VW  CVM 1.1 
Two different homomorphisms from E to {1,1} can give rise to patterns with exactly the same
symmetries and antisymmetries. For example, with
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 twocenters; the
patterns seem to have the same symmetry. This
is because the composition
The thing that matters is that half of the twocenters 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
satisfies
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 P_{1} (from
E' to {1,1})
and P_{2} (from E to {1,1}) give rise to the same wallpaper type if, and only
if,
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.

