAL computer algebra system to try extending the kind of thing Delaunay had done. (And by 1970, Delaunay’s 7th-order calculations had been extended to 20th order.)
When I myself started to work on computer algebra in 1976 (primarily for computations in particle physics), I’d certainly heard about CAMAL, but I didn’t know what it had been used for (beyond vaguely “celestial mechanics”). And as a practicing theoretical physicist in the late 1970s, I have to say that the “problem of the Moon” that had been so prominent in the 1700s and 1800s had by then fallen into complete obscurity.
I remember for example in 1984 asking a certain Martin Gutzwiller, who was talking about quantum chaos, what his main interest actually was. And when he said “the problem of the Moon”, I was floored; I didn’t know there still was any problem with the Moon. As it turns out, in writing this post I found out that Gutzwiller was actually the person who took over from Eckert and spent nearly two decades working on trying to improve the computations of the position of the Moon.
Traditional approaches to the three-body problem come very much from a mathematical way of thinking. But modern computational thinking immediately suggests a different approach. Given the differential equations for the three-body problem, why not just directly solve them? And indeed in the Wolfram Language there’s a built-in function NDSolve for numerically solving systems of differential equations.
So what happens if one just feeds in equations for a three-body problem? Well, here are the equations:
Now as an example let’s set the masses to random values:
And let’s define the initial position and velocity for each body to be random as well:
Now we can just use NDSolve to get the solutions (it gives them as implicit approximate numerical functions of 
):
And now we can plot them. And now we’ve got a solution to a three-body problem, just like that!
Well, obviously this is using the Wolfram Language and a huge tower of modern technology. But would it have been possible even right from the beginning for people to generate direct numerical solutions to the three-body problem, rather than doing all that algebra? Back in the 1700s, Euler already knew what’s now called Euler’s method for finding approximate numerical solutions to differential equations. So what if he’d just used that method to calculate the motion of the Moon?
The method relies on taking a sequence of discrete steps in time. And if he’d used, say, a step size of a minute, then he’d have had to take 40,000 steps to get results for a month, but he should have been able to successfully reproduce the position of the Moon to about a percent. If he’d tried to extend to 3 months, however, then he would already have had at least a 10% error.
Any numerical scheme for solving differential equations in practice eventually builds up some kind of error—but the more one knows about the equations one’s solving, and their expected solutions, the more one’s able to preprocess and adapt things to minimize the error. NDSolve has enough automatic adaptivity built into it that it’ll do pretty well for a surprisingly long time on a typical three-body problem. (It helps that the Wolfram Language and NDSolve can handle numbers with arbitrary precision, not just machine precision.)
But if one looks, say, at the total energy of the three-body system—which one can prove from the equations should stay constant—then one will typically see an error slowly build up in it. One can avoid this if one effectively does a change of variables in the equations to “factor out” energy. And one can imagine doing a whole hierarchy of algebraic transformations that in a sense give the numerical scheme as much help as possible.
And indeed since at least the 1980s that’s exactly what’s been done in practical work on the three-body problem, and the Earth-Moon-Sun system. So in effect it’s a mixture of the traditional algebraic approach from the 1700s and 1800s, together with modern numerical computation.
OK, so what’s involved in solving the real problem of the Earth-Moon-Sun system? The standard three-body problem gives a remarkably good approximation to the physics of what’s happening. But it’s obviously not the whole story.
For a start, the Earth isn’t the only planet in the solar system. And if one’s trying to get sufficiently accurate answers, one’s going to have to take into account the gravitational effect of other planets. The most important is Jupiter, and its typical effect on the orbit of the Moon is at about the 
level—sufficiently large that for example Brown had to take it into account in his tables.
The next effect is that the Earth isn’t just a point mass, or even a precise sphere. Its rotation makes it bulge at the equator, and that affects the orbit of the Moon at the 
level.
Orbits around the Earth ultimately depend on the full mass distribution and gravitational field of the Earth (which is what Sputnik-1 was nominally launched to map)—and both this, and the reverse effect from the Moon, come in at the 
level. At the 
level there are then effects from tidal deformations (“solid tides”) on the Earth and Moon, as well as from gravitational redshift and other general relativistic phenomena.
To predict the position of the Moon as accurately as possible one ultimately has to have at least some model for these various effects.
But there’s a much more immediate issue to deal with: one has to know the initial conditions for the Earth, Sun and Moon, or in other words, one has to know as accurately as possible what their positions and velocities were at some particular time.
And conveniently enough, there’s now a really good way to do that, because Apollo 11, 14 and 15 all left laser retroreflectors on the Moon. And by precisely timing how long it takes a laser pulse from the Earth to round-trip to these retroreflectors, it’s now possible in effect to measure the position of the Moon to millimeter accuracy.
OK, so how do modern analogs of the Babylonian ephemerides actually work? Internally they’re dealing with the equations for all the significant bodies in the solar system. They do symbolic preprocessing to make their numerical work as easy as possible. And then they directly solve the differential equations for the system, appropriately inserting models for things like the mass distribution in the Earth.
They start from particular measured initial conditions, but then they repeatedly insert new measurements, trying to correct the parameters of the model so as to optimally reproduce all the measurements they have. It’s very much like a typical machine learning task—with the training data here being observations of the solar system (and typically fitting just being least squares).
But, OK, so there’s a model one can run to figure out something like the position of the Moon. But one doesn’t want to have to explicitly do that every time one needs to get a result; instead one wants in effect just to store a big table of pre-computed results, and then to do something like interpolation to get any particular result one needs. And indeed that’s how it’s done today.
Back in the 1960s NASA started directly solving differential equations for the motion of planets. The Moon was more difficult to deal with, but by the 1980s that too was being handled in a similar way. Ongoing data from things like the lunar retroreflectors was added, and all available historical data was inserted as well.
The result of all this was the JPL Development Ephemeris (JPL DE). In addition to new observations being used, the underlying system gets updated every few years, for example to get what’s needed for some spacecraft going to some new place in the solar system. (The latest is DE441—that follows DE432, which was built for going to Pluto.)
But so how is the actual ephemeris delivered? Well, for every thousand years covered, the ephemeris has about 100 megabytes of results, given as coefficients for Chebyshev polynomials, which are convenient for interpolation. And for any given quantity in any given coordinate system over a particular period of time, one accesses the appropriate parts of these results.
In Wolfram Language, it’s all packaged up into the function AstroPosition—which here gives the position of the Moon in coordinates relative to the equator of the Earth right now:
OK, but so how does one find an eclipse? Well, it’s an iterative process. Start with an approximation, perhaps from the saros cycle. Then interpolate the ephemeris and look at the result. Then keep iterating until one finds out just when the Moon will be in the appropriate position.
But actually there’s some more to do. Because what’s originally computed are the positions of the barycenters (centers of mass) of the various bodies. But now one has to figure out how the bodies are oriented.
The Earth rotates, and we know its rate quite precisely. But the Moon is basically locked with the same face pointing to the Earth, except that in practice there are small “librations” where the Moon wobbles a little back and forth—and these turn out to be particularly troublesome to predict.
OK, so let’s say one knows where the Earth, Moon and Sun are. How does one then figure out what type of eclipse will happen, and where on the Earth the eclipse will actually hit? Well, there’s some further geometry to do. And here’s the beginning of what’s involved:
Basically the Moon generates a cone of shadow, and then the question is how this cone intersects the Earth. If the tip of the cone is inside the Earth, that means there’ll be a region of total shadow (“umbra”) on the Earth—and a total eclipse. (If the tip is above the Earth, there’ll be an annular eclipse, in which there’s a “ring of sun” visible around the Moon.)
By the way, the more complete geometry is like this (again not to scale)
where now we’ve included the penumbra in which only part of the Sun is shadowed by the Moon. In the particular case shown, the umbra cone “misses the Earth”, so there’s no total eclipse, but there’s still a partial eclipse where part of the Sun is shadowed.
OK, but let’s say there’s going to be a total eclipse. To see where on the Earth the region of totality will be, we have to work out where the “cone of total shadow” (i.e. umbra) will intersect the surface of the Earth. It’s a somewhat complicated 3D geometry problem:
It’s easiest to understand what happens by looking at things from the position of the Sun. The light gray region is the penumbra, and the little black dot is the region of totality (i.e. the umbra):
As the Earth and Moon move in their orbits, the region of shadow will move relative to the Earth:
But now there’s another part of the story, which is the rotation of the Earth. And if we include that, we’ll see that the region of totality (at least in this case) traces out a kind of S-shaped curve on the surface of the Earth:
All this tricky geometry got figured out in 1824 by Friedrich Bessel, who introduced what are now called the Besselian elements—eight variables that specify the location, orientation and aperture of the umbra and penumbra cones, as well as the orientation of the Earth, as a function of time. And for any given eclipse, the whole story of its appearance and trajectory on the surface of the Earth is determined by its Besselian elements.
OK, so now we know what the trajectory of an eclipse will be. But how do we figure out at what time the eclipse will actually reach a given point on Earth? Well, first we have to be clear on our definition of time. And there’s an immediate issue with the speed of light and special relativity. What does it mean to say that the positions of the Earth and Sun are such-and-such at such-and-such a time? Because it takes light about 8 minutes to get to the Earth from the Sun, we only get to see where the Sun was 8 minutes ago, not where it is now.
And what we need is really a classic special relativity setup. We essentially imagine that the solar system is filled with a grid of clocks that have been synchronized by light pulses. And what a modern ephemeris does is to quote the results for positions of bodies in the solar system relative to the times on those clocks. (General relativity implies that in different gravitational fields the clocks will run at different rates, but for our purposes this is a tiny effect. But what isn’t a tiny effect is including retardation in the equations for the 
-body problem—making them become delay differential equations.)
But now there’s another issue. If one’s observing the eclipse, one’s going to be using some timepiece (phone?) to figure out what time it is. And if it’s working properly that timepiece should show official “civil time” that’s based on UTC—which is what NTP internet time is synchronized to. But the issue is that UTC has a complicated relationship to the time used in the astronomical ephemeris.
The starting point is what’s called UT1: a definition of time in which one day is the average time it takes the Earth to rotate once relative to the Sun. But the point is that this average time isn’t constant, because the rotation of the Earth is gradually slowing down, primarily as a result of interactions with the Moon. But meanwhile, UTC is defined by an atomic clock whose timekeeping is independent of any issues about the rotation of the Earth.
There’s a convention for keeping UT1 aligned with UTC: if UT1 is going to get more than 0.9 seconds away from UTC, then a leap second is added to UTC. One might think this would be a tiny effect, but actually, since 1972, a total of 27 leap seconds have been added (as specified in the Wolfram Language by GeoOrientationData):
Exactly when a new leap second will be needed is unpredictable; it depends on things like what earthquakes have occurred. But we need to account for leap seconds if we’re going to get the time of the eclipse correct to the second relative to UTC or internet time.
There are a few other effects that are also important in the precise observed timing of the eclipse. The most obvious is geo elevation. In doing astronomical computations, the Earth is assumed to be an ellipsoid. (There are many different definitions, corresponding to different geodetic “datums”—and that’s an issue in defining things like “sea level”, but it’s not relevant here.) But if you’re at a different height above the ellipsoid, the cone of shadow from the eclipse will reach you at a different time. And the size of this effect can be as much as 0.3 seconds for every 1000 feet of height.
All of the effects we’ve talked about we’re readily able to account for. But there is one remaining effect that’s a bit more difficult. Right at the beginning or end of totality one typically sees points of light on the rim of the Moon. Known as Baily’s beads, these are the result of rays of light that make it to us between mountains on the Moon. Figuring out exactly when all these rays are extinguished requires taking geo elevation data for the Moon, and effectively doing full 3D ray tracing. And in doing this, one ends with the rather bizarre conclusion that the region of shadow on the earth isn’t a perfect circle; instead it’s roughly a polygon, each of whose edges is associated with a particular Baily’s bead. The effect of this can last as long as a second, and can cause the precise edge of totality to move by as much as a mile. (One can also imagine effects having to do with the corona of the Sun, which is constantly changing.)
But in the end, even though the shadow of the Moon on the Earth moves at more than 1000 mph, modern science successfully makes it possible to compute when the shadow will reach a particular point on Earth to an accuracy of perhaps a second. And that’s what our precisioneclipse.com website is set up to do.
I saw my first partial solar eclipse more than 50 years ago. And I’ve seen one total solar eclipse before in my life—in 1991. It was the longest eclipse (6 minutes 53 seconds) that’ll happen for more than a century.
There was a certain irony to my experience, though, especially in view of our efforts now to predict the exact arrival time of next week’s eclipse. I’d chartered a plane and flown to a small airport in Mexico (yes, that’s me on the left with the silly hat)—and my friends and I had walked to a beautiful deserted beach, and were waiting under a cloudless sky for the total eclipse to begin.
I felt proud of how prepared I was—with maps marking to the minute when the eclipse should arrive. But then I realized: there we were, out on a beach with no obvious signs of modern civilization—and nobody had brought any properly set timekeeping device (and in those days my cellphone was just a phone, and didn’t even have signal there).
And so it was that I missed seeing a demonstration of an impressive achievement of science. And instead I got to experience the eclipse pretty much the way people throughout history have experienced eclipses—even if I did know that the Moon would continue gradually eating into the Sun and eventually cover it, and that it wouldn’t make the world end.
There’s always something sobering about astronomical events, and about realizing just how tiny human scales are compared to them. Billions of eclipses have happened over the course of the Earth’s history. Recorded history has covered only a few thousand of them. On average, there’s an eclipse at any given place on Earth roughly every 400 years; in Jackson, WY, where I’m planning to see next week’s eclipse, it turns out the next total eclipse will be 727 years from now—in 2744.
In earlier times, civilizations built giant monuments to celebrate the motions of the Sun and Moon. Today, for the eclipse next week, what we’re making is a website. But that website builds on one of the great epics of human intellectual history—stretching back to the earliest times of systematic science, and encompassing contributions from a remarkable cross-section of the most celebrated scientists and mathematicians from past centuries.
It’ll be about 9538 days since the eclipse I saw in 1991. The Moon will have traveled some 500 million miles around the Earth, and the Earth some 15 billion miles around the Sun. But now—in a remarkable triumph of science—we’re computing to the second when they’ll be lined up again.
In the days leading up to August 21, 2017, millions of people accessed our precisioneclipse.com website—with their geoIPs increasingly concentrating near the path of totality. I had traveled to Wyoming and—with a couple of hours to spare—found a place with a clear view across a valley. And this now being 2017, I tweeted:
But would our carefully done computations actually be accurate? The eclipse was going to make landfall on the Oregon coast, and, conveniently, we had a spotter right there. From where I was, the partial eclipse was well underway. And then I got a text: yes, totality in Oregon had come at the predicted second. It would take the shadow of the Moon a little under 20 minutes to reach me.
Unlike in 1991, I and everyone else had a cellphone with a precise clock—and was able to access our precisioneclipse.com site:
While I was waiting I was making little images of the crescent Sun with my fingers—repeating something I’d first noticed more than 50 years earlier when I saw my first partial eclipse at the age of 6:
A few minutes before the eclipse, I started to see a strange shimmering (invisible on any video I took): shadow bands, a strange and poorly understood eclipse phenomenon. And then, there it came, sweeping across the valley: totality. Arriving right at the predicted second:
I had set up a camera to capture a video of the eclipse, and a little later that day I did an analysis of it—and, since earlier in 2017 I’d started routinely doing livestreams, I livestreamed it:
At the end, I published my notebook to the Wolfram Cloud, and it’s still there:
And now six years have passed. Much has happened in our human world. But the Moon has just inexorably continued in its orbit. And 2422 days later, it will once again line up to create a total eclipse…
It’s taken millennia to get to the point where it’s possible to accurately compute eclipses. But now—as a tiny part of making “everything in the world” computable—computation about eclipses is just a built-in feature of the Wolfram Language.
The core function is SolarEclipse. By default, SolarEclipse tells us the time of the next solar eclipse from now:
It can also tell us the next solar eclipse from any time more than 10,000 years in the past and future:
By default, SolarEclipse tells us about all solar eclipses, including partial ones. But we can request only total eclipses (or annular ones, etc.):
SolarEclipse immediately lets you compute nearly 100 properties of an eclipse. The most basic property is the type of eclipse; this tells us that the next eclipse from now will be total:
And here are the types of all eclipses for the rest of the decade:
This gives a timeline of these eclipses; notice that most eclipses are separated by 5 or 6 months, but there’s one pair (in 2029) that is just a month apart:
Another basic property of an eclipse is its magnitude: how much of the diameter of the disk of the Sun is covered by the Moon. The next eclipse is total, so the magnitude is greater than 1:
Looking at the sequence of eclipses for the remainder of the decade, we see which ones are total and which are not:
Here’s the path of totality for the upcoming total eclipse:
And here’s where the eclipse is partial:
This is easier to understand with a different geo projection:
There are all sorts of special points and lines associated with these regions. This gives the “point of maximum eclipse” (i.e. essentially the place where the eclipse lasts longest):
And this gives the precise time (converted to my current time zone) of the maximum eclipse:
This gives the time of maximum eclipse in the time zone of the point of maximum eclipse:
At the point of maximum eclipse, this gives the duration of the umbra (i.e. the time of totality):
And here’s a map of where on the Earth the umbra is at the time of maximum eclipse:
Zooming out to a range of 500 miles makes it easier to tell where this is:
This shows the position of the umbra every minute for the hour after the time of maximum eclipse:
Here’s a summary map of the eclipse, including for example (in red) the points of first and last contact of the penumbra:
Instead of looking at what amounts to the shadow of the Moon on the Earth, we can ask what we’d see in the sky. During a total eclipse the Moon will completely cover the Sun. But here’s what happens just 15 minutes before the time of maximum eclipse:
Here’s a map of the path of totality for all the total eclipses for the next 50 years:
Over the course of 500 years there are lots of total eclipses:
Although it’s not terribly obvious there, there’s actually a lot of regularity in these paths. In particular, as we discussed previously, similar eclipses occur in “saros series”, separated by a time of about 1 saros, or roughly 18 years. Here are the paths of eclipses that appear for the 10 saroses after the next eclipse:
Each successive eclipse in the saros series is systematically about 120° to the west of the previous one, and a little south (or north, depending on the series). The series continues like this until the eclipse paths hit one of the poles, at which point the series ends:
Any given eclipse is in a saros series. The next eclipse is in series 139 (the numbering scheme for these series was set in 1955—with series 0 chosen, quite arbitrarily, to be the one that starts just after 3000 BC):
There are 71 eclipses in this saros series
running from the year 1501 to the year 2763 (a span of 1263 years):
Not all these eclipses are total, however. But if we plot the magnitudes of all the eclipses, we see that the partial ones appear only at the ends of the saros series:
If we look at the next few eclipses, we’ll see that they are part of all sorts of different saros series:
Here are dates for eclipses in a sequence of different saros series:
Right now there are 40 saros series active:
Any given eclipse could be specified by its “index number” within its saros series. But in the mid-20th century it was realized that there’s a more convenient and robust way to label eclipses, using a combination of “saros number” and what’s called “inex number”.
As we discussed previously, for an eclipse to occur, a new moon must happen at a time when the Moon is close to the plane of the ecliptic. The average time between new moons is the so-called synodic month:
Meanwhile, the average time between the Moon’s crossings of the plane of the ecliptic is half the so-called draconic month:
Given a particular eclipse, the time before an approximate “repeat eclipse” will correspond to a coincidence between an integer multiple of the synodic month, and of half the draconic month. And to figure out when these coincidences will occur is essentially a question of number theory.
Let’s compute the continued fraction expansion:
From this we can derive a sequence of rational approximations:
These approximations get progressively better:
The 8th one is 484/223—and this corresponds to the saros cycle, which reflects the close similarity of 223 synodic months and 242 draconic months:
But now let’s look at the 9th rational approximation: 777/358. This reflects the coincidence:
And now this coincidence defines another cycle—which is the inex cycle. There are lots of other cycles one can identify—but all the common ones can be expressed as linear combinations of the saros and inex cycles.
We saw above how eclipses occur in saros series. But we now see that they will also occur in inex series. And a convenient way to specify an eclipse is to say what saros series and what inex series it appears in. The saros and inex series are numbered according to when they started, with the 0th saros series by convention being the one that spans:
With this setup, the April 2024 eclipse can then be specified by its saros and inex numbers:
But how does this fit in with other eclipses? Here’s a plot of the inex and saros numbers of all eclipses between 1000 AD and 3000 AD (with the April 2024 eclipse indicated in red):
Each saros series shows up as a vertical line of eclipses, and each inex series as a horizontal line. The finite overall date range for the picture leads to the diagonal cutoffs on each side.
For each saros, inex number pair there’ll be some kind of eclipse. But most of the eclipses won’t be total. Here’s where total eclipses occur in the saros, inex plane:
And here’s a complete color-coded “panorama” of different kinds of eclipses:
(Note that—as we discussed above—the eclipses at the ends of saros series are partial.)
By the way, when we’re discussing eclipses in the future, there’s a subtlety to mention. By computing the overall motion of the Earth and Moon we can work out when there’ll be an eclipse, and where in space the cone of shadow associated with it will be. But it’s a different question where geographically that shadow will land on the Earth because that depends on the rotation of the Earth—which isn’t precisely predictable, as illustrated by the somewhat haphazard way in which leap seconds have had to be added to align “universal” UTC time with time based on the rotation of the Earth and the length of a day. Is it a large effect? Over the past 500 years the discrepancy has been about 176 seconds; over the next 500 years it could easily be as much as 1000 seconds. Meanwhile, for an eclipse on the equator each second corresponds to a change of 0.29 miles in where the eclipse will be total (the change is less farther from the equator). In what we’ve shown here we’ve made some standard assumptions about how the rotation of the Earth is slowing down—but ultimately this isn’t predictable, making the locations of eclipses 500 years from now uncertain in the east-west direction by as much as several hundred miles.
