The Sandpile That Draws Itself

Take an endless grid of squares and put some grains of sand on one of them. A square holding four or more grains is unstable. It topples: it loses four grains and gives one to each of the four squares it touches along an edge. That may make a neighbour unstable, so the neighbour topples in turn, and the grains go on moving until no square holds more than three. The pile that is left is stable, and it is the finished pile.

For example, four grains on one square take a single toppling: the middle square empties and each of its four neighbours holds one. Sixteen grains take the nine topplings of the table above. Pour sixteen thousand grains and the finished pile is a picture: squares holding 0, 1, 2 or 3 grains, laid out in arcs and stripes with a hollow in the middle, unchanged if you reflect the grid or turn it a quarter. Nothing in the rule draws a picture or mentions symmetry.

The laboratory has five views of this. The pour shows the pile toppling in waves, and you can step, run and scrub through it. The four sizes view puts finished piles of about a thousand, four thousand, sixteen thousand and sixty-five thousand grains side by side at the same width. The order view finishes the same pour a second way, toppling one randomly chosen square at a time, and compares the two piles square by square. The avalanche view drops grains one at a time on a small grid with a swallowing edge. The identity view draws a particular pile from the algebra of sandpiles, which is explained in the Proof layer.

The sizes go up to 131,072 grains. Past a few thousand grains a pour takes more rounds than any screen can show, so the page keeps a few dozen frames, always the first and the last, and says which round each one is. The rule can be changed to eight neighbours, and the ground can be paved with threes, to see which parts of the picture belong to the rule.

Three kinds of statement are made on this page and they are labelled as such. That the finished pile does not depend on the order of topplings is proved. That the pictures look like scaled copies of one another as the pile grows is seen and measured, with a weaker proved result behind it. That avalanches have no typical size is observed in one small simulation.

Two ideas do most of the work.

A square that holds four is a square that cannot stay put. Everything it gives away lands on neighbours that may themselves be full, so grains pushed out are often pushed straight back, and a square near the middle can topple many times before the pile is done. That is why sixteen grains take nine topplings and a thousand take far more than a thousand.

Toppling one square never stops another from toppling. If two squares are both full and you topple the first, the second is still full, because the first only gave it grains. So any two legal topplings can be done in either order, and the result of doing both is the same. The page turns this small fact into a proof in the last layer but it is worth holding in mind first.

It is the reason the page can say something that sounds too strong: you may topple the full squares in any order you like, in waves or one at a time or in a random order, and the finished pile is always the same one, and every square will have toppled the same number of times. The order view lets you check that on a pile of thousands of squares rather than take it on trust.

The same fact explains the symmetry. The pile before toppling is its own mirror image, so if you mirror an order of topplings you get a legal order for the same pile. Both end at the same stable pile, so the finished pile equals its own mirror image.

What the same fact does not explain is the picture itself. Why the arcs and stripes come out as they do is not something that follows from the order not mattering, and the next layers say how much of it is understood.

The sentence worth carrying away is this one.

Because the order of topplings cannot matter, the finished pile is determined by the pile you start with, and not by how anyone computed it.

That is what makes the sandpile a mathematical object and not a simulation. Two programs that topple in different orders must agree, and the page lets you watch them agree. It is also what makes it possible to add piles together: pour one pile onto another, let the sum settle, and the answer is a well-defined pile. On a grid whose edge swallows grains this gives an actual group law on the piles that can recur, and that group has an identity, a pile that changes nothing it is added to. The identity view draws it. It is a strikingly intricate picture, found by a recipe that mentions no picture.

What the idea does not give you is any account of the pictures. The next piece of the story is the one that is only partly understood. As you pour more grains, the finished piles look like coarser and finer versions of one another, and the four sizes view lets you see it. Pegden and Smart proved that this is true in a weak sense, for averages over regions, and nobody has a complete description of the detailed picture that is as short as the rule. This page shows the likeness and measures it, and it does not claim to explain it.

The avalanche view is a different part of the same story. On a small grid with a swallowing edge, drop grains one at a time. The grid fills to about two grains a square and stays there, and from then on a single grain can set off an avalanche of any size. Bak, Tang and Wiesenfeld, who introduced the model, claimed there is no typical size. This page shows what a simulation of one grid does and leaves the claim about the model to its sources.

A pile is a function $\eta :{\mathbb{Z}}^2\to \{0,1,2,\dots \}$ with finitely many non-zero values. Write $x\sim y$ when two squares share an edge. A square $x$ is unstable if $\eta (x)\ge 4$. Toppling $x$ gives the pile $\eta -\Delta {\delta }_x$, where ${\delta }_x$ is $1$ at $x$ and $0$ elsewhere and $\Delta$ is the lattice Laplacian: $(\Delta f)(x)=4f(x)-{\sum }_{y\sim x}f(y)$. For eight neighbours, replace $4$ by $8$ and the sum by one over all eight surrounding squares. If $u(x)$ counts how often $x$ has toppled over a whole sequence, the pile at the end is $\eta -\Delta u$.

