VW - CVM 1.1

Classification and Naming

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 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.)

For example, consider the function

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.

Recipes for Negation The Algebra of Wallpapers Negating Isometries A Motivating Example