The function SolarEclipse in the Wolfram Language immediately tells us when eclipses occur. But we can also deduce this information from other “lower-level” functions. In particular, we know that an eclipse (of at least some kind) occurs if the angular separation between the Sun and the Moon in the sky is smaller than the total of their apparent radii (or about 0.5°). The angular separation depends on where you are on the Earth. Let’s pick a location from which we know an eclipse will be visible:
Now let’s plot angular separation for each hour over the course of the next year:
The minima occur once per lunar month, near the time of the new moons. Changing the plot range, we can see that these minima are different in different (lunar) months:
Let’s look at early April in more detail, now sampling every minute:
And what we see here is that the angular separation goes to zero—reflecting the fact that there’s a total eclipse. What are those little glitches? They’re a consequence of the fact that the apparent positions of the Sun and Moon change when they’re close to the horizon because of refraction in the atmosphere. Not including refraction gives a smoother curve:
We’ve been looking at the angular separation between the centers of the disks of the Sun and Moon in the sky. But what about the actual positions in the sky? Here’s the astro position of the Sun (in local horizon coordinates) at the time and place of maximum eclipse:
And here now is the almost-exactly-equal result for the Moon:
What will it actually look like in the sky? Here’s now a graphic showing the disks of the Sun and Moon 15 minutes before the time of maximum eclipse:
Here’s a sequence of images 15 minutes apart:
I saw the 2017 eclipse from a rather scenic spot near Jackson, Wyoming:
Specifically (according to Wolfram|Alpha accessed through my phone) I was at geo position 43.5125°N 110.6506°W—at an elevation of 7526 ft:
As a check, here’s the expected (ground) elevation at that lat-lon—definitely within bounds, particularly considering I was holding the phone about 5 feet off the ground, etc.:
Defining my location as
the predicted arrival time of the eclipse at that location was then:
And indeed that’s exactly what our precisioneclipse.com site told me at the time:
What would happen at the “moment of totality”? Here’s where the shadow of the Moon was predicted to be (where I was standing is indicated by a red dot; the whole picture is 100 miles across):
Fifteen seconds earlier, the shadow of the Moon would have just “crossed” the row of mountains I could see:
Looking (in exaggerated 3D) at the terrain, here’s the “umbral cone” at the moment of totality for me:
And it was moving at a speed
which is effectively a vector difference of the rotation speed of the Earth
corrected for latitude
and the orbital velocity of the Moon:
Shown at 5-second intervals for 30 seconds, here’s how the edge of the umbra was moving just before it reached me:
That day, I had brought some not-very-high-tech “equipment” to record the eclipse:
The main video I captured—which was from the iPad—is now stored for posterity in the Wolfram Data Repository:
It was 11 minutes long. Sampling frames from it we get:
To get more of a sense of the eclipse, we can pick out the center column of the video, and arrange it in time:
Notice that the “band of totality” appears slightly further to the left at the top of the image—reflecting the fact that totality reaches the mountains in the distance (at the top of the image) slightly earlier than it reaches the “foreground” at the bottom of the image.
To be more quantitative, we can measure the mean intensity of the bottom part of the image as a function of time (where here we’ve aligned with the timestamp that records the start time of the video):
We see in this “light curve” a big dip during the period of totality. There’s also a little dip earlier from someone walking in front of the camera. And we also see lots of little glitches that we’ll discuss later.
But what should we expect this light curve to look like? Well, we can predict what the obscuration of the Sun around the time of totality should be—and but for the 20% effect of “limb darkening”, this should give the intensity of sunlight:
Rescaling values somewhat, we can compare the curves:
And zooming in on the minute around the beginning of totality, we see:
And, yes indeed, the observed onset of totality seems to agree basically to the second with our prediction!
But even though there’s this agreement, the overall shapes of the observed and predicted light curves definitely aren’t the same. And, yes, this is a story of experimental science. And, arguably, of a mistake I made—of using consumer electronics, optimized for consumer purposes, to make a quantitative scientific measurement. You see, as I now realize, an iPad by default always tries to “get a good picture” by maintaining the brightness of an image independent of overall light level. And while for “consumer purposes” that’s usually the right thing to do, it definitely confuses things if one’s trying to measure the light curve for an eclipse.
And indeed if we look at our “measured light curve” it’s very flat until the period of totality. In other words, the iPad succeeded in maintaining the same image brightness until there just wasn’t enough light at all, at which point the image “faded to black”. (At the end of totality, the iPad gradually realized “yes, there’s more light now”, and the measured light curve slowly climbs back up.)
But what’s with all the glitches we see? They’re already visible in our “video time collage” above. And one thought might be that they’re an actual eclipse phenomenon—perhaps associated with the “shadow bands” that I did indeed see just before this eclipse. But the characteristic shimmering associated with such shadow bands—while very difficult to capture on video, perhaps because actual images aren’t being formed—happens much faster than the glitches we’re seeing.
Looking in a bit more detail, we see that there are upward glitches in the period before totality, and downward ones after. Zooming in on a couple of minutes of the “before” period and a couple of minutes of the “after” period, we see:
And, yes, we can validate that this is an “instrumental phenomenon” by doing a simple experiment—using the very same iPad as in 2017—and continuously sliding a piece of cardboard in front of a light and then capturing video and measuring the light curve for this in-my-basement-style “model eclipse”:
We see the very same kind of glitches as from the actual eclipse video. And presumably in both cases they’re reflections of the automatic exposure control system used by the iPad. (If the individual frames of the video had EXIF metadata, we might be able to see that explicitly, but as it is, there’s only EXIF data for the whole video.) I don’t know in detail how this iPad’s exposure control system works, but what we’re seeing is a kind of overshooting-and-correction that’s very common in all sorts of control systems. If we knew in advance everything that happens in the video, then maybe we could avoid the glitches. But if we’re going to try to maintain light level on an ongoing basis during the recording of the video (perhaps by adjusting the gamma correction that determines how raw sensor values are translated to pixel values), then control theory most likely implies that glitches are inevitable.
So many years later, I still remember it well. I was 6 years old (almost 7), walking the couple of blocks to school (yes, on my own, which kids in England in those days did). It happened when I was walking under a tree (and, yes, I still remember exactly where). There were lots of dappled patches of light on the ground. And something looked strange about them. And suddenly I realized what it was—and I still have an image of it in my mind today. All the patches of light had the same bite taken out of them. And it didn’t take me long to look up at the Sun, and see that it too had a bite taken out of it.
Being already something of a science enthusiast, I’d heard of eclipses, and realized this must be one. I arrived at school a few minutes later, and regaled the other kids with my “discovery”. But despite the obviousness (at least to me) of what was going on, I wasn’t widely believed. And, yes, that was a very educational experience. But that’s a different story…
So what was that eclipse? Well, it was the one in May 1966:
It was a partial eclipse—visible from England:
The geo location of my “discovery tree” was 51.7636° N 1.2558° W. So now we can compute the magnitude of the eclipse there that morning:
I believe school started at 9am, so what I saw was an eclipse with rough magnitude:
The Sun (and Moon) were about 50° above the horizon:
And the Moon was poking into the disk of the Sun:
Thirty minutes later the Moon had poked a little further into the disk of the Sun. But after a bit more than a hour, the whole Sun was back, and the eclipse was over:
And it would be 25 years before I’d see another eclipse—though this time a total one.
By the way, this is me back at the time of my first eclipse—captured in a long-before-those-were-popular “selfie”, taken with a film camera and manual focus, and, it seems, a lot of concentration (and, yes, I think I still make the same strange expression when I’m concentrating hard today):
Version 14.0 of Wolfram Language and Mathematica is available immediately both on the desktop and in the cloud. See also more detailed information on Version 13.1, Version 13.2 and Version 13.3.
Today we celebrate a new waypoint on our journey of nearly four decades with the release of Version 14.0 of Wolfram Language and Mathematica. Over the two years since we released Version 13.0 we’ve been steadily delivering the fruits of our research and development in .1 releases every six months. Today we’re aggregating these—and more—into Version 14.0.
It’s been more than 35 years now since we released Version 1.0. And all those years we’ve been continuing to build a taller and taller tower of capabilities, progressively expanding the scope of our vision and the breadth of our computational coverage of the world:
Version 1.0 had 554 built-in functions; in Version 14.0 there are 6602. And behind each of those functions is a story. Sometimes it’s a story of creating a superalgorithm that encapsulates decades of algorithmic development. Sometimes it’s a story of painstakingly curating data that’s never been assembled before. Sometimes it’s a story of drilling down to the essence of something to invent new approaches and new functions that can capture it.
And from all these pieces we’ve been steadily building the coherent whole that is today’s Wolfram Language. In the arc of intellectual history it defines a broad, new, computational paradigm for formalizing the world. And at a practical level it provides a superpower for implementing computational thinking—and enabling “computational X” for all fields X.
To us it’s profoundly satisfying to see what has been done over the past three decades with everything we’ve built so far. So many discoveries, so many inventions, so much achieved, so much learned. And seeing this helps drive forward our efforts to tackle still more, and to continue to push every boundary we can with our R&D, and to deliver the results in new versions of our system.
Our R&D portfolio is broad. From projects that get completed within months of their conception, to projects that rely on years (and sometimes even decades) of systematic development. And key to everything we do is leveraging what we have already done—often taking what in earlier years was a pinnacle of technical achievement, and now using it as a routine building block to reach a level that could barely even be imagined before. And beyond practical technology, we’re also continually going further and further in leveraging what’s now the vast conceptual framework that we’ve been building all these years—and progressively encapsulating it in the design of the Wolfram Language.
We’ve worked hard all these years not only to create ideas and technology, but also to craft a practical and sustainable ecosystem in which we can systematically do this now and into the long-term future. And we continue to innovate in these areas, broadening the delivery of what we’ve built in new and different ways, and through new and different channels. And in the past five years we’ve also been able to open up our core design process to the world—regularly livestreaming what we’re doing in a uniquely open way.
And indeed over the past several years the seeds of essentially everything we’re delivering today in Version 14.0 has been openly shared with the world, and represents an achievement not only for our internal teams but also for the many people who have participated in and commented on our livestreams.
Part of what Version 14.0 is about is continuing to expand the domain of our computational language, and our computational formalization of the world. But Version 14.0 is also about streamlining and polishing the functionality we’ve already defined. Throughout the system there are things we’ve made more efficient, more robust and more convenient. And, yes, in complex software, bugs of many kinds are a theoretical and practical inevitability. And in Version 14.0 we’ve fixed nearly 10,000 bugs, the majority found by our increasingly sophisticated internal software testing methods.
Even after all the work we’ve put into the Wolfram Language over the past several decades, there’s still yet another challenge: how to let people know just what the Wolfram Language can do. Back when we released Version 1.0 I was able to write a book of manageable size that could pretty much explain the whole system. But for Version 14.0—with all the functionality it contains—one would need a book with perhaps 200,000 pages.
And at this point nobody (even me!) immediately knows everything the Wolfram Language does. Of course one of our great achievements has been to maintain across all that functionality a tightly coherent and consistent design that results in there ultimately being only a small set of fundamental principles to learn. But at the vast scale of the Wolfram Language as it exists today, knowing what’s possible—and what can now be formulated in computational terms—is inevitably very challenging. And all too often when I show people what’s possible, I’ll get the response “I had no idea the Wolfram Language could do that!”
So in the past few years we’ve put increasing emphasis into building large-scale mechanisms to explain the Wolfram Language to people. It begins at a very fine-grained level, with “just-in-time information” provided, for example, through suggestions made when you type. Then for each function (or other construct in the language) there are pages that explain the function, with extensive examples. And now, increasingly, we’re adding “just-in-time learning material” that leverages the concreteness of the functions to provide self-contained explanations of the broader context of what they do.
By the way, in modern times we need to explain the Wolfram Language not just to humans, but also to AIs—and our very extensive documentation and examples have proved extremely valuable in training LLMs to use the Wolfram Language. And for AIs we’re providing a variety of tools—like immediate computable access to documentation, and computable error handling. And with our Chat Notebook technology there’s also a new “on ramp” for creating Wolfram Language code from linguistic (or visual, etc.) input.
But what about the bigger picture of the Wolfram Language? For both people and AIs it’s important to be able to explain things at a higher level, and we’ve been doing more and more in this direction. For more than 30 years we’ve had “guide pages” that summarize specific functionality in particular areas. Now we’re adding “core area pages” that give a broader picture of large areas of functionality—each one in effect covering what might otherwise be a whole product on its own, if it wasn’t just an integrated part of the Wolfram Language:
But we’re going even much further, building whole courses and books that provide modern hands-on Wolfram-Language-enabled introductions to a broad range of areas. We’ve now covered the material of many standard college courses (and quite a lot besides), in a new and very effective “computational” way, that allows immediate, practical engagement with concepts:
All these courses involve not only lectures and notebooks but also auto-graded exercises, as well as official certifications. And we have a regular calendar of everyone-gets-together-at-the-same-time instructor-led peer Study Groups about these courses. And, yes, our Wolfram U operation is now emerging as a significant educational entity, with many thousands of students at any given time.
In addition to whole courses, we have “miniseries” of lectures about specific topics:
And we also have courses—and books—about the Wolfram Language itself, like my Elementary Introduction to the Wolfram Language, which came out in a third edition this year (and has an associated course, online version, etc.):
In a somewhat different direction, we’ve expanded our Wolfram Summer School to add a Wolfram Winter School, and we’ve greatly expanded our Wolfram High School Summer Research Program, adding year-round programs, middle-school programs, etc.—including the new “Computational Adventures” weekly activity program.
And then there’s livestreaming. We’ve been doing weekly “R&D livestreams” with our development team (and sometimes also external guests). And I myself have also been doing a lot of livestreaming (232 hours of it in 2023 alone)—some of it design reviews of Wolfram Language functionality, and some of it answering questions, technical and other.
The list of ways we’re getting the word out about the Wolfram Language goes on. There’s Wolfram Community, that’s full of interesting contributions, and has ever-increasing readership. There are sites like Wolfram Challenges. There are our Wolfram Technology Conferences. And lots more.
We’ve put immense effort into building the whole Wolfram technology stack over the past four decades. And even as we continue to aggressively build it, we’re putting more and more effort into telling the world about just what’s in it, and helping people (and AIs) to make the most effective use of it. But in a sense, everything we’re doing is just a seed for what the wider community of Wolfram Language users are doing, and can do. Spreading the power of the Wolfram Language to more and more people and areas.
The machine learning superfunctions Classify and Predict first appeared in Wolfram Language in 2014 (Version 10). By the next year there were starting to be functions like ImageIdentify and LanguageIdentify, and within a couple of years we’d introduced our whole neural net framework and Neural Net Repository. Included in that were a variety of neural nets for language modeling, that allowed us to build out functions like SpeechRecognize and an experimental version of FindTextualAnswer. But—like everyone else—we were taken by surprise at the end of 2022 by ChatGPT and its remarkable capabilities.
Very quickly we realized that a major new use case—and market—had arrived for Wolfram|Alpha and Wolfram Language. For now it was not only humans who’d need the tools we’d built; it was also AIs. By March 2023 we’d worked with OpenAI to use our Wolfram Cloud technology to deliver a plugin to ChatGPT that allows it to call Wolfram|Alpha and Wolfram Language. LLMs like ChatGPT provide remarkable new capabilities in reproducing human language, basic human thinking and general commonsense knowledge. But—like unaided humans—they’re not set up to deal with detailed computation or precise knowledge. For that, like humans, they have to use formalism and tools. And the remarkable thing is that the formalism and tools we’ve built in Wolfram Language (and Wolfram|Alpha) are basically a broad, perfect fit for what they need.
We created the Wolfram Language to provide a bridge from what humans think about to what computation can express and implement. And now that’s what the AIs can use as well. The Wolfram Language provides a medium not only for humans to “think computationally” but also for AIs to do so. And we’ve been steadily doing the engineering to let AIs call on Wolfram Language as easily as possible.
But in addition to LLMs using Wolfram Language, there’s also now the possibility of Wolfram Language using LLMs. And already in June 2023 (Version 13.3) we released a major collection of LLM-based capabilities in Wolfram Language. One category is LLM functions, that effectively use LLMs as “internal algorithms” for operations in Wolfram Language:
In typical Wolfram Language fashion, we have a symbolic representation for LLMs: LLMConfiguration[…] represents an LLM with its various parameters, promptings, etc. And in the past few months we’ve been steadily adding connections to the full range of popular LLMs, making Wolfram Language a unique hub not only for LLM usage, but also for studying the performance—and science—of LLMs.
You can define your own LLM functions in Wolfram Language. But there’s also the Wolfram Prompt Repository that plays a similar role for LLM functions as the Wolfram Function Repository does for ordinary Wolfram Language functions. There’s a public Prompt Repository that so far has several hundred curated prompts. But it’s also possible for anyone to post their prompts in the Wolfram Cloud and make them publicly (or privately) accessible. The prompts can define personas (“talk like a [stereotypical] pirate”). They can define AI-oriented functions (“write it with emoji”). And they can define modifiers that affect the form of output (“haiku style”).
In addition to calling LLMs “programmatically” within Wolfram Language, there’s the new concept (first introduced in Version 13.3) of “Chat Notebooks”. Chat Notebooks represent a new kind of user interface, that combines the graphical, computational and document features of traditional Wolfram Notebooks with the new linguistic interface capabilities brought to us by LLMs.
The basic idea of a Chat Notebook—as introduced in Version 13.3, and now extended in Version 14.0—is that you can have “chat cells” (requested by typing ‘) whose content gets sent not to the Wolfram kernel, but instead to an LLM:
You can use “function prompts”—say from the Wolfram Prompt Repository—directly in a Chat Notebook:
And as of Version 14.0 you can also knit Wolfram Language computations directly into your “conversation” with the LLM:
(You type \ to insert Wolfram Language, very much like the way you can use <* … *> to insert Wolfram Language into external evaluation cells.)
One thing about Chat Notebooks is that—as their name suggests—they really are centered around “chatting”, and around having a sequential interaction with an LLM. In an ordinary notebook, it doesn’t matter where in the notebook each Wolfram Language evaluation is requested; all that’s relevant is the order in which the Wolfram kernel does the evaluations. But in a Chat Notebook the “LLM evaluations” are always part of a “chat” that’s explicitly laid out in the notebook.
A key part of Chat Notebooks is the concept of a chat block: type ~ and you get a separator in the notebook that “starts a new chat”:
Chat Notebooks—with all their typical Wolfram Notebook editing, structuring, automation, etc. capabilities—are very powerful just as “LLM interfaces”. But there’s another dimension as well, enabled by LLMs being able to call Wolfram Language as a tool.
At one level, Chat Notebooks provide an “on ramp” for using Wolfram Language. Wolfram|Alpha—and even more so, Wolfram|Alpha Notebook Edition—let you ask questions in natural language, then have the questions translated into Wolfram Language, and answers computed. But in Chat Notebooks you can go beyond asking specific questions. Instead, through the LLM, you can just “start chatting” about what you want to do, then have Wolfram Language code generated, and executed:
The workflow is typically as follows. First, you have to conceptualize in computational terms what you want. (And, yes, that step requires computational thinking—which is a very important skill that too few people have so far learned.) Then you tell the LLM what you want, and it’ll try to write Wolfram Language code to achieve it. It’ll typically run the code for you (but you can also always do it yourself)—and you can see whether you got what you wanted. But what’s crucial is that Wolfram Language is intended to be read not only by computers but also by humans. And particularly since LLMs actually usually seem to manage to write pretty good Wolfram Language code, you can expect to read what they wrote, and see if it’s what you wanted. If it is, you can take that code, and use it as a “solid building block” for whatever larger system you might be trying to set up. Otherwise, you can either fix it yourself, or try chatting with the LLM to get it to do it.
One of the things we see in the example above is the LLM—within the Chat Notebook—making a “tool call”, here to a Wolfram Language evaluator. In the Wolfram Language there’s now a whole mechanism for defining tools for LLMs—with each tool being represented by an LLMTool symbolic object. In Version 14.0 there’s an experimental version of the new Wolfram LLM Tool Repository with some predefined tools:
In a default Chat Notebook, the LLM has access to some default tools, which include not only the Wolfram Language evaluator, but also things like Wolfram documentation search and Wolfram|Alpha query. And it’s common to see the LLM go back and forth trying to write “code that works”, and for example sometimes having to “resort” (much like humans do) to reading the documentation.
Something that’s new in Version 14.0 is experimental access to multimodal LLMs that can take images as well as text as input. And when this capability is enabled, it allows the LLM to “look at pictures from the code it generated”, see if they’re what was asked for, and potentially correct itself:
The deep integration of images into Wolfram Language—and Wolfram Notebooks—yields all sorts of possibilities for multimodal LLMs. Here we’re giving a plot as an image and asking the LLM how to reproduce it:
Another direction for multimodal LLMs is to take data (in the hundreds of formats accepted by Wolfram Language) and use the LLM to guide its visualization and analysis in the Wolfram Language. Here’s an example that starts from a file data.csv in the current directory on your computer:
One thing that’s very nice about using Wolfram Language directly is that everything you do (well, unless you use RandomInteger, etc.) is completely reproducible; do the same computation twice and you’ll get the same result. That’s not true with LLMs (at least right now). And so when one uses LLMs it feels like something more ephemeral and fleeting than using Wolfram Language. One has to grab any good results one gets—because one might never be able to reproduce them. Yes, it’s very helpful that one can store everything in a Chat Notebook, even if one can’t rerun it and get the same results. But the more “permanent” use of LLM results tends to be “offline”. Use an LLM “up front” to figure something out, then just use the result it gave.
One unexpected application of LLMs for us has been in suggesting names of functions. With the LLM’s “experience” of what people talk about, it’s in a good position to suggest functions that people might find useful. And, yes, when it writes code it has a habit of hallucinating such functions. But in Version 14.0 we’ve actually added one function—DigitSum—that was suggested to us by LLMs. And in a similar vein, we can expect LLMs to be useful in making connections to external databases, functions, etc. The LLM “reads the documentation”, and tries to write Wolfram Language “glue” code—which then can be reviewed, checked, etc., and if it’s right, can be used henceforth.
Then there’s data curation, which is a field that—through Wolfram|Alpha and many of our other efforts—we’ve become extremely expert at over the past couple of decades. How much can LLMs help with that? They certainly don’t “solve the whole problem”, but integrating them with the tools we already have has allowed us over the past year to speed up some of our data curation pipelines by factors of two or more.
If we look at the whole stack of technology and content that’s in the modern Wolfram Language, the overwhelming majority of it isn’t helped by LLMs, and isn’t likely to be. But there are many—sometimes unexpected—corners where LLMs can dramatically improve heuristics or otherwise solve problems. And in Version 14.0 there are starting to be a wide variety of “LLM inside” functions.
