Symmetry Groups
The actual symmetry group should be distinguished from the wallpaper group. Wallpaper groups are collections of symmetry groups. There are 17 of these collections, but for each collection there are infinitely many symmetry groups, in the sense of actual groups of isometries. These depend, apart from the wallpaper group, on a number of parameters for the translation vectors, the orientation and position of the reflection axes and rotation centers.
The numbers of degrees of freedom are:
- 6 for p2
- 5 for pmm, pmg, pgg, and cmm
- 4 for the rest.
However, within each wallpaper group, all symmetry groups are algebraically isomorphic.
Some symmetry group isomorphisms:
- p1: Z2
- pm: Z × D∞
- pmm: D∞ × D∞.
Read more about this topic: Wallpaper Group
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