VW - CVM 1.1

Eigenfunctions with 3-Fold Rotations

A wallpaper function with group p3.
Armed with the lattice coordinates of the previous section, it is easy to find the eigenfunctions of the Laplacian that are periodic with respect to a special lattice we call the hexagonal lattice. We call it this because it is the only lattice that permits rotations through 120 degrees. The hexagons are formed from three lattice cells together.

The translations in the hexagonal lattice are (up to a scalar multiple):

 Ö __ t1 = 2 p (1, 0)   and   t2 = 2 p (-1/2, 3 /2)

The lattice coordinates are:

 Ö __ Ö __ X = x + y / 3 and   Y = 2 y / 3

In terms of these coordinates, cos(nX+mY) and sin(nX+mY) fill out the set of eigenfunctions periodic with respect to the hexagonal lattice.

It remains to investigate invariance under other desired symmetries. In this case, these would be the various symmetries in the five groups whose translational subgroup amounts to the hexagonal lattice. Since all five contain a rotation of 120 degrees, we begin with this.

Call R3 the rotation through 120 degrees counterclockwise about the origin. In terms of the lattice coordinates, R3(X,Y) = (-Y,X-Y). If f(x) is, for instance, cos(nX+mY), then f(R3 x) = cos(mY-(n+m)X). Thus, cos(nX+mY) is not rotationally invariant; but, surprisingly (or not), the rotated function is also an eigenfunction with the same eigenvalue, namely (2/3)(n2 + m2 + (n+m)2).

To construct a single rotationally symmetric eigenfunction, we bundle together three eigenfunctions with the same eigenvalue, and define

C(n,m) = [cos(nX+mY) + cos(mX-(n+m)Y) + cos(-(n+m)X+nY)]/3.

Here the division by 3 is mostly for aesthetic purposes, to remind us that we are actually averaging a function over the action of a group of order three . The function S(n,m) is defined similarly.

Again, the reader may now like to take a break to experiment with designing wallpaper patterns with 3-fold symmetry . Any combination of multiples of C(n,m) and S(n,m) will result in a wallpaper function with group at least p3. Remember in your experimentation that n and m need not be positive.

The general results for groups with hexagonal lattice will be postponed until after the discussion of negating isometries , but perhaps the reader has noticed by experiment what happens if only terms like C(n,m) are included. This is the recipe for wallpaper functions with group p6.

Communications in Visual Mathematics, vol 1, no 1, August 1998.