Shape

280 II Seeing How It Works
Indeterminate rules are really useful, but this trick alone doesn’t work for quadrilaterals
because they needn’t be geometrically similar. That’s why squares worked. With quadrilaterals,
I still have to define an indefinite number of rules to get the shapes I want. I
can avoid this embarrassment easily enough if I use indeterminate rules—including
my rule to move a point on a line—in another way. But then the intuitive idea of
inscribing quadrilaterals in quadrilaterals in a recursive process is nearly lost. I have to
do almost everything line by line. I want to be able to define rules as naturally as I can,
so that I can see how they work without having to think about it. Schemas make this
possible—they describe shapes in rules without getting in the way when I calculate.
Let’s rehearse the easy technicalities once more. A schema
x fi y
is a pair of variables x and y that take shapes—or whatever there is in one of my
algebras—as values. These are given in an assignment g that satisfies a given predicate.
Whenever g is used, a rule
gðxÞ fi gðyÞ
is defined. The rule applies in the usual way. In effect, this allows for shapes and their
relations to vary within rules, and extends the transformations under which rules apply.
It provides a nice way to express many intuitive ideas about shapes and how to
change them as calculating goes on. It lets me do what I want with quadrilaterals in a
natural way.
(I’ve shown, in outline, that there’s an algorithm to find the transformations
under which a rule applies to a shape. But is this also the case for assignments and
schemas? What kinds of predicates allow for this, and what kinds of predicates don’t?
These are still open questions. Their answers are crucial to what I hope can be accomplished
calculating.)
This