Shape

312 III Using It to Design
(3) design is drawing
and I like to think that the corollary
(4) design is calculating when you don’t know what you’re going to see and do
next
follows, as well. This completes the transition from the mostly verbal discussion in the
introduction through the unfolding visual argument in parts I and II to the mostly
visual presentation now. It’s going from calculating by counting to calculating by
seeing with the shifting (subjective) viewpoints this implies.
Statement 4 pretty much sums up how I want to approach design with shapes
and rules. I’ve been trying to show that calculating with shapes and rules is inclusive
enough to deal with anything that might come up in design when it’s done visually.
That’s the reason embedding and transformations are needed to apply rules. First, the
embedding relation—what you see is there if you can trace it out, no matter what has
gone on before. Second, the transformations—what you see is like given examples
of what to look for, maybe things that were noticed in the past and used. And
together—embedding and transformations interact as rules are tried to calculate with
shapes.
The details aren’t too far from drawing when you pay attention to what you’re
seeing and doing. The trick is to slow this process down a little bit to describe what’s
going on in a mechanical way. It’s easy to say and, more important, easy to see. The
beginning isn’t much—start with any shape
C
Intuitively, there are lines on paper, but C can also include points, planes, solids,
labels, weights, etc. Shapes can have any dimension you please, both in terms of basic
elements and how they’re combined. In drawings, lines are one dimensional and
located on planes that are two. It all depends on the algebra in which you calculate.
Then a rule
A fi B
that shows two shapes A and B applies to C whenever
tðAÞ a C
that is to say, there’s a transformation t such that the shape tðAÞ is part of (embedded
in) C. The rule A fi B is an example of what I want to see and do, and tðAÞ is some part
of C that catches my eye because it looks like A. The result of this is another shape
ðC
tðAÞÞ þ tðBÞ
The shape tðAÞ is taken away from C, and the shape tðBÞ that looks like B is the new
part that’s added back. This is the same drawing with pencil and paper, although
replacing parts with new ones isn’t set out explicitly. The beauty of the process—with