Shape

265 Classifying Rules with Transformations
Of course, not every shape in the left side of a rule contains a pair of points
where maximal lines intersect. The empty shape is one of these shapes, and K is, too.
Moreover, there are shapes with parallel lines and shapes with lines that intersect at a
common point—for example, these
Nonetheless, for parallel lines there are singular cases where rules apply determinately.
Then lines are just the right length and are separated by just the right amount. This
complicates things a little without changing much. Indeterminate rules are easy to
identify and have many uses, even if ad hoc decisions are needed to apply them. Earlier,
I tried the rule
to get random shapes containing points. And I’ve used indeterminate rules to calculate
with axes, and to extend lines. I’ll show some other examples, too. Only now, there’s
more to say about rules and transformations. Surprisingly, the right side of a rule affects
the way the rule is applied. How does this work?
The rule
that inscribes a square in a square is determinate. The symmetry of the square in its left
side lets it apply under eight distinct transformations—four rotations and four reflections.
But for each transformation, the results are exactly the same. No matter how I
replace a square
I get the shape