Shape

82 I What Makes It Visual?
they provide trying to find what you want. Combinatorial play goes just so far. But
shapes fuse and divide as rules are tried. Sagacity makes a difference, along with learning
that pairs shapes in rules.)
Let’s agree that a line l is embedded in any other line lA if l and lA are identical
or—allowing more than embedding for points—l is a segment of lA
In general, a shape M is part of any shape S if the maximal lines of M are embedded
in maximal lines of S. (I was drawn to this neat relation by necessity. When I started
using a typewriter—I was eleven or so—I contrived an easy way to check my work
against an original or if I had to retype a page to correct mistakes without making
more. Proofreading was hard. It took too long, and mine was unreliable. So I would
embed one page in another one. I would tape the first to a window and move my
copy over it. This let me see what was on both pages at once. If things that were supposed
to be the same lined up, I knew my typing was OK. Creative designers use tracing
paper this way. Who would have guessed that it’s just using rules?) Whatever I can
see in S—anything I can trace—is one of its parts. This leads straight to calculating in
the algebras of shapes in part II, and is almost exactly what James has in mind when he
tells how reasoning works. For the rule M fi P and the shape S, if I can see M in S, then
I can replace it with another shape P by subtracting M from S and adding P. I’ll add by
drawing shapes together, so that their maximal lines fuse. And I’ll subtract by adding
shapes and then erasing one, so that segments of maximal lines can be removed. Remarkably,
the embedding relation implies nearly everything I need in order to do all
of this with axiomatic precision. I also need transformations of some kind to complete
the correspondence between M and P and S in the way I did for Evans’s kind of rules.
But I’ll skip the details for now because they’re not too hard to fill in, even if they have
a host of important consequences. It’s enough to know that transformations are there,
so that different shapes can be alike.
With lines and this embedding relation, Evans’s rule to define triangles is simply
the identity
I don’t have to say what a triangle is in terms of constituents that are already given. I
only have to draw it. It makes no sense at all to have the rule
Three lines fi Triangle