1. If one order of legal topplings of $\eta$ ends at a stable pile, every legal order ends, they all end at the same stable pile, and they all give the same $u$. Proved, in the next layer.
2. That some order ends for every finite pile on the grid is standard and is cited, not reproved; every pour this page runs ended.
3. For a pour of any number of grains on one square of bare ground, the finished pile is unchanged by the eight symmetries of the square. Proved, from statement 1.
4. For the four-neighbour rule, from 1,024 grains to 131,072, the radius of the disc with the same area as the occupied squares lies between $0.35\sqrt{g}$ and $0.39\sqrt{g}$ for $g$ grains, falling slowly with $g$. Observed, in the five pours the laboratory runs.
5. The rescaled finished piles converge in the weak-star sense as $g\to \infty$. Proved by Pegden and Smart, cited and not reproduced. Individual squares are not covered.
6. On a $32\times 32$ grid with a swallowing edge, avalanche sizes fall roughly along a straight line on logarithmic axes with fitted slope about $-1.29$. Observed, in seeds 1, 2 and 3.
7. For a square grid with an open edge, settling $2c$ (where $c$ is three on every square), then settling $2c-s$ for the result $s$, gives the identity of the sandpile group. Proved, given the standard structure theory, which is cited.

Statements 4 and 6 are measurements, and they are the weakest on the list. Statement 1 is the strongest.

Statement 1, the order does not matter. Take two sequences of legal topplings of the same pile $\eta$, the first $a_1,\dots ,a_m$ ending at a stable pile, and the second $b_1,b_2,\dots$ any legal sequence. Write $v(x)$ for how often $x$ topples in the first and $w_k(x)$ for how often it has toppled after the first $k$ topplings of the second. I claim $w_k\le v$ for every $k$.

This is true for $k=0$. Suppose it holds after $k$ topplings and the next toppling is at $x$. If $w_k(x)=v(x)$, then the grains at $x$ just before it topples are

so $x$ is not unstable (for eight neighbours, read 8 where this has 4, 7 where it has 3, and sum over the eight surrounding squares), which is a contradiction. Hence $w_k(x)<v(x)$ and $w_{k+1}\le v$. ∎

The second sequence therefore topples no square more often than the first, and cannot be longer than $m$ topplings, so it ends. If it ends at a stable pile, run the same argument with the roles reversed to get $v\le w$, so the two toppling counts are equal and the final piles, which are $\eta -\Delta v$ and $\eta -\Delta w$, are equal. Nothing about four or eight was used, only that a toppling takes the threshold from one square and gives one grain to each of its neighbours.

Statement 3, symmetry. Let $\sigma$ be a symmetry of the square, a reflection or a quarter turn about the pour square. If $a_1,\dots ,a_m$ is a legal order for a pile $\eta$ with $\eta \circ \sigma =\eta$, then $\sigma (a_1),\dots ,\sigma (a_m)$ is a legal order for the same pile, and it ends at the stable pile reflected by $\sigma$. By statement 1 it ends at the same pile as the first. So the finished pile equals its own image under $\sigma$. ∎

Statement 7, the identity. On a grid with a swallowing edge, let $c$ be the pile with three grains on every square and let $s$ be $2c$ settled. Then $2c-s$ is a pile with no negative squares, because $s\le 3$ everywhere, and $2c-s=c+(c-s)$, so settling it gives a pile reached from $c$ by adding grains. Such piles are the recurrent ones. Also $2c-s$ differs from $0$ only by topplings, since $2c$ and $s$ differ by topplings. The standard theory says there is exactly one recurrent pile in each class of piles that differ by topplings, and the class of $0$ is the identity of the group. So settling $2c-s$ gives the identity. This uses that theory, which is cited and not reproduced here; the proof that it is a group and that each class has one such pile is in the sources. The page checks the result by computer instead of leaning on that citation alone: on grids 1, 2 and 3 squares across, every recurrent pile was listed two independent ways and the recipe's pile fixes every one of them, and on the larger grids it fixes itself and the pile of threes.

What is not proved here. That some order of topplings ends for every finite pile on the grid is cited. The likeness between piles of different sizes is not proved. Pegden and Smart's theorem is a weak limit and not a description of the pictures. The slope of the avalanche counts is a measurement of one small grid and is not an exponent for the model.

What this laboratory computes. Pours of 16 to 131,072 grains on a grid that is widened until nothing reaches its edge, in exact integer arithmetic with no floating point in the rule. The largest pour takes about 132,000 synchronous rounds and a few hundred million topplings, which is why the frames are thinned and why that pour is computed once and served as a file rather than on your device. The avalanche grid is 32 squares across with 40,000 grains per run, the identity grids go up to 151 squares across, and everything is deterministic: the same address always draws the same picture.

What the pictures do and do not show. The four sizes view shows piles that resemble one another and a radius per root grain that drifts down slowly: it is still moving at 131,072 grains, so it is not shown to have a limit. A weak limit of the pictures is proved. A description of the detailed picture that explains the arcs and stripes is further work, and the paper of Levine, Pegden and Smart on Apollonian structure in the limit is the place to read about it. This page does not reproduce it and does not claim it is the end of the story.

What avalanches show. On one small grid the avalanche sizes spread over more than eight doublings and fall roughly along a line on logarithmic axes, which is how having no typical scale looks. That is a simulation. It cannot tell a true power law from something that merely looks like one over this range, and the exponent of the model is not the slope fitted here. A larger grid, more seeds and a careful statistical treatment are what a real estimate would need, and the sources say more.

Other rules. The eight-neighbour rule and the floor of threes are included to show what breaks when the rule changes and what does not. Stability, symmetry and independence of order survive, because they follow from the kind of rule. The pictures do not, and the page claims nothing about how they scale.

Read this next to the chaos game. Both pages draw a picture that is fixed while some part of the process that makes it is free. In the chaos game the random corner choices are free and the set the dots land on is fixed. In the sandpile the order of the topplings is free and the finished pile is fixed, and here the reason is a short argument that fits on a page. The Langton ant and Pascal's triangle sit nearby for the opposite reasons: structure appears and nobody can say in a few lines why.