Shape

128 I What Makes It Visual?
Here, g assigns regular polygons to x and z, so that the sides of the one have midpoints
that are the vertices of the other, for example, like so
Still, this kind of detail isn’t always necessary. I’m apt to take shortcuts whenever I can.
In the schema
x fi x þ tðxÞ
and others like it, I conflate the two variables x and y and the predicate. More technically,
I should have said that x is any shape and y ¼ x þ tðxÞ. And the schema is developed
further in the predicate above, in which x is a polygon and the transformation t is
carefully spelled out. In like fashion,
x fi tðxÞ
provides a summary of the schema x fi y, where x is a shape and y is a transformation
t of x. Ift is the identity, then rules are identities themselves. And I can use rotations
and translations as I did above. Or perhaps there’s a boundary operator b to define rules
that change lines to points and back again. If I write
bðxÞ fi x
as the inverse of the schema
x fi bðxÞ
in which x is a shape made up of lines, I get the rule
There’s more about this in part II. But now, I want to show something different
that goes to the heart of what it means to calculate visually. For this, it’s enough
to know what the variables x and y and the assignment g do. The main issue isn’t
technical.
For every predicate, there are indefinitely many others equivalent to it. It’s clear
that shapes made up of lines or higher-dimensional elements can be described in indefinitely
many ways in terms of their different parts. It might go like this
x is the sum of the parts . . . and y is the sum of the parts . . . such that . . .