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Informal Discussion of Anti-Symmetries

[cell diagram for pgg]
The pgg cell.

Recall that the symmetries for any wallpaper group can all be represented as compositions of a few, selected isometries, called the generators for the group [Link]. For example, the group pgg [Image] is generated by the half-turn Rc and the horizontal glide B shown in the diagram at the right.

Now what happens when we introduce anti-symmetries into a wallpaper function based on this group? If some isometries negate the values of our wallpaper function, then others may be forced to do so also.

For example, if we decide to make a function having B as a symmetry but with Rc as an anti-symmetry, then Bv would have to be an anti-symmetry because Bv = Rc B. Since B preserves the sign and Rc reverses it, the net effect of Bv is a negation of the sign of the wallpaper function. Thus Bv is forced to be an anti-symmetry.

The picture below shows this, with negating symmetries drawn in green. Every positive glide reflection is horizontal, and thus derived from B. All the half-turns negate the pattern, as they are all "cousins" of Rc. All the vertical glide reflections are negative.

[pgg' wallpaper]
[Link]
A pgg' wallpaper with its fundamental cell outlined. Negating isometries are in green.

There are two groups at play here. If you ignore color, you would say that the symmetry group is pgg. If color is taken into account, so that the half-turns no longer count as symmetries, then the group would be pg.

The precise mathematical formulation of this idea involves pairs of groups in this type of relation. The intuitive idea behind the group homomorphisms, central to the formal discussion, is this:

Take a wallpaper group, E, and tag some elements with +1 and the others with -1. This has to be done in a consistent way so that when you compose symmetries, the tag for the composition of the two symmetries is the product of their tags. If you do this, the set tagged by +1 also is a wallpaper group called G. Of course these groups are infinite, but in some sense G has to be exactly half of E.

In the example above, E, the extended symmetry group, is pgg, while G, the actual symmetry group, is pg.

You may wish to skip the details of the proof and go on to the classification result [skip]. It lists all the possible E,G pairs and gives a name to the corresponding pattern type. The pattern above is called a pgg'. The name is supposed to make sense because one of the glides in pgg is negating and so gets a prime, while the other is positive.


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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 9:26:18 AM
Comments to: CVM@maa.org