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Recipes for Negation

Recall that there are two groups associated with a wallpaper pattern: The group G of symmetries of the pattern, and the extended group of symmetries, E, that includes the anti-symmetries as well [Back].

When we start to work more algebraically, the extended group E will take primary consideration. Here, however, we think in terms of extending the symmetries of a function by starting with the class of functions invariant under G and finding special classes with anti-symmetries in E.

For each of the lattice types, we use the technique of lattice waves [Back] to identify the building blocks of functions invariant with respect to that lattice. Then we list the possible pattern types for each lattice, along with a recipe for constructing that type.

The technique of averaging over a group [Back] is often the key to a given recipe. Parity restrictions on the frequencies are derived from simple trigonometric identities.

Each section begins by naming symmetries relevant to that lattice type. We maintain consistent notation, so that R always is some kind of rotation, M a reflection, L a glide reflection, with t, v, h and d being various kinds of translations.

The recipes can be viewed in any order, although the rhombic [Link] lattice recipes depend on the ones for the rectangular [Link] lattice. For maximal symmetry in your wallpaper, use the square cell [Link] or hexagonal [Link] recipes. Try out your own wallpaper functions in the interactive wallpaper-design laboratory [Skip].

[Next] The General Lattice
[Next] The Rectangular Lattice
[Next] The Rhombic Lattice
[Next] The Square Lattice
[Next] The Hexagonal Lattice
[Skip] Interactive Wallpaper-Design Lab

[Next] The General Lattice
[Skip] Telling the Types Apart
[Up] Negating Isometries
[Prev] Classification and Naming

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 9:27:23 AM
Comments to: CVM@maa.org