Shape

180 II Seeing How It Works
meaning. Before I calculate, all shapes are the same—each is a single, unitary thing, a
simple it. And after I start, units change freely as I go on. They’re different every time I
apply a rule.
This contrast can be traced to embedding. For points, it’s identity, so that each
contains only itself. This is what units require, and it makes counting possible. But for
all of the other basic elements embedding doesn’t end. There are indefinitely many of
the same kind embedded in each one—lines in lines, planes in planes, and solids in
solids. This is the crucial difference between points and lines, etc. It explains the properties
of the algebras I’m going to define for shapes in which basic elements fuse and
divide. And it’s how rules deal with ambiguity as calculating goes on, so that seeing
never stops.
Yet the contrast isn’t categorical. I have reciprocal models of calculating, but each
includes the other in its own way. My algebras of shapes begin with the counting
model. It’s a special case of seeing when embedding and identity coincide. And I can
deal with the seeing model via counting—and approximate it in computer implementations.
These are telling equivalencies that show how complex things containing
units are related to shapes without definite parts.
Whether or not there are units is the question used to distinguish verbal and
visual expression—Susanne Langer’s discursive and presentational forms of symbolism
I mentioned in the introduction. It goes like this—
verbal expression (discursive forms) : units, symbols, or points
< visual expression (presentational forms) : lines, planes, or solids
Verbal and visual expression aren’t incompatible, though, as these relationships are
sometimes thought to imply. Shapes with points and shapes made up of lines,
planes, or solids aren’t incommensurable. People calculate all the time. And art and
design whenever they’re visual are as much a matter of calculating as counting. This
is what algebras of shapes and calculating in them show. The claim is figuratively
embedded if not literally rooted in the following details.
Shapes in Algebras and Algebras in Rows
Three things go together to define algebras of shapes. First, there are the shapes themselves
that are made up of basic elements. Second, there’s the part relation for shapes
that includes the Boolean operations. And third, there are the Euclidean transformations.
The algebras are enumerated up to three dimensions—of course, there are
more—in this series