|VW - CVM 1.1|
C(n,m) = [cos(nX+mY) + cos(mX-(n+m)Y) + cos(-(n+m)X+nY)] / 3
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 three-fold 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 R3 cannot be negating, as its order is odd.
|The hexagonal lattice cell|
We need notation for only two mirrors: Mx denotes reflection about the x-axis, while My is reflection about the y-axis. 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 My is given precedence as the primary, or default mirror. The other mirror gets the designation 1m. Thus, p3m' has My as an antisymmetry, while in p31m', Mx is negating. To gain more experience with this distinction, visit the page that tests your ability to tell p31m from p3m1 .
|type||G||E||Recipe for this type and remarks|
|p3||p3||p3||C and S terms appear;
|p6'||p3||p6||only S terms appear;
R3 positive, R6 negative
S(n,m) - S(n,-(n+m))
R3 positive, Mx negative
S(n,m) - S(-n,(n+m))
R3 positive, My negative
S(n,m) + S(n,-(n+m))
R3, Mx positive
R3, Mx positive, R6 negative
S(n,m) + S(-n,(n+m))
R3, My positive
R3, My positive, R6 negative
|p6||p6||p6||only C terms appear;
R6 positive, Mx, My negative
R6, Mx, My positive