Shape

175 Back to Basics—Elements and Embedding
and lines are coincident with planes
But this definition may be too narrow. Lines that extend past the boundaries of
planes
aren’t entirely coincident with them, even though they contain lines that are. Only
how about cases like this
where a line isn’t coincident with a plane but can be divided into segments, so that
each segment is coincident with the plane? (I can also say this for any basic element
in terms of embedding. If I stick with lines, it goes like this: every line embedded
in the line has a line embedded in it that’s coincident with the plane.) Moreover, it
would be useful for every basic element to be coincident with itself, and for the first
term in a series of basic elements, each term coincident with the next, to be coincident
with the last term. Once transitivity—that’s the third amendment—is in place,
points are also coincident with planes in various ways, either inside, on an edge, or at
a corner
And with the final two amendments, I get a partial order. What’s more, coincidence
lets me define touching in another way. Two basic elements touch if there’s a single
basic element coincident with both. There’s more here than in my initial definition,
since basic elements aren’t distinguished by kind—lines and planes can touch. This
line touches three squares, each in a different way