An example is TextSummarize, which is a function we’ve considered adding for many versions—but now, thanks to LLMs, can finally implement to a useful level:
The main LLMs that we’re using right now are based on external services. But we’re building capabilities to allow us to run LLMs in local Wolfram Language installations as soon as that’s technically feasible. And one capability that’s actually part of our mainline machine learning effort is NetExternalObject—a way of representing symbolically an externally defined neural net that can be run inside Wolfram Language. NetExternalObject allows you, for example, to take any network in ONNX form and effectively treat it as a component in a Wolfram Language neural net. Here’s a network for image depth estimation—that we’re here importing from an external repository (though in this case there’s actually a similar network already in the Wolfram Neural Net Repository):
Now we can apply this imported network to an image that’s been encoded with our built-in image encoder—then we’re taking the result and visualizing it:
It’s often very convenient to be able to run networks locally, but it can sometimes take quite high-end hardware to do so. For example, there’s now a function in the Wolfram Function Repository that does image synthesis entirely locally—but to run it, you do need a GPU with at least 8 GB of VRAM:
By the way, based on LLM principles (and ideas like transformers) there’ve been other related advances in machine learning that have been strengthening a whole range of Wolfram Language areas—with one example being image segmentation, where ImageSegmentationComponents now provides robust “content-sensitive” segmentation:
When Mathematica 1.0 was released in 1988, it was a “wow” that, yes, now one could routinely do integrals symbolically by computer. And it wasn’t long before we got to the point—first with indefinite integrals, and later with definite integrals—where what’s now the Wolfram Language could do integrals better than any human. So did that mean we were “finished” with calculus? Well, no. First there were differential equations, and partial differential equations. And it took a decade to get symbolic ODEs to a beyond-human level. And with symbolic PDEs it took until just a few years ago. Somewhere along the way we built out discrete calculus, asymptotic expansions and integral transforms. And we also implemented lots of specific features needed for applications like statistics, probability, signal processing and control theory. But even now there are still frontiers.
And in Version 14 there are significant advances around calculus. One category concerns the structure of answers. Yes, one can have a formula that correctly represents the solution to a differential equation. But is it in the best, simplest or most useful form? Well, in Version 14 we’ve worked hard to make sure it is—often dramatically reducing the size of expressions that get generated.
Another advance has to do with expanding the range of “pre-packaged” calculus operations. We’ve been able to do derivatives ever since Version 1.0. But in Version 14 we’ve added implicit differentiation. And, yes, one can give a basic definition for this easily enough using ordinary differentiation and equation solving. But by adding an explicit ImplicitD we’re packaging all that up—and handling the tricky corner cases—so that it becomes routine to use implicit differentiation wherever you want:
Another category of pre-packaged calculus operations new in Version 14 are ones for vector-based integration. These were always possible to do in a “do-it-yourself” mode. But in Version 14 they are now streamlined built-in functions—that, by the way, also cover corner cases, etc. And what made them possible is actually a development in another area: our decade-long project to add geometric computation to Wolfram Language—which gave us a natural way to describe geometric constructs such as curves and surfaces:
Related functionality new in Version 14 is ContourIntegrate:
Functions like ContourIntegrate just “get the answer”. But if one’s learning or exploring calculus it’s often also useful to be able to do things in a more step-by-step way. In Version 14 you can start with an inactive integral
and explicitly do operations like changing variables:
Sometimes actual answers get expressed in inactive form, particularly as infinite sums:
And now in Version 14 the function TruncateSum lets you take such a sum and generate a truncated “approximation”:
Functions like D and Integrate—as well as LineIntegrate and SurfaceIntegrate—are, in a sense, “classic calculus”, taught and used for more than three centuries. But in Version 14 we also support what we can think of as “emerging” calculus operations, like fractional differentiation:
What are the primitives from which we can best build our conception of computation? That’s at some level the question I’ve been asking for more than four decades, and what’s determined the functions and structures at the core of the Wolfram Language.
And as the years go by, and we see more and more of what’s possible, we recognize and invent new primitives that will be useful. And, yes, the world—and the ways people interact with computers—change too, opening up new possibilities and bringing new understanding of things. Oh, and this year there are LLMs which can “get the intellectual sense of the world” and suggest new functions that can fit into the framework we’ve created with the Wolfram Language. (And, by the way, there’ve also been lots of great suggestions made by the audiences of our design review livestreams.)
One new construct added in Version 13.1—and that I personally have found very useful—is Threaded. When a function is listable—as Plus is—the top levels of lists get combined:
But sometimes you want one list to be “threaded into” the other at the lowest level, not the highest. And now there’s a way to specify that, using Threaded:
In a sense, Threaded is part of a new wave of symbolic constructs that have “ambient effects” on lists. One very simple example (introduced in 2015) is Nothing:
Another, introduced in 2020, is Splice:
An old chestnut of Wolfram Language design concerns the way infinite evaluation loops are handled. And in Version 13.2 we introduced the symbolic construct TerminatedEvaluation to provide better definition of how out-of-control evaluations have been terminated:
In a curious connection, in the computational representation of physics in our recent Physics Project, the direct analog of nonterminating evaluations are what make possible the seemingly unending universe in which we live.
But what is actually going on “inside an evaluation”, terminating or not? I’ve always wanted a good representation of this. And in fact back in Version 2.0 we introduced Trace for this purpose:
But just how much detail of what the evaluator does should one show? Back in Version 2.0 we introduced the option TraceOriginal that traces every path followed by the evaluator:
But often this is way too much. And in Version 14.0 we’ve introduced the new setting TraceOriginal→Automatic, which doesn’t include in its output evaluations that don’t do anything:
This may seem pedantic, but when one has an expression of any substantial size, it’s a crucial piece of pruning. So, for example, here’s a graphical representation of a simple arithmetic evaluation, with TraceOriginal→True:
And here’s the corresponding “pruned” version, with TraceOriginal→Automatic:
(And, yes, the structures of these graphs are closely related to things like the causal graphs we construct in our Physics Project.)
In the effort to add computational primitives to the Wolfram Language, two new entrants in Version 14.0 are Comap and ComapApply. The function Map takes a function f and “maps it” over a list:
Comap does the “mathematically co-” version of this, taking a list of functions and “comapping” them onto a single argument:
Why is this useful? As an example, one might want to apply three different statistical functions to a single list. And now it’s easy to do that, using Comap:
By the way, as with Map, there’s also an operator form for Comap:
Comap works well when the functions it’s dealing with take just one argument. If one has functions that take multiple arguments, ComapApply is what one typically wants:
Talking of “co-like” functions, a new function added in Version 13.2 is PositionSmallest. Min gives the smallest element in a list; PositionSmallest instead says where the smallest elements are:
One of the important objectives in the Wolfram Language is to have as much as possible “just work”. When we released Version 1.0 strings could be assumed just to contain ordinary ASCII characters, or perhaps to have an external character encoding defined. And, yes, it could be messy not to know “within the string itself” what characters were supposed to be there. And by the time of Version 3.0 in 1996 we’d become contributors to, and early adopters of, Unicode, which provided a standard encoding for “16-bits’-worth” of characters. And for many years this served us well. But in time—and particularly with the growth of emoji—16 bits wasn’t enough to encode all the characters people wanted to use. So a few years ago we began rolling out support for 32-bit Unicode, and in Version 13.1 we integrated it into notebooks—in effect making strings something much richer than before:
And, yes, you can use Unicode everywhere now:
Back when Version 1.0 was released, a megabyte was a lot of memory. But 35 years later we routinely deal with gigabytes. And one of the things that makes practical is computation with video. We first introduced Video experimentally in Version 12.1 in 2020. And over the past three years we’ve been systematically broadening and strengthening our ability to deal with video in Wolfram Language. Probably the single most important advance is that things around video now—as much as possible—“just work”, without “creaking” under the strain of handling such large amounts of data.
We can directly capture video into notebooks, and we can robustly play video anywhere within a notebook. We’ve also added options for where to store the video so that it’s conveniently accessible to you and anyone else you want to give access to it.
There’s lots of complexity in the encoding of video—and we now robustly and transparently support more than 500 codecs. We also do lots of convenient things automatically, like rotating portrait-mode videos—and being able to apply image processing operations like ImageCrop across whole videos. In every version, we’ve been further optimizing the speed of some video operation or another.
But a particularly big focus has been on video generators: programmatic ways to produce videos and animations. One basic example is AnimationVideo, which produces the same kind of output as Animate, but as a Video object that can either be displayed directly in a notebook, or exported in MP4 or some other format:
AnimationVideo is based on computing each frame in a video by evaluating an expression. Another class of video generators take an existing visual construct, and simply “tour” it. TourVideo “tours” images, graphics and geo graphics; Tour3DVideo (new in Version 14.0) tours 3D geometry:
A very powerful capability in Wolfram Language is being able to apply arbitrary functions to videos. One example of how this can be done is VideoFrameMap, which maps a function across frames of a video, and which was made efficient in Version 13.2:
And although Wolfram Language isn’t intended as an interactive video editing system, we’ve made sure that it’s possible to do streamlined programmatic video editing in the language, and for example in Version 14.0 we’ve added things like transition effects in VideoJoin and timed overlays in OverlayVideo.
With every new version of Wolfram Language we add new capabilities to extend yet further the domain of the language. But we also put a lot of effort into something less immediately visible: making existing capabilities faster, stronger and sleeker.
And in Version 14 two areas where we can see some examples of all these are dates and quantities. We introduced the notion of symbolic dates (DateObject, etc.) nearly a decade ago. And over the years since then we’ve built many things on this structure. And in the process of doing this it’s become clear that there are certain flows and paths that are particularly common and convenient. At the beginning what mattered most was just to make sure that the relevant functionality existed. But over time we’ve been able to see what should be streamlined and optimized, and we’ve steadily been doing that.
In addition, as we’ve worked towards new and different applications, we’ve seen “corners” that need to be filled in. So, for example, astronomy is an area we’ve significantly developed in Version 14, and supporting astronomy has required adding several new “high-precision” time capabilities, such as the TimeSystem option, as well as new astronomy-oriented calendar systems. Another example concerns date arithmetic. What should happen if you want to add a month to January 30? Where should you land? Different kinds of business applications and contracts make different assumptions—and so we added a Method option to functions like DatePlus to handle this. Meanwhile, having realized that date arithmetic is involved in the “inner loop” of certain computations, we optimized it—achieving a more than 100x speedup in Version 14.0.
Wolfram|Alpha has been able to deal with units ever since it was first launched in 2009—now more than 10,000 of them. And in 2012 we introduced Quantity to represent quantities with units in the Wolfram Language. And over the past decade we’ve been steadily smoothing out a whole series of complicated gotchas and issues with units. For example, what does 
.
At first our priority with Quantity was to get it working as broadly as possible, and to integrate it as widely as possible into computations, visualizations, etc. across the system. But as its capabilities have expanded, so have its uses, repeatedly driving the need to optimize its operation for particular common cases. And indeed between Version 13 and Version 14 we’ve dramatically sped up many things related to Quantity, often by factors of 1000 or more.
Talking of speedups, another example—made possible by new algorithms operating on multithreaded CPUs—concerns polynomials. We’ve worked with polynomials in Wolfram Language since Version 1, but in Version 13.2 there was a dramatic speedup of up to 1000x on operations like polynomial factoring.
In addition, a new algorithm in Version 14.0 dramatically speeds up numerical solutions to polynomial and transcendental equations—and, together with the new MaxRoots options, allows us, for example, to pick off a few roots from a degree-one-million polynomial
or to find roots of a transcendental equation that we could not even attempt before without pre-specifying bounds on their values:
Another “old” piece of functionality with recent enhancement concerns mathematical functions. Ever since Version 1.0 we’ve set up mathematical functions so that they can be computed to arbitrary precision:
But in recent versions we’ve wanted to be “more precise about precision”, and to be able to rigorously compute just what range of outputs are possible given the range of values provided as input:
But every function for which we do this effectively requires a new theorem, and we’ve been steadily increasing the number of functions covered—now more than 130—so that this “just works” when you need to use it in a computation.
Trees are useful. We first introduced them as basic objects in the Wolfram Language only in Version 12.3. But now that they’re there, we’re discovering more and more places they can be used. And to support that, we’ve been adding more and more capabilities to them.
One area that’s advanced significantly since Version 13 is the rendering of trees. We tightened up the general graphic design, but, more importantly, we introduced many new options for how rendering should be done.
For example, here’s a random tree where we’ve specified that for all nodes only 3 children should be explicitly displayed: the others are elided away:
Here we’re adding several options to define the rendering of the tree:
By default, the branches in trees are labeled with integers, just like parts in an expression. But in Version 13.1 we added support for named branches defined by associations:
Our original conception of trees was very centered around having elements one would explicitly address, and that could have “payloads” attached. But what became clear is that there were applications where all that mattered was the structure of the tree, not anything about its elements. So we added UnlabeledTree to create “pure trees”:
Trees are useful because many kinds of structures are basically trees. And since Version 13 we’ve added capabilities for converting trees to and from various kinds of structures. For example, here’s a simple Dataset object:
You can use ExpressionTree to convert this to a tree:
And TreeExpression to convert it back:
We’ve also added capabilities for converting to and from JSON and XML, as well as for representing file directory structures as trees:
In Version 1.0 we had integers, rational numbers and real numbers. In Version 3.0 we added algebraic numbers (represented implicitly by Root)—and a dozen years later we added algebraic number fields and transcendental roots. For Version 14 we’ve now added another (long-awaited) “number-related” construct: finite fields.
Here’s our symbolic representation of the field of integers modulo 7:
And now here’s a specific element of that field
which we can immediately compute with:
But what’s really important about what we’ve done with finite fields is that we’ve fully integrated them into other functions in the system. So, for example, we can factor a polynomial whose coefficients are in a finite field:
We can also do things like find solutions to equations over finite fields. So here, for example, is a point on a Fermat curve over the finite field GF(173):
And here is a power of a matrix with elements over the same finite field:
A major new capability added since Version 13 is astro computation. It begins with being able to compute to high precision the positions of things like planets. Even knowing what one means by “position” is complicated, though—with lots of different coordinate systems to deal with. By default AstroPosition gives the position in the sky at the current time from your Here location:
But one can instead ask about a different coordinate system, like global galactic coordinates:
And now here’s a plot of the distance between Saturn and Jupiter over a 50-year period:
In direct analogy to GeoGraphics, we’ve added AstroGraphics, here showing a patch of sky around the current position of Saturn:
And this now shows the sequence of positions for Saturn over the course of a couple of years—yes, including retrograde motion:
There are many styling options for AstroGraphics. Here we’re adding a background of the “galactic sky”:
And here we’re including renderings for constellations (and, yes, we had an artist draw them):
Something specifically new in Version 14.0 has to do with extended handling of solar eclipses. We always try to deliver new functionality as fast as we can. But in this case there was a very specific deadline: the total solar eclipse visible from the US on April 8, 2024. We’ve had the ability to do global computations about solar eclipses for some time (actually since soon before the 2017 eclipse). But now we can also do detailed local computations right in the Wolfram Language.
So, for example, here’s a somewhat detailed overall map of the April 8, 2024, eclipse:
Now here’s a plot of the magnitude of the eclipse over a few hours, complete with a little “rampart” associated with the period of totality:
And here’s a map of the region of totality every minute just after the moment of maximum eclipse:
We first introduced computable data on biological organisms back when Wolfram|Alpha was released in 2009. But in Version 14—following several years of work—we’ve dramatically broadened and deepened the computable data we have about biological organisms.
So for example here’s how we can figure out what species have cheetahs as predators:
And here are pictures of these:
Here’s a map of countries where cheetahs have been seen (in the wild):
We now have data—curated from a great many sources—on more than a million species of animals, as well as most of the plants, fungi, bacteria, viruses and archaea that have been described. And for animals, for example, we have nearly 200 properties that are extensively filled in. Some are taxonomic properties:
Some are physical properties:
Some are genetic properties:
Some are ecological properties (yes, the cheetah is not the apex predator):
It’s useful to be able to get properties of individual species, but the real power of our curated computable data shows up when one does larger-scale analyses. Like here’s a plot of the lengths of genomes for organisms with the longest ones across our collection of organisms:
Or here’s a histogram of the genome lengths for organisms in the human gut microbiome:
And here’s a scatterplot of the lifespans of birds against their weights:
Following the idea that cheetahs aren’t apex predators, this is a graph of what’s “above” them in the food chain:
We began the process of introducing chemical computation into the Wolfram Language in Version 12.0, and by Version 13 we had good coverage of atoms, molecules, bonds and functional groups. Now in Version 14 we’ve added coverage of chemical formulas, amounts of chemicals—and chemical reactions.
Here’s a chemical formula, that basically just gives a “count of atoms”:
Now here are specific molecules with that formula:
Let’s pick one of these molecules:
Now in Version 14 we have a way to represent a certain quantity of molecules of a given type—here 1 gram of methylcyclopentane:
ChemicalConvert can convert to a different specification of quantity, here moles:
And here a count of molecules:
But now the bigger story is that in Version 14 we can represent not just individual types of molecules, and quantities of molecules, but also chemical reactions. Here we give a “sloppy” unbalanced representation of a reaction, and ReactionBalance gives us the balanced version:
And now we can extract the formulas for the reactants:
We can also give a chemical reaction in terms of molecules:
But with our symbolic representation of molecules and reactions, there’s now a big thing we can do: represent classes of reactions as “pattern reactions”, and work with them using the same kinds of concepts as we use in working with patterns for general expressions. So, for example, here’s a symbolic representation of the hydrohalogenation reaction:
Now we can apply this pattern reaction to particular molecules:
Here’s a more elaborate example, in this case entered using a SMARTS string:
Here we’re applying the reaction just once:
And now we’re doing it repeatedly
in this case generating longer and longer molecules (which in this case happen to be polypeptides):
Every minute of every day, new data is being added to the Wolfram Knowledgebase. Much of it is coming automatically from real-time feeds. But we also have a very large-scale ongoing curation effort with humans in the loop. We’ve built sophisticated (Wolfram Language) automation for our data curation pipeline over the years—and this year we’ve been able to increase efficiency in some areas by using LLM technology. But it’s hard to do curation right, and our long-term experience is that to do so ultimately requires human experts being in the loop, which we have.
So what’s new since Version 13.0? 291,842 new notable current and historical people; 264,467 music works; 118,538 music albums; 104,024 named stars; and so on. Sometimes the addition of an entity is driven by the new availability of reliable data; often it’s driven by the need to use that entity in some other piece of functionality (e.g. stars to render in AstroGraphics). But more than just adding entities there’s the issue of filling in values of properties of existing entities. And here again we’re always making progress, sometimes integrating newly available large-scale secondary data sources, and sometimes doing direct curation ourselves from primary sources.
A recent example where we needed to do direct curation was in data on alcoholic beverages. We have very extensive data on hundreds of thousands of types of foods and drinks. But none of our large-scale sources included data on alcoholic beverages. So that’s an area where we need to go to primary sources (in this case typically the original producers of products) and curate everything for ourselves.
So, for example, we can now ask for something like the distribution of flavors of different varieties of vodka (actually, personally, not being a consumer of such things, I had no idea vodka even had flavors…):
But beyond filling out entities and properties of existing types, we’ve also steadily been adding new entity types. One recent example is geological formations, 13,706 of them:
So now, for example, we can specify where T. rex have been found
and we can show those regions on a map:
PDEs are hard. It’s hard to solve them. And it’s hard to even specify what exactly you want to solve. But we’ve been on a multi-decade mission to “consumerize” PDEs and make them easier to work with. Many things go into this. You need to be able to easily specify elaborate geometries. You need to be able to easily define mathematically complicated boundary conditions. You need to have a streamlined way to set up the complicated equations that come out of underlying physics. Then you have to—as automatically as possible—do the sophisticated numerical analysis to efficiently solve the equations. But that’s not all. You also often need to visualize your solution, compute other things from it, or run optimizations of parameters over it.
It’s a deep use of what we’ve built with Wolfram Language—touching many parts of the system. And the result is something unique: a truly streamlined and integrated way to handle PDEs. One’s not dealing with some (usually very expensive) “just for PDEs” package; what we now have is a “consumerized” way to handle PDEs whenever they’re needed—for engineering, science, or whatever. And, yes, being able to connect machine learning, or image computation, or curated data, or data science, or real-time sensor feeds, or parallel computing, or, for that matter, Wolfram Notebooks, to PDEs just makes them so much more valuable.
We’ve had “basic, raw NDSolve” since 1991. But what’s taken decades to build is all the structure around that to let one conveniently set up—and efficiently solve—real-world PDEs, and connect them into everything else. It’s taken developing a whole tower of underlying algorithmic capabilities such as our more-flexible-and-integrated-than-ever-before industrial-strength computational geometry and finite element methods. But beyond that it’s taken creating a language for specifying real-world PDEs. And here the symbolic nature of the Wolfram Language—and our whole design framework—has made possible something very unique, that has allowed us to dramatically simplify and consumerize the use of PDEs.
It’s all about providing symbolic “construction kits” for PDEs and their boundary conditions. We started this about five years ago, progressively covering more and more application areas. In Version 14 we’ve particularly focused on solid mechanics, fluid mechanics, electromagnetics and (one-particle) quantum mechanics.
Here’s an example from solid mechanics. First, we define the variables we’re dealing with (displacement and underlying coordinates):
Next, we specify the parameters we want to use to describe the solid material we’re going to work with:
Now we can actually set up our PDE—using symbolic PDE specifications like SolidMechanicsPDEComponent—here for the deformation of a solid object pulled on one side:
And, yes, “underneath”, these simple symbolic specifications turn into a complicated “raw” PDE:
Now we are ready to actually solve our PDE in a particular region, i.e. for an object with a particular shape:
And now we can visualize the result, which shows how our object stretches when it’s pulled on:
The way we’ve set things up, the material for our object is an idealization of something like rubber. But in the Wolfram Language we now have ways to specify all sorts of detailed properties of materials. So, for example, we can add reinforcement as a unit vector in a particular direction (say in practice with fibers) to our material:
Then we can rerun what we did before
but now we get a slightly different result:
Another major PDE domain that’s new in Version 14.0 is fluid flow. Let’s do a 2D example. Our variables are 2D velocity and pressure:
Now we can set up our fluid system in a particular region, with no-slip conditions on all walls except at the top where we assume fluid is flowing from left to right. The only parameter needed is the Reynolds number. And instead of just solving our PDEs for a single Reynolds number, let’s create a parametric solver that can take any specified Reynolds number:
Now here’s the result for Reynolds number 100:
But with the way we’ve set things up, we can as well generate a whole video as a function of Reynolds number (and, yes, the Parallelize speeds things up by generating different frames in parallel):
Much of our work in PDEs involves catering to the complexities of real-world engineering situations. But in Version 14.0 we’re also adding features to support “pure physics”, and in particular to support quantum mechanics done with the Schrödinger equation. So here, for example, is the 2D 1-particle Schrödinger equation (with 
):
Here’s the region we’re going to be solving over—showing explicit discretization:
Now we can solve the equation, adding in some boundary conditions:
And now we get to visualize a Gaussian wave packet scattering around a barrier:
Systems engineering is a big field, but it’s one where the structure and capabilities of the Wolfram Language provide unique advantages—that over the past decade have allowed us to build out rather complete industrial-strength support for modeling, analysis and control design for a wide range of types of systems. It’s all an integrated part of the Wolfram Language, accessible through the computational and interface structure of the language. But it’s also integrated with our separate Wolfram System Modeler product, that provides a GUI-based workflow for system modeling and exploration.
