Shape

173 Back to Basics—Elements and Embedding
I can go on to classify different ways of touching. For example, two basic elements
touch externally when they’re both embedded in a common element and touch
without overlapping, as here for discrete lines
and in these twin cases for pairs of triangles
Of course, twins aren’t always identical. In the first case, another plane is defined—
boundary elements overlap—as this division into four triangles confirms
while in the second case, the bow tie still contains separate basic elements. And likewise,
one basic element is tangentially embedded in another if the first is embedded
in the second and there’s a basic element that touches both externally. For lines, it’s
and for planes, it’s
(But what happens if two basic elements are said to touch externally without both
being embedded in a third? With three lines, I could have the first embedded in the
second and cut by the third—schematically, like this
—so that embedding and tangential embedding are the same.) Moreover, I can define
embedding, overlap, and discrete in terms of touching. One basic element is embedded
in another whenever every basic element that touches the first also touches the second.
I used the same idea to get embedding from overlap.