Shape

214 II Seeing How It Works
This has important implications for the way rules work when I calculate. A rule
applies under a transformation that makes one shape part of another shape. Everything
is fine if the rule is given in the algebra U 00 . It’s always determinate—if it applies,
it does so under a finite number of transformations. In fact, the identity transformation
is the only one. But in the other algebras U ii , a rule is always indeterminate—it
applies to every nonempty shape in infinitely many ways, as I’ve already described.
This makes it hard, if not impossible, to control how the rule is used. It can be applied
haphazardly to change shapes anywhere there’s an embedded basic element—and
that’s everywhere. Indeterminate rules are a burden. They appear to be purposeless
when they can be applied so freely. Still, indeterminate rules have some important
uses that I’ll get to later. They let me do things that seem beyond the reach of rules.
I like to think that shapes made up of lines in the plane—a few pencil strokes on
a scrap of paper—are all that’s ever required to study shapes and how to calculate with
them. This is how I started playing around with shapes, and it’s hard to stop because
it’s easy and fun, and it works so well. Nonetheless, there are other reasons for shapes
with lines that aren’t personal. In particular, they make sense historically and practically.
On the one hand, for example, lines are key in Leon Battista Alberti’s famous account
of architecture—
Let us therefore begin thus: the whole matter of building is composed of lineaments and structure.
All the intent and purpose of lineaments lies in finding the correct, infallible way of joining and
fitting together those lines and angles which define and enclose the surfaces of the building. It
is the function and duty of lineaments, then, to prescribe an appropriate place, exact numbers,
a proper scale, and a graceful order for whole buildings and for each of their constituent parts,
so that the whole form and appearance of the building may depend on the lineaments alone.
Nor do lineaments have anything to do with material, but they are of such a nature that we
may recognize the same lineaments in several different buildings that share one and the same
form, that is, when the parts, as well as the siting and order, correspond with one another in their
every line and angle. It is quite possible to project whole forms in the mind without any recourse
to the material, by designating and determining a fixed orientation and conjunction of the various
lines and angles. Since that is the case, let lineaments be the precise and correct outline, conceived
in the mind, made up of lines and angles, and perfected in the learned intellect and
imagination.
And it’s the same today—try to imagine designing without lines. They let you find out
what to do, and record the results of seeing and doing. Moreover, Alberti’s idea of lines
and angles comes up again a little later on: it’s implicit in my use of spatial relations
to define rules for design. Then, on the other hand, the technology of lines is uncomplicated
and accessible to adults and children alike with minimal training. Pencil and
paper are enough. Yet an algebraic reason supersedes anything I can find either in my
personal experience, in history, or in technology. There are compelling formal arguments
for drawing with lines.
Shapes containing lines arranged in the plane suit my interests because their
algebra U 12