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The General Lattice

Recall that we use lattice coordinates, X and Y, when we describe lattice waves. These refer to the skewed coordinates along the directions of translation [Back].

The most general function periodic with respect to any given lattice is a sum of terms of the form:

anm cos(nX) cos(mY) + bnmcos(nX) sin(mY) + cnmsin(nX) cos(mY) + dnmsin(nX) sin(mY)

Superimposing these will give a periodic function. How can additional symmetries be achieved?

We define several useful isometries that are important here: h is a horizontal half-translation, d a diagonal half-translation, v a vertical half-translation and R is a half-turn about the origin.

[isometries in primitive cell]
The general lattice cell

Each of these can be used as a negating symmetry to create a function invariant under p1 but with additional anti-symmetries. Some turn out to be algebraically equivalent. For instance, forcing h to be negating is algebraically the same as making d negating [Skip], so these are not traditionally thought of as being different.

Furthermore, we would not want to create functions with h, v, or d as actual symmetries, because these would then be symmetric in a finer lattice (one with a smaller cell). If the goal were to reduce the lattice, why not just start with a smaller one to begin with?

Thus there is only one new pattern type that can be formed from the group p1. For similar reasons, there is only one type that grows out of p2, the one where R is negative.

Table: Symmetry types in general lattice cell
type G E Recipe for this type and remarks
p1 [Image] p1 p1 No additional symmetries;
use general parity, terms of all types
p'b1 [Image] p1 p1 n is odd;
new cell is half of old
p2' [Image] p1 p2 only sin-cos and cos-sin terms are used;
R negative
p2 [Image] p2 p2 only cos-cos and sin-sin terms appear
p'b2 [Image] p2 p2 p2 recipe, n + m is odd; negative half-turns;
new cell half of old

[Next] The Rectangular Lattice
[Skip] Experiment interactively with Wallpaper Design
[Up] Recipes for Negation
[UpUp] Negating Isometries

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: Sep 30, 2003 9:46:38 AM
Comments to: CVM@maa.org