Shape

202 II Seeing How It Works
They’re independent with respect to symmetric difference—their product is the empty
shape. But this isn’t so for the product of their boundaries
It’s a single point. As a result, bðxÞ l bðyÞ
is missing a point that’s in bðx l yÞ
More generally, common pieces of boundary elements in the maximal elements in the
symmetric difference of x and y are lost in the symmetric difference of their boundaries.
But it’s easy to fix this when shapes are divided into parts according to whether
or not their maximal elements are embedded in common ones.
The following equivalence relation does the trick.
Let two basic elements be coembedded whenever they’re embedded in a common
element.
The effect of this is clear in the shape A
that’s divided into three different parts A 1 , A 2 , and A 3
(This is a nice way to store maximal elements in a computer. After all, it’s how an
experienced draftsman would normally draw the parts of a shape made up of lines.
But notice that the relation doesn’t organize points, as it does basic elements of higher
dimension—lines and planes and solids—when j is more than i.) Then two other
facts—in addition to the fact that symmetric difference works for coembedded
elements—are used to exploit this kind of division. First, the symmetric difference of