Shape

329 What the Thinking Eye Sees
But be careful. Inverses may not do what you expect. For example, the inverse
x þ xA fi x
of the schema x fi x þ xA doesn’t work for quadrilaterals and pentagons. When I use
the rules it defines, I’m left with an ugly gap
Yet beauty is in the eye of the beholder—perhaps this outcome is good for something
else. And I can always go on from anything I have with shapes and rules.
Working through Klee’s drawing in this way isn’t meant to recapitulate what
he did—no one knows for sure, and it’s no good relying on what he says. Rather, I’m
trying to show that schemas are up to Klee’s kind of designs. The rules I need are
included in a handful of schemas that are easy to use. And the rules themselves are
easy to apply to shapes in terms of embedding and transformations. Embedding
ensures there’s no letter of the law—I can calculate with triangles and see quadrilaterals
and pentagons—and general transformations allow for as much irregularity as I
want—parametric variation of any kind is possible. Rules and freedom go hand in
hand—the one implies the other.
I’m staking the plausibility of the metaphor ‘‘design is calculating’’ as a heuristic
and equivalence squarely on my success with schemas, especially as I’m able to use
them more and more to calculate with shapes and to make designs in different styles
from different places and times. This is another kind of freedom, and it provides a
very practical kind of proof that calculating works. There’s seeing and doing, and that’s
all it takes when rules apply to shapes. I guess ‘‘design is calculating’’ really is more of a
thesis than a theorem. The evidence adds up, but not to logical certainty, just to empirical
belief.