The puzzling beauty of Abelian sandpiles

sandpile.jpg

Pour real sand, a grain at a time, onto a flat surface and the result is a rather dull pyramidal shape. but in the mathematical world, the result is a little  different.

The image above is produced using a theoretical model called an Abelian sandpile model. It is  produced by dropping some 200,000 grains onto a single point on flat grid and distributing them according to a set of toppling rules.

For example, a point on the grid can hold no more than three grains. If a fourth lands, all the grains are redistributed, avalanche style, to surrounding points.

This is a relatively new discipline–Abelian sandpile models were only discovered in 1990 by Deepak Dhar at the Tata Institute of Fundamental Research in Mumbai, so people are still trying to characterise them.

This pattern was produced Dhar and colleagues who are obviously captivated by its beauty and complexity but puzzled by how to analyse it. They say that simpler, related patterns seem to have an eightfold symmetry. But this one has them stumped.  “It has not been possible to characterize [this pattern] so far,” they say.

That looks like an interesting puzzle. There’s work here for anyone who needs it.

Ref: arxiv.org/abs/0808.1732: Pattern Formation in Growing Sandpiles

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