Shape

284 II Seeing How It Works
applies in the usual way to define a series of shapes
C Þ ðC
tðAÞÞ þ tðBÞ Þ Þ D
that starts with the shape C and finishes with the shape D. When the rule A fi B is
tried, a transformation t of A is taken away and the same transformation t of B is added
back. But this is only the algebraic mechanism for applying the rule. What’s interesting
about the series is what I can say about it as a continuous process, especially when
parts are resolved in surprising ways. For example, I can calculate so
to turn a pair of touching chevrons into a pair of squares and back again, so that the
pair of chevrons is reflected in between. The rule
that translates a chevron, and the rule
that translates a square, are used for this purpose. It looks hard—chevrons and squares
stay the same when they’re moved—but it works. What parts do I need to account for
the change, so that there’s no break or inconsistency? How are the parts the rules pick
out related? When do chevrons become squares, and vice versa? What does it mean for
this process to be continuous?
I’m going to build on a configurational (combinatorial) idea that’s part of normal
experience to answer these questions. It’s the very idea I’ve been against as a way of
handling shapes to calculate. Still, the idea works when it comes to describing what
happens as rules are applied. Keep this foremost in mind—there’s a huge difference between
describing shapes and calculating with them. The latter doesn’t depend on the