Shape

274 II Seeing How It Works
produce the shape
in fifteen iterations. Only this isn’t necessary. Cellular automata are abstract symbol
systems like Turing machines, and what they do can be presented in many ways. You
can pretty much design the visual effects you want. My two examples, first with the
game of life and then with strings of cells, begin to show this when the graphic devices
used in each are switched. Or change the shape just shown, so that rows of cells are
lined up to the left. There are new rules for this, and they’re easy to define—in fact, I
can use rules that apply to my original rules as shapes to get the new rules I want. Try
it. Then after seven iterations, the result looks like this
Key (abstract) relationships stay the same, but what do you see? To find out when
there’s a fractal or ‘‘self-similar’’ pattern as intended, and not a checkerboard design—
although this is interesting in its own way—apply the rule
that divides plane triangles and their boundaries, and use identities from the schema
x fi x, so that x is one such triangle and its boundary, or any shape determined by
the rule. These identities show exactly what it means to be self-similar, and they root
the idea in seeing. What I’ve done lets me describe what I’m doing now and frame
what I do later—learning and experience really matter. I can also see in novel ways—
for example, let x be a polygon or A, B, C—or go on to something entirely different.
Past experience and new are equally transferable in identities and in the left sides of
other rules. What’s useful here is useful there once it’s transformed and embedded
again—learning and sagacity go hand in hand in creative activity. And for shapes and
rules, all of this works with nary a hitch.
(Cellular automata are intriguing because they can be self-similar and can have
other ‘‘emergent’’ properties; that is to say, they can form global patterns in time that