Neutral and yet fractal?

(M.Sc. Lothar Dirks)

Take a finite chessboard and let grains of sand fall down onto it at random. As soon as there are four grains of sand in one of the squares, they are distributed to the four adjacent squares (at the boundary of the board the grains of sand fall down). This dynamic behavior describes a simple two-dimensional version of the Bak-Tang-Wiesenfeld model.
The recurrent states of this dynamical system form a group whose neutral element forms interesting fractal structures (see figure). The aim of this project is to program the Bak-Tang-Wiesenfeld model for different graphs and to investigate the neutral element for these.

Basic knowledge in the fields of calculus and linear algebra is recommended, as well as some elementary programming experience. The duration of the work within the research group should be at least four weeks. The project includes a written paper and a presentation in a seminar.