Shape

164 II Seeing How It Works
Back to Basics—Elements and Embedding
At the very least, shapes are made up of basic elements of a single kind—either points,
lines, planes, or solids. Here are some examples of the first three
and these are drawings—just shapes containing lines and planes—of solids
Basic elements are readily described with the linear relationships of coordinate geometry.
It’s easy to extend this repertoire to include other curves and exotic surfaces,
especially when these are also described analytically—for example, when lines are represented
as conics to include segments of circles, ellipses, parabolas, and hyperbolas.
But my results are pretty much the same whether or not I allow further kinds of basic
elements. I’ll stick with the ones that show what I need to. A little later, I’ll say why
straight lines alone are more than enough to see how shapes work—both as mathematics
and as the stuff of design. It has nothing to do with approximation—using a
lot of tiny segments to make curves. The real leap is to get over points and what they
imply for shapes and rules.
Some of the key properties of basic elements are summarized in table 3. Points
are undivided units. They’re simple and yield nothing to analysis. But in contrast,
I can cut lines, planes, and solids into discrete pieces—line segments, triangles, and
tetrahedrons—so that any two of these pieces are connected in a finite sequence in
which successive ones have a common boundary element—a point, a line (edge),
or a plane (face). The construction is always possible. Still, it seems worth doing
only for planes and solids that aren’t triangles or tetrahedrons themselves. This is just
Table 3
Properties of Basic Elements
Basic element Dimension Boundary Content Embedding
point 0 none none identity
line 1 two points length partial order
plane 2 three or more lines area partial order
solid 3 four or more planes volume partial order