|VW - CVM 1.1|
If G is any of the wallpaper groups, we have said
that a function f is G-invariant if
In addition to showing how to construct functions invariant with respect to
every wallpaper group G , we have also given recipes for creating functions with given
anti-symmetry . Recall that if k is an isometry of the Euclidean
plane and if
At first glance, one might assume that a G-invariant function could have a large collection of unrelated negating symmetries, complicating the proposed classification. For example, in our motivating example , there are several parallel negating mirrors, numerous negating half-turns, but also a negating diagonal half-translation. Need we consider patterns with some but not all of these negating symmetries? It turns out that in classifying patterns, we need consider the case with only a single negating symmetry.
Proposition: Suppose f is G-invariant and k-negating. Then every negating symmetry of f has the form kg, where g is in G.
Proof: First observe that the inverse of k is a
Similarly, it is easy to show that the product of two negating symmetries is a positive symmetry of f. Thus if k* is any negating symmetry, composing k* with k-1 must result in a positive symmetry, namely an element g of G. Transposing shows that k* is kg.
Further observations simplify the picture. If f is
G-invariant and k-negating, then the function |f|
The situation becomes elegant when stated in terms of algebra:
Theorem: G is normal in E and the quotient E/G is cyclic of order 2, unless f has no negating symmetries in which case E and G are the same.
Thus, in classifying wallpaper functions with a given invariance group in mind, one need only look for a single negating symmetry to generate whatever others the function may have.
The algebraic details follow . The classification result proceeds with an exhaustive list of all possibilities, identifying any two that are equivalent .