VW  CVM 1.1 

A wallpaper function with group p3. 
The translations in the hexagonal lattice are (up to a scalar multiple):
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t_{1} = 2 p (1, 0) and t_{2} = 2 p (1/2,  3  /2) 
The lattice coordinates are:
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X = x + y /  3  and Y = 2 y /  3 
In terms of these coordinates,
It remains to investigate invariance under other desired symmetries. In this case, these would be the various symmetries in the five groups whose translational subgroup amounts to the hexagonal lattice. Since all five contain a rotation of 120 degrees, we begin with this.
Call R_{3} the rotation through 120 degrees counterclockwise
about the origin. In terms of the lattice coordinates,
To construct a single rotationally symmetric eigenfunction, we bundle together three eigenfunctions with the same eigenvalue, and define
Here the division by 3 is mostly for aesthetic purposes, to remind us that we are actually averaging a function over the action of a group of order three . The function S(n,m) is defined similarly.
Again, the reader may now like to take a break to experiment with designing wallpaper patterns with 3fold symmetry . Any combination of multiples of C(n,m) and S(n,m) will result in a wallpaper function with group at least p3. Remember in your experimentation that n and m need not be positive.
The general results for groups with hexagonal lattice will be postponed until after the discussion of negating isometries , but perhaps the reader has noticed by experiment what happens if only terms like C(n,m) are included. This is the recipe for wallpaper functions with group p6.

