Shape

208 II Seeing How It Works
boundary. Parts are disconnected. (Topological boundaries aren’t the boundaries I’ve
been describing so far that show how shapes in different algebras are related. Now the
boundary of a shape is part of it and not a shape in an algebra of lower dimension.)
This is a formal way of saying what seeing confirms all the time—there’s no preferred
way to divide the shape into parts. Any division is possible. None is better or worse
than any other without ad hoc reasons that depend on how rules are used. This is
where meaning begins, and going on is how it changes and grows. Shapes have meaning
because I calculate. There’s no meaning until I do.
Going on, the atomless algebras U ij are defined when i is more than zero
Each of these algebras contains infinitely many shapes made up of a finite number of
maximal lines, planes, or solids. But every nonempty shape has infinitely many parts.
The empty shape is the zero in the algebra. There’s no universal element and hence
no complements. (A universal shape can’t be formed with lines, planes, or solids, as
this would fill all of an unbounded space. Moreover in a bounded space where i is
less than j, there would be infinitely many basic elements. It doesn’t work either
way.) This gives a generalized Boolean algebra, in the way this was defined for
points. Only this time, the properties of sums and products are symmetric—for both,
infinite ones may or may not be defined. When infinite sums are formed, basic elements
fuse. But this alone isn’t enough—for example, the sum of all singleton shapes
isn’t a shape.
In an algebra U ij when i isn’t zero, a nonempty shape and its parts form an
infinite Boolean algebra. But the algebra isn’t complete—infinite sums and products
needn’t determine shapes. It’s worth seeing how this works, at least to emphasize
once more that shapes are finite through and through. Every shape is the sum of the
set of its parts. Only there are subsets of this set that don’t have sums. Let’s suppose
that the shape is a single line
Then the series of shapes containing the first quarter of the line, the fifth eighth, the
thirteenth sixteenth, and so on