Shape

267 Classifying Rules with Transformations
Table 10
Classification of Rules in terms of Lagrange’s Theorem
Rule Subgroup Number of cosets
0, 90, 180, 270, I, II, III, IV 1
0, 90, 180, 270 2
0, 180, I, III 2
0, 180 4
0, I 4
0 8
Rules behave in at least six different ways according to the six subgroups of the symmetry
group of the square given in table 10. The symmetry group of the square has other
subgroups that are isomorphic to these. One way to see this is in terms of equivalencies
among the four axes of the square. There may be ten subgroups (the axes are all different),
possibly eight (the horizontal axis and the vertical one are the same, and so are
the diagonal axes), or the six in table 10 (the axes are all the same).
The rule
has the symmetry properties of the first rule I show in table 10, and therefore it acts
like an identity with only one distinct use. The eight transformations in the symmetry
group of the square in the left side of the rule are equivalent with respect to the shape
in its right side. Try it yourself to make sure. And its also worthwhile to try this for the
rule