Shape

170 II Seeing How It Works
while these three lines are discrete
and these five squares are, too
Points never touch when they’re discrete, but touching is allowed for discrete lines,
planes, and solids either at points, lines, or planes. It’s worth repeating over and over
again—it was evident for meets earlier—that basic elements of different kinds are
always separate. In particular, no basic element has content with respect to any basic
element of higher dimension. This is why Euler was sure that points don’t add up to
make lines. Points aren’t lines, or planes or solids. And this extends to lines and planes.
Content is never zero.
It’s easy to see that both of these relations—overlap and discrete—include
embedding.
One basic element is embedded in another if every basic element that overlaps the
first also overlaps the second, and conversely, if every basic element discrete from the
second is discrete from the first.
For overlapping lines, it looks like this
and for discrete lines, like this
Whatever relation I use—embedding, overlap, or discrete—it comes to exactly the
same thing. Any one implies the other two. But what about touching? I’ve used it
without a definition. Points touch when they’re identical, and more generally, other
basic elements do when they overlap. Only lines, planes, and solids can also touch if
they’re discrete. Before I can say how this works, I need to say a little something about
boundaries.