VW - CVM 1.1

# Type Equivalence

Two different homomorphisms from E to {1,-1} can give rise to patterns with exactly the same symmetries and antisymmetries. For example, with E = p2 = {t1, t2, R}, where t1 and t2 are independent translations and R is a half-turn, consider the two homomorphisms

P1(t1) = -1, P1(t2) = 1, P1(R) = 1
and
P2(t1) = -1, P2(t2) = 1, P2(R)= -1.

These are different homomorphisms, but every function whose symmetries arise from either one can be seen to have an alternating grid of positive and negating two-centers; the patterns seem to have the same symmetry. This is because the composition t1 R is a half-turn, which must be positive if both t1 and R are negating.

The thing that matters is that half of the two-centers are negating and the others are positive. Finding an isomorphism of E that carries one homomorphism to the other solves the apparent problem.

The isomorphism

i(tj) = tj, i(R) = t1 R

satisfies P1(i(k)) = P2(k) for all k in E.

We need to be one step more precise. Many of these infinite groups can be isomorphic to subgroups of themselves. The isomorphism playing the role of i in the formula above will not literally go from a single group of isometries to itself.

Thus we say that if E and E' are isomorphic to the same wallpaper group, and hence to each other via the isomorphism i, then the homomorphisms P1 (from E' to {1,-1}) and P2 (from E to {1,-1}) give rise to the same wallpaper type if, and only if, P1 o i = P2.

A wallpaper type is an equivalence class of homomorphisms under this sense of equivalence. A function has a given wallpaper type if there is a homomorphism in that equivalence class for which the function satisfies the equation above.

Our next task is to count these equivalence classes.

Communications in Visual Mathematics, vol 1, no 1, August 1998.