Martin von Gagern (see post on The Grammar of Ornament) has made a tool for drawing patterns with different wallpaper groups. It’s cross-platform and very intuitive and I’ve used it successfully with students in examples classes. Below is a screenshot of the drawing area with default settings: and the following image is a screenshot with the cell and symmetry elements marked:
“The Grammar of Ornament” by Owen Jones (1809 – 1874) is a work in which the author collects elements of design and typical motifs from a wide range of cultures. A site dedicated to the book can be found here and a full digitised version at The Grammar of Ornament: University of Wisconsin Digital Library. There is a also set of images from The Grammar of Ornament on Flickr. Since many of the designs are repeating patterns, they can be used to teach the concept of wallpaper group symmetry (and, by extension, space group symmetry); Martin von Gagern has used them to make some very beautiful illustrations of these groups on his Morenaments page. Some of the motifs are also used to illustrate the Wikipedia entry on wallpaper groups. I have used an Egyptian design from The Grammar of Ornament to illustrate the concept of primitive and non-primitive unit cells. Lattice points (those points in identical environments, assuming that the pattern extends infinitely) are marked using blue dots. A primitive unit cell (or, more properly, unit mesh) contains only one unique lattice point, whereas a non-primitive cell contains more than one lattice point. The choice of a non-primitive cell usually reflects the overall symmetry of the structure; here, for example, the square shape of the overall pattern can be seen more clearly from the non-primitive cell. One of the great advantages of using The Grammar of Ornament for teaching is that the patterns come from so many different ages and cultures, making it a great inclusive tool for international and multicultural classes.
The tiling patterns above were produced using Repper, a patterning/tessellation program that has a selection of wallpaper groups available (as well as some other effects). The program allows you to load an image and select part of it, to which it applies its symmetry transforms. As well as some very straightforward symmetry groups with visible edges (p1, pm, pg, cm and others), there are a set of ‘seamless’ groups that give much prettier results as can be seen above. The original image was of a forget-me-not flower and the various patterns were produced by moving the selection window and changing the symmetry. There’s a limited, trial version of the software available on the Web here. Repper isn’t primarily intended as a tool for teaching symmetry but it does give aesthetically very appealing results which can help in gaining students’ attention and interest at the beginning of a course or unit. Also, it’s great fun for anyone who has ever enjoyed playing with a kaleidoscope!