Chip firing is a process on a graph. Initially, stacks of chips are placed on the vertices of the graph. Each vertex can “fire” its chips if it has at least one chip for each neighbor. When a vertex fires, it distributes one of its chips to each neighbor respectively. This process is repeated until no vertex has enough chips to fire anymore, which we will call a “final configuration”.
The process on a one-dimensional graph, i.e. a line of connected vertices, is completely understood and well-studied.
In the two-dimensional setting, on a checkerboard-graph, an interesting setting is given by a single vertex with a large number of chips. As this vertex and subsequently other vertices fire, intriguing patterns start to emerge. “Patterns” in this case refers to a coloring of the vertices by the number of chips they have in the final configuration. Surprisingly, these patterns depend on the number of chips initially placed on the other vertices. For instance, if only the vertex at the origin has a large stack of chips and no other vertex has any chips, the final configuration looks like a truncated circle. If each vertex starts with a “debt” of two chips, i.e. with -2, the final configuration is a square with four “leaves” inside.
The goal of the project is to investigate 3D chip firing on a cubical lattice in 3D. We will illustrate the final configuration of chip firing processes with different initial numbers of chips. Furthermore, we will investigate crosssections of the final configurations and compare them to the two-dimensional setup.
- Wesley Pegden, Gallery of 2-dimensional sandpiles, https://www.math.cmu.edu/~wes/sand.html
- Caroline J. Clivans, The Mathematics of Chip Firing, CRC Press, https://www.crcpress.com/The-Mathematics-of-Chip-Firing/Klivans/p/book/9781138634091