Random Explorations in Automata Theory
Gary J. Shannon
Mar. 2, 2003

Tubular and Tubular Toroidal Automata

Conventional cellular automata are often configured so that the cells at the top and bottom act as if they are connected and the cells on the left and right edges do the same.  This prevents the boundary problems encountered when the grid suddenly ends.  This is called toroidal connection since it represents the same topology as if the grid were drawn on the surface of a torus or donut.

In addition to the usual large grid with roughly equal size in both dimensions it is interesting to see what happens when the grid size is severely restricted in one dimension.  Imagine a grid that is 100 cells wide but only 4 cells high.  Wrapped around a tube so that the top cells connect with the bottom cells such an automaton has some interesting properties not found in more spacious arenas.  In order to avoid boundary problems we can also connect the two ends of the tube together and form a skinny torus that more closely resembles a bicycle tire than a chocolate donut with candy sprinkles.

Using two states plus a third virtual state and the simple rule that an empty cell turns to state 1 when it has exactly 2 neighbors, we can put a photon on a tube with circumference of 4 and connected as a torus with a circumference of 10.  

Animation 1. Photon on a Torus.

What is not immediately obvious from looking at the animation is that when the two separate dots are moving along the top and bottom rows they are not actually separate dots, but because of the way the tube is connected they are really another photon made of two vertically adjacent cells. By viewing the automaton as nodes in a matrix rather than cells in a grid we can see that when two photons collide they spawn a new photon on the opposite side of the tube.

Animation 2. The toroidal grid as a matrix.

Perhaps the clearest way to observe the behavior of this particular atomaton is to make give the torus a larger diameter and flatten it out into a ring as in animation 3.

Animation 3. The torus flattened.

== UNDER CONSTRUCTION. Much more to come ==