Shape

195 Shapes in Algebras and Algebras in Rows
These transformations also form a group under composition. The properties of this
group characterize Euclidean geometry. I use this group later to describe the behavior
of rules when they’re used to calculate with shapes.
That does all of it. I have shapes, parts, sums, and differences, and various kinds
of transformations—some Boolean stuff and some Euclidean stuff—along with basic
elements. This is what it takes to make seeing work. It blends a lot of well-known
mathematics to handle shapes and rules. Let’s look at it again all together. A few illustrations
are enough to fix the algebras U ij visually. And in fact, this is just the kind of
presentation they’re meant to support. Once the algebras are defined, what you see is
what you get. There’s no need to represent shapes in symbols to calculate with them,
or for any other reason. Look at the shapes in the algebra U 02 . This one
is an arrangement of eight points in the plane. It’s the boundary of the shape
in the algebra U 12 that contains lines in the plane. The eight longest lines of the shape
are the maximal ones. And in turn, the shape is the boundary of the shape
in the algebra U 22 . The four largest triangles in the shape are maximal planes. They’re
a good example of how this works. The triangles are discrete, and their boundary elements
are, too, even though they touch externally. Points aren’t lines and aren’t in the
boundaries of planes.
The properties of basic elements are extended to shapes in table 7. The index i
is varied to reflect the facts in table 3. The algebras are linked via shapes and their
boundaries—I’ll say more about this in the next section. But for now, notice that an
algebra U ij contains shapes that are boundaries of shapes in the algebra U iþ1 j only if i
and j are different. That’s why my examples ended at U 22 . The algebras U ii don’t include
shapes that are boundaries of shapes, and they’re the only algebras for which
this is so. How could they?—when i ¼ j, j is one dimension too low. Of course, boundaries
aren’t the only way to distinguish these algebras. They also have special properties
in terms of embedding and the transformations. And I’ll describe these, as well,
only yet farther on.