Click on any pattern to see what it looks like when tiled. For an outline explanation see the bottom of the page. Some of the tiles in this gallery can be viewed in a slide show.
Mathematically speaking, wallpaper designs that can fill an arbitrary sized flat region by repetitions of a single tile in more than one direction fall into seventeen different categories depending on any other symmetries they may have. A symmetry of a pattern is a one to one transformation that moves every point to another point that looks the same as the original point. The combination of any two symmetries is another symmetry, so the symmetries of any pattern form what mathematicians call a group. The 17 patterns fall into five families depending on the shape of the tile: in general the more symmetries the pattern has the more restricted is the shape of the tile.
There are many excellent sources of information on the web about these patterns. Here are a few links to other sites that you may find interesting:
The various groups can be categorised by by the shape of the lattice on which the tiles are placed. The lattice determines what symmetries are possible. A few of the patterns above are color symetries, where the real symmetry of the design is disguised through the use of different colors.