|VW - CVM 1.1|
As a preview of the classification of patterns with anti-symmetries, which we carry out in the next section , 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
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.)
shown at the right. Here, G is pmm and E is cmm. The two names for this pattern type are p'cmm and cmm5.
The negating diagonal half-
Recipes for Negation The Algebra of Wallpapers Negating Isometries 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