A British Train Station
A week ago a new train station, named “Cambridge North”, opened in Cambridge, UK. Normally such an event would be far outside my sphere of awareness. (I think I last took a train to Cambridge in .) But last week people started sending me pictures of the new train station, wondering if I could identify the pattern on it:
The system evolves line by line from the top, determining the color of each cell according to the rule underneath. This particular cellular automata I called “”, because the bit pattern in the rule corresponds to the number 182 in binary. There are altogether like this, and this is what all of them do:
Many of them show fairly simple behavior. But the huge surprise I got when I first ran all these cellular automata in the early 1980s is that even though all the rules are very simple to state, some of them generate very complex behavior. The first in the list that does that—and still my favorite example—is :
And, yes, it’s remarkable that starting from one black cell at the top, and just repeatedly following a simple rule, it’s possible to get all this complexity. I think it’s actually an example of a , that’s central to how complexity gets made in nature, as well as to how we can get a new level of technology. And in fact, I think it’s important enough that I spent writing a (that just ) based on it.
And for years I’ve actually had rule 30 on my business cards:
But back to the Cambridge North train station. Its pattern is obviously not completely random. But if it was made by a rule, what kind of rule? Could it be a cellular automaton?
I zoomed in on a photograph of the pattern:
Suddenly, something seemed awfully familiar: the triangles, the stripes, the L shapes. Wait a minute… it couldn’t actually be my favorite rule of all time, rule 30?
Clearly the pattern is tipped 45° from how I’d usually display a cellular automaton. And there are black triangles in the photograph, not white ones like in rule 30. But if one black-white inverts the rule (so it’s now ), one gets this:
And, yes, it’s the same kind of pattern as in the photograph! But if it’s rule 30 (or rule 135) what’s its initial condition? Rule 30 can actually be —because it can be hard (maybe even ) to reconstruct its initial condition.
But, OK, if it’s my favorite rule, I wondered if maybe it’s also my favorite initial condition—a single black cell. And, yes, it is! The train station pattern comes exactly from the (inverted) right-hand edge of my favorite rule 30 pattern!
It’s a little trickier to pull out precisely the section of the pattern that’s used. Here’s the code (the PlotRange is what determines the part of the pattern that’s shown):
OK, so where is this pattern actually used at the train station? Everywhere!
It’s made of perforated aluminum. You can actually look through it, reminiscent of an old latticed window. From inside, the pattern is left-right reversed—so if it’s from outside, it’s from inside. And at night, the pattern is black-white inverted, because there’s light coming from inside—so from the outside it’s “rule 135 by day, and rule 30 at night”.
What are some facts about the rule 30 pattern? It’s about it (and that’s interesting in itself—and closely related to the fundamental phenomenon of ). But, for example—like, say, the —many aspects of it seem random. And, for instance, black and white squares appear to occur with —meaning that at the train station the panels let in about 50% of the outside light.
If one looks at sequences of n cells, it seems that all 2n configurations will occur on average with equal frequency. But not everything is random. And so, for example, if one looks at 3×2 blocks of cells, only ever occur. (Maybe some people waiting for trains will figure out which blocks are missing…)
At the train station, there are lots of identical panels. But rule 30 is actually an inexhaustible source of new patterns. So what would happen if one just continued the evolution, and rendered it on successive panels? Here’s the result. It’s a pity about the hint of periodicity on the right-hand edge, and the big triangle on panel 5 (which might be a safety problem at the train station).
Fifteen more steps in from the edge, there’s no hint of that anymore:
And starting from random initial conditions, rather than a single black cell, things again look different:
And here are a few more rules, started from random initial conditions:
Here’s a website (made in a couple of minutes with a tiny piece of Wolfram Language code) that lets you experiment (including with larger rule numbers, based on longer-range rules). (And if you want to explore more systematically, here’s a Wolfram Notebook to try.)
It’s amazing what’s out there in the computational universe of possible programs. There’s an infinite range of possible patterns. But it’s cool that the Cambridge North train station uses my all-time favorite discovery in the computational universe—rule 30! And it looks great!
The Bigger Picture
But can one generate richer forms algorithmically? Before I discovered rule 30, I’d always assumed that any form generated from simple rules would always somehow end up being obviously simple. But rule 30 was a big shock to my intuition—and from it I realized that actually in the computational universe of all possible rules, it’s actually very easy to get rich and complex behavior, even from simple underlying rules.
And what’s more, the patterns that are generated often have remarkable visual interest. Here are a few produced by cellular automata (now with 3 possible colors for each cell, rather than 2):
There’s an amazing diversity of forms. And, yes, they’re often complicated. But because they’re based on simple underlying rules, they always have a certain logic to them: in a sense each of them tells a definite “algorithmic story”.
One thing that’s notable about forms we see in the computational universe is that they often look a lot like forms we see in nature. And I don’t think that’s a coincidence. Instead, I think what’s going on is that rules in the computational universe capture the essence of laws that —whether in physics, biology or wherever. And maybe there’s a certain familiarity or comfort associated with forms in the computational universe that comes from their similarity to forms we’re used to in nature.
But is what we get from the computational universe art? When we pick out something like rule 30 for a particular purpose, what we’re doing is conceptually a bit like photography: we’re not creating the underlying forms, but we are selecting the ones we choose to use.
In the computational universe, though, we can be more systematic. Given some aesthetic criterion, we can automatically search through perhaps even millions or billions of possible rules to find optimal ones: in a sense automatically “discovering art” in the computational universe.
We did an experiment on this for music back in 2007: . And what’s remarkable is that even by sampling fairly small numbers of rules (cellular automata, as it happens), we’re able to produce all sorts of interesting short pieces of music—that often seem .
From a practical point of view, automatic discovery in the computational universe is important because it allows for mass customization. It makes it easy to be “original” (and “creative”)—and to find something different every time, or to fit constraints that have never been seen before (say, a pattern in a complicated geometric region).
The Cambridge North train station uses a particular rule from the computational universe to make what amounts to an ornamental pattern. But one can also use . And one can even imagine a building in which everything—from overall massing down to details of moldings—is completely determined by something close to a single rule.
One might assume that such a building would somehow be minimalist and sterile. But the remarkable fact is that this doesn’t have to be true—and that instead there are plenty of rich, almost “organic” forms to be “mined” from the computational universe.
Ever since I started writing about one-dimensional cellular automata back in the early 1980s, there’s been all sorts of interesting art done with them. Lots of different rules have been used. Sometimes they’ve been what I called “” rules that have a particularly organic look. But often it’s been other rules—and rule 30 has certainly made its share of appearances—whether it’s on floors, shirts, tea cosies, kinetic installations, or, recently, (with the knitting machine actually running the cellular automaton):