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Informal Discussion of Symmetries

Each of the wallpaper groups is generated by a short list of isometries. Even though a group may contain many isometries, only a small number are needed to produce them all, since if a picture has certain symmetries, then it may be forced to have others.

For example, consider the cell diagram for the group pgg [Image] below.

[cell diagram for pgg]
The pgg cell.

The group pgg is generated by the half-turn RC and the horizontal glide B labeled in the diagram. You can't have these two symmetries without having Bv as well. This is because Bv is the result of performing B and then RC in succession. That is, Bv = RC B, the composition of the two isometries on the right [Link].

A wallpaper group must contain translations in two independent directions [Link]; why aren't any translations listed among the generators? It's because the they can be expressed as B2 and Bv2. Hence all the isometries in the group pgg can be expressed as compositions of the two transformations B and RC.


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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 9:26:23 AM
Comments to: CVM@maa.org