Shape

244 II Seeing How It Works
to limit the possibilities and thus guarantee the results I want. First, this makes lines
behave like units—a point and a line together
determine a perpendicular
and thus a constant scaling factor. And then, the point and the line break the fourfold
symmetry of the line
so that rules apply in definite ways. (Turn this around and ask what a line does for a
point that’s invariant under indefinitely many transformations—rotations, reflections,
and changes in scale. Points and lines interact with one another reciprocally when
rules are tried. There’s nothing to tell them apart that implies that points constrain
lines or vice versa. Geometrically at least, basic elements are the same.) But apply the
rule
and see what happens
or other rules like it. (I count two hundred and fifty-six rules that can each be used for
something different, and in combination to produce more. There are four distinct left
sides and sixty-four distinct right sides, and no transformation matches left to left and
right to right at the same time. But this is profligate—a few rules and other tricks do
similar kinds of things. I can begin to think about this by moving points around with
the pair of rules
to simulate what my other rules do. My new rules are defined in terms of generators for
the symmetry group of the line. And the idea is good for other shapes with different
symmetries.)