# Wallpaper Groups

### Symmetry group 3 (pm)

This is the first group that contains reflections. The axes of reflection are parallel to one axis of translation and perpendicular to the other axis of translation. There are no rotations or glide reflections. A fundamental region for the translation group is a rectangle, and one can be chosen that is split by an axis of reflection so that one of the half rectangles forms a fundamental region for the symmetry group.

Reflections are usually easy to see. Also called "bilateral symmetries," humans seem to be designed to recognize them, although it helps if the axis of reflection is vertical. In the example, the axes of reflection are inclined, so it's a little harder to see them. They're drawn as red lines in the image to the right.

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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/