Shape

194 II Seeing How It Works
how it worked finding parts—using tracing paper to see what was there independent
of what I drew—and replacing them in a recursive process. The part relation and a
few transformations—they’re given below—were what I used to find parts. And then
subtracting and adding—that’s erasing and drawing—and the same transformations
let me replace the parts I found. So, what I did with rules told me what I needed to do
with algebras. They’re merely a summary of how things turned out. I didn’t start with
them or anything like them—that’s preposterous. But that they’re so nice in the end
gives added support to my approach. The mathematics is too neat for rules not to
work like this. In some ways, though, my algebras are profligate. In the theory of formal
languages and automata—it’s the mathematical basis for linguistics—algebras
(monoids) are defined for strings of symbols using a single associative operation—
concatenation. This explains strings, but it doesn’t explain how rules apply. For that,
you need other operations like sum and difference. The rules in a generative grammar
need more than concatenation to work. To avoid this kind of problem for shapes, I
started with shapes and rules at the same time. The two had to work together—the former
without the latter left seeing a mystery. Without rules it was all magic.
The transformations that make rules work move shapes around, turn them over,
and make them bigger and smaller. They’re operations on shapes that change them
into geometrically similar ones. They distribute over the Boolean operations, and may
include, for example, translation
rotation
reflection
and scale