Shape

206 II Seeing How It Works
made up of three lines. Are the lines that define its complement
pure white or really gray? But this is a concrete problem without serious abstract implications.
It’s interesting, though, that a defining perceptual phenomenon finds its way
so naturally into mathematics. And figure-ground-like relationships come up in other
ways, as well, whenever complements are involved.) Shapes with points form a generalized
Boolean algebra. It’s a relatively complemented (table 6, fact 7), distributive
(fact 6) lattice when the operations are sum, product, and difference, or a Boolean ring
with symmetric difference and product. There’s a zero in the algebra—it’s the empty
shape—but no universal element because the universal shape that includes all points
can’t be defined. As a lattice, the algebra isn’t complete—every product is always
defined, while no infinite sum ever is. Distinct points don’t fuse in sums the way basic
elements of other kinds do. The finite subsets of an infinite set, say, the numbers in the
counting sequence 0, 1, 2, 3. . . , define a comparable algebra.
Some of my technical jargon may be hard to take. Still, it’s what the mathematics
is about, and it repays the effort to track down definitions for terms that aren’t familiar.
Both the formalist and the artist need to meet each other halfway, so that they can join
at more than a boundary. There’s much to be gained when each side heeds the other.
After all, I’ve been trying to show what calculating would be like if Turing had been a
painter. I try to use formal terminology only when it’s helpful. Without it here, it
would be difficult if not impossible to make the rigorous contrasts that are implicit in
my series of algebras. At the very least, my appeal to these ideas should make it clear
that there’s no reason to be flaky when you’re talking about art and design. And at
the same time, I hope it’s also clear that a formal presentation doesn’t diminish the
expressive potential of drawings and the like. Shapes are always there to see with all
of their possibilities when rules are tried.
Algebras of shapes made up of points come up again when I consider algebras
of decompositions—they have a lot to do with spatial relations and set grammars. A
decomposition is a finite set of parts (shapes) that add up to make a shape. It gives the
structure of the shape by showing just how it’s divided, and how these divisions interact.
The decomposition may have special properties, so that parts are related in some
way. It may be a Boolean algebra on its own, a topology, a hierarchy, or something
else. For example, suppose a singleton part contains a single basic element. These are
atoms in the algebras U 0 j but don’t work this way if i isn’t zero. The set of singleton