Shape

89 There Is No Vocabulary—Well, Almost
There Is No Vocabulary—Well, Almost
Shapes containing points are numerically distinct. But shapes made up of basic elements
of higher dimension don’t have this property. For example, Evans’s shape
has at least eight lines, and any number of lines more. Of course, I can cook up special
ways to divide shapes—always with the concomitant loss of other parts—so that basic
elements are numerically distinct. I can use maximal lines in the way I have or I might
also choose maximal lines and the different segments they define when they intersect.
The total number of segments for a line in that’s divided n times and the line itself is
the sum of the numbers from 1 to n þ 1, as here for n ¼ 2
Counting like this provides a nice way to describe shapes that’s especially useful when
you’re looking for polygons. But it’s already pretty clear that it isn’t visual. It gives the
twenty-four lines Evans uses so effectively to resolve triangles. Each maximal line is
divided once in half to define three segments. But this seems profligate. Maybe the sixteen
halves—the ‘‘minimal’’ elements—are all that I need. They’re the segments I got
when I applied the identity
everywhere I could. But let’s try something else where there may be more than basic
elements—maximal, minimal, or otherwise.
Suppose I look at the lowest-level constituents in Evans’s approach as symbols or
units—basic elements or not—in a given vocabulary, and that I want to arrange them,
so that it’s always the same what’s there. How can I use a vocabulary of shapes if they’re
taken this way when they’re combined? How do the shapes look, and what kinds
of arrangements are possible? Can I be certain that I can distinguish the shapes I give
for symbols visually, with no ambiguity? This isn’t always so. For example, this shape