Shared with System Modeler are large collections of domain-specific modeling libraries. And, for example, since Version 13, we’ve added libraries in areas such as battery engineering, hydraulic engineering and aircraft engineering—as well as educational libraries for mechanical engineering, thermal engineering, digital electronics, and biology. (We’ve also added libraries for areas such as business and public policy simulation.)
A typical workflow for systems engineering begins with the setting up of a model. The model can be built from scratch, or assembled from components in model libraries—either visually in Wolfram System Modeler, or programmatically in the Wolfram Language. For example, here’s a model of an electric motor that’s turning a load through a flexible shaft:
Once one’s got a model, one can then simulate it. Here’s an example where we’ve set one parameter of our model (the moment of inertia of the load), and we’re computing the values of two others as a function of time:
A new capability in Version 14.0 is being able to see the effect of uncertainty in parameters (or initial values, etc.) on the behavior of a system. So here, as an example, we’re saying the value of the parameter is not definite, but is instead distributed according to a normal distribution—then we’re seeing the distribution of output results:
The motor with flexible shaft that we’re looking at can be thought of as a “multidomain system”, combining electrical and mechanical components. But the Wolfram Language (and Wolfram System Modeler) can also handle “mixed systems”, combining analog and digital (i.e. continuous and discrete) components. Here’s a fairly sophisticated example from the world of control systems: a helicopter model connected in a closed loop to a digital control system:
This whole model system can be represented symbolically just by:
And now we compute the input-output response of the model:
Here’s specifically the output response:
But now we can “drill in” and see specific subsystem responses, here of the zero-order hold device (labeled ZOH above)—complete with its little digital steps:
But what if we want to design the control systems ourselves? Well, in Version 14 we can now apply all our Wolfram Language control systems design functionality to arbitrary system models. Here’s an example of a simple model, in this case in chemical engineering (a continuously stirred tank):
Now we can take this model and design an LQG controller for it—then assemble a whole closed-loop system for it:
Now we can simulate the closed-loop system—and see that the controller succeeds in bringing the final value to 0:
Graphics have always been an important part of the story of the Wolfram Language, and for more than three decades we’ve been progressively enhancing and updating their appearance and functionality—sometimes with help from advances in hardware (e.g. GPU) capabilities.
Since Version 13 we’ve added a variety of “decorative” (or “annotative”) effects in 2D graphics. One example (useful for putting captions on things) is Haloing:
Another example is DropShadowing:
All of these are specified symbolically, and can be used throughout the system (e.g. in hover effects, etc). And, yes, there are many detailed parameters you can set:
A significant new capability in Version 14.0 is convenient texture mapping. We’ve had low-level polygon-by-polygon textures for a decade and a half. But now in Version 14.0 we’ve made it straightforward to map textures onto whole surfaces. Here’s an example wrapping a texture onto a sphere:
And here’s wrapping the same texture onto a more complicated surface:
A significant subtlety is that there are many ways to map what amount to “texture coordinate patches” onto surfaces. The documentation illustrates new, named cases:
And now here’s what happens with stereographic projection onto a sphere:
Here’s an example of “surface texture” for the planet Venus
and here it’s been mapped onto a sphere, which can be rotated:
Here’s a “flowerified” bunny:
Things like texture mapping help make graphics visually compelling. Since Version 13 we’ve also added a variety of “live visualization” capabilities that automatically “bring visualizations to life”. For example, any plot now by default has a “coordinate mouseover”:
As usual, there’s lots of ways to control such “highlighting” effects:
One might say it’s been two thousand years in the making. But four years ago (Version 12) we began to introduce a computable version of Euclid-style synthetic geometry.
The idea is to specify geometric scenes symbolically by giving a collection of (potentially implicit) constraints:
We can then generate a random instance of geometry consistent with the constraints—and in Version 14 we’ve considerably enhanced our ability to make sure that geometry will be “typical” and non-degenerate:
But now a new feature of Version 14 is that we can find values of geometric quantities that are determined by the constraints:
Here’s a slightly more complicated case:
And here we’re now solving for the areas of two triangles in the figure:
We’ve always been able to give explicit styles for particular elements of a scene:
Now one of the new features in Version 14 is being able to give general “geometric styling rules”, here just assigning random colors to each element:
Our goal with Wolfram Language is to make it as easy as possible to express oneself computationally. And a big part of achieving that is the coherent design of the language itself. But there’s another part as well, which is being able to actually enter Wolfram Language input one wants—say in a notebook—as easily as possible. And with every new version we make enhancements to this.
One area that’s been in continuous development is interactive syntax highlighting. We first added syntax highlighting nearly two decades ago—and over time we’ve progressively made it more and more sophisticated, responding both as you type, and as code gets executed. Some highlighting has always had obvious meaning. But particularly highlighting that is dynamic and based on cursor position has sometimes been harder to interpret. And in Version 14—leveraging the brighter color palettes that have become the norm in recent years—we’ve tuned our dynamic highlighting so it’s easier to quickly tell “where you are” within the structure of an expression:
On the subject of “knowing what one has”, another enhancement—added in Version 13.2—is differentiated frame coloring for different kinds of visual objects in notebooks. Is that thing one has a graphic? Or an image? Or a graph? Now one can tell from the color of frame when one selects it:
An important aspect of the Wolfram Language is that the names of built-in functions are spelled out enough that it’s easy to tell what they do. But often the names are therefore necessarily quite long, and so it’s important to be able to autocomplete them when one’s typing. In 13.3 we added the notion of “fuzzy autocompletion” that not only “completes to the end” a name one’s typing, but also can fill in intermediate letters, change capitalization, etc. Thus, for example, just typing lll brings up an autocompletion menu that begins with ListLogLogPlot:
A major user interface update that first appeared in Version 13.1—and has been enhanced in subsequent versions—is a default toolbar for every notebook:
The toolbar provides immediate access to evaluation controls, cell formatting and various kinds of input (like inline cells, 
, hyperlinks, drawing canvas, etc.)—as well as to things like 
cloud publishing, 
documentation search and 
“chat” (i.e. LLM) settings.
Much of the time, it’s useful to have the toolbar displayed in any notebook you’re working with. But on the left-hand side there’s a little tiny 
that lets you minimize the toolbar:
In 14.0 there’s a Preferences setting that makes the toolbar come up minimized in any new notebook you create—and this in effect gives you the best of both worlds: you have immediate access to the toolbar, but your notebooks don’t have anything “extra” that might distract from their content.
Another thing that’s advanced since Version 13 is the handling of “summary” forms of output in notebooks. A basic example is what happens if you generate a very large result. By default only a summary of the result is actually displayed. But now there’s a bar at the bottom that gives various options for how to handle the actual output:
By default, the output is only stored in your current kernel session. But by pressing the Iconize button you get an iconized form that will appear directly in your notebook (or one that can be copied anywhere) and that “has the whole output inside”. There’s also a Store full expression in notebook button, which will “invisibly” store the output expression “behind” the summary display.
If the expression is stored in the notebook, then it’ll be persistent across kernel sessions. Otherwise, well, you won’t be able to get to it in a different kernel session; the only thing you’ll have is the summary display:
It’s a similar story for large “computational objects”. Like here’s a Nearest function with a million data points:
By default, the data is just something that exists in your current kernel session. But now there’s a menu that lets you save the data in various persistent locations:
There are many ways to run the Wolfram Language. Even in Version 1.0 we had the notion of remote kernels: the notebook front end running on one machine (in those days essentially always a Mac, or a NeXT), and the kernel running on a different machine (in those days sometimes even connected by phone lines). But a decade ago came a major step forward: the Wolfram Cloud.
There are really two distinct ways in which the cloud is used. The first is in delivering a notebook experience similar to our longtime desktop experience, but running purely in a browser. And the second is in delivering APIs and other programmatically accessed capabilities—notably, even at the beginning, a decade ago, through things like APIFunction.
The Wolfram Cloud has been the target of intense development now for nearly 15 years. Alongside it have also come Wolfram Application Server and Wolfram Web Engine, which provide more streamlined support specifically for APIs (without things like user management, etc., but with things like clustering).
All of these—but particularly the Wolfram Cloud—have become core technology capabilities for us, supporting many of our other activities. So, for example, the Wolfram Function Repository and Wolfram Paclet Repository are both based on the Wolfram Cloud (and in fact this is true of our whole resource system). And when we came to build the Wolfram plugin for ChatGPT earlier this year, using the Wolfram Cloud allowed us to have the plugin deployed within a matter of days.
Since Version 13 there have been quite a few very different applications of the Wolfram Cloud. One is for the function ARPublish, which takes 3D geometry and puts it in the Wolfram Cloud with appropriate metadata to allow phones to get augmented-reality versions from a QR code of a cloud URL:
On the Cloud Notebook side, there’s been a steady increase in usage, notably of embedded Cloud Notebooks, which have for example become common on Wolfram Community, and are used all over the Wolfram Demonstrations Project. Our goal all along has been to make Cloud Notebooks be as easy to use as simple webpages, but to have the depth of capabilities that we’ve developed in notebooks over the past 35 years. We achieved this some years ago for fairly small notebooks, but in the past couple of years we’ve been going progressively further in handling even multi-hundred-megabyte notebooks. It’s a complicated story of caching, refreshing—and dodging the vicissitudes of web browsers. But at this point the vast majority of notebooks can be seamlessly deployed to the cloud, and will display as immediately as simple webpages.
It’s been possible to call external code from Wolfram Language ever since Version 1.0. But in Version 14 there are important advances in the extent and ease with which external code can be integrated. The overall goal is to be able to use all the power and coherence of the Wolfram Language even when some part of a computation is done in external code. And in Version 14 we’ve done a lot to streamline and automate the process by which external code can be integrated into the language.
Once something is integrated into the Wolfram Language it just becomes, for example, a function that can be used just like any other Wolfram Language function. But what’s underneath is necessarily quite different for different kinds of external code. There’s one setup for interpreted languages like Python. There’s another for C-like compiled languages and dynamic libraries. (And then there are others for external processes, APIs, and what amount to “importable code specifications”, say for neural networks.)
Let’s start with Python. We’ve had ExternalEvaluate for evaluating Python code since 2018. But when you actually come to use Python there are all these dependencies and libraries to deal with. And, yes, that’s one of the places where the incredible advantages of the Wolfram Language and its coherent design are painfully evident. But in Version 14.0 we now have a way to encapsulate all that Python complexity, so that we can deliver Python functionality within Wolfram Language, hiding all the messiness of Python dependencies, and even the versioning of Python itself.
As an example, let’s say we want to make a Wolfram Language function Emojize that uses the Python function emojize within the emoji Python library. Here’s how we can do that:
And now you can just call Emojize in the Wolfram Language and—under the hood—it’ll run Python code:
The way this works is that the first time you call Emojize, a Python environment with all the right features is created, then is cached for subsequent uses. And what’s important is that the Wolfram Language specification of Emojize is completely system independent (or as system independent as it can be, given vicissitudes of Python implementations). So that means that you can, for example, deploy Emojize in the Wolfram Function Repository just like you would deploy something written purely in Wolfram Language.
There’s very different engineering involved in calling C-compatible functions in dynamic libraries. But in Version 13.3 we also made this very streamlined using the function ForeignFunctionLoad. There’s all sorts of complexity associated with converting to and from native C data types, managing memory for data structures, etc. But we’ve now got very clean ways to do this in Wolfram Language.
As an example, here’s how one sets up a “foreign function” call to a function RAND_bytes in the OpenSSL library:
Inside this, we’re using Wolfram Language compiler technology to specify the native C types that will be used in the foreign function. But now we can package this all up into a Wolfram Language function:
And we can call this function just like any other Wolfram Language function:
Internally, all sorts of complicated things are going on. For example, we’re allocating a raw memory buffer that’s then getting fed to our C function. But when we do that memory allocation we’re creating a symbolic structure that defines it as a “managed object”:
And now when this object is no longer being used, the memory associated with it will be automatically freed.
And, yes, with both Python and C there’s quite a bit of complexity underneath. But the good news is that in Version 14 we’ve basically been able to automate handling it. And the result is that what gets exposed is pure, simple Wolfram Language.
But there’s another big piece to this. Within particular Python or C libraries there are often elaborate definitions of data structures that are specific to that library. And so to use these libraries one has to dive into all the—potentially idiosyncratic—complexities of those definitions. But in the Wolfram Language we have consistent symbolic representations for things, whether they’re images, or dates or types of chemicals. When you first hook up an external library you have to map its data structures to these. But once that’s done, anyone can use what’s been built, and seamlessly integrate with other things they’re doing, perhaps even calling other external code. In effect what’s happening is that one’s leveraging the whole design framework of the Wolfram Language, and applying that even when one’s using underlying implementations that aren’t based on the Wolfram Language.
A single line (or less) of Wolfram Language code can do a lot. But one of the remarkable things about the language is that it’s fundamentally scalable: good both for very short programs and very long programs. And since Version 13 there’ve been several advances in handling very long programs. One of them concerns “code editing”.
Standard Wolfram Notebooks work very well for exploratory, expository and many other forms of work. And it’s certainly possible to write large amounts of code in standard notebooks (and, for example, I personally do it). But when one’s doing “software-engineering-style work” it’s both more convenient and more familiar to use what amounts to a pure code editor, largely separate from code execution and exposition. And this is why we have the “package editor”, accessible from File > New > Package/Script. You’re still operating in the notebook environment, with all its sophisticated capabilities. But things have been “skinned” to provide a much more textual “code experience”—both in terms of editing, and in terms of what actually gets saved in .wl files.
Here’s typical example of the package editor in action (in this case applied to our GitLink package):
Several things are immediately evident. First, it’s very line oriented. Lines (of code) are numbered, and don’t break except at explicit newlines. There are headings just like in ordinary notebooks, but when the file is saved, they’re stored as comments with a certain stylized structure:
It’s still perfectly possible to run code in the package editor, but the output won’t get saved in the .wl file:
One thing that’s changed since Version 13 is that the toolbar is much enhanced. And for example there’s now “smart search” that is aware of code structure:
You can also ask to go to a line number—and you’ll immediately see whatever lines of code are nearby:
In addition to code editing, another set of features new since Version 13 of importance to serious developers concern automated testing. The main advance is the introduction of a fully symbolic testing framework, in which individual tests are represented as symbolic objects
and can be manipulated in symbolic form, then run using functions like TestEvaluate and TestReport:
In Version 14.0 there’s another new testing function—IntermediateTest—that lets you insert what amount to checkpoints inside larger tests:
Evaluating this test, we see that the intermediate tests were also run:
The Wolfram Function Repository has been a big success. We introduced it in 2019 as a way to make specific, individual contributed functions available in the Wolfram Language. And now there are more than 2900 such functions in the Repository.
The nearly 7000 functions that constitute the Wolfram Language as it is today have been painstakingly developed over the past three and a half decades, always mindful of creating a coherent whole with consistent design principles. And now in a sense the success of the Function Repository is one of the dividends of all that effort. Because it’s the coherence and consistency of the underlying language and its design principles that make it feasible to just add one function at a time, and have it really work. You want to add a function to do some very specific operation that combines images and graphs. Well, there’s a consistent representation of both images and graphs in the Wolfram Language, which you can leverage. And by following the principles of the Wolfram Language—like for the naming of functions—you can create a function that’ll be easy for Wolfram Language users to understand and use.
Using the Wolfram Function Repository is a remarkably seamless process. If you know the function’s name, you can just call it using ResourceFunction; the function will be loaded if it’s needed, and then it’ll just run:
If there’s an update available for the function, it’ll give you a message, but run the old version anyway. The message has a button that lets you load in the update; then you can rerun your input and use the new version. (If you’re writing code where you want to “burn in” a particular version of a function, you can just use the ResourceVersion option of ResourceFunction.)
If you want your code to look more elegant, just evaluate the ResourceFunction object
and use the formatted version:
And, by the way, pressing the + then gives you more information about the function:
An important feature of functions in the Function Repository is that they all have documentation pages—that are organized pretty much like the pages for built-in functions:
But how does one create a Function Repository entry? Just go to File > New > Repository Item > Function Repository Item and you’ll get a Definition Notebook:
We’ve optimized this to be as easy to fill in as possible, minimizing boilerplate and automatically checking for correctness and consistency whenever possible. And the result is that it’s perfectly realistic to create a simple Function Repository item in under an hour—with the main time spent being in the writing of good expository examples.
When you press Submit to Repository your function gets sent to the Wolfram Function Repository review team, whose mandate is to ensure that functions in the repository do what they say they do, work in a way that is consistent with general Wolfram Language design principles, have good names, and are adequately documented. Except for very specialized functions, the goal is to finish reviews within a week (and sometimes considerably sooner)—and to publish functions as soon as they are ready.
There’s a digest of new (and updated) functions in the Function Repository that gets sent out every Friday—and makes for interesting reading (you can subscribe here):
The Wolfram Function Repository is a curated public resource that can be accessed from any Wolfram Language system (and, by the way, the source code for every function is available—just press the Source Notebook button). But there’s another important use case for the infrastructure of the Function Repository: privately deployed “resource functions”.
It all works through the Wolfram Cloud. You use the exact same Definition Notebook, but now instead of submitting to the public Wolfram Function Repository, you just deploy your function to the Wolfram Cloud. You can make it private so that only you, or some specific group, can access it. Or you can make it public, so anyone who knows its URL can immediately access and use it in their Wolfram Language system.
This turns out to be a tremendously useful mechanism, both for group projects, and for creating published material. In a sense it’s a very lightweight but robust way to distribute code—packaged into functions that can immediately be used. (By the way, to find the functions you’ve published from your Wolfram Cloud account, just go to the DeployedResources folder in the cloud file browser.)
(For organizations that want to manage their own function repository, it’s worth mentioning that the whole Wolfram Function Repository mechanism—including the infrastructure for doing reviews, etc.—is also available in a private form through the Wolfram Enterprise Private Cloud.)
So what’s in the public Wolfram Function Repository? There are a lot of “specialty functions” intended for specific “niche” purposes—but very useful if they’re what you want:
There are functions that add various kinds of visualizations:
Some functions set up user interfaces:
Some functions link to external services:
Some functions provide simple utilities:
There are also functions that are being explored for potential inclusion in the core system:
There are also lots of “leading-edge” functions, added as part of research or exploratory development. And for example in pieces I write (including this one), I make a point of having all pictures and other output be backed by “click-to-copy” code that reproduces them—and this code quite often contains functions either from the public Wolfram Function Repository or from (publicly accessible) private deployments.
Paclets are a technology we’ve used for more than a decade and a half to distribute updated functionality to Wolfram Language systems in the field. In Version 13 we began the process of providing tools for anyone to create paclets. And since Version 13 we’ve introduced the Wolfram Language Paclet Repository as a centralized repository for paclets:
What is a paclet? It’s a collection of Wolfram Language functionality—including function definitions, documentation, external libraries, stylesheets, palettes and more—that can be distributed as a unit, and immediately deployed in any Wolfram Language system.
The Paclet Repository is a centralized place where anyone can publish paclets for public distribution. So how does this relate to the Wolfram Function Repository? They are interestingly complementary—with different optimization and different setups. The Function Repository is more lightweight, the Paclet Repository more flexible. The Function Repository is for making available individual new functions, that independently fit into the whole existing structure of the Wolfram Language. The Paclet Repository is for making available larger-scale pieces of functionality, that can define a whole framework and environment of their own.
The Function Repository is also fully curated, with every function being reviewed by our team before it is posted. The Paclet Repository is an immediate-deployment system, without pre-publication review. In the Function Repository every function is specified just by its name—and our review team is responsible for ensuring that names are well chosen and have no conflicts. In the Paclet Repository, every contributor gets their own namespace, and all their functions and other material live inside that namespace. So, for example, I contributed the function RandomHypergraph to the Function Repository, which can be accessed just as ResourceFunction["RandomHypergraph"]. But if I had put this function in a paclet in the Paclet Repository, it would have to be accessed as something like PacletSymbol["StephenWolfram/Hypergraphs", "RandomHypergraph"].
PacletSymbol, by the way, is a convenient way of “deep accessing” individual functions inside a paclet. PacletSymbol temporarily installs (and loads) a paclet so that you can access a particular symbol in it. But more often one wants to permanently install a paclet (using PacletInstall), then explicitly load its contents (using Needs) whenever one wants to have its symbols available. (All the various ancillary elements, like documentation, stylesheets, etc. in a paclet get set up when it is installed.)
What does a paclet look like in the Paclet Repository? Every paclet has a home page that typically includes an overall summary, a guide to the functions in the paclet, and some overall examples of the paclet:
Individual functions typically have their own documentation pages:
Just like in the main Wolfram Language documentation, there can be a whole hierarchy of guide pages, and there can be things like tutorials.
Notice that in examples in paclet documentation, one often sees constructs like 
. These represent symbols in the paclet, presented in forms like PacletSymbol["WolframChemistry/ProteinVisualization", "AmidePlanePlot"] that allow these symbols to be accessed in a “standalone” way. If you directly evaluate such a form, by the way, it’ll force (temporary) installation of the paclet, then return the actual, raw symbol that appears in the paclet:
So how does one create a paclet suitable for submission to the Paclet Repository? You can do it purely programmatically, or you can start from File > New > Repository Item > Paclet Repository Item, which launches what amounts to a whole paclet creation IDE. The first step is to specify where you want to assemble your paclet. You give some basic information
then a Paclet Resource Definition Notebook is created, from which you can give function definitions, set up documentation pages, specify what you want your paclet’s home page to be like, etc.:
There are lots of sophisticated tools that let you create full-featured paclets with the same kind of breadth and depth of capabilities that you find in the Wolfram Language itself. For example, Documentation Tools lets you construct full-featured documentation pages (function pages, guide pages, tutorials, …):
Once you’ve assembled a paclet, you can check it, build it, deploy it privately—or submit it to the Paclet Repository. And once you submit it, it will automatically get set up on the Paclet Repository servers, and within just a few minutes the pages you’ve created describing your paclet will show up on the Paclet Repository website.
So what’s in the Paclet Repository so far? There’s a lot of good and very serious stuff, contributed both by teams at our company and by members of the broader Wolfram Language community. In fact, many of the 134 paclets now in the Paclet Repository have enough in them that there’s a whole piece like this that one could write about them.
