# 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* below.

The group *pgg* is generated by the half-turn
*R*_{C} and the horizontal glide
*B* labeled in the diagram. You can't have these two symmetries
without having *B*_{v} as well. This is because
*B*_{v} is the result of performing *B* and then
*R*_{C} in succession. That is, *B*_{v} =
*R*_{C} B, the composition of the two isometries on the
right .

A wallpaper group must contain translations in two
independent directions ; why aren't any translations listed among the generators?
It's because the they can be expressed as *B*^{2} and
*B*_{v}^{2}. Hence all the isometries in the group
*pgg* can be expressed as compositions of the two transformations
*B* and *R*_{C}.

*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

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