Shape

205 Boolean Divisions
shape, and sums of atoms produce all of the other shapes. No two shapes are the same
unless their atoms are. A shape either is empty or has an atom as a part. There are two
kinds of atomic algebras depending on the value of the index j.
The only Boolean algebra of shapes is U 00
It contains exactly two shapes—the empty shape and an atom made up of a single
point. In fact, it’s the only finite algebra of shapes, and it’s the only complete one in
the sense that shapes are defined in all possible sums and products. The algebra is the
same as the Boolean algebra for true and false used to evaluate logical expressions.
The other atomic algebras U 0 j are defined when j is more than zero
Each of these algebras contains infinitely many shapes that all have a finite number of
points and a finite number of parts. But since there are infinitely many points, there’s
no universal shape that includes all of the others, and so no complements. (It’s easy to
get around this for points and for basic elements of higher dimension if I look at
shapes and their complements in a kind of figure-ground relationship. For example,
here
the shape is either black or white, either the figure containing three points or all of the
other points in the ground. And the same finite representation can be used to distinguish
both cases by saying what is or isn’t in the shape—the shape is three points or
all points with the exception of three
This works without a hitch for points for any value of j, and for lines, etc., when i
equals j. Otherwise, a little squinting is helpful. Try it for the shape