One category of things you’ll find in the Paclet Repository are snapshots of our ongoing internal development projects—many of which will eventually become built-in parts of the Wolfram Language. A good example of this is our LLM and Chat Notebook functionality, whose rapid development and deployment over the past year was made possible by the use of the Paclet Repository. Another example, representing ongoing work from our chemistry team (AKA WolframChemistry in the Paclet Repository) is the ChemistryFunctions paclet, which contains functions like:
And, yes, this is interactive:
Or, also from WolframChemistry:
Another “development snapshot” is DiffTools—a paclet for making and viewing diffs between strings, cells, notebooks, etc.:
A major paclet is QuantumFramework—which provides the functionality for our Wolfram Quantum Framework
and delivers broad support for quantum computing (with at least a few connections to multiway systems and our Physics Project):
Talking of our Physics Project, there are over 200 functions supporting it that are in the Wolfram Function Repository. But there are also paclets, like WolframInstitute/Hypergraph:
An example of an externally contributed package is Automata—with more than 250 functions for doing computations related to finite automata:
Another contributed paclet is FunctionalParsers, which goes from a symbolic parser specification to an actual parser, here being used in a reverse mode to generate random “sentences”:
Phi4Tools is a more specialized paclet, for working with Feynman diagrams in 
field theory:
And, as another example, here’s MaXrd, for crystallography and x-ray scattering:
As just one more example, there’s the Organizer paclet—a utility paclet for making and manipulating organizer notebooks. But unlike the other paclets we’ve seen here, it doesn’t expose any Wolfram Language functions; instead, when you install it, it puts a palette in your Palettes list:
As of today, Version 14 is finished, and out in the world. So what’s next? We have lots of projects underway—some already with years of development behind them. Some extend and strengthen what’s already in the Wolfram Language; some take it in new directions.
One major focus is broadening and streamlining the deployment of the language: unifying the way it’s delivered and installed on computers, packaging it so it can be efficiently integrated into other standalone applications, etc.
Another major focus is expanding the handling of very large amounts of data by the Wolfram Language—and seamlessly integrating out-of-core and lazy processing.
Then of course there’s algorithmic development. Some is “classical”, directly building on the towers of functionality we’ve developed over the decades. Some is more “AI based”. We’ve been creating heuristic algorithms and meta-algorithms ever since Version 1.0—increasingly using methods from machine learning. How far will neural net methods go? We don’t know yet. We’re routinely using them in things like algorithm selection. But to what extent can they help in the heart of algorithms?
I’m reminded of something we did back in 1987 in developing Version 1.0. There was a long tradition in numerical analysis of painstakingly deriving series approximations for particular cases of mathematical functions. But we wanted to be able to compute hundreds of different functions to arbitrary precision for any complex values of their arguments. So how did we do it? We generalized from series to rational approximations—and then, in a very “machine-learning-esque” way—we spent months of CPU time systematically optimizing these approximations. Well, we’ve been trying to do the same kind of thing again—though now over more ambitious domains—and now using not rational functions but large neural nets as our basis.
We’ve also been exploring using neural nets to “control” precise algorithms, in effect making heuristic choices which either guide or can be validated by the precise algorithms. So far, none of what we’ve produced has outperformed our existing methods, but it seems plausible that fairly soon it will.
We’re doing a lot with various aspects of metaprogramming. There’s the project of
getting LLMs to help in the construction of Wolfram Language code—and in giving comments on it, and in analyzing what went wrong if the code didn’t do what one expected. Then there’s code annotation—where LLMs may help in doing things like predicting the most likely type for something. And there’s code compilation. We’ve been working for many years on a full-scale compiler for the Wolfram Language, and in every version what we have becomes progressively more capable. We’ve been doing some level of automatic compilation in particular cases (particularly ones involving numerical computation) for more than 30 years. And eventually full-scale automatic compilation will be possible for everything. But as of now some of the biggest payoffs from our compiler technology have been for our internal development, where we can now get optimal down-to-the-metal performance simply by compiled (albeit carefully written) Wolfram Language code.
One of the big lessons of the surprising success of LLMs is that there’s potentially more structure in meaningful human language than we thought. I’ve long been interested in creating what I’ve called a “symbolic discourse language” that gives a computational representation of everyday discourse. The LLMs haven’t explicitly done that. But they encourage the idea that it should be possible, and they also provide practical help in doing it. And whether the goal is to be able to represent narrative text, or contracts, or textual specifications, it’s a matter of extending the computational language we’ve built to encompass more kinds of concepts and structures.
There are typically several kinds of drivers for our continued development efforts. Sometimes it’s a question of continuing to build a tower of capabilities in some known direction (like, for example, solving PDEs). Sometimes the tower we’ve built suddenly lets us see new possibilities. Sometimes when we actually use what we’ve built we realize there’s an obvious way to polish or extend it—or to “double down” on something that we can now see is valuable. And then there are cases where things happening in the technology world suddenly open up new possibilities—like LLMs have recently done, and perhaps XR will eventually do. And finally there are cases where new science-related insights suggest new directions.
I had assumed that our Physics Project would at best have practical applications only centuries hence. But in fact it’s become clear that the correspondence it’s defined between physics and computation gives us quite immediate new ways to think about aspects of practical computation. And indeed we’re now actively exploring how to use this to define a new level of parallel and distributed computation in the Wolfram Language, as well as to represent symbolically not only the results of computations but also the ongoing process of computation.
One might think that after nearly four decades of intense development there wouldn’t be anything left to do in developing the Wolfram Language. But in fact at every level we reach, there’s ever more that becomes possible, and ever more that can we see might be possible. And indeed this moment is a particularly fertile one, with an unprecedentedly broad waterfront of possibilities. Version 14 is an important and satisfying waypoint. But there are wonderful things ahead—as we continue our long-term mission to make the computational paradigm achieve its potential, and to build our computational language to help that happen.
We call it perception. We call it measurement. We call it analysis. But in the end it’s about how we take the world as it is, and derive from it the impression of it that we have in our minds.
We might have thought that we could do science “purely objectively” without any reference to observers or their nature. But what we’ve discovered particularly dramatically in our Physics Project is that the nature of us as observers is critical even in determining the most fundamental laws we attribute to the universe.
But what ultimately does an observer—say like us—do? And how can we make a theoretical framework for it? Much as we have a general model for the process of computation—instantiated by something like a Turing machine—we’d like to have a general model for the process of observation: a general “observer theory”.
Central to what we think of as an observer is the notion that the observer will take the raw complexity of the world and extract from it some reduced representation suitable for a finite mind. There might be zillions of photons impinging on our eyes, but all we extract is the arrangement of objects in a visual scene. Or there might be zillions of gas molecules impinging on a piston, yet all we extract is the overall pressure of the gas.
In the end, we can think of it fundamentally as being about equivalencing. There are immense numbers of different individual configurations for the photons or the gas molecules—that are all treated as equivalent by an observer who’s just picking out the particular features needed for some reduced representation.
There’s in a sense a certain duality between computation and observation. In computation one’s generating new states of a system. In observation, one’s equivalencing together different states.
That equivalencing must in the end be implemented “underneath” by computation. But in observer theory what we want to do is just characterize the equivalencing that’s achieved. For us as observers it might in practice be all about how our senses work, what our biological or cultural nature is—or what technological devices or structures we’ve built. But what makes a coherent concept of observer theory possible is that there seem to be general, abstract characterizations that capture the essence of different kinds of observers.
It’s not immediately obvious that anything suitable for a finite mind could ever be extracted from the complexity of the world. And indeed the Principle of Computational Equivalence implies that computational irreducibility (and its multicomputational generalization) will be ubiquitous. But within computational irreducibility there must always be slices of computational reducibility. And it’s these slices of reducibility that an observer must try to pick out—and that ultimately make it possible for a finite mind to develop a “useful narrative” about what happens in the world, that allows it to make decisions, predictions, and so on.
How “special” is what an observer does? At its core it’s just about taking a large set of possible inputs, and returning a much smaller set of possible outputs. And certainly that’s a conceptual idea that’s appeared in many fields under many different names: a contractive mapping, reduction to canonical form, a classifier, an acceptor, a forgetful functor, evolving to an attractor, extracting statistics, model fitting, lossy compression, projection, phase transitions, renormalization group transformations, coarse graining and so on. But here we want to think not about what’s “mathematically describable”, but instead about what in general is actually implemented—say by our senses, our measuring devices, or our ways of analyzing things.
At an ultimate level, everything that happens can be thought of as being captured by the ruliad—the unique object that emerges as the entangled limit of all possible computations. And in a vast generalization of ideas like that our brains—like any other material thing—are made of atoms, so too any observer must be embedded as some kind of structure within the ruliad. But a key concept of observer theory is that it’s possible to make conclusions about an observer’s impression of the world just by knowing about the capabilities—and assumptions—of the observer, without knowing in detail what the observer is “like inside”.
And so it is, for example, that in our Physics Project we seem to be able to derive—essentially from the structure of the ruliad—the core laws of twentieth-century physics (general relativity, quantum mechanics and the Second Law) just on the basis of two features of us as observers: that we’re computationally bounded, and that we believe we’re persistent in time (even though “underneath” we’re made of different atoms of space at every successive moment). And we can expect that if we were to include other features of us as observers (for example, that we believe there are persistent objects in the world, or that we believe we have free will) then we’d be able to derive more aspects of the universe as we experience it—or of natural laws we attribute to it.
But the notion of observers—and observer theory—isn’t limited purely to “physical observers”. It applies whenever we try to “get an impression” of something. And so, for example, we can also operate as “mathematical observers”, sampling the ruliad to build up conclusions about mathematical laws. Some features of us as physical observers—like the computational boundedness associated with the finiteness of our minds—inevitably carry over to us as mathematical observers. But other features do not. But the point of observer theory is to provide a general framework in which we can characterize observers—and then see the consequences of those characterizations for the impressions or conclusions observers will form.
As humans we have senses like sight, hearing, touch, taste, smell and balance. And through our technology we also have access to a few thousand other kinds of measurements. So how basically do all these work?
The vast majority in effect aggregate a large number of small inputs to generate some kind of “average” output—which in the case of measurements is often specified as a (real) number. In a few cases, however, there’s instead a discrete choice between outputs that’s made on the basis of whether the total input exceeds a threshold (think: distributed consensus schemes, weighing balances, etc.)
But in all cases what’s fundamentally happening is that lots of different input configurations are all being equivalenced—or, more operationally, the dynamics of the system essentially make all equivalenced states evolve to the same “attractor state”.
As an example, let’s consider measuring the pressure of a gas. There are various ways to do this. But a very direct one is just to have a piston, and see how much force is exerted by the gas on this piston. So where does this force come from? At the lowest level it’s the result of lots of individual molecules bouncing off the surface of the piston, each transferring a tiny amount of momentum to it. If we looked at the piston at an atomic scale, we’d see it temporarily deform from each molecular impact. But the crucial point is that at a large scale the piston moves together, as a single rigid object—aggregating the effects of all those individual molecular impacts.
But why does it work this way? Essentially it’s because the intermolecular forces inside the piston are much stronger than the forces associated with molecules in the gas. Or, put more abstractly, there’s more coupling and coherence “inside the observer” than between the observer and what it’s observing.
We see the same basic pattern over and over again. There’s some form of transduction that couples the individual elements of what’s being observed to the observer. Then “within the observer” there’s something that in essence aggregates all these small effects. Sometimes that aggregation is “directly numerical”, as in the addition of lots of small momentum transfers. But sometimes it’s instead more explicitly like evolution to one attractor rather than another.
Consider, for example, the case of vision. An array of photons fall on the photoreceptor cells on our retinas, generating electrical signals transmitted through nerve fibers to our brains. Within the brain there’s then effectively a neural net that evolves to different attractors depending on what one’s looking at. Most of the time a small change in input image won’t affect what attractor one evolves to. But—much like with a weighing balance—there’s an “edge” at which even a small change can lead to a different output.
One can go through lots of different types of sensory systems and measuring devices. But the basic outline seems to always be the same. First, there’s a coupling between what is being sensed or measured and the thing that’s doing the sensing or measuring. Quite often that coupling involves transducing from one physical form to another—say from light to electricity, or from force to position. Sometimes then the crucial step of equivalencing different detailed inputs is achieved by simple “numerical aggregation”, most often by accumulation of objects (atoms, raindrops, etc.) or physical effects (forces, currents, etc.). But sometimes the equivalencing is instead achieved by a more obviously dynamical process.
It could amount to simple amplification, in which, say, the presence of a small element of input (say an individual particle) “tips over” some metastable system so that it goes into a certain final state. Or it could be more like a neural net where there’s a more complicated translation defined by hard-to-describe borders between basins of attraction leading to different attractors.
But, OK, so what’s the endpoint of a process of observation? Ultimately for us humans it’s an impression created in our minds. Of course that gets into lots of slippery philosophical issues. Yes, each of us has an “inner experience” of what’s going on in our mind. But anything else is ultimately an extrapolation. We make the assumption that other human minds also “see what we see”, but we can never “feel it from the inside”.
We can of course make increasingly detailed measurements—say of neural activity—to see how similar what’s going on is between one brain and another. But as soon as there’s the slightest structural—or situational—difference between the brains, we really can’t say exactly how their “impressions” will compare.
But for our purposes in constructing a general “observer theory” we’re basically going to make the assumption (or, in effect, “philosophical approximation”) that whenever a system does enough equivalencing, that’s tantamount to it “acting like an observer”, because it can then act as a “front end” that takes the “incoherent complexity of the world” and “collimates it” to the point where a mind will derive a definite impression from it.
Of course, there’s still a lot of subtlety here. There has to be “just enough equivalencing” and not too much. For example, if all inputs were always equivalenced to the same output, there’d be nothing useful observed. And in the end there’s somehow got to be some kind of match between the compression of input achieved by equivalencing, and the “capacity” of the mind that’s ultimately deriving an impression from it.
A crucial feature of anything that can reasonably be called a mind is that “something’s got to be going on in there”. It can’t be, for example, that the internal state of the system is fixed. There has to be some internal dynamics—some computational process that we can identify as the ongoing operation of the mind.
At an informational level we might say that there has to be more information processing going on inside than there is flow of information from the outside. Or, in other words, if we’re going to be meaningful “observers like us” we can’t just be bombarded by input we don’t process; we have to have some capability to “think about what we’re seeing”.
All of this comes back to the idea that a crucial feature of us as observers is that we are computationally bounded. We do computation; that’s why we can have an “inner sense of things going on”. But the amount of computation we do is tiny compared to the computation going on in the world around us. Our experience represents a heavily filtered version of “what’s happening outside”. And the essence of “being an observer like us” is that we’re effectively doing lots of equivalencing to get to that filtered version.
But can we imagine a future in which we “expand our minds”? Or perhaps encounter some alien intelligence with a fundamentally “less constrained mind”? Well, at some point there’s an issue with this. Because in a sense the idea that we have a coherent existence relies on us having “limited minds”. For without such constraints there wouldn’t be a coherent “self” that we could identify—with coherent inner experience.
Let’s say we’re shown some system—say in nature—“from the outside”. Can we tell if “there’s an observer in there”? Ultimately not, because in a sense we’d have to be “inside that observer” and be able to experience the impression of the world that it’s getting. But in much the same way as we extrapolate to believing that, say, other human minds are experiencing things like we’re experiencing, so also we can potentially extrapolate to say what we might think of as an observer.
And the core idea seems to be that an “observer” should be a subsystem whose “internal states” are affected by the rest of the system, but where many “external states” lead to the same internal state—and where there is rich dynamics “within the observer” that in effect operates only on its internal states. Ultimately—following the Principle of Computational Equivalence—both the outside and the inside of the “observer subsystem” can be expected to be equivalent in the computations they’re performing. But the point is that the coupling from outside the subsystem to inside effectively “coarse grains” what’s outside, so that the “inner computation” is operating on a much-reduced set of elements.
Why should any such “observer subsystems” exist? Presumably at some level it’s inevitable from the presence of pockets of computational reducibility within arbitrary computationally irreducible systems. But more important for us is that our very existence—and the possibility of our coherent inner experience—depends on us “operating as observers”. And—almost as a “self-fulfilling prophecy”—our behavior tends to perpetuate our ability to successfully do this. For example, we can think of us as choosing to put ourselves in situations and environments where we can “predict what’s going to happen” well enough to “survive as observers”. (At a mundane practical level we might do this by not living in places subject to unpredictable natural forces—or by doing things like building ourselves structures that shelter us from those forces.)
We’ve talked about observers operating by compressing the complexities of the world to “inner impressions” suitable for finite minds. And in typical situations that we describe as perception and measurement, the main way this happens is by fairly direct equivalencing of different states. But in a sense there’s a higher-level story that relies on formalization—and in essence computation—and that’s what we usually call “analysis”.
Let’s say we have some intricate structure—perhaps some nested, fractal pattern. A direct rendering of all the pixels in this pattern ultimately won’t be something well suited for a “finite mind”. But if we gave rules—or a program—for generating the pattern we’d have a much more succinct representation of it.
But now there’s a problem with computational irreducibility. Yes, the rules determine the pattern. But to get from these rules to the actual pattern can require an irreducible amount of computation. And to “reverse engineer the pattern” to find the rules can require even more computation.
Yes, there are particular cases—like repetitive and simple nested patterns—where there’s enough immediate computational reducibility that a computationally bounded system (or observer) can fairly easily “do the analysis” and “get the compression”. But in general it’s hard. And indeed in a sense it’s the whole mission of science to pick away at the problem, and try to find more ways to “reduce the complexities of the world” to “human-level narratives”.
Computational irreducibility limits the extent to which this can be successful. But the inevitable existence of pockets of reducibility even within computational irreducibility guarantees that progress can always in principle be made. As we invent more kinds of measuring devices we can extend our domain as observers. And the same is true when we invent more methods of analysis, or identify more principles in science.
But the overall picture remains the same: what’s crucial to “being an observer” is equivalencing many “states of the world”, either through perceiving or measuring only specific aspects of them, or through identifying “simplified narratives” that capture them. (In effect, perception and measurement tend to do “lossy compression”; analysis is more about “lossless compression” where the equivalencing is effectively not between possible inputs but between possible generative rules.)
Our view of the world is ultimately determined by what we observe of it. We take what’s “out there in the world” and in effect “construct our perceived reality” by our operation as observers. Or, in other words, insofar as we have a narrative about “what’s going on in the world”, that’s something that comes from our operation as observers.
And in fact from our Physics Project we’re led to an extreme version of this—in which what’s “out there in the world” is just the whole ruliad, and in effect everything specific about our perceived reality must come from how we operate as observers and thus how we sample the ruliad.
But long before we get to this ultimate level of abstraction, there are lots of ways in which our nature as observers “builds” our perceived reality. Think about any material substance—like a fluid. Ultimately it’s made up of lots of individual molecules “doing their thing”. But observers like us aren’t seeing those molecules. Instead, we’re aggregating things to the point where we can just describe the system as a fluid, that operates according to the “narrative” defined by the laws of fluid mechanics.
But why do things work this way? Ultimately it’s the result of the repeated story of the interplay between underlying computational irreducibility, and the computational boundedness of us as observers. At the lowest level the motion of the molecules is governed by simple rules of mechanics. But the phenomenon of computational irreducibility implies that to work out the detailed consequences of “running these rules” involves an irreducible amount of computational work—which is something that we as computationally bounded observers can’t do. And the result of this is that we’ll end up describing the detailed behavior of the molecules as just “random”. As I’ve discussed at length elsewhere, this is the fundamental origin of the Second Law of thermodynamics. But for our purposes here the important point is that it’s what makes observers like us “construct the reality” of things like fluids. Our computational boundedness as observers makes us unable to trace all the detailed behavior of molecules, and leaves us “content” to describe fluids in terms of the “narrative” defined by the laws of fluid mechanics.
Our Physics Project implies that it’s the same kind of story with physical space. For in our Physics Project, space is ultimately “made” of a network of relations (or connections) between discrete “atoms of space”—that’s progressively being updated in what ends up being a computationally irreducible way. But we as computationally bounded observers can’t “decode” all the details of what’s happening, and instead we end up with a simple “aggregate” narrative, that turns out to correspond to continuum space operating according to the laws of general relativity.
The way both coherent notions of “matter” (or fluids) and spacetime emerge for us as observers can be thought of as a consequence of the equivalencing we do as observers. In both cases, there’s immense and computationally irreducible complexity “underneath”. But we’re ignoring most of that—by effectively treating different detailed behaviors as equivalent—so that in the end we get to a (comparatively) “simple narrative” more suitable for our finite minds. But we should emphasize that what’s “really going on in the system” is something much more complicated; it’s just that we as observers aren’t paying attention to that, so our perceived reality is much simpler.
OK, but what about quantum mechanics? In a sense that’s an extreme test of our description of how observers work, and the extent to which the operation of observers “constructs their perceived reality”.
In our Physics Project the underlying structure (hypergraph) that represents space and everything in it is progressively being rewritten according to definite rules. But the crucial point is that at any given stage there can be lots of ways this rewriting can happen. And the result is that there’s a whole tree of possible “states of the universe” that can be generated. So given this, why do we ever think that definite things happen in the universe? Why don’t we just think that there’s an infinite tree of branching histories for the universe?
Well, it all has to do with our nature as observers, and the equivalencing we do. At an immediate level, we can imagine looking at all those different possible branching paths for the evolution of the universe. And the key point is that even though they come from different paths of history, two states can just be the same. Sometimes it’ll be obvious that they’re same; sometimes one might have to determine, say, whether two hypergraphs are isomorphic. But the point is that to any observer (at least one that isn’t managing to look at arbitrary “implementation details”), the states will inevitably be considered equivalent.
But now there’s a bigger point. Even though “from the outside” there might be a whole branching and merging multiway graph of histories for the universe, observers like us can’t trace that. And in fact all we perceive is a single thread of history. Or, said another way, we believe that we have a single thread of experience—something closely related to our belief that (despite the changing “underlying elements” from which we are made) we are somehow persistent in time (at least during the span of our existence).
But operationally, how do we go from all those underlying branches of history to our perceived single thread of history? We can think of the states on different threads of history as being related by what we call a branchial graph, that joins states that have immediate common ancestors. And in the limit of many threads, we can think of these different states as being laid out “branchial space”. (In traditional quantum mechanics terms, this layout defines a “map of quantum entanglements”—with each piece of common ancestry representing an entanglement between states.)
