Shape

235 Trying It Out
The rule finds squares and inscribes smaller ones. In the first series, the rule is applied
under a different transformation each time it’s tried to pick out the initial square, or
the square inscribed most recently. But in the second series, equivalent transformations
with respect to the right side of the rule (the details are given later, in table 10)
are used in the first and second steps. The same square can be distinguished repeatedly
without producing new results.
I can modify the rule
with a point in its left side and a point in its right side
so that the rule has to apply to the most recently added square
Points and labeled points are really good for this, and provide the kind of logic and
control that’s useful in calculating.
Playing around with the rule
lets me introduce and use a telling asymmetry. If I tilt and stretch the smaller square
my rule applies only under rotations as it did before