Shape

245 Spatial Relations and Rules
Modifying rules according to the symmetry of the shapes they contain is a useful
ploy. More generally, though, I can vary the properties of the shapes in rules with
basic elements, labels, weights, and so on, to get the outcomes I like. It’s a lot like
composing—there’s symmetry, scale, balance, rhythm, color, etc. I’m still surprised at
all the things rules do as they apply—with transformations, parts, and adding and subtracting
in two easy formulas. That’s all it takes to see and do—just calculating with
shapes and rules.
Spatial Relations and Rules
There’s more to calculating with shapes than symmetrical patterns and fractals or combining
shapes and their boundaries. In fact, there are more generic schemas to define
rules that include the previous ones with transformations and the boundary operator.
And these new schemas are pedagogically effective—they make adding and subtracting
explicit in rules before they’re applied.
For any two shapes A and B, I can define addition rules in terms of the schema
x fi A þ B
and subtraction rules in terms of its inverse
A þ B fi x
where the variable x has either A or B as its value. (Evidently, addition rules—and likewise
subtraction rules—are the same whenever the rules A fi B and B fi A are.) The
shapes A and B define a spatial relation that determines how to combine shapes. They
show how conjugate parts are arranged after I add one, or if I want to take one away
and leave the other. But there’s a caveat. A spatial relation is an equivalence relation
for decompositions—two are the same when one is a transformation of the other.
This is a nice way to describe how shapes go together to define rules in schemas. In
fact, it provides a general model of calculating that includes Turing machines. Still,
the spatial relation isn’t preserved in the rules it defines. Descriptions have no lasting
value, only heuristic appeal. They’re something to use that helps to get started. Then
you’re on your own—you’re free to see anything you choose. There’s a lot more to
shapes once they fuse than there is when they’re kept apart in spatial relations. (That’s
calculating with sets in set grammars, not with shapes in shape grammars.)
Let’s see how my twin schemas work. The shape
gives some good examples when it’s divided into two squares to define a spatial relation.
This keeps to the schema x fi x þ tðxÞ and its inverse, but nothing is lost in the
simplification. There are still four distinct rules—two for addition