A cyclic cellular automaton is a kind of cellular automaton rule developed by David Griffeath and studied by several other cellular automaton researchers. In this system, each cell remains unchanged until some neighboring cell has a modular value exactly one unit larger than that of the cell itself, at which point it copies its neighbor's value. One-dimensional cyclic cellular automata can be interpreted as systems of interacting particles, while cyclic cellular automata in higher dimensions exhibit complex spiraling behavior. Rules As with any cellular automaton, the cyclic cellular automaton consists of a regular grid of cells in one or more dimensions. The cells can take on any of states, ranging from to . The first generation starts out with random states in each of the cells. In each subsequent generation, if a cell has a neighboring cell whose value is the successor of the cell's value, the cell is "consumed" and takes on the succeeding value. (Note that is the successor of ; see also modular arithmetic.) More general forms of this type of rule also include a threshold parameter, and only allow a cell to be consumed when the number of neighbors with the successor value exceeds this threshold. One dimension The one-dimensional cyclic cellular automaton has been extensively studied by Robert Fisch, a student of Griffeath. Starting from a random configuration with n = 3 or n = 4, this type of rule can produce a pattern which, when presented as a time-space diagram, shows growing triangles of values competing for larger regions of the grid. The boundaries between these regions can be viewed as moving particles which collide and interact with each other. In the three-state cyclic cellular automaton, the boundary between regions with values i and i + 1 (mod n) can be viewed as a particle that moves either leftwards or rightwards depending on the ordering of the regions; when a leftward-moving particle collides with a rightward-moving one, they annihilate each other, leaving two fewer particles in the system. This type of ballistic annihilation process occurs in several other cellular automata and related systems, including Rule 184, a cellular automaton used to model traffic flow. In the n = 4 automaton, the same two types of particles and the same annihilation reaction occur. Additionally, a boundary between regions with values i and i + 2 (mod n) can be viewed as a third type of