The wallpaper atlas

The 17 wallpaper groups, and over each one every animation group