Two-Dimensional CA Examples

Here are a few examples of patterns grown by two-dimensional CA. Click the small picture for a larger version.

First, the gasket can be grown from an initial condition consisting of a single live cell. Can you find rules to grow gaskets in different orientations?

From a single live cell, this rule grow a pattern of concentric diamonds that fill to a maximum density, then empty and refill. The far right picture is a time record: the middle cross-section of the two-dimensional picture (horizontally) vs time (vertically, generations increase downward). Note a familiar pattern.

Another example, grown from a single live cell.

Now three examples, grown from a single live cell, for Moore nbhd CA. In the first, the central region freezes and the growth occurs only along the periphery. The others continue to change throughout.

Now we give just a hint of sensitivity in two-dimensional CA. First, here is the pattern evolving from a single live cell for this rule.

Delete the part of the rule under 0 and a homogeneous pattern grows from a single live cell. From an initial condition of two live cells separated by about 1/4 the state space size, the pattern on the far right grows.

Now we consider several patterns grown from random initial distributions of live and dead cells. First, the "majority rules" von Neumann and Moore CA. (Do you see why these CA are called majority rules?) The Moore CA pattern freezes completely; the von Neumann pattern oscillates at the "checkerboard" regions.

Here is another CA that evolves from random initial distributions of live and dead cells to an organized pattern. This is an example of "self-organization" in CA.

This is another self-organizing CA. The second picture is generated using the same rule, but with a different initial distribution of live and dead cells. The pictures differ in fine detail, but not in general form.

Return to 4. C. Examples of Cellular Automaton Patterns