Shape

237 Trying It Out
with a point on a symmetry axis of each of the squares is also enough
Notice, however, that the rule
doesn’t work—the centers of both squares are the same and so don’t distinguish them.
Of course, there are other places for points with different results, depending on the
squares and how they’re related, and the symmetries involved. Try the left side of
the first rule and the right side of the second one to get what I’ve done—at least in a
special case. And all of this can be extended with labels, for example, for rhythmic
patterns of rotations and reflections. Things get interesting fast.
I can also describe the rule
in a nice way, and its successors with points, too. A transformation of the square in the
left side of the rule is part of the shape in its right side. The rule is recursive—I can
apply it again in new places. But more generally for any shape x, I can define rules in
terms of the schema
x fi x þ tðxÞ
This gives parts of symmetrical patterns when the transformations t are the generators
of a symmetry group. I can calculate with scalene right triangles using the rule
to get a shape with cyclic symmetry