What would happen if you dropped a billion grains of sand on top of each other and let them cascade into a stable…

What would happen if you dropped a billion grains of sand on top of each other and let them cascade into a stable pattern following a simple mathematical rule? Find out more below. (The answer is in the picture.)

Originally shared by Richard Green

The sandpile model with a billion grains of sand

This picture by Wesley Pegden shows an example of a stable configuration in the abelian sandpile model on a square lattice. This consists of a rectangular square array of a large number of pixels. Each pixel has one of four possible colours (blue, cyan, yellow and maroon) corresponding to the numbers 0, 1, 2 and 3 respectively. These numbers should be thought of as representing stacks of tokens, often called chips, which in this case might be grains of sand.

Despite its intricate fractal structure, this picture is generated by a simple iterative process, as follows. If a vertex of the grid (i.e., one of pixels) holds at least 4 chips, it is allowed to fire, meaning that it transfers one chip to each of its neighbours to the north, south, east and west. The boundary of the grid can be thought of as the edge of a cliff, meaning that any chips that cross the boundary will fall off and be lost. If no vertices can fire in a particular chip configuration, then the configuration is called stable. For example, the configuration in the picture is stable, because no pixel holds 4 or more chips.

One of the key theorems about this particular sandpile model is that any chip configuration will become stable after firing various vertices a finite number of times. More surprisingly, the ultimate stable configuration obtained does not depend on the order in which the vertices were fired. The irrelevance of the order in which the vertices are fired is why the model is called “abelian”.

If we start with 2^{30} chips, all placed on the same pixel, and we then perform firings repeatedly until a stable configuration is reached, the resulting stable configuration is the one shown in the picture. (The number 2^{30} is just over a billion.) It is clear from the symmetrical nature of the firing rules that the resulting picture will be symmetric under rotation by a right angle or mirror-reflection, but the fractal-like structures are much more surprising.

Relevant links

Wesley Pegden is an Assistant Professor of Mathematics at Carnegie Mellon University. His home page includes an interactive zoomable version of this image: http://www.math.cmu.edu/~wes/sand.html

The page also allows you to generate corresponding images on other lattices, including a triangular lattice with six colours, and a hexagonal lattice with three colours. You can also change the number of chips, like Dr Evil from Austin Powers. (One billion chips. No, one million chips.)

The article The Amazing, Autotuning Sandpile by Jordan Ellenberg appeared in Nautilus in April 2015: http://nautil.us/issue/23/dominoes/the-amazing-autotuning-sandpile

The article discusses this picture, and also includes some details from it, including a close-up of the centre.

I have posted about the sandpile model before, here: https://plus.google.com/101584889282878921052/posts/QezmLcTCTMJ

The other post includes more technical details, and describes how to construct a group out of certain sandpiles. A surprising feature of this is that the identity element of the group has a very complicated appearance.

David Perkinson is a Professor of Mathematics at Reed College. He has a gallery of images relating to abelian sandpiles: http://people.reed.edu/~davidp/sand/gallery/gallery.html

(Found via Cliff Pickover (@pickover) on Twitter.)

#mathematics #scienceeveryday