Shape

266 II Seeing How It Works
Why? The reason can be given in terms of Lagrange’s famous theorem for subgroups,
and provides another way to classify rules.
Let two transformations be equivalent with respect to a shape if they change it
identically. In this example
a clockwise rotation of 90 degrees about the point O
and a reflection across the axis I
are equivalent relative to the rectangle. Both the rotation and the reflection do exactly
the same thing—just look. And this gives a general result.
If a rule A fi B has the property that the symmetry group of the shape A is partitioned
into q classes with respect to the shape B, then the rule can be used in q distinct
ways. And if the symmetry group of A has n transformations, then q divides n without
remainder. The symmetry group of A has a subgroup containing n=q transformations
with cosets given by the classes defined by B. Conversely, there’s a rule for every subgroup
of the symmetry group of A that behaves as the subgroup describes. The distinct
uses of the rule A fi B show the symmetry properties of A, and these properties classify
every rule with A in its left side.
This is nicely illustrated with a square in the algebra U 12 for clockwise rotations
and for reflections named by these axes