Shape

303 Background
seeing isn’t lost in my algorithms for rules. Still, I like to stress my visual intuitions
more than algorithms. There’s an imbalance today—rote counting passes for thinking
far too often, with preposterous results. Everything is numbers and symbols without
shapes. Seeing doesn’t need to be explained, it needs to be used.
I can always see things in new ways when I calculate with shapes made up of
lines, planes, and solids. I can draw to figure things out, and to see what I can do.
This is my kind of design. But shapes with points—and likewise decompositions or
representations in computer models—limit what I can see and how I can change my
mind. A census of points doesn’t offset the loss. The last thing I want to do while I’m
calculating or thinking is to count. Showing how I calculate to see—that’s what shapes
and rules do—shows that calculating and thinking needn’t be combinatorial. There’s
an alternative—I can always see something new. Counting is a bad habit that’s hard
to break, so count me out.
Background
James Gips and I published the original idea for shapes and rules in 1972. 1 That
paper also describes the weights—shaded areas—in the five decompositions for the
shape
on page 290. However, instead of algebras like W 22 in which everything takes care of
itself, we had an algorithm for ‘‘levels’’ (overlapping areas) and ad hoc ‘‘painting rules’’
given in Boolean expressions. Many of the formal devices I use go back to my book
Pictorial and Formal Aspects of Shape and Shape Grammars. 2 There, I cover embedding—
including parts (‘‘subshapes’’), representing shapes with maximal elements, and reduction
rules—rules and calculating in the algebra U 12 , calculating in parallel in direct
products of this algebra, algorithms for computer implementations of shapes and rules,
and equivalence with Turing machines. In fact, it’s where I first used the beginning
entries in Paul Klee’s Pedagogical Sketchbook—I show them on pages 251–252—to link
visual reasoning and calculating. 3 There’s substantially more, too, that I haven’t
described here on sets (‘‘languages’’) of shapes and how they’re closed with respect to
various operations on shapes and sets.
I introduced spatial relations and the schema
x fi A þ B
in an early paper—‘‘Two Exercises in Formal Composition.’’ 4 The problem of defining
styles from scratch (design synthesis in exercise 1) and from known examples (analysis
using identities in exercise 2) is framed and discussed there. The first example of rules
that generate designs in a given style—Chinese ice-ray lattices—was described in my