Symmetry group 4 (pg)
This is the first group that contains glide-reflections. The direction of the glide reflection is parallel to one axis of translation and perpendicular to the other axis of translation. There are neither rotations nor regular reflections. The lattice is rectanglular, and a rectangular fundamental region for the translation group can be chosen that is split by an axis of a glide reflection so that one of the half rectangles forms a fundamental region for the symmetry group.
Half-turns were hard to see, but glide-reflections are even harder to see. A glide-reflecion is composed of a reflection across an axis and a translation along the axis. Here, the axis of the glide reflection is drawn in magenta. The green arrows on either side of it suggest the glide reflection. If you reflect across the magenta axis, then translate along it (either up or down), one green arrow will fall onto another green arrow on the other side of the axis.
On the right you see the pattern marked up with lines for the axes of all the glide reflections. Some lie midway between the dark blue patches, but others lie along them.
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© 1994, 1997.
David E. Joyce
Department of Mathematics and Computer Science
Worcester, MA 01610
These files are located at http://aleph0.clarku.edu/~djoyce/wallpaper/