Symmetry in the Sandbox
Nature often has a delightful tendency towards self-organization.
As the symmetry of the final configuration suggests, the end state of the abelian sandpile doesn’t depend on the order in which you carry out the topples. That’s what makes it “abelian,” which is mathese for “doing this, then that, is the same thing as doing that, then this.” Addition is abelian: Adding 2, then adding 3 is the same as adding 3, then adding 2. Most operations aren’t abelian. Unlock your car and pull the door handle, and the door opens; pull the door first and then unlock the car, and you get a different result: closed door, unlocked car. So the abelian nature of the sandpile is a pleasant surprise.
What happens if you pile a lot of sand — say, a million grains — on one dot, and let the sand flow until the toppling settles down to stability? You might imagine you’d end up with a big smooth pile of sand, with a big area near the center of dots maxed out with three grains of sand…The hoped-for smoothing isn’t happening. Instead, these crazy fractal patterns persist. Near the center, an intricate pattern, like the inside of a dome inlaid with latticework that somehow looks geometric and random at once; and near the edge of the heap, triangular islands of more consistent behavior, interlocking in regular patterns.
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The abelian sandpile model doesn’t even try to capture the behavior of actual physical materials. Rather, all of its complexity emerges from an abstraction, a simple deterministic algorithm you can describe in five lines of code. It’s reminiscent of John Conway’s “Game of Life,” which also produces rich complexity from a very simple ruleset. The abelian sandpile, like the Game of Life, is a cellular automaton: that is, a mini-universe whose condition can be completely described in the kind of discrete language computers find agreeable. In the sandpile, each dot on the grid gets a number between 0 and 4, and a simple set of rules set the values of adjacent dots. In the Game of Life, the state is even simpler: Each dot is either “alive” or “dead,” 1 or 0.
But there’s a difference. The Game of Life can be coaxed into complex behavior, but it tends to take a little work. In this respect, it’s typical among cellular automata. The sandpile, on the other hand, seems to direct itself automatically toward complex behavior, without any special effort to set up just the right initial conditions. It does this by seeking out a so-called critical threshold, around which complex behavior tends to be found. You’re familiar with the idea of thresholds from nature. Water at a high temperature is a disorganized liquid; when the temperature crosses a certain critical value, the water undergoes a sharp transition, crystallizing into ice. For the sandpile, the analogue of temperature is density: How much sand is there per dot? Too much sand and the pile is unstable, essentially one long avalanche. Too little, and the sand quickly settles into a stable state. How much is too much? The answer is unexpectedly simple: An average of exactly 2.125 grains per dot is the critical threshold, the dividing line between quiet and chaos.
Remarkably, a sand pile on a finite table — where any sand reaching the rim falls off the edge and disappears — tunes itself to 2.125 grains per dot. Say the table starts empty, and you drop sand, grain by grain, in the center. For a while, the pattern of sand expands, looking a lot like Pegden’s pictures above (which depict an infinite table.) You drop a grain, the sand settles, there’s one more grain on the table. But once the sand gets to the rim, the story changes. The pile approaches an equilibrium, where sand drops off the edge at the same rate you add sand to the center, and the density holds steady at the critical value. Of course, there may be local fluctuations, denser and less dense patches that come and go as the system evolves; but on average over the whole table, the number of grains per dot will hover around 2.125.
What if you start with a finite table as dense with sand as possible, three grains at each spot? That configuration is stable. But only in a very fragile sense. Drop one grain, anywhere on the table, and a massive avalanche begins, not ceasing until the density goes down to 2.125 again. And what happens once the table reaches threshold density? Then the sand pile is poised at its most interesting state. Avalanches keep happening, but they don’t create a state of constant universal tumble; rather, they come in waves, with frequent smaller avalanches punctuated by rare table-spanning catastrophes. The distribution of avalanches at threshold density seems to obey a power law: The frequency of avalanches is inversely proportional to their size. There’s constant activity, but the activity is somehow organized and structured. What’s more, the san dpile doesn’t have to be fine-tuned in order to show off its complex behavior; it auto tunes. This is the phenomenon called “self-organized criticality.” Wherever the system starts, it finds its way to the critical threshold, and stays there, as long as new sand keeps getting added at a constant rate…
This looks like a living process to me. And that’s no coincidence. The notion of self-organized criticality is one popular way to think about how the rich structures of life might have emerged from simple systems that automatically seek the critical threshold. Some biologists see self-organized criticality as a potential unified theory for complex biological behavior, which governs the way a flock of birds moves in sync just as genetic information governs the development of the individual birds. “Living systems,” wrote biological theorist Stuart Kauffman, “exist in the solid regime near the edge of chaos, and natural selection achieves and sustains such a poised state.” As does the sand pile. It isn’t life, of course, not really. But it’s lively, isn’t it?