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Classification and Naming

[example of negating wallpaper]

As a preview of the classification of patterns with anti-symmetries, which we carry out in the next section [Skip], imagine the image above drained of colors or shadings, with only the shapes present, as if in a coloring book waiting to be filled in. That pattern is invariant under the group cmm.

One name for that pattern-type emphasizes this fact. The notation of Grunbaum [Gr], calls this a cmm[2]3 motif, meaning that it is third in a list of five possible two-colorings of cmm patterns. Shubnikov's "rational" notation [Sh] calls this a p'cmg motif, showing that pmg symmetry is present but enhanced by a negating translation toward the center of the cell.

Whichever nomenclature one uses, the main thing to understand before going on is this: in patterns with negating symmetries, one must keep track of the group of symmetries, along with a larger group that includes the anti-symmetries as well. We use Shubnikov's notation here, as it indicates this pair of groups in a natural way.

This leads to the following notation. Given a function f, let us call G the largest group of isometries under which f is invariant, while E will be the largest group of isometries of the function |f|. (E is a mnemonic for the extended symmetry group of G; it includes all symmetries and anti-symmetries of f.)

[ negating pc'mm wallpaper]
For example, consider the function

cos(2X) cos(3Y) + cos(0X) cos(1Y) + cos(1X) cos(2Y)

shown at the right. Here, G is pmm and E is cmm. The two names for this pattern type are p'cmm and cmm[2]5.

The negating diagonal half-translation occurs because the sum of the frequencies is odd in every term. In constructing this type we refer to extending from pmm symmetry to include certain anti-symmetries from cmm.

[Next] Recipes for Negation
[Skip] The Algebra of Wallpapers
[Up] Negating Isometries
[Prev] A Motivating Example

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 9:27:21 AM
Comments to: CVM@maa.org