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The Hexagonal Lattice

We have already done most of the work for the hexagonal lattice in an example in the first section [Back], and there is little to add to it now. We use the abbreviations

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.

[isometries in hex lattice]
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 [Skip].

Table: Symmetry types in cells of hex lattice
type G E Recipe for this type and remarks
p3 [Image] p3 p3 C and S terms appear;
R3 positive
p6' [Image] p3 p6 only S terms appear;
R3 positive, R6 negative
p31m' [Image] p3 p31m C(n,m) - C(n,-(n+m)) and
S(n,m) - S(n,-(n+m))
R3 positive, Mx negative
p3m' [Image] p3 p3m1 C(n,m) - C(-n,(n+m)) and
S(n,m) - S(-n,(n+m))
R3 positive, My negative
p31m [Image] p31m p31m C(n,m) + C(n,-(n+m)) and
S(n,m) + S(n,-(n+m))
R3, Mx positive
p6'm'm [Image] p31m p6m S(n,m) + S(n,-(n+m))
R3, Mx positive, R6 negative
p3m1 [Image] p3m1 p3m1 C(n,m) + C(-n,(n+m)) and
S(n,m) + S(-n,(n+m))
R3, My positive
p6'mm' [Image] p3m1 p6m S(n,m) + S(-n,(n+m))
R3, My positive, R6 negative
p6 [Image] p6 p6 only C terms appear;
R6 positive
p6m'm' [Image] p6 p6m C(n,m) - C(n,-(n+m))
R6 positive, Mx, My negative
p6m [Image] p6m p6m C(n,m) + C(n,-(n+m))
R6, Mx, My positive

[Next] Telling the Types Apart
[Skip] Experiment interactively with Wallpaper Design
[Up] Recipes for Negation
[UpUp] Negating Isometries
[Prev] The Square Lattice

Communications in Visual Mathematics, vol 1, no 1, August 1998.
Copyright © 1998, The Mathematical Association of America. All rights reserved.
Created: 08 Jul 1998 --- Last modified: 18 Aug 1998 23:59:59
Comments to: CVM@maa.org