Shape

85 What Makes Calculating Visual?
Identities are interesting rules. Rules are supposed to change things, but identities
don’t. Whenever they’re used to calculate—let’s apply one or more to a shape S a
number of times—the result is a monotonous series that looks like this
S Þ S Þ Þ S
In each step (S Þ S), another part of S is resolved—that’s the part I see—and then
nothing else happens. The shape doesn’t change. It stays the same. At least that’s how
it looks. Identities are constructively empty. They’re useless! And it’s standard practice
to discard them. I’ve seen doctoral students do so with relish. There’s something liberating
about getting rid of unnecessary stuff. But this may be shortsighted—even rash. It
misses what identities do. There’s more to them than idle repetition. Identities are
observational devices. They’re all I need to divide the shape S with respect to what I
see. If I record the various parts they pick out as they’re tried, I can define topologies
for S. These show its constituents and how they change as I calculate.
Suppose I apply the identity
to the shape
so that large triangles are picked out in a clockwise fashion in this four-step series
If I take the triangles the identity resolves—remember, these are the triangles I see—I
can use them to define constituents. I’m going to calculate some more to explain how
I’ve been calculating. There are a number of ways to do this. For example, I can work
out sums and products, or I can add complements as well. Complements give Boolean
algebras for the shape. They’re a special kind of topology with atoms that provide a
neat inventory of constituents.
The first time I try the identity to calculate, I get a Boolean algebra with two atoms