Shape

212 II Seeing How It Works
topology and connected in the other. Among other things, the topology of the point
set confuses the boundary of the shape that’s not part of it—a shape in the algebra
U i 1 j —and the topological boundary of the shape that is. Points are too small to distinguish
between boundaries as parts and limits. To get around this, point sets are ‘‘regular’’
when they’re shapes, and Boolean operations are ‘‘regularized.’’ This seems an
artificial way to handle lines, planes, and solids. Of course, shapes made up of points
are point sets with topologies in which parts are disconnected. This and my gloss on
individuals reinforce what I’ve already said. The switch from identity to embedding
may seem modest—it’s going from zero to one—but it has telling consequences for
shapes and how rules work.
Euclidean Embeddings
The Euclidean transformations augment the Boolean operations in the algebras of
shapes U ij with additional operators. They’re defined for basic elements and extend
easily to shapes. Any transformation of the empty shape is the empty shape. And
a transformation of a nonempty shape contains the transformation of each of the
basic elements in the shape. This relies on the underlying recursion implicit in tables
3 and 7 in which boundaries of basic elements are transformed until new points are
defined.
The universe of all shapes, no matter what kind of basic elements are involved,
can be described as a set containing certain specific shapes, and other shapes formed
using the sum operation and the transformations. For points and lines, the universe is
defined with the empty shape and a shape that contains a single point or a single line.
Points are geometrically similar, and so are lines. But this relationship doesn’t carry
over to planes and solids. As a result, more shapes are needed at the start. All shapes
with a single triangle or a single tetrahedron do the trick, because triangles and tetrahedrons
make the other planes and solids. For a point in zero dimensions, the identity is
the only transformation. So there are exactly two shapes—the empty shape and the
point itself. In all other cases, there are indefinitely many distinct shapes that come
with any number of basic elements. Every nonempty shape is geometrically similar to
indefinitely many other shapes. There’s plenty of opportunity to move shapes around
and to increase or decrease their size without changing the relationships between their
basic elements—for example, the angles defined by lines.
The algebras of shapes U ij also have a Euclidean classification that refines the
Boolean classification I’ve given. Together these classifications form an interlocking
taxonomy. The algebras U ii on the diagonal provide the main Euclidean division