In physical space—whether we’re looking at molecules in a fluid or atoms of space—we can think of us operating as observers who are physically large enough to span many underlying discrete elements, so that what we end up observing is just some kind of aggregate, averaged result. And it’s very much the same kind of thing in branchial space: we as observers tend to be large enough in branchial space to be spread across an immense number of branches of history, so that what we observe is just aggregate, averaged results across all those branches.
There’s lots of detailed complexity in what happens on different branches, just like there is in what happens to different molecules, or different atoms of space. And the reason is that there’s inevitably computational irreducibility, or, in this case, more accurately, multicomputational irreducibility. But as computationally bounded observers we just perceive aggregate results that “average out” the “underlying apparent randomness” to give a consistent single thread of experience.
And effectively this is what happens in the transition from quantum to classical behavior. Even though there are many possible detailed (“quantum”) threads of history that an object can follow, what we perceive corresponds to a single consistent “aggregate” (“classical”) sequence of behavior.
And this is typically true even at the level of our typical observation of molecules and chemical processes. Yes, there are many possible threads of history for, say, a water molecule. But most of our observations aggregate things to the point where we can talk about a definite shape for the molecule, with definite “chemical bonds”, etc.
But there is a special situation that actually looms large in typical discussions of quantum mechanics. We can think of it as the result of doing measurements that aren’t “aggregating threads of history to get an average”, but are instead doing something more like a weighing balance, always “tipping” one way or the other. In the language of quantum computing, we might say that we’re arranging things to be able to “measure a single qubit”. In terms of the equivalencing of states, we might say that we’re equivalencing lots of underlying states to specific canonical states (like “spin up” and “spin down”).
Why do we get one outcome rather than another? Ultimately we can think of it as all depending on the details of us as observers. To see this, let’s start from the corresponding question in physical space. We might ask why we observe some particular thing happening. Well, in our Physics Project everything about “what happens” is deterministic. But there’s still the “arbitrariness” of where we are in physical space. We’ll always basically see the same laws of physics, but the particulars of what we’ll observe depend on where we are, say on the surface of the Earth versus in interstellar space, etc.
Is there a “theory” for “where we are”? In some sense, yes, because we can go back and see why the molecules that make us up landed up in the particular place where they did. But what we can’t have an “external theory” for is just which molecules end up making up “us”, as we experience ourselves “from inside”. In our view of physics and the universe, it’s in some sense the only “ultimately subjective” thing: where our internal experience is “situated”.
And the point is that basically—even though it’s much less familiar—the same thing is going on at the level of quantum mechanics. Just as we “happen” to be at a certain place in physical space, so we’re at a certain place in branchial space. Looking back we can trace how we got here. But there’s no a priori way to determine “where our particular experience will be situated”. And that means we can’t know what the “local branchial environment” will be—and so, for example, what the outcome of “balance-like” measurements will be.
Just as in traditional discussions of quantum mechanics, the mechanics of doing the measurement—which we can think of as effectively equivalencing many underlying branches of history—will have an effect on subsequent behavior, and subsequent measurements.
But let’s say we look just at the level of the underlying multiway graph—or, more specifically, the multiway causal graph that records causal connections between different updating events. Then we can identify a complicated web of interdependence between events that are timelike, spacelike and branchlike separated. And this interdependence seems to correspond precisely to what’s expected from quantum mechanics.
In other words, even though the multiway graph is completely determined, the arbitrariness of “where the observer is” (particularly in branchial space), combined with the inevitable interdependence of different aspects of the multiway (causal) graph, seems sufficient to reproduce the not-quite-purely-probabilistic features of quantum mechanics.
In making observations in physical space, it’s common to make a measurement at one place or time, then make another measurement at another place or time, and, for example, see how they’re related. But in actually doing this, the observer will have to move from one place to the other, and persist from one time to another. And in the abstract it’s not obvious that that’s possible. For example, it could be that an observer won’t be able to move without changing—or, in other words, that “pure motion” won’t be possible for an observer. But in effect this is something we as observers assume about ourselves. And indeed, as I’ve discussed elsewhere, this is a crucial part of why we perceive spacetime to operate according to the laws of physics we know.
But what about in branchial space? We have much less intuition for this than for physical space. But we still effectively believe that pure motion is possible for us as observers in branchial space. It could be—like an observer in physical space, say, near a spacetime singularity—that an observer would get “shredded” when trying to “move” in branchial space. But our belief is that typically nothing like that happens. At some level being at different locations in branchial space presumably corresponds to picking different bases for our quantum states, or effectively to defining our experiments differently. And somehow our belief in the possibility of pure motion in branchial space seems related to our belief in the possibility of making arbitrary sequences choices in sets of experiments we do.
We might have thought that the only thing ultimately “out there” for us to observe would be our physical universe. But actually there are important situations where we’re essentially operating not as observers of our familiar physical universe, but instead of what amount to abstract universes. And what we’ll see is that the ideas of observer theory seem to apply there too—except that now what we’re picking out and reducing to “internal impressions” are features not of the physical world but of abstract worlds.
Our Physics Project in a sense brings ideas about the physical and abstract worlds closer—and the concept of the ruliad ultimately leads to a deep unification between them. For what we now imagine is that the physical universe as we perceive it is just the result of the particular kind of sampling of the ruliad made by us as certain kinds of observers. And the point is that we as observers can make other kinds of samplings, leading to what we can describe as abstract universes. And one particularly prominent example of this is mathematics, or rather, metamathematics.
Imagine starting from all possible axioms for mathematics, then constructing the network of all possible theorems that can be derived from them. We can consider this as forming a kind of “metamathematical universe”. And the particular mathematics that some mathematician might study we can then think of as the result of a “mathematical observer” observing that metamathematical universe.
There are both close analogies and differences between this and the experience of a physical observer in the physical universe. Both ultimately correspond to samplings of the ruliad, but somewhat different ones.
In our Physics Project we imagine that physical space and everything in it is ultimately made up of discrete elements that we identify as “atoms of space”. But in the ruliad in general we can think of everything being made up of “pure atoms of existence” that we call emes. In the particular case of physics we interpret these emes as atoms of space. But in metamathematics we can think of emes as corresponding to (“subaxiomatic”) elements of symbolic structures—from which things like axioms or theorems can be constructed.
A central feature of our interaction with the ruliad for physics is that observers like us don’t track the detailed behavior of all the various atoms of space. Instead, we equivalence things to the point where we get descriptions that are reduced enough to “fit in our minds”. And something similar is going on in mathematics.
We don’t track all the individual subaxiomatic emes—or usually in practice even the details of fully formalized axioms and theorems. Instead, mathematics typically operates at a much higher and “more human” level, dealing not with questions like how real numbers can be built from emes—or even axioms—but rather with what can be deduced about the properties of mathematical objects like real numbers. In a physics analogy to the behavior of a gas, typical human mathematics operates not at the “molecular” level of individual emes (or even axioms) but rather at the “fluid dynamics” level of “human-accessible” mathematical concepts.
In effect, therefore, a mathematician is operating as an observer who equivalences many detailed configurations—ultimately of emes—in order to form higher-level mathematical constructs suitable for our computationally bounded minds. And while at the outset one might have imagined that anything in the ruliad could serve as a “possible mathematics”, the point is that observers like us can only sample the ruliad in particular ways—leading to only particular possible forms for “human-accessible” mathematics.
It’s a very similar story to the one we’ve encountered many times in thinking about physics. In studying gases, for example, we could imagine all sorts of theories based on tracking detailed molecular motions. But for observers like us—with our computational boundedness—we inevitably end up with things like the Second Law of thermodynamics, and the laws of fluid mechanics. And in mathematics the main thing we end up with is “higher-level mathematics”—mathematics that we can do directly in terms of typical textbook concepts, rather than constantly having to “drill down” to the level of axioms, or emes.
In physics we’re usually particularly concerned with issues like predicting how things will evolve through time. In mathematics it’s more about accumulating what can be considered true. And indeed we can think of an idealized mathematician as going through the ruliad and collecting in their minds a “bag” of theorems (or axioms) that they “consider to be true”. And given such a collection, they can essentially follow the “entailment paths” defined by computations in the ruliad to find more theorems to “add to their bag”. (And, yes, if they put in a false theorem then—because a false premise in the standard setup of logic implies everything—they’ll end up with an “infinite explosion of theorems”, that won’t fit in a finite mind.)
In observing the physical universe, we talk about our different possible senses (like vision, hearing, etc.) or different kinds of measuring devices. In observing the metamathematical universe the analogy is basically different possible kinds of theories or abstractions—say, algebraic vs. geometrical vs. topological vs. categorical, etc. (with new approaches being like new kinds of measuring devices).
Particularly when we think in terms of the ruliad we can expect a certain kind of ultimate unity in the metamathematical universe—but different theories and different abstractions will pick up different aspects of it, just as vision and hearing pick up different aspects of the physical universe. But in a sense observer theory gives us a global way to talk about this, and to characterize what kinds of observations observers like us can make—whether of the physical universe or the metamathematical one.
In physics we’ve then seen in our Physics Project how this allows us to find general laws that describe our perception of the physical world—and that turn out to reproduce the core known laws of physics. In mathematics we’re not as familiar with the concept of general laws, though the very fact that higher-level mathematics is possible is presumably in essence such a law, and perhaps the kinds of regularities seen in areas like category theory are others—as are the inevitable dualities we expect to be able to identify between different fields of mathematics. All these laws ultimately rely on the structure of the ruliad. But the crucial point is that they’re not talking about the “raw ruliad”; instead they’re talking about just certain samplings of the ruliad that can be done by observers like us, and that lead to certain kinds of “internal impressions” in terms of which these laws can be stated.
Mathematics represents a certain kind of abstract setup that’s been studied in a particularly detailed way over the centuries. But it’s not the only kind of “abstract setup” we can imagine. And indeed there’s even a much more familiar one: the use of concepts—and words—in human thinking and language.
We might imagine that at some time in the distant past our forebears could signify, say, rocks only by pointing at individual ones. But then there emerged the general notion of “rock”, captured by a word for “rock”. And once again this is a story of observers and equivalences. When we look at a rock, it presumably produces all sorts of detailed patterns of neuron firings in our brains, different for each particular rock. But somehow—presumably essentially through evolution to an attractor in the neural net in our brains—we equivalence all these patterns to extract our “inner impression” of the “concept of a rock”.
In the typical tradition of quantitative science we tend to be interested in doing measurements that lead to things like numerical results. But in representing the world using language we tend to be interested instead in creating symbolic structures that involve collections of discrete words embedded in a grammatical framework. Such linguistic descriptions don’t capture every detail; in a typical observer kind of way they broadly equivalence many things—and in a sense reduce the complexity of the world to a description in terms of a limited number of discrete words and linguistic forms.
Within any given person’s brain there’ll be “thoughts” defined by patterns of neuron firings. And the crucial role of language is to provide a way to robustly “package up” those thoughts, and for example represent them with discrete words, so they can be communicated to another person—and unpacked in that person’s brain to produce neuron firings that reproduce what amount to those same thoughts.
When we’re dealing with something like a numerical measurement we might imagine that it could have some kind of absolute interpretation. But words are much more obviously an “arbitrary basis” for communication. We could pick a different specific word (say from a different human language) but still “communicate the same thing”. All that’s required is that everyone who’s using the word agrees on its meaning. And presumably that normally happens because of shared “social” history between people who use a given word.
It’s worth pointing out that for this to work there has to be a certain separation of scales. The collective impression of the meaning of a word may change over time, but that change has to be slow compared to the rate at which the word is used in actual communication. In effect, the meaning of a word—as we humans might understand it—emerges from the aggregation of many individual uses.
In the abstract, there might not be any reason to think that there’d be a way to “understand words consistently”. But it’s a story very much like what we’ve encountered in both physics and mathematics. Even though there are lots of complicated individual details “underneath”, we as observers manage to pick out features that are “simple enough for us to understand”. In the case of molecules in a gas that might be the overall pressure of the gas. And in the case of words it’s a stable notion of “meaning”.
Put another way, the possibility of language is another example of observer theory at work. Inside our brains there are all sorts of complicated neuron firings. But somehow these can be “packaged up” into things like words that form “human-level narratives”.
There’s a certain complicated feedback loop between the world as we experience it and the words we use to describe it. We invent words for things that we commonly encounter (“chair”, “table”, …). Yet once we have a word for something we’re more able to form thoughts about it, or communicate about it. And that in turn makes us more likely to put instances of it in our environment. In other words, we tend to build our environment so that the way we have of making narratives about it works well—or, in effect, so our inner description of it can be as simple as possible, and it can be as predictable to us as possible.
We can view our experience of physics and of mathematics as being the result of us acting as physical observers and mathematical observers. Now we’re viewing our experience of the “conceptual universe” as being the result of us acting as “conceptual observers”. But what’s crucial is that in all these cases, we have the same intrinsic features as observers: computational boundedness and a belief in persistence. The computational boundedness is what makes us equivalence things to the point where we can have symbolic descriptions of the world, for example in terms of words. And the belief in persistence is what lets those words have persistent meanings.
And actually these ideas extend beyond just language—to paradigms, and general ways of thinking about things. When we define a word we’re in effect defining an abstraction for a class of things. And paradigms are somehow a generalization of this: ways of taking lots of specifics and coming up with a uniform framework for them. And when we do this, we’re in effect making a classic observer theory move—and equivalencing lots of different things to produce an “internal impression” that’s “simple enough” to fit in our finite minds.
Our tendency as observers is always to believe that we can separate our “inner experience” from what’s going on in the “outside world”. But in the end everything is just part of the ruliad. And at the level of the ruliad we as observers are ultimately “made of the same stuff” as everything else.
But can we imagine that we can point at one part of the ruliad and say “that’s an observer”, and at another part and say “that’s not”? At least to some extent the answer is presumably yes—at least if we restrict ourselves to “observers like us”. But it’s a somewhat subtle—and seemingly circular—story.
For example, one core feature of observers like us is that we have a certain persistence, or at least we believe we have a certain persistence. But, inevitably, at the level of the “raw ruliad”, we’re continually being made from different atoms of existence, i.e. different emes. So in what sense are we persistent? Well, the point is that an observer can equivalence those successive patterns of emes, so that what they observe is persistent. And, yes, this is at least on the face of it circular. And ultimately to identify what parts of the ruliad might be “persistent enough to be observers”, we’ll have to ground this circularity in some kind of further assumption.
What about the computational boundedness of observers like us, which forces us to do lots of equivalencing? At some level that equivalencing must be implemented by lots of different states evolving to the same states. But once again there’s circularity, because even to define what we mean by “the same states” (“Are isomorphic graphs the same?”, etc.) we have to be imagining certain equivalencing.
So how do we break out of the circularity? The key is presumably the presence of additional features that define “observers like us”. And one important class of such features has to do with scale.
We’re neither tiny nor huge. We involve enough emes that consistent averages can emerge. Yet we don’t involve so many emes that we span anything but an absolutely tiny part of the whole ruliad.
And actually a lot of our experience is determined by “our size as observers”. We’re large enough that certain equivalencing is inevitable. Yet we’re small enough that we can reasonably think of there being many choices for “where we are”.
The overall structure of the ruliad is a matter of formal necessity; there’s only one possible way for it to be. But there’s contingency in our character as observers. And for example in a sense there’s a fundamental constant of nature as we perceive it, which is our extent in the ruliad, say measured in emes (and appropriately projected into physical space, branchial space, etc.).
And the fact that this extent is small compared to the whole ruliad means that there are “many possible observers”—who we can think of as existing at different positions in the ruliad. And those different observers will look at the ruliad from different “points of view”, and thus develop different “internal impressions” of “perceived reality”.
But a crucial fact central to our Physics Project is that there are certain aspects of that perceived reality that are inevitable for observers like us—and that correspond to core laws of physics. But when it gets to more specific questions (“What does the night sky look like from where you are?”, etc.) different observers will inevitably have different versions of perceived reality.
So is there a way to translate from one observer to another? Essentially that’s a story of motion. What happens when an observer at one place in the ruliad “moves” to another place? Inevitably, the observer will be “made of different emes” if it’s at a different place. But will it somehow still “be the same”? Well, that’s a subtle question, that depends both on the background structure of the ruliad, and the nature of the observer.
If the ruliad is “too wild” (think: spacetime near a singularity) then the observer will inevitably be “shredded” as it “moves”. But computational irreducibility implies a certain overall regularity to most of the ruliad, making “pure motion” at least conceivable. But to achieve “pure motion” the observer still has to be “made of” something that is somehow robust—essentially some “lump of computational reducibility” that can “predictably survive” the underlying background of computational irreducibility.
In spacetime we can identify such “lumps” with things like black holes, and particles like electrons, photons, etc. (and, yes, in our models there’s probably considerable commonality between black holes and particles). It’s not yet clear quite what the analog is in branchial space, though a very simple example might involve persistence of qubits. And in rulial space, one kind of analog is the very notion of concepts. For in effect concepts (as represented for example by words) are the analog of particles in rulial space: they are the robust structures that can move across rulial space and “maintain their identity”, carrying “the same thoughts” to different minds.
So what does all this mean for what can constitute an observer in the ruliad? Observers in effect leverage computational reducibility to extract simplified features that can “fit in finite minds”. But observers themselves must also embody computational reducibility in order to maintain their own persistence and the persistence of the features they extract. Or in other words, observers must in a sense always correspond to “patches of regularity” in the ruliad.
But can any patch of regularity in the ruliad be thought of as an observer? Probably not usefully so. Because another feature of observers like us is that we are connected in some kind of collective “social” framework. Not only do we individually form internal impressions in our minds, but we also communicate these impressions. And indeed without such communication we wouldn’t, for example, be able to set up things like coherent languages with which to describe things.
A key implication of our Physics Project and the concept of the ruliad is that we perceive the universe to be the way we do because we are the way we are as observers. And the most fundamental aspect of observers like us is that we’re doing lots of equivalencing to reduce the “complexity of the world” to “internal impressions” that “fit into our minds”. But just what kinds of equivalencing are we actually doing? At some level a lot of that is defined by the things we believe—or assume—about ourselves and the way we interact with the world.
A very central assumption we make is that we’re somehow “stable observers” of a changing “outside world”. Of course, at some level we’re actually not “stable” at all: we’re built up from emes whose configuration is changing all the time. But our belief in our own stability—and, in effect, our belief in our “persistence in time”—makes us equivalence those configurations. And having done that equivalencing we perceive the universe to operate in a certain way, that turns out to align with the laws of physics we know.
But actually there’s more than just our assumption of persistence in time. For example, we also have an assumption of persistence in space: we assume that—at least on reasonably short timescales—we’re consistently “observing the universe from the same place”, and not, say, “continually darting around”. The network that represents space is continually changing “around us”. But we equivalence things so that we can assume that—in a first approximation—we are “staying in the same place”.
Of course, we don’t believe that we have to stay in exactly the same place all the time; we believe we’re able to move. And here we make what amounts to another “assumption of stability”: we assume that pure motion is possible for us as observers. In other words, we assume that we can “go to different places” and still be “the same us”, with the same properties as observers.
At the level of the “raw ruliad” it’s not at all obvious that such assumptions can be consistently made. But as we discussed above, the fact that for observers like us they can (at least to a good approximation) is a reflection of certain properties of us as observers—in particular of our physical scale, being large in terms of atoms of space but small in terms of the whole universe.
Related to our assumption about motion is our assumption that “space exists”—or that we can treat space as something coherent. Underneath, there’s all sorts of complicated dynamics of changing patterns of emes. But on the timescales at which we experience things we can equivalence these patterns to allow us to think of space as having a “coherent structure”. And, once again, the fact that we can do this is a consequence of physical scales associated with us as observers. In particular, the speed of light is “fast enough” that it brings information to us from the local region around us in much less time than it takes our brain to process it. And this means that we can equivalence all the different ways in which different pieces of information reach us, and we can consistently just talk about the state of a region of space at a given time.
Part of our assumption that we’re “persistent in time” is that our thread of experience is—at least locally—continuous, with no breaks. Yes, we’re born and we die—and we also sleep. But we assume that at least on scales relevant for our ongoing perception of the world, we experience time as something continuous.
More than that, we assume that we have just a single thread of experience. Or, in other words, that there’s always just “one us” going through time. Of course, even at the level of neurons in our brains all sorts of activity goes on in parallel. But somehow in our normal psychological state we seem to concentrate everything so that our “inner experience” follows just one “thread of history”, on which we can operate in a computationally bounded way, and form definite memories and have definite sequences of thoughts.
We’re not as familiar with branchial space as with physical space. But presumably our “fundamental assumption of stability” extends there as well. And when combined with our basic computational boundedness it then becomes inevitable that (as we discussed above) we’ll conflate different “quantum paths of history” to give us as observers a definite “classical thread of inner experience”.
Beyond “stability”, another very important assumption we implicitly make about ourselves is what amounts to an assumption of “independence”. We imagine that we can somehow separate ourselves off from “everything else”. And one aspect of this is that we assume we’re localized—and that most of the ruliad “doesn’t matter to us”, so that we can equivalence all the different states of the “rest of the ruliad”.
But there’s also another aspect of “independence”: that in effect we can choose to do “whatever we want” independent of the rest of the universe. And this means that we assume we can, for example, essentially “do any possible experiment”, make any possible measurement—or “go anywhere we want” in physical or branchial space, or indeed rulial space. We assume that we effectively have “free will” about these things—determined only by our “inner choices”, and independent of the state of the rest of the universe.
Ultimately, of course, we’re just part of the ruliad, and everything we do is determined by the structure of the ruliad and our history within it. But we can view our “belief of freedom” as a reflection of the fact that we don’t know a priori where we’ll be located in the ruliad—and even if we did, computational irreducibility would prevent us from making predictions about what we will do.
Beyond our assumptions about our own “independence from the rest of the universe”, there’s also the question of independence between different parts of what we observe. And quite central to our way of “parsing the world” is our typical assumption that we can “think about different things separately”. In other words, we assume it’s possible to “factor” what we see happening in the universe into independent parts.
In science, this manifests itself in the idea that we can do “controlled experiments” in which we study how something behaves in isolation from everything else. It’s not self-evident that this will be possible (and indeed in areas like ethics it might fundamentally not be), but we as observers tend to implicitly assume it.
And actually, we normally go much further. Because we typically assume that we can describe—and think about—the world “symbolically”. In other words, we assume that we can take all the complexity of the world and represent at least the parts of it that we care about in terms of discrete symbolic concepts, of the kind that appear in human (or computational) language. There’s lots of detail in the world that our limited collection of symbolic concepts doesn’t capture, and effectively “equivalences out”. But the point is that it’s this symbolic description that normally seems to form the backbone of the “inner narrative” we have about the world.
