Abelian Sand Piles
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Abelian sandpiles are a single celled automoton or "0-player game" similar to Conway's game of life. With the sand piles, the game is governed in the following way:
Every cell in a grid is given a value. This could be any positive integer (including zero) If the value of any cell exceeds 3 then that cell loses 4 and each cell around it gains 1. This is repeated until all cells in the grid have value less than 4.So for example, a 5x5 grid with a single stack of 16 "grains of sand" in the center will go through the following iterations before settling:
Here a red pixel represents a cell with value greater than 3 and a blue pixel represents a cell with value less than or equal to three with 1 being the lightest and 3 the darkest. So the process terminates when there are no more red pixels.
Abelian sand piles have lots of intriguing mathematical processes (they don't call them Abelian for nothing). For example, large stacks in the middle tend to follow predictable patterns which mirror the structure that the curvatures of circles in Apolonian circle packings exhibit. Truly an out there result due to Lionel Levine, Wesley Pegden and Charles K. Smart. Try it yourself!
