Shape

325 What the Thinking Eye Sees
is defined in the schema x fi x þ tðxÞ, and it evidently produces a pinwheel that’s cyclically
symmetric
when calculating goes like this
Not a bad design, easy to get, and a little something like Klee’s palm-leaf umbrella.
Nonetheless, everything seems far too regular in the way you’d expect when recursively
applying a single rule that adds two congruent right triangles hypotenuse to
side. But this is readily fixed in the new schema
x fi x þ xA
where x and xA are both triangles, and not always similar ones x and tðxÞ. (The schema
is another version of the schema x fi A þ B for spatial relations. I showed how to use it
for arbitrary shapes A and B, along with its inverse A þ B fi x, in several examples in
part II.) The rule
is defined in my new schema, and so is the alternative rule
In fact, I can show how all of the triangles in the palm-leaf umbrella are related—no
surprise. This sort of parametric variation in rules is already allowed in my schema for
quadrilaterals that includes the rule