VW  CVM 1.1 
C(n,m) = [cos(nX+mY) + cos(mX(n+m)Y) + cos((n+m)X+nY)] / 3
and
S(n,m) = [sin(nX+mY) + sin(mX(n+m)Y) + sin((n+m)X+nY)] / 3
as we did there. These are the building blocks of every function with threefold invariant symmetry.
In the table, we use these shorthands to list the terms, or combinations of terms, that achieve the desired pattern. Recall that parity considerations don't apply to this lattice type. Another thing that makes this table mercifully short is that R_{3} cannot be negating, as its order is odd.
The hexagonal lattice cell 
We need notation for only two mirrors: M_{x} denotes reflection about the xaxis, while M_{y} is reflection about the yaxis. When a wallpaper pattern has one of these and the rotational symmetry, it necessarily picks up many other mirrors.
In the naming of the types, the mirror M_{y} is given precedence as the primary, or default mirror. The other mirror gets the designation 1m. Thus, p3m' has M_{y} as an antisymmetry, while in p31m', M_{x} is negating. To gain more experience with this distinction, visit the page that tests your ability to tell p31m from p3m1 .

