Shape

203 Boundaries of Shapes Are Shapes
two shapes is their sum, so long as no maximal element of one is coembedded with
any maximal element of the other. In this case, the shapes have an empty product.
But further, basic elements are maximal if they’re not coembedded. So, second, the
boundary of the symmetric difference of the two shapes is the sum of their boundaries.
I can combine parts in one way if their basic elements are coembedded, and in another
way if they’re not, to get the result I want. Now suppose I take the boundary of the
symmetric difference of the shape A
with three parts, and another shape B
that’s also divided into three parts B 1 , B 2 , and B 3
according to my equivalence relation. Then, I get
bðA l BÞ ¼bðA 1 ÞþbðB 1 ÞþðbðA 2 Þ l bðB 2 ÞÞþðbðA 3 Þ l bðB 3 ÞÞ
instead of
bðA l BÞ ¼bðAÞ l bðBÞ
The symmetric difference of A and B is
and both shapes have the same boundary