Shape

316 III Using It to Design
that I draw or point to in the usual way to show what they are. But no matter what rule
is defined, it applies exactly as my two formulas describe. Still, it may be worthwhile to
say this for schemas directly. The schema x fi y applies to the shape C whenever an
assignment g and a transformation t define a shape tðgðxÞÞ that’s part of C
tðgðxÞÞ a C
The result of this is another shape
ðC
tðgðxÞÞÞ þ tðgðyÞÞ
produced by replacing tðgðxÞÞ with the shape tðgðyÞÞ. In most of the examples that
follow, I’ll be using schemas. I’ll usually define them with one or two of the rules they
determine under specific assignments with a short verbal description of the predicate
involved, just as I’ve been doing. This avoids the need for a lot of technical details
and lets me present everything visually. We’ll just agree that predicates can be filled
in as necessary. And I promise not to do anything where the details are mysterious.
A useful way to explain my formulas for schemas is to notice that the composition
t g in the expressions tðgðxÞÞ and tðgðyÞÞ generalizes the transformations. What it
means for shapes to look alike can range very broadly and be decided in all sorts of different
ways. I tend to show examples that depend only on Euclidean transformations—
for example, shapes are alike if they’re copies of the square
either because they’re congruent
or because they’re geometrically similar