Shape

167 Back to Basics—Elements and Embedding
Phenomena of this kind are found in fractals. Their approximations—as in my
two series for squares—produce some nice visual effects. But the phenomena themselves
have nothing to do with basic elements or the shapes they make. Basic elements
aren’t defined in the limit, but rather here and now. Everything about them is finite,
except that I can divide lines, planes, and solids anywhere I want. Still, this is the potential
infinite of Aristotle and not the actual infinite of sets where members are given
all at once. I can see just a finite number of divisions at any time—and these may come
and go as rules are used to calculate. Basic elements are the stuff of shapes and practical
design before they’re material for mathematics. Drawing a line in a single stroke with a
pencil on a sheet of paper and tracing segments one by one are the twin examples I
keep in mind.
Embedding is the main relation I use to describe basic elements, and by implication
the shapes they combine to make. The only point that’s embedded in a point is
the point itself. Embedding is identity. But, more generally, every basic element of
dimension greater than zero has another basic element of the same kind embedded in
it—lines are embedded in lines
—schematically, like this
—and planes in planes
etc. Additionally, I can always embed these basic elements in other ones that are bigger.
Content is finite for separate elements, but there are always other elements with more.
The embedding relation is a partial order. Most of the time, I take standard mathematical
devices like this for granted, and use them without definition. But to start, it
may be worth rehearsing the kind of ideas involved. A partial order satisfies three conditions.
In particular, embedding is (1) reflexive, (2) antisymmetric, and (3) transitive.
This isn’t much help, so let’s try it in everyday language. Then the conditions go like
this—(1) every basic element is embedded in itself, (2) two basic elements, each
embedded in the other, are identical, and (3) if three basic elements are such that
each is embedded in the next, then the first basic element is embedded in the last.
Drawing this doesn’t show much. It’s a problem with most abstractions that becomes
more acute as I go on. That’s what I like about shapes and their parts—they’re always
there to see. Even so, transitivity can look like this for lines
at least schematically.