VW - CVM 1.1

The Hexagonal Lattice

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 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: M_x denotes reflection about the x-axis, while M_y 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 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 .

Table: Symmetry types in cells of hex lattice

type	G	E	Recipe for this type and remarks
p3	p3	p3	C and S terms appear; R3 positive
p6'	p3	p6	only S terms appear; R3 positive, R6 negative
p31m'	p3	p31m	C(n,m) - C(n,-(n+m)) and S(n,m) - S(n,-(n+m)) R3 positive, Mx negative
p3m'	p3	p3m1	C(n,m) - C(-n,(n+m)) and S(n,m) - S(-n,(n+m)) R3 positive, My negative
p31m	p31m	p31m	C(n,m) + C(n,-(n+m)) and S(n,m) + S(n,-(n+m)) R3, Mx positive
p6'm'm	p31m	p6m	S(n,m) + S(n,-(n+m)) R3, Mx positive, R6 negative
p3m1	p3m1	p3m1	C(n,m) + C(-n,(n+m)) and S(n,m) + S(-n,(n+m)) R3, My positive
p6'mm'	p3m1	p6m	S(n,m) + S(-n,(n+m)) R3, My positive, R6 negative
p6	p6	p6	only C terms appear; R6 positive
p6m'm'	p6	p6m	C(n,m) - C(n,-(n+m)) R6 positive, Mx, My negative
p6m	p6m	p6m	C(n,m) + C(n,-(n+m)) R6, Mx, My positive

Telling the Types Apart Experiment interactively with Wallpaper Design Recipes for Negation Negating Isometries The Square Lattice