Abelian Sandpiles

Imagine a large grid of piles of sand. Every pile has four neighboring piles, and around the grid is a large pit. Whenever a pile has four grains or more, it topples, spilling one grain each into each of the four neighboring piles. We start with every pile empty except for the centermost one, which we fill with a very large number of grains. It immediately spills one grain into the four neighboring piles, and this process repeats until no pile has four grains or more. Amazingly, if we have more than one pile that will topple, the order in which we pick which to topple first has no bearing on the final outcome. Because of this, the model is called Abelian.


Piles with no grains are colored black, piles with one colored blue, those with two colored purple, and those with three colored yellow. Any pile with more than three grains is colored white.

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