The Rules of Chip-Firing

The figure on the Spatial Systems Laboratory home-page shows a four-coloring of the cells of a 198-by-198 square grid. You should think of this grid as having a number of indistinguishable tokens, called "chips", on each grid-cell. Grid-cells (also called "sites") that are colored gray, red, blue, or green have 0, 1, 2, or 3 chips, respectively. Thus the picture represents one of many possible "chip-configurations".

In these other configurations, a site that has 4 or more chips is unstable; at some point, it "fires", sending 1 chip to each of its four neighbors. (When one or two of the neighbors isn't there, as happens at the boundary of the grid, the chips that were sent to nonexistent neighbors disappear from the board.)

An important fact is that if a chip-configuration with one or more unstable sites is allowed to equilibrate by repeated firings, it will ultimately reach a stable configuration, and moreover the stable configuration that is reached is INDEPENDENT of the order in which unstable sites fire.

Certain stable chip-configurations have the additional feature of being "recurrent". A configuration X is recurrent if and only if it has the property that, for any configuration Y, it is possible to add chips to Y so that the resulting system, when allowed to equilibrate, turns into X.

It can be shown that the set of recurrent chip-configurations is an abelian group under the obvious kind of addition, where the chips of one grid get added to the chips of the other grid at the corresponding site and the system then equilibrates.

The specific chip-configuration that appears on the home-page is nothing other than the IDENTITY ELEMENT for this group. Note that this configuration has a lot of geometrical structure that is not at all apparent from its algebraic definition! What's more, if one uses an n-by-n square grid for any value of n, the same picture comes up (with minor differences if n is odd).

One of the goals of the Spatial Systems Laboratory is to figure out more about this (possibly fractal) structure.