Shape

171 Back to Basics—Elements and Embedding
Basic elements that aren’t points—either lines, planes, or solids—have boundaries.
The boundary of any such basic element contains a finite number of basic elements,
all of dimension one less than the original basic element itself. The boundary
of every line
is just two points
while the boundary of a plane
contains three or more lines
More precisely, I should say maximal lines. Otherwise, there’s nothing definite to
count. Maximal elements are used to define shapes uniquely. And in fact, the boundary
of any basic element that has a boundary is a shape. But before I deal with the
details of this, let’s try touching.
Points touch whenever they’re identical. And basic elements of the same kind—
two lines, planes, or solids—touch if there are basic elements embedded in each
with boundary elements that touch. This is evident for basic elements that overlap.
Any shared element has a boundary element that does the trick. For a pair of lines, I
have
and for two squares
But the recursion is more general than this. It’s clear in a few easy examples. When
lines touch