Shape

169 Back to Basics—Elements and Embedding
isn’t defined for lines because meets aren’t always defined. In fact, it’s the same for
lines, planes, and solids—meets are determined only if these elements overlap. There’s
no other way for lines, planes, and solids to share an element and its content—shared
boundaries alone won’t do. For example, meets don’t exist for three lines embedded in
a fourth in this way
I can go on to consider embedding—again in a non-Aristotelian way—in terms
of its properties on all basic elements of a given kind instead of just those basic elements
embedded in a given one. This sort of generalization is especially useful when I
define shapes and their algebras. But, for now, look at what happens with lines. I still
have least upper bounds for lines that are collinear. However, joins in this case may or
may not be defined for infinite sets of lines. The lines embedded in a given line have a
least upper bound—the line itself. And the lines in this geometric series
have a least upper bound
However, the lines in this series of equal segments
don’t, because there’s no longest line. Content—now it’s length—can increase
without limit. Collinearity, however, is just a special case. Joins for lines that aren’t
collinear
aren’t defined.
Even though I can get by with embedding alone, other relations on basic elements
are helpful. In fact, I’ve used two of them a few times already—discrete and
overlap.
Two basic elements overlap if there’s a common basic element embedded in both.
Otherwise, they’re discrete.
Two points overlap only when they’re the same. But, for lines and planes, there are
other possibilities. These lines overlap
and so do these pairs of squares