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Combining Negating Isometries

If G is any of the wallpaper groups, we have said that a function f is G-invariant if f(g x) = f(x) for all x in R2 and g in G [Back]. We modify this slightly to insist that this equation should hold for all x only when g belongs to G. Thus we reserve the language of G-invariance for situations where f is not invariant under any group larger than G.

In addition to showing how to construct functions invariant with respect to every wallpaper group G [Back], we have also given recipes for creating functions with given anti-symmetry [Back]. Recall that if k is an isometry of the Euclidean plane and if f(k x) = -f(x) for all x in R2, then we say that f is k-negating and call k a negating symmetry of f [Back].

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 [Back], 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 negating symmetry:

f(x) = f(k k-1x) = - f(k-1x),
(since k is negating) so
f(k-1x) = - f(x).

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| given by |f|(x) = |f(x)| is both G-invariant and k-invariant and therefore invariant under the group generated by G and k. We call this group the extended invariance group of f, and usually refer to it as E. Since E is a group of isometries containing two independent translations, E must be one of the 17 wallpaper groups, a fact that further simplifies our classification.

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 [Link]. The classification result proceeds with an exhaustive list of all possibilities, identifying any two that are equivalent [Skip].

[Next] Group Homomorphisms
[Skip] Why there are 63 Types
[Up] The Algebra of Wallpapers
[Prev] Informal Discussion of Anti-Symmetries

Communications in Visual Mathematics, vol 1, no 1, August 1998.
Copyright © 1998, The Mathematical Association of America. All rights reserved.
Created: 08 Jul 1998 --- Last modified: Sep 29, 1998 1:07:42 PM
Comments to: CVM@maa.org