Wallpaper Groups

Symmetry group 17 (p6m)

This most complicated group has rotations of order 2, 3, and 6 as well as reflections. The axes of reflection meet at all the centers of rotation. At the centers of the order-6 rotations, six reflection axes meet and are inclined at 30° to one another. The lattice is hexagonal. A fundamental region for the symmetry group is one-twelfth of an equilateral triangle for the lattice.

There are glide reflections in this group as well, but the image is pretty cluttered even without their axes being drawn. The axes of the glide reflections are midway between parallel reflection axes, so they pass through the centers of the half-turns (black dots). They go in all six directions just as the reflections axes do.

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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/