There’s another implicit assumption that’s being made here, however. And that’s that there’s some kind of stability in the symbolic concepts we’re using. Yes, any particular mind might parse the world using a particular set of symbolic concepts. But we make the implicit assumption that there are other minds out there that work like ours. And this makes us imagine that there can be some form of “objective reality” that’s just “always out there”, to be sampled by whatever mind might happen to come along.
Not only, therefore, do we assume our own stability as observers; we also assume a certain stability to what we perceive of “everything that’s out there”. Underneath, there’s all the wildness and complexity of the ruliad. But we assume that we can successfully equivalence things to the point where all we perceive is something quite stable—and something that we can describe as ultimately governed by consistent laws.
It could be that every part of the universe just “does its own thing”, with no overall laws tying everything together. But we make the implicit assumption that, no, the universe—at least as far as we perceive it—is a more organized and consistent place. And indeed it’s that assumption that makes it feasible for us to operate as observers like us at all, and to even imagine that we can usefully reduce the complexity of the world to something that “fits in our finite minds”.
What resources does it take for an observer to make an observation? In most of traditional science, observation is at best added as an afterthought, and no account is taken of the process by which it occurs. And indeed, for example, in the traditional formalism of quantum mechanics, while “measurement” can have an effect on a system, it’s still assumed to be an “indivisible act” without any “internal process”.
But in observer theory, we’re centrally talking about the process of observation. And so it makes sense to try asking questions about the resources involved in this process.
We might start with our own everyday experience. Something happens out in the world. What resources—and, for example, how much time—does it take us to “form an impression of it”? Let’s say that out in the world a cat either comes into view or it doesn’t. There are signals that come to our brain from our eyes, effectively carrying data on each pixel in our visual field. Then, inside our brain, these signals are processed by a succession of layers of neurons, with us in the end concluding either “there’s a cat there”, or “there’s not”.
And from artificial neural nets we can get a pretty good idea of how this likely works. And the key to it—as we discussed above—is that there’s an attractor. Lots of different detailed configurations of pixels all evolve either to the “cat” or “no cat” final state. The different configurations have been equivalenced, so that only a “final conclusion” survives.
The story is a bit trickier though. Because “cat” or “no cat” really isn’t the final state of our brain; hopefully it’s not the “last thought we have”. Instead, our brain will continue to “think more thoughts”. So “cat”/”no cat” is at best some kind of intermediate waypoint in our process of thinking; an instantaneous conclusion that we’ll continue to “build on”.
And indeed when we consider measuring devices (like a piston measuring the pressure of a gas) we similarly usually imagine that they will “come to an instantaneous conclusion”, but “continue operating” and “producing more data”. But how long should we wait for each intermediate conclusion? How long, for example, will it take for the stresses generated by a particular pattern of molecules hitting a piston to “dissipate out”, and for the piston to be “ready to produce more data”?
There are lots of specific questions of physics here. But if our purpose is to build a formal observer theory, how should we think about such things? There is something of an analogy in the formal theory of computation. An actual computational system—say in the physical world—will just “keep computing”. But in formal computation theory it’s useful to talk about computations that halt, and about functions that can be “evaluated” and give a “definite answer”. So what’s the analog of this in observer theory?
Instead of general computations, we’re interested in computations that effectively “implement equivalences”. Or, put another way, we want computations that “destroy information”—and that have many incoming states but few outgoing ones. As a practical matter, we can either have the outgoing states explicitly represent whole equivalence classes, or they can just be “canonical representatives”—like in a network where at each step each element takes on whatever the “majority” or “consensus” value of its neighbors was.
But however it works, we can still ask questions about what computational resources were involved. How many steps did it take? How many elements were involved?
And with the idea that observers like us are “computationally bounded”, we expect limitations on these resources. But with this formal setup we can start asking just how far an observer like us can get, say in “coming to a conclusion” about the results of some computationally irreducible process.
An interesting case arises in putative quantum computers. In the model implied by our Physics Project, such a “quantum computer” effectively “performs many computations in parallel” on the separate branches of a multiway system representing the various threads of history of the universe. But if the observer tries to “come to a conclusion” about what actually happened, they have to “knit together” all those threads of history, in effect by implementing equivalences between them.
One could in principle imagine an observer who’d just follow all the quantum branches. But it wouldn’t be an observer like us. Because what seems to be a core feature of observers like us is that we believe we have just a single thread of experience. And to maintain that belief, our “process of observation” must equivalence all the different quantum branches.
How much “effort” will that be? Well, inevitably if a thread of history branched, our equivalencing has to “undo that branching”. And that suggests that the number of “elementary equivalencings” will have to be at least comparable to the number of “elementary branchings”—making it seem that the “effort of observation” will tend to be at least comparable to reduction of effort associated with parallelism in the “underlying quantum process”.
In general it’s interesting to compare the “effort of observation” with the “effort of computation”. With our concept of “elementary equivalencings” we have a way to measure both in terms of computational operations. And, yes, both could in principle be implemented by something like a Turing machine, though in practice the equivalencings might be most conveniently modeled by something like string rewriting.
And indeed one can often go much further, talking not directly in terms of equivalencings, but rather about processes that show attractors. There are different kinds of attractors. Sometimes—as in class 1 cellular automata—there are just a limited number of static, global fixed points (say, either all cells black or all cells white). But in other cases—such as class 3 cellular automata—the number of “output states” may be smaller than the number of “input states” but there may be no computationally simple characterization of them.
“Observers like us”, though, mostly seem to make use of the fixed points. We try to “symbolicize the world”, taking all the complexities “out there”, and reducing them to “discrete conclusions”, that we might for example describe using the discrete words in a language.
There’s an immediate subtlety associated with attractors of any kind, though. Typical physics is reversible, in the sense that any process (say two molecules scattering from each other) can run equally well forwards and backwards. But in an attractor one goes from lots of possible initial states to a smaller number of “attractor” final states. And there are two basic ways this can happen, even when there’s underlying reversibility. First, the system one’s studying can be “open”, in the sense that effects can “radiate” out of the region that one’s studying. And second, the states the system gets into can be “complicated enough” that, say, a computationally bounded observer will inevitably equivalence them. And indeed that’s the main thing that’s happening, for example, when a system “reaches thermodynamic equilibrium”, as described by the Second Law.
And actually, once again, there’s often a certain circularity. One is trying to determine whether an observer has “finished observing” and “come to a conclusion”. But one needs an observer to make that determination. Can we tell if we’ve finished “forming a thought”? Well, we have to “think about it”—in effect by forming another thought.
Put another way: imagine we are trying to determine whether a piston has “come to a conclusion” about pressure in a gas. Particularly if there’s microscopic reversibility, the piston and things around it will “continue wiggling around”, and it’ll “take an observer” to determine whether the “heat is dissipated” to the point where one can “read out the result”.
But how do we break out of what seems like an infinite regress? The point is that whatever mind is ultimately forming the impression that is “the observation” is inevitably the final arbiter. And, yes, this could mean that we’d always have to start discussing all sorts of details about photoreceptors and neurons and so on. But—as we’ve discussed at length—the key point that makes a general observer theory possible is that there are many conclusions that can be drawn for large classes of observers, quite independent of these details.
But, OK, what happens if we think about the raw ruliad? Now all we have are emes and elementary events updating the configuration of them. And in a sense we’re “fishing out of this” pieces that represent observers, and pieces that represent things they’re observing. Can we “assess the cost of observation” here? It really depends on the fundamental scale of what we consider to be observers. And in fact we might even think of our scale as observers (say measured in emes or elementary events) as defining a “fundamental constant of nature”—at least for the universe as we perceive it. But given this scale, we can for example ask for there to develop “consensus across it”, or at least for “every eme in it to have had time to communicate with every other”.
In an attempt to formalize the “cost of observation” we’ll inevitably have to make what seem like arbitrary choices, just as we would in setting up a scheme to determine when an ongoing computational process has “generated an answer”. But if we assume a certain boundedness to our choices, we can expect that we’ll be able to draw definite conclusions, and in effect be able to construct an analog of computational complexity theory for processes of observation.
My goal here has been to explore some of the key concepts and principles needed to create a framework that we can call observer theory. But what I’ve done is just the beginning, and there is much still to be done in fleshing out the theory and investigating its implications.
One important place to start is in making more explicit models of the “mechanics of observation”. At the level of the general theory, it’s all about equivalencing. But how specifically is that equivalencing achieved in particular cases? There are many thousands of kinds of sensors, measuring devices, analysis methods, etc. All of these should be systematically inventoried and classified. And in each case there’s a metamodel to be made, that clarifies just how equivalencing is achieved, and, for example, what separation of physical (or other) scales make it possible.
Human experience and human minds are the inspiration—and ultimate grounding—for our concept of an observer. And insofar as neural nets trained on what amounts to human experience have emerged as somewhat faithful models for what human minds do, we can expect to use them as a fairly detailed proxy for observers like us. So, for example, we can imagine exploring things like quantum observers by studying multiway generalizations of neural nets. (And this is something that becomes easier if instead of organizing their data into real-number weights we can “atomize” neural nets into purely discrete elements.)
Such investigations of potentially realistic models provide a useful “practical grounding” for observer theory. But to develop a general observer theory we need a more formal notion of an observer. And there is no doubt a whole abstract framework—perhaps using methods from areas like category theory—that can be developed purely on the basis of our concept of observers being about equivalencing.
But to understand the connection of observer theory to things like science as done by us humans, we need to tighten up what it means to be an “observer like us”. What exactly are all the general things we “believe about ourselves”? As we discussed above, many we so much take for granted that it’s challenging for us to identify them as actually just “beliefs” that in principle don’t have to be that way.
But I suspect that the more we can tighten up our definition of “observers like us”, the more we’ll be able to explain why we perceive the world the way we do, and attribute to it the laws and properties we do. Is there some feature of us as observers, for example, that makes us “parse” the physical world as being three-dimensional? We could represent the same data about what’s out there by assigning a one-dimensional (“space-filling”) coordinate to everything. But somehow observers like us don’t do that. And instead, in effect, we “probe the ruliad” by sampling it in what we perceive as 3D slices. (And, yes, the most obvious coarse graining just considers progressively larger geodesic balls, say in the spatial hypergraphs that appear in our Physics Project—but that’s probably at best just an approximation to the sampling observers like us do.)
As part of our Physics Project we’ve discovered that the structure of the three main theories of twentieth-century physics (statistical mechanics, general relativity and quantum mechanics) can be derived from properties of the ruliad just by knowing that observers like us are computationally bounded and believe we’re persistent in time. But how might we reach, say, the Standard Model of particle physics—with all its particular values of parameters, etc.? Some may be inevitable, given the underlying structure of our theory. But others, one suspects, are in effect reflections of aspects of us as observers. They are “derivable”, but only given our particular character—or beliefs—as observers. And, yes, presumably things like the “constant of nature” that characterizes “our size in emes” will appear in the laws we attribute to the universe as we perceive it.
And, by the way, these considerations of “observers like us” extend beyond physical observers. Thus, for example, as we tighten up our characterization of what we’re like as mathematical observers, we can expect that this will constrain the “possible laws of our mathematical universe”. We might have thought that we could “pick whatever axioms we want”, in effect sampling the ruliad to get any mathematics we want. But, presumably, observers like us can’t do this—so that questions like “Is the continuum hypothesis true?” can potentially have definite answers for any observers like us, and for any coherent mathematics that we build.
But in the end, do we really have to consider observers whose characteristics are grounded in human experience? We already reflexively generalize our own personal experiences to those of other humans. But can we go further? We don’t have the internal experience of being a dog, an ant colony, a computer, or an ocean. And typically at best we anthropomorphize such things, trying to reduce the behavior we perceive in them to elements that align with our own human experience.
But are we as humans just stuck with a particular kind of “internal experience”? The growth of technology—and in particular sensors and measuring devices—has certainly expanded the range of inputs that can be delivered to our brains. And the growth of our collective knowledge about the world has expanded our ways of representing and thinking about things. Right now those are basically our only ways of modifying our detailed “internal experience”. But what if we were to connect directly—and internally—into our brains?
Presumably, at least at first, we’d need the “neural user interface” to be familiar—and we’d be forced into, for example, concentrating everything into a single thread of experience. But what if we allowed “multiway experience”? Well, of course our brains are already made up of billions of neurons that each do things. But it seems to be a core feature of human experience that we concentrate those things to give a single thread of experience. And that seems to be an essential feature of being an “observer like us”.
That kind of concentration also happens in a flock of birds, an ant colony—or a human society. In all these cases, each individual organism “does their thing”. But somehow collective “decisions” get made, with many different detailed situations getting equivalenced together to leave only the “final decision”. So that means that from the outside, the system behaves as we would expect of an “observer like us”. Internally, that kind of “observer behavior” is happening “above the experience” of each single individual. But still, at the level of the “hive mind” it’s behavior typical of an observer like us.
That’s not to say, though, that we can readily imagine what it’s like to be a system like this, or even to be one of its parts. And in the effort to explore observer theory an important direction is to try to imagine ourselves having a different kind of experience than we do. And from “within” that experience, try to see what kind of laws would we attribute, say, to the physical universe.
In the early twentieth century, particularly in the context of relativity and quantum mechanics, it became clear that being “more realistic” about the observer was crucial in moving forward in science. Things like computational irreducibility—and even more so, our Physics Project—take that another step.
One used to imagine that science should somehow be “fundamentally objective”, and independent of all aspects of the observer. But what’s become clear is that it’s not. And that the nature of us as observers is actually crucial in determining what science we “experience”. But the crucial point is that there are often powerful conclusions that can be drawn even without knowing all the details of an observer. And that’s a central reason for building a general observer theory—in effect to give an objective way of formally and robustly characterizing what one might consider to be the subjective element in science.
There are no doubt many precursors of varying directness that can be found to the things I discuss here; I have not attempted a serious historical survey. In my own work, a notable precursor from 2002 is Chapter 10 of A New Kind of Science, entitled “Processes of Perception and Analysis”. I thank many people involved with our Wolfram Physics Project for related discussions, including Xerxes Arsiwalla, Hatem Elshatlawy and particularly Jonathan Gorard.
It’s all about systems where there can in effect be many possible paths of history. In a typical standard computational system like a cellular automaton, there’s always just one path, defined by evolution from one state to the next. But in a multiway system, there can be many possible next states—and thus many possible paths of history. Multiway systems have a central role in our Physics Project, particularly in connection with quantum mechanics. But what’s now emerging is that multiway systems in fact serve as a quite general foundation for a whole new “multicomputational” paradigm for modeling.
My objective here is twofold. First, I want to use multiway systems as minimal models for growth processes based on aggregation and tiling. And second, I want to use this concrete application as a way to develop further intuition about multiway systems in general. Elsewhere I have explored multiway systems for strings, multiway systems based on numbers, multiway Turing machines, multiway combinators, multiway expression evaluation and multiway systems based on games and puzzles. But in studying multiway systems for aggregation and tiling, we’ll be dealing with something that is immediately more physical and tangible.
When we think of “growth by aggregation” we typically imagine a “random process” in which new pieces get added “at random” to something. But each of these “random possibilities” in effect defines a different path of history. And the concept of a multiway system is to capture all those possibilities together. In a typical random (or “stochastic”) model one’s just tracing a single path of history, and one imagines one doesn’t have enough information to say which path it will be. But in a multiway system one’s looking at all the paths. And in doing so, one’s in a sense making a model for the “whole story” of what can happen.
The choice of a single path can be “nondeterministic”. But the whole multiway system is deterministic. And by studying that “deterministic whole” it’s often possible to make useful, quite general statements.
One can think of a particular moment in the evolution of a multiway system as giving something like an ensemble of states of the kind studied in statistical mechanics. But the general concept of a multiway system, with its discrete branching at discrete steps, depends on a level of fundamental discreteness that’s quite unfamiliar from traditional statistical mechanics—though is perfectly straightforward to define in a computational, or even mathematical, way.
For aggregation it’s easy enough to set up a minimal discrete model—at least if one allows explicit randomness in the model. But a major point of what we’ll do here is to “go above” that randomness, setting up our model in terms of a whole, deterministic multiway system.
What can we learn by looking at this whole multiway system? Well, for example, we can see whether there’ll always be growth—whatever the random choices may be—or whether the growth will sometimes, or even always, stop. And in many practical applications (think, for example, tumors) it can be very important to know whether growth always stops—or through what paths it can continue.
A lot of what we’ll at first do here involves seeing the effect of local constraints on growth. Later on, we’ll also look at effects of geometry, and we’ll study how objects of different shapes can aggregate, or ultimately tile.
The models we’ll introduce are in a sense very minimal—combining the simplest multiway structures with the simplest spatial structures. And with this minimality it’s almost inevitable that the models will show up as idealizations of all sorts of systems—and as foundations for good models of these systems.
At first, multiway systems can seem rather abstract and difficult to grasp—and perhaps that’s inevitable given our human tendency to think sequentially. But by seeing how multiway systems play out in the concrete case of growth processes, we get to build our intuition and develop a more grounded view—that will stand us in good stead in exploring other applications of multiway systems, and in general in coming to terms with the whole multicomputational paradigm.
It’s the ultimate minimal model for random discrete growth (often called the Eden model). On a square grid, start with one black cell, then at each step randomly attach a new black cell somewhere onto the growing “cluster”:
After 10,000 steps we might get:
But what are all the possible things that can happen? For that, we can construct a multiway system:
A lot of these clusters differ only by a trivial translation; canonicalizing by translation we get
or after another step:
If we also reduce out rotations and reflections we get
or after another step:
The set of possible clusters after t steps are just the possible polyominoes (or “square lattice animals”) with t cells. The number of these for successive t is
growing roughly like kt for large t, with k a little larger than 4:
By the way, canonicalization by translation always reduces the number of possible clusters by a factor of t. Canonicalization by rotation and reflection can reduce the number by a factor of 8 if the cluster has no symmetry (which for large clusters becomes increasingly likely), and by a smaller factor the more symmetry the cluster has, as in:
With canonicalization, the multiway graph after 7 steps has the form
and it doesn’t look any simpler with alternative rendering:
If we imagine that at each step, cells are added with equal probability at every possible position on the cluster, or equivalently that all outgoing edges from a given cluster in the uncanonicalized multiway graph are followed with equal probability, then we can get a distribution of probabilities for the distinct canonical clusters obtained—here shown after 7 steps:
One feature of the large random cluster we saw at the beginning is that it has some holes in it. Clusters with holes start developing after 7 steps, with the smallest being:
This cluster can be reached through a subset of the multiway system:
And in fact in the limit of large clusters, the probability for there to be a hole seems to approach 1—even though the total fraction of area covered by holes approaches 0.
One way to characterize the “space of possible clusters” is to create a branchial graph by connecting every pair of clusters that have a common ancestor one step back in the multiway graph:
The connectedness of all these graphs reflects the fact that with the rule we’re using, it’s always possible at any step to go from one cluster to another by a sequence of delete-one-cell/add-one-cell changes.
The branchial graphs here also show a 4-fold symmetry resulting from the symmetry of the underlying lattice. Canonicalizing the states, we get smaller branchial graphs that no longer show any such symmetry:
With the rule we’ve been discussing so far, a new cell to be attached can be anywhere on a cluster. But what if we limit growth, by requiring that new cells must have certain numbers of existing cells around them? Specifically, let’s consider rules that look at the 
neighbors around any given position, and allow a new cell there only if there are specified numbers of existing cells in the neighborhood.
Starting with a cross of black cells, here are some examples of random clusters one gets after 20 steps with all possible rules of this type (the initial “4” designates that these are 4-neighbor rules):
Rules that don’t allow new cells to end up with just one existing neighbor can only fill in corners in their initial conditions, and can’t grow any further. But any rule that allows growth with only one existing neighbor produces clusters that keep growing forever. And here are some random examples of what one can get after 10,000 steps:
The last of these is the unconstrained (Eden model) rule we already discussed above. But let’s look more carefully at the first case—where there’s growth only if a new cell will end up with exactly one neighbor. The canonicalized multiway graph in this case is:
The possible clusters here correspond to polyominoes that are “always one cell wide” (i.e. have no 2×2 blocks), or, equivalently, have perimeter 2t + 2 at step t. The number of such canonicalized clusters grows like:
This is an increasing fraction of the total number of polyominoes—implying that most large polyominoes take this “spindly” form.
A new feature of a rule with constraints is that not all locations around a cluster may allow growth. Here is a version of the multiway system above, with cells around each cluster annotated with green if new growth is allowed there, and red if it never can be:
In a larger random cluster, we can see that with this rule, most of the interior is “dead” in the sense that the constraint of the rule allows no further growth there:
By the way, the clusters generated by this rule can always be directly represented by their “skeleton graphs”:
Looking at random clusters for all the (grow-with-1-neighbor) rules above, we see different patterns of holes in each case:
There are altogether five types of cells being distinguished here, reflecting different neighbor configurations:
Here’s a sample cluster generated with the 4:{1,3} rule:
Cells indicated with 
already have too many neighbors, and so can never be added to the cluster. Cells indicated with 
have exactly the right number of neighbors to be added immediately. Cells indicated with 
don’t currently have the right number of neighbors to grow, but if neighbors are filled in, they might be able to be added. Sometimes it will turn out that when neighbors of cells get filled in, they will actually prevent the cell from being added (so that it becomes )—and in the particular case shown here that happens with the 2×2 blocks of cells.
The multiway graphs from the rules shown here are all qualitatively similar, but there are detailed differences. In particular, at least for many of the rules, an increasing number of states are “missing” relative to what one gets with the grow-in-all-cases 4:{1,2,3,4} rule—or, in other words, there are an increasing number of polyominoes that can’t be generated given the constraints:
The first polyomino that can’t be reached (which occurs at step 4) is:
At step 6 the polyominoes that can’t be reached for rules 4:{1,3} and 4:{1,3,4} are
while for 4:{1} and 4:{1,4} the additional polyomino
can also not be reached.
At step 8, the polyomino
is reachable with 4:{1} and 4:{1,3} but not with 4:{1,4} and 4:{1,3,4}.
