Wallpaper Groups

Symmetry group 2 (p2)

This group differs only from the first group in that it contains 180° rotations, that is, rotations of order 2, also called "half-turns." As in all symmetry groups there are translations, but there are no reflections or glide-reflections. The two translation axes may be inclined at any angle to each other. The lattice is a parallelogrammatic. A fundamental region for the symmetry group is half of a parallelogram that is a fundamental region for the translation group.

It may be a little difficult to see the half-turns, so their fixed points are covered with black dots in this image. Half-turn symmetries are subtile and aren't very easy to find. Note that in this example (as in all patterns with this symmetry) there are four essentially different centers of half-turns.

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© 1994, 1997.

David E. Joyce
Department of Mathematics and Computer Science
Clark University
Worcester, MA 01610

These files are located at http://aleph0.clarku.edu/~djoyce/wallpaper/