Shape

33 Trying to Be Clear
Points aren’t parts of (embedded in) lines, planes, or solids because points have no
extension—neither length, area, or volume. And in the same way, lines aren’t parts of
planes or solids—there’s no area or volume—nor are planes parts of solids. This idea is
the heart of William Blake’s spirited defense of ‘‘Republican Art.’’
I know too well that a great majority of Englishmen are fond of The Indefinite which they Measure
by Newton’s Doctrine of the Fluxions of an Atom, A Thing that does not Exist. These are Politicians
& think that Republican Art is Inimical to their Atom. For a Line or Lineament is not
formed by Chance: a Line is a Line in its Minutest Subdivisions: Strait or Crooked It is Itself &
Not Intermeasurable with or by any Thing Else.
Newton’s atoms are bits—points or infinitesimals—and they aren’t in Blake’s lines.
Every subdivision of a line contains only lines—and planes and solids are alike in
exactly the same way. Basic elements can only be divided into basic elements of just
the same kind.
It’s Blake versus Newton, art versus science, visual versus verbal, seeing versus
counting—shapes versus bits includes it all. But this isn’t good and evil or the Yankees
and the Red Sox. Shapes and bits provide a way to blend these stark dichotomies.
There’s a nifty kind of equivalence between them that calculating with shapes helps
to reveal. Dealing with shapes as if they were bits is a good practical idea that’s also an
example of calculating with your eyes. Yes, it can be visual, but then sometimes it’s
not. I’m going to tell you when and how. It’s easy once you see how embedding lets
you calculate with shapes without using symbols. Then everything falls into place. If I
can calculate with shapes, then I can say how to put lines on paper in order to see
what’s there. I can design.
Trying to Be Clear
The rubric for the previous subsection—how I stopped counting and started to see—
may need a little more explanation than I’ve given it. The explanation is implicit
in what I’ve been showing you. But sometimes it pays to say exactly what you mean,
at least if you can. Calculating with symbols has a lot to do with counting. You
can always say how many symbols you’ve got. Whenever you add another one you
don’t already have, there’s one more, and whenever you subtract one, there’s one
less. Only shapes don’t work this way. Seeing takes over—shapes fuse and divide as
they’re combined. It’s what happens when embedding isn’t identity. If I have three
squares
and add one more