Shape

32 Introduction: Tell Me All About It
The trouble with bits—including points—is that they’re not the same as the
things they combine to make. This worries me, because it gets in the way when I
want to calculate with shapes. I’m going to say a lot about it in terms of embedding.
But maybe others share my qualms. In fact, I have some pretty extraordinary company.
Leonhard Euler was the first to classify polyhedra according to their vertices, edges,
and faces, and thus to show how basic elements are related in shapes. (Miss H——
knew about this.) Points (vertices) are boundaries of lines. Then lines (edges) are boundaries
of planes, and planes (faces) are boundaries of solids. These relationships
were surprising at the time—this was a radically new way to describe polyhedra in
terms of geometric entities of finite extent and increasing dimension, and with sensible
properties.
These new elements are tactile, they are differences in texture. If you hold a model polyhedron
you feel its flat faces, the ridges where they meet, and the sharp points at the corners.
But Albrecht Dürer more than two hundred years earlier had described polyhedra in
terms of their ‘‘nets’’
and so distinguished vertices, edges, and faces that are undeniably tactile, too. Just
think of using your hands to fold a net to make a polyhedron. Maybe the hand is
as perceptive as the eye. After all, I can trace parts with my fingers when they’re
embedded. Is this another way to calculate? There are vertices for Braille—it uses
symbols—and also edges, faces, and polyhedra that allow for more. So I can calculate
with shapes via hand or eye—they’re reciprocally related in what I see and in what I
draw and make. Sensory experience is sensory experience, in one modality or another.
But let’s get back to the point, and the higher elements that take up space. Euler wasn’t
keen on bits.
Only admit this proposition, bodies are compounded of simple beings, that is, of parts which have
no extension, and you are entangled. With all your might, then, resist this assertion: every compound
being is made up of simple beings; and though you may not be able directly to prove the fallacy,
the absurd consequences which immediately result, would be sufficient to overthrow it.