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]](../../../buttons/image.gif)
below.
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]](../../../buttons/link.gif)
.
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 B2 and
Bv2. Hence all the isometries in the group
pgg can be expressed as compositions of the two transformations
B and RC.
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
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