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 particle, that remains stationary. A collision between a moving and a stationary particle results in a single moving particle moving in the opposite direction.
However, for n ≥ 5, random initial configurations tend to stabilize quickly rather than forming any non-trivial long-range dynamics. Griffeath has nicknamed this dichotomy between the long-range particle dynamics of the n = 3 and n = 4 automata on the one hand, and the static behavior of the n ≥ 5 automata on the other hand, "Bob's dilemma", after Bob Fisch.
Read more about this topic: Cyclic Cellular Automaton
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