Shape

273 Classifying Rules with Parts
where the apex is at 1/n for n ¼ 3; 4; 5 ... is enough for as many distinct symbols and
states as I wish to have. Then I might define the rule
There are other possibilities, too. Stanislaw Ulam invented a device that’s used
widely to model physical phenomena of all sorts. Cellular automata are points with
labels or weights. They’re located in a grid—it’s usually square—that has one or more
dimensions. (Otherwise, points are solipsistic.) Neighborhood relations are specified in
rules that are applied under translations. The rules work in parallel to change every
point at the same time, with the sum of the individual outcomes as the overall result.
The best-known rules of this kind are the ones for John Conway’s game of life. In
words, it goes like this—
Survival
Death
Birth
Stasis
If an occupied cell has two or three neighbors, it survives.
If an occupied cell has four or more neighbors, it dies from overcrowding.
If an occupied cell has one or no neighbors, it dies from isolation.
If an unoccupied cell has exactly three neighbors, it becomes occupied.
If an unoccupied cell has less than three neighbors or four or more, it stays
unoccupied.
—so that all 512 rules are divided into four equivalence classes. And for shapes, the 56
rules for birth are similar to this one
with black and white points—dots—at the centers of white grid cells. Colors are
weights—the part relation ðaÞ is identity, the sum of two is black unless both are
white, and the difference of two is white. Or maybe I have cellular automata defined
on a line of evenly spaced points. I can record their history by stacking strings of cells,
and I can give explicit rules for this purpose. These eight rules