Shape

126 I What Makes It Visual?
What Schemas Show
How descriptions of shapes change while I calculate is highlighted in another way
when I use schemas to define rules. I’ve shown one schema already
x fi tðxÞ
without giving a general definition. That’s because the idea is nearly transparent. But
sometimes, a little formality is useful.
A rule schema
x fi y
is a pair of variables x and y that take shapes as values. These are given in an assignment
g that satisfies a predicate. Whenever g is used, a rule
gðxÞ fi gðyÞ
is defined. The rule then applies as usual, as if it were given explicitly from the start.
This works to express a host of intuitive ideas about shapes and how to change them.
Schemas can be very general. In fact, I can give one for all rules. I only have to
say that x and y are shapes. Or schemas can be very restricted. Just a single rule is
defined if x and y are constants—maybe the triangle
and the square
to give the rule
But usually schemas are more in the middle, so that shapes have useful descriptions.
How to decide this and whether or not it makes a difference are hard to say. If you’re
interested in explaining everything of a certain kind once and for all, then it’s nice to
be clear about what schemas do. But if you take a historical view that looks at particular
ways of calculating and how they go on, then there are other ways—topologies, etc.—
to describe what’s happening relative to rules that needn’t be described themselves,
and how these rules are used. Most of what I was told in school seems to support the
first option. It isn’t easy to ignore. And most work on shape grammars—including my