Published 2025-12-9
I can't remember where I first saw them, but ever since, I have been unable to forget them: abelian sandpiles. I'm far from the only one. They're remarkably simple, yet produce lovely symmetric patterns. I loved them so much that I adorned the title banner of this blog with an animation of an abelian sandpile. But what exactly are abelian sandpiles? How do they work? And how many pretty, mesmerizing pictures can we make with these things?
Let's start with an explanation. An abelian sandpile lives on a grid. On each grid cell there can be any number of grains of sand. But if there are four or more grains of sand on a single cell, then the grains topple over into the four neighbouring cells. Toppling might cause other grid cells to have four or more grains of sand, so they also must topple. This repeats until all cells have three grains of sand or fewer, at which point the abelian sandpile is said to be stable. If a cell topples on the edge of the grid, then a grain "falls off" the edge and only the neighbours of the cell that are in the grid gain a grain of sand. This ensures that toppling always finishes in a final stable sandpile. Try adding grains to the grid below to see how the rules work.
Reset
Repeatedly adding grains of sand to the center gives a smaller version of the animation that lives on the top of my blog. If you try this, then it doesn't take long before you start seeing the some grid configurations repeat.
Notice how when there are many neighbouring cells with three grains of sand, adding one more grain of sand causes a cascade of toppling and the final stable pattern is hard to predict. You might wonder how we should handle cases where there are multiple grid cells that need to be toppled. Since toppling one cell affects its neighbours then we need to be careful about the order we topple cells. Or do we?
This brings me neatly to the "Abelian" term. From here on, I will be referring to abelian sandpiles as just sandpiles. In the context of Group Theory, an abelian group is both associative and commutative. In "English", this means that order doesn't matter. This is precisely analogous to addition of numbers. When you are summing a set of numbers, no matter in which order you add them together, you will get the same result.
As it turns out, the same is true of toppling cells in our sandpile, which is part of the reason why it carries the name "Abelian". This might seem like a quirky observation that I can use to simplify my toppling implementation. Which is true. However, it also gives us a connection to a rich field of Mathematics: namely "Abstract Algebra" & "Group Theory". We will use this later to generate a nice pattern, but for now let's simply focus on the fact that the toppling order is irrelevant. I won't be proving that toppling order is irrelevant, but I will at least demonstrate it.
In the sandpile widget above, you can build a sandpile by adding one grain at a time, and toppling is done eagerly as soon as a cell has more than 4 grains of sand. Since I've now claimed that toppling order is irrelevant, we can consider a different way of building a sandpile. We can add all the grains of sand at the beginning, allowing cells to temporarily have 4 grains of sand or more. Then, when we're done, we can topple all the cells that have 4 grains or more. You can try this in the widget below. On the left you can place all the sand you want and on the right you'll see the usual view of what the sandpile looks like when toppling is done eagerly. When you're done adding sand, you can press "Topple" to topple all the sand in a random order. At the end, the left sandpile and the right sandpile will be equal.
Topple Reset
... continue reading