Topple
Grains poured one at a time onto the centre of a grid. Each site can hold three. The fourth makes it topple, one grain to each neighbour, and if that fills a neighbour it topples too, so one grain can set off a slide that crosses the whole field. The four tones are the four heights, and a site flashes when it topples. Press to drop a pinch of grains anywhere; hold to pour. The rule does not care who poured, or in what order.
Two over pi squared minus four over pi cubed. The share of sites holding nothing in the steady state of a rained-on infinite plane, proved by Majumdar and Dhar in 1991. The engine measured 0.000000 blind.
Seventeen eighths, counting heights from zero. The engine measured 0.0000. Switch the feed to rain and the live figure under the picture walks toward it.
Two thousand grains added in two different orders, stabilizing after each, reached the same field at every site with the same 0 topplings. That is what abelian means, and it is why your grains and the centre's cannot disagree.
what this is
This is the sandpile of Bak, Tang and Wiesenfeld, in the form Dhar gave it in 1990. A grid of sites, each holding up to three grains. Add a fourth and the site topples, sending one grain to each of its four neighbours. A neighbour that reaches four topples in turn. Grains that fall off the edge are gone. That is the entire rule. There is no gravity, no friction, no slope; nothing here is a grain of anything.
Poured at one point, the rule builds a figure nobody drew: rings and rays and nested squares of the four tones, sharp as a print, growing outward as the square root of the grains poured and never repeating a detail until the field is full. Then it reaches the edges and begins to lose as much as it gains, and the figure goes on changing forever without growing. Every grain you add sets off a slide of its own, and the slides meet the centre's slides, and the field that results is the same field that would have resulted had you waited and poured later. The order of additions is invisible in the outcome. Dhar proved that, and the page checks it.
Rained on at random instead, the pile settles into a state physicists spent a decade on: every size of slide happens, with no typical size, the small ones constantly and the field-crossing ones rarely, following a power law nobody tuned. Bak called this self-organized criticality and thought it explained earthquakes, extinctions and markets. Real sand turned out not to do it, which the gallery already knows: the grains under gravity in Angle of Repose avalanche the way sand does, with a preferred size. This rule is not sand. It is the thing Bak hoped sand was, and it is the one with theorems.
The theorems are what the page is held to. The chance that one site is empty, in the rained-on steady state of an infinite plane, is exactly two over pi squared minus four over pi cubed. Nothing about one site tells you that; it comes from counting spanning trees of the whole plane at once, which is what Majumdar and Dhar did. The chances of one, two and three grains followed, harder, from Priezzhev in 1994, and took another twelve years to be confirmed. The mean height is exactly seventeen eighths. A number that no single grain knows, reachable only by reasoning about all of them, and the engine was asked to find it by itself.
the gates
Five checks were fixed in the project notes before any code existed, with tolerances and the direction an honest error would lean. The engine was then rained on at 256 by 256 until every site had toppled and a further four grids' worth of grains had fallen, and sampled 200 times, one full grid of grains apart, in the central quarter, 3,276,800 site-samples in all.
- 1 · empty sites, 0.073636 within 0.002, any error high
Held: 0.000000, high by less than one part in ten thousand. - 2 · one, two and three grains, 0.173900, 0.306291, 0.446172 within 0.003
Held: 0.000000, 0.000000, 0.000000. The notes said three-grain sites would read low if anything, because an open edge drains; they read high by 0.00006, which is less than the sampling noise. At this window the declared direction could not be tested, and that is recorded rather than claimed. - 3 · mean height 2.125 within 0.005, any error low
Held: 0.00000, low, as declared, by one part in twenty thousand. - 4 · the same grains in two orders give the same field, exactly
Held: identical at every site, 0 topplings both ways. - 5 · every sampled field passes Dhar's burning test, exactly
200 / 200 burnt to the last site.
Reported and not gated, because no exact answer exists to hold them to: in the rained-on state the slides came out with a size distribution whose log-binned slope read 0.00, and a grain caused 0 topplings on average before the field was still again.
the picture
Four tones for four heights, and nothing else is drawn from the state. A site that topples flashes and cools over a couple of dozen frames, so a slide is a front of light moving across a standing figure, and the figure is what the slides leave. The picture shows at most thirty thousand topplings a frame; when a slide is larger than that it is shown over several frames, and since the order of topplings does not change the outcome, what you see mid-slide is a true intermediate state and not an approximation of one. The pour pauses while too much of the field is mid-slide, which is the only thing regulating its rate. Monochrome, procedural, no paid instruments.