Of some note is that none of the rules that exclude polyominoes can reach:
What happens if one considers diagonal as well orthogonal neighbors, giving a total of 8 neighbors around a cell? There are 256 possible rules in this case, corresponding to the possible subsets of Range[8]. Here are samples of what they do after 200 steps, starting from an initial 
cluster:
Two cases that at least initially show growth here are (the “8” designates that these are 8-neighbor rules):
In the {2} case, the multiway graph begins with:
One might assume that every branch in this graph would continue forever, and that growth would never “get stuck”. But it turns out that after 9 steps the following cluster is generated:
And with this cluster, no further growth is possible: no positions around the boundary have exactly 2 neighbors. In the multiway graph up to 10 steps, it turns out this is the only “terminal cluster” that can be generated—out of a total of 1115 possible clusters:
So how is that terminal cluster reached? Here’s the fragment of multiway graph that leads to it:
If we don’t prune off all the ways to “go astray”, the fragment appears as part of a larger multiway graph:
And if one follows all paths in the unpruned (and uncanonicalized) multiway graph at random (i.e. at each step, one chooses each branch with equal probability), it turns out that the probability of ever reaching this particular terminal cluster is just:
(And the fact that this number is fairly small implies that the system is far from confluent; there are many paths that, for example, don’t converge to the fixed point corresponding to this terminal cluster.)
If we keep going in the evolution of the multiway system, we’ll reach other terminal clusters; after 12 steps the following have appeared:
For the {3} rule above, the multiway system takes a little longer to “get going”:
Once again there are terminal clusters where the system gets stuck; the first of them appears at step 14:
And also once again the terminal cluster appears as an isolated node in the whole multiway system:
The fragment of multiway graph that leads to it is:
So far we’ve been finding terminal clusters by waiting for them to appear in the evolution of the multiway system. But there’s another approach, similar to what one might use in filling in something like a tiling. The idea is that every cell in a terminal cluster must have neighbors that don’t allow further growth. In other words, the terminal cluster must consist of certain “local tiles” for which the constraints don’t allow growth. But what configurations of local tiles are possible? To determine this, we turn the matching conditions for the tiles into logical expressions whose variables are True and False depending on whether particular positions in the template do or do not contain cells in the cluster. By solving the satisfiability problem for the combination of these logical expressions, one finds configurations of cells that could conceivably correspond to terminal clusters.
Following this procedure for the {2} rules with regions of up to 6×6 cells we find:
But now there’s an additional constraint. Assuming one starts from a connected initial cluster, any subsequent cluster generated must also be connected. Removing the non-connected cases we get:
So given these terminal clusters, what initial conditions can lead to them? To determine this we effectively have to invert the aggregation process—giving in the end a multiway graph that includes all initial conditions that can generate a given terminal cluster. For the smallest terminal cluster we get:
Our 4-cell “T” initial condition appears here—but we see that there are also even smaller 2-cell initial conditions that lead to the same terminal cluster.
For all the terminal clusters we showed before, we can construct the multiway graphs starting with the minimal initial clusters that lead to them:
For terminal clusters like
there’s no nontrivial multiway system to show, since these clusters can only appear as initial conditions; they can never be generated in the evolution.
There are quite a few small clusters that can only appear as initial conditions, and do not have preimages under the aggregation rule. Here are the cases that fit in a 3×3 region:
The case of the {3} rule is fairly similar to the {2} rule. The possible terminal clusters up to 5×5 are:
However, most of these have only a fairly limited set of possible preimages:
For example we have:
And indeed beyond the (size-17) example we already showed above, no other terminal clusters that can be generated from a T initial condition appear here. Sampling further, however, additional terminal clusters appear (beginning at size 25):
The fragments of multiway graphs for the first few of these are:
We’ve seen above that for the rules we’ve been investigating, terminal clusters are quite rare among possible states in the multiway system. But what happens if we just evolve at random? How often will we wind up with a terminal cluster? When we say “evolve at random”, what we mean is that at each step we’re going to look at all possible positions where a new cell could be added to the cluster that exists so far, and then we’re going to pick with equal probability at which of these to actually add the new cell.
For the 8:{3} rule something surprising happens. Even though terminal clusters are rare in its multiway graph, it turns out that regardless of its initial conditions, it always eventually reaches a terminal cluster—though it often takes a while. And here, for example, are a few possible terminal clusters, annotated with the number of steps it took to reach them (which is also equal to the number of cells they contain):
The distribution of the number of steps to termination seems to be very roughly exponential (here based on a sample of 10,000 random cases)—with mean lifetime around 2300 and half-life around 7400:
Here’s an example of a large terminal cluster—that takes 21,912 steps to generate:
And here’s a map showing when growth in different parts of this cluster occurred (with blue being earliest and red being latest):
This picture suggests that different parts of the cluster “actively grow” at different times, and if we look at a “spacetime” plot of where growth occurs as a function of time, we can confirm this:
And indeed what this suggests is that what’s happening is that different parts of the cluster are at first “fertile”, but later inevitably “burn out”—so that in the end there are no possible positions left where growth can occur.
But what shapes can the final terminal clusters form? We can get some idea by looking at a “compactness measure” (of the kind often used to study gerrymandering) that roughly gives the standard deviation of the distances from the center of each cluster to each of the cells in it. Both “very stringy” and “roughly circular” clusters are fairly rare; most clusters lie somewhere in between:
If we look not at the 8:{3} but instead at the 8:{2} rule, things are very different. Once again, it’s possible to reach a terminal cluster, as the multiway graph shows. But now random evolution almost never reaches a terminal cluster, and instead almost always “runs away” to generate an infinite cluster. The clusters generated in this case are typically much more “compact” than in the 8:{3} case
and this is also reflected in the “spacetime” version:
In building up our clusters so far, we’ve always been assuming that cells are added sequentially, one at a time. But if two cells are far enough apart, we can actually add them “simultaneously”, in parallel, and end up building the same cluster. We can think of the addition of each cell as being an “event” that updates the state of the cluster. Then—just like in our Physics Project, and other applications of multicomputation—we can define a causal graph that represents the causal dependencies between these events, and then foliations of this causal graph tell us possible overall sequences of updates, including parallel.
As an example, consider this sequence of states in the “always grow” 4:{1,2,3,4} rule—where at each step the cell that’s new is colored red (and we’re including the “nothing” state at the beginning):
Every transition between successive states defines an event:
There’s then causal dependence of one event on another if the cell added in the second event is adjacent to the one added in the first event. So, for example, there are causal dependencies like
and
where in the second case additional “spatially separated” cells have been added that aren’t involved in the causal dependence. Putting all the causal dependencies together, we get the complete causal graph for this evolution:
We can recover our original sequence of states by picking a particular ordering of these events (here indicated by the positions of the cells they add):
This path has the property that it always follows the direction of causal edges—and we can make that more obvious by using a different layout for the causal graph:
But in general we can use any ordering of events consistent with the causal graph. Another ordering (out of a total of 40,320 possibilities in this case) is
which gives the sequence of states
with the same final cluster configuration, but different intermediate states.
But now the point is that the constraints implied by the causal graph do not require all events to be applied sequentially. Some events can be considered “spacelike separated” and so can be applied simultaneously. And in fact, any foliation of the causal graph defines a certain sequence for applying events—either sequentially or in parallel. So, for example, here is one particular foliation of the causal graph (shown with two different renderings for the causal graph):
And here is the corresponding sequence of states obtained:
And since in some slices of this foliation multiple events happen “in parallel”, it’s “faster” to get to the final configuration. (As it happens, this foliation is like a “cosmological rest frame foliation” in our Physics Project, and involves the maximum possible number of events happening on each slice.)
Different foliations (and there are a total of 678,972 possibilities in this case) will give different sequences of states, but always the same final state:
Note that nothing we’ve done here depends on the particular rule we’ve used. So, for example, for the 8:{2} rule with sequence of states
the causal graph is:
It’s worth commenting that everything we’ve done here has been for particular sequences of states, i.e. particular paths in the multiway graph. And in effect what we’re doing is the analog of classical spacetime physics—tracing out causal dependencies in particular evolution histories. But in general we could look at the whole multiway causal graph, with events that are not only timelike or spacelike separated, but also branchlike separated. And if we make foliations of this graph, we’ll end up not only with “classical” spacetime states, but also “quantum” superposition states that would need to be represented by something like multispace (in which at each spatial position, there is a “branchial stack” of possible cell values).
So far we’ve been considering aggregation processes in two dimensions. But what about one dimension? In 1D, a “cluster” just consists of a sequence of cells. The simplest rule allows a cell to be added whenever it’s adjacent to a cell that’s already there. Starting from a single cell, here’s a possible random evolution according to such a rule, shown evolving down the page:
We can also construct the multiway system for this rule:
Canonicalizing the states gives the trivial multiway graph:
But just like in the 2D case things get less trivial if there are constraints on growth. For example, assume that before placing a new cell we count the number of cells that lie either distance 1 or distance 2 away. If the number of allowed cells can only be exactly 1 we get behavior like:
The corresponding multiway system is
or after canonicalization:
The number of distinct sequences after t steps here is given by
which can be expressed in terms of Fibonacci numbers, and for large t is about 
.
The rule in effect generates all possible Morse-code-like sequences, consisting of runs of either 2-cell (“long”) black blocks or 1-cell (“short”) black blocks, interspersed by “gaps” of single white cells.
The branchial graphs for this system have the form:
Looking at random evolutions for all possible rules of this type we get:
The corresponding canonicalized multiway graphs are:
The rules we’ve looked at so far are purely totalistic: whether a new cell can be added depends only on the total number of cells in its neighborhood. But (much like, for example, in cellular automata) it’s also possible to have rules where whether one can add a new cell depends on the complete configuration of cells in a neighborhood. Mostly, however, such rules seem to behave very much like totalistic ones.
Other generalizations include, for example, rules with multiple “colors” of cells, and rules that depend either on the total number of cells of different colors, or their detailed configurations.
The kind of analysis we’ve done for 2D and 1D aggregation systems can readily be extended to 3D. As a first example, consider a rule in which cells can be added along each of the 6 coordinate directions in a 3D grid whenever they are adjacent to an existing cell. Here are some typical examples of random clusters formed in this case:
Taking successive slices through the first of these (and coloring by “age”) we get:
If we allow a cell to be added only when it is adjacent to just one existing cell (corresponding to the rule 6:{1}) we get clusters that from the outside look almost indistinguishable
but which have an “airier” internal structure:
Much like in 2D, with 6 neighbors, there can’t be unbounded growth unless cells can be added when there is just one cell in the neighborhood. But in analogy to what happens in 2D, things get more complicated when we allow “corner adjacency” and have a 26-cell neighborhood.
If cells can be added whenever there’s at least one adjacent cell, the results are similar to the 6-neighbor case, except that now there can be “corner-adjacent outgrowths”
and the whole structure is “still airier”:
Little qualitatively changes for a rule like 26:{2} where growth can occur only with exactly 2 neighbors (here starting with a 3D dimer):
But the general question of when there is growth, and when not, is quite complicated and subtle. In particular, even with a specific rule, there are often some initial conditions that can lead to unbounded growth, and others that cannot.
Sometimes there is growth for a while, but then it stops. For example, with the rule 26:{9}, one possible path of evolution from a 3×3×3 block is:
The full multiway graph in this case terminates, confirming that no unbounded growth is ever possible:
With other initial conditions, however, this rule can grow for longer (here shown every 10 steps):
And from what one can tell, all rules 26:{n} lead to unbounded growth for 
, and do not for 
.
So far, we’ve been looking at “filling in cells” in grids—in 2D, 1D and 3D. But we can also look at just “placing tiles” without a grid, with each new tile attaching edge to edge to an existing tile.
For square tiles, there isn’t really a difference:
And the multiway system is just the same as for our original “grow anywhere” rule on a 2D grid:
Here’s now what happens for triangular tiles:
The multiway graph now generates all polyiamonds (triangular polyforms):
And since equilateral triangles can tessellate in a regular lattice, we can think of this—like the square case—as “filling in cells in a lattice” rather than just “placing tiles”. Here are some larger examples of random clusters in this case:
Essentially the same happens with regular hexagons:
The multiway graph generates all polyhexes:
Here are some examples of larger clusters—showing somewhat more “tendrils” than the triangular case:
And in an “effectively lattice” case like this we could also go on and impose constraints on neighborhood configurations, much as we did in earlier sections above.
But what happens if we consider shapes that do not tessellate the plane—like regular pentagons? We can still “sequentially place tiles” with the constraint that any new tile can’t overlap an existing one. And with this rule we get for example:
Here are some “randomly grown” larger clusters—showing all sorts of irregularly shaped interstices inside:
(And, yes, generating such pictures correctly is far from trivial. In the “effectively lattice” case, coincidences between polygons are fairly easy to determine exactly. But in something like the pentagon case, doing so requires solving equations in a high-degree algebraic number field.)
The multiway graph, however, does not show any immediately obvious differences from the ones for “effectively lattice” cases:
It makes it slightly easier to see what’s going on if we riffle the results on the last step we show:
The branchial graphs in this case have the form:
Here’s a larger cluster formed from pentagons:
And remember that the way this is built is sequentially to add one pentagon at each step by testing every “exposed edge” and seeing in which cases a pentagon will “fit”. As in all our other examples, there is no preference given to “external” versus “internal” edges.
Note that whereas “effectively lattice” clusters always eventually fill in all their holes, this isn’t true for something like the pentagon case. And in this case it appears that in the limit, about 28% of the overall area is taken up by holes. And, by the way, there’s a definite “zoo” of at least small possible holes, here plotted with their (logarithmic) probabilities:
So what happens with other regular polygons? Here’s an example with octagons (and in this case the limiting total area taken up by holes is about 35%):
And, by the way, here’s the “zoo of holes” in this case:
With pentagons, it’s pretty clear that difficult-to-resolve geometrical situations will arise. And one might have thought that octagons would avoid these. But there are still plenty of strange “mismatches” like
that aren’t easy to characterize or analyze. By the way, one should note that any time a “closed hole” is formed, the vectors corresponding to the edges that form its boundary must sum to zero—in effect defining an equation.
When the number of sides in the regular polygon gets large, our clusters will approximate circle packings. Here’s an example with 12-gons:
But of course because we’re insisting on adding one polygon at a time, the resulting structure is much “airier” than a true circle packing—of the kind that would be obtained (at least in 2D) by “pushing on the edges” of the cluster.
In the previous section we considered “sequential tilings” constructed from regular polygons. But the methods we used are quite general, and can be applied to sequential tilings formed from any shape—or shapes (or, at least, any shapes for which “attachment edges” can be identified).
As a first example, consider a domino or dimer shape—which we assume can be oriented both vertically and horizontally:
Here’s a somewhat larger cluster formed from dimers:
Here’s the canonicalized multiway graph in this case:
And here are the branchial graphs:
So what about other polyomino shapes? What happens when we try to sequentially tile with these—effectively making “polypolyominoes”?
Here’s an example based on an L-shaped polyomino:
Here’s a larger cluster
and here’s the canonicalized multiway graph after just 1 step
and after 2 steps:
The only other 3-cell polyomino is the tromino:
(For dimers, the limiting fraction of area covered by holes seems to be about 17%, while for L and tromino polyominoes, it’s about 27%.)
Going to 4 cells, there are 5 possible polyominoes—and here are samples of random clusters that can be built with them (note that in the last case shown, we require only that “subcells” of the 2×2 polyomino must align):
The corresponding multiway graphs are:
Continuing for more steps in a few cases:
Some polyominoes are “more awkward” to fit together than others—so these typically give clusters of “lower density”:
So far, we’ve always considered adding new polyominoes so that they “attach” on any “exposed edge”. And the result is that we can often get long “tendrils” in our clusters of polyominoes. But an alternative strategy is to try to add polyominoes as “compactly” as possible, in effect by adding successive “rings” of polyominoes (with “older” rings here colored bluer):
In general there are many ways to add these rings, and eventually one will often get stuck, unable to add polyominoes without leaving holes—as indicated by the red annotation here:
Of course, that doesn’t mean that if one was prepared to “backtrack and try again”, one couldn’t find a way to extend the cluster without leaving holes. And indeed for the polyomino we’re looking at here it’s perfectly possible to end up with “perfect tilings” in which no holes are left:
In general, we could consider all sorts of different strategies for growing clusters by adding polyominoes “in parallel”—just like in our discussion of causal graphs above. And if we add polyominoes “a ring at a time” we’re effectively making a particular choice of foliation—in which the successive “ring states” turn out be directly analogous to what we call “generational states” in our Physics Project.
If we allow holes (and don’t impose other constraints), then it’s inevitable that—just with ordinary, sequential aggregation—we can grow an unboundedly large cluster of polyominoes of any shape, just by always attaching one edge of each new polyomino to an “exposed” edge of the existing cluster. But if we don’t allow holes, it’s a different story—and we’re talking about a traditional tiling problem, where there are ultimately cases where tiling is impossible, and only limited-size clusters can be generated.
As it happens, all polyominoes with 6 or fewer cells do allow infinite tilings. But with 7 cells the following do not:
It’s perfectly possible to grow random clusters with these polyominoes—but they tend not to be at all compact, and to have lots of holes and tendrils:
So what happens if we try to grow clusters in rings? Here are all the possible ways to “surround” the first of these polyominoes with a “single ring”:
And it turns out in every single case, there are edges (indicated here in red) where the cluster can’t be extended—thereby demonstrating that no infinite tiling is possible with this particular polyomino.
By the way, much like we saw with constrained growth on a grid, it’s possible to have “tiling regions” that can extend only a certain limited distance, then always get stuck.
It’s worth mentioning that we’ve considered here the case of single polyominoes. It’s also possible to consider being able to add a whole set of possible polyominoes—“Tetris style”.
We’ve looked at polyominoes—and shapes like pentagons—that don’t tile the plane. But what about shapes that can tile the plane, but only nonperiodically? As an example, let’s consider Penrose tiles. The basic shapes of these tiles are
though there are additional matching conditions (implicitly indicated by the arrows on each tile), which can be enforced either by putting notches in the tiles or by decorating the tiles:
Starting with these individual tiles, we can build up a multiway system by attaching tiles wherever the matching rules are satisfied (note that all edges of both tiles are the same length):
So how can we tell that these tiles can form a nonperiodic tiling? One approach is to generate a multiway system in which at successive steps we surround clusters with rings in all possible ways:
Continuing for another step we get:
Notice that here some of the branches have died out. But the question is what branches exist that will continue forever, and thus lead to an infinite tiling? To answer this we have to do a bit of analysis.
The first step is to see what possible “rings” can have formed around the original tile. And we can read all of these off from the multiway graph:
But now it’s convenient to look not at possible rings around a tile, but instead at possible configurations of tiles that can surround a single vertex. There turns out to be the following limited set:
The last two of these configurations have the feature that they can’t be extended: no tile can be added on the center of their “blue sides”. But it turns out that all the other configurations can be extended—though only to make a nested tiling, not a periodic one.
And a first indication of this is that larger copies of tiles (“supertiles”) can be drawn on top of the first three configurations we just identified, in such a way that the vertices of the supertiles coincide with vertices of the original tiles:
And now we can use this to construct rules for a substitution system:
Applying this substitution system builds up a nested tiling that can be continued forever:
But is such a nested tiling the only one that is possible with our original tiles? We can prove that it is by showing that every tile in every possible configuration occurs within a supertile. We can pull out possible configurations from the multiway system—and then in each case it turns out that we can indeed find a supertile in which the original tile occurs:
And what this all means is that the only infinite paths that can occur in the multiway system are ones that correspond to nested tilings; all other paths must eventually die out.
The Penrose tiling involves two distinct tiles. But in 2022 it was discovered that—if one’s allowed to flip the tile over—just a single (“hat”) tile is sufficient to force a nonperiodic tiling:
The full multiway graph obtained from this tile (and its flip-over) is complicated, but many paths in it lead (at least eventually) to “dead ends” which cannot be further extended. Thus, for example, the following configurations—which appear early in the multiway graph—all have the property that they can’t occur in an infinite tiling:
In the first case here, we can successively add a few rings of tiles:
But after 7 rings, there is a “contradiction” on the boundary, and no further growth is possible (as indicated by the red annotations):
Having eliminated cases that always lead to “dead ends” the resulting simplified multiway graph effectively includes all joins between hat tiles that can ultimately lead to surviving configurations:
Once again we can define a supertile transformation
where the region outlined in red can potentially overlap another supertile. Now we can construct a multiway graph for the supertile (in its “bitten out” and full variant)
and can see that there is a (one-to-one) map from the multiway graph for the original tiles and for these supertiles:
And now from this we can tell that there can be arbitrarily large nested tilings using the hat tile:
Tucked away on page 979 of my 2002 book A New Kind of Science is a note (written in 1995) on “Generalized aggregation models”:
And in many ways the current piece is a three-decade-later followup to that note—using a new approach based on multiway systems.
In A New Kind of Science I did discuss multiway systems (both abstractly, and in connection with fundamental physics). But what I said about aggregation was mostly in a section called “The Phenomenon of Continuity” which discussed how randomness could on a large scale lead to apparent continuity. That section began by talking about things like random walks, but went on to discuss the same minimal (“Eden model”) example of “random aggregation” that I give here. And then, in an attempt to “spruce up” my discussion of aggregation, I started looking at “aggregation with constraints”. In the main text of the book I gave just two examples:
But then for the footnote I studied a wider range of constraints (enumerating them much as I had cellular automata)—and noticed the surprising phenomenon that with some constraints the aggregation process could end up getting stuck, and not being able to continue.
For years I carried around the idea of investigating that phenomenon further. And it was often on my list as a possible project for a student to explore at the Wolfram Summer School. Occasionally it was picked, and progress was made in various directions. And then a few years ago, with our Physics Project in the offing, the idea arose of investigating it using multiway systems—and there were Summer School projects that made progress on this. Meanwhile, as our Physics Project progressed, our tools for working with multiway systems greatly improved—ultimately making possible what we’ve done here.
By the way, back in the 1990s, one of the many topics I studied for A New Kind of Science was tilings. And in an effort to determine what tilings were possible, I investigated what amounts to aggregation under tiling constraints—which is in fact even a generalization of what I consider here:
First and foremost, I’d like to thank Brad Klee for extensive help with this piece, as well as Nik Murzin for additional help. (Thanks also to Catherine Wolfram, Christopher Wolfram and Ed Pegg for specific pointers.) I’d like to thank various Wolfram Summer School students (and their mentors) who’ve worked on aggregation systems and their multiway interpretation in recent years: Kabir Khanna 2019 (mentors: Christopher Wolfram & Jonathan Gorard), Lina M. Ruiz 2021 (mentors: Jesse Galef & Xerxes Arsiwalla), Pietro Pepe 2023 (mentor: Bob Nachbar). (Also related are the Summer School projects on tilings by Bowen Ping 2023 and Johannes Martin 2023.)
Games and Puzzles as Multicomputational Systems
The Physicalization of Metamathematics and Its Implications for the Foundations of Mathematics
Multicomputation with Numbers: The Case of Simple Multiway Systems
Multicomputation: A Fourth Paradigm for Theoretical Science
Combinators: A Centennial View—Updating Schemes and Multiway Systems
