VW  CVM 1.1 
Recall that we use lattice coordinates, X and Y, when we describe lattice waves. These refer to the skewed coordinates along the directions of translation .
The most general function periodic with respect to any given lattice is a sum of terms of the form:
a_{nm} cos(nX) cos(mY)+ b_{nm}cos(nX) sin(mY) + c_{nm}sin(nX) cos(mY) + d_{nm}sin(nX) sin(mY)
Superimposing these will give a periodic function. How can additional symmetries be achieved?
We define several useful isometries that are important here: h is a horizontal halftranslation, d a diagonal halftranslation, v a vertical halftranslation and R is a halfturn about the origin.
The general lattice cell 
Each of these can be used as a negating symmetry to create a function invariant under p1 but with additional antisymmetries. Some turn out to be algebraically equivalent. For instance, forcing h to be negating is algebraically the same as making d negating , so these are not traditionally thought of as being different.
Furthermore, we would not want to create functions with h, v, or d as actual symmetries, because these would then be symmetric in a finer lattice (one with a smaller cell). If the goal were to reduce the lattice, why not just start with a smaller one to begin with?
Thus there is only one new pattern type that can be formed from the group p1. For similar reasons, there is only one type that grows out of p2, the one where R is negative.

