A cellular automaton that makes beautiful little patterns

One of the things I pulled off of my old floppy disks (“Making floppy disk images under OS X”, 2019-07-26) was a program called “PATTERNS.C”. Its timestamp is 2000-09-24, or right about the start of eleventh grade for me. What does it do? Well, first I’ll explain it mathematically, and then I’ll show you what it looks like live.

It’s a cellular automaton that runs in a 31-by-31 grid of cells. Each of these cells can be in either of two states: “live” or “dead.” At each time-step: each cell with an odd number of live neighbors becomes “live”; each cell with an even number of live neighbors becomes “dead.”

The “neighbors” of a cell are its Von Neumann neighborhood (just the four cardinal directions, no diagonals).

In Stephen Wolfram’s idiosyncratic notation, this is “the five-cell outer-totalistic cellular automaton with code \(\tilde{C}\) = 204.” It appears that Wolfram even mentions this specific automaton by name, on page 824 of

A New Kind of Science(2002).

In “PATTERNS.C”, the grid does not wrap; instead, cells at the edge of the grid act as if their missing neighbors are “dead.” This will be important in a minute.

In “PATTERNS.C”, I actually used four states: the “dead” state and three different “live” states. The different “live” states are used to track the “age” of a living cell. On each turn in which a cell doesn’t move into the “dead” state, it moves to the next greater “live” state; unless it has already reached state 3, in which case it just remains there. The extra states don’t much change the mathematical properties of the automaton; they just add some visual interest.

Now for the sad part. No matter what pattern of cells you start with, if you run the automaton for exactly 32 time-steps, you’ll wind up with a grid consisting entirely of “dead” cells. Each living cell in the starting configuration sends out “waves” that bounce off the boundaries of the grid and ultimately cancel each other out.

Now for the happy part! If you stop after 31 time-steps, the configuration displays surprising, intricate, even beautiful symmetry.

The vintage-2000 source code is here, and the Javascript/Canvas port embedded in the iframe above is also downloadable here.