Shape

165 Back to Basics—Elements and Embedding
geometry—points are congruent and lines are similar, but planes and solids need be
neither. It’s easy to divide a hexagonal plane
in three different ways
But the bow tie (quadrilateral?)
isn’t a basic element. No matter what divisions I try to make, there are going to be triangles
that aren’t connected in the right way—they don’t share an edge, or begin and
end a sequence of triangles in which successive ones do.
The way I construct lines, planes, and solids from segments, triangles, and tetrahedrons
implies that they have finite, nonzero content, that is to say, extension
measured by length, area, or volume. Yet reciprocal relationships aren’t so easy. In
particular, finite content needn’t imply finite boundaries. Lines are no problem—each
is bounded by a pair of distinct points—but what about planes and solids? Does finite
content guarantee a finite number of basic elements in every boundary, or that the
total content of the basic elements in every boundary is finite? The answer is no both
times, and it’s worth seeing why to emphasize that shapes are finite through and
through.
Begin with the four lines
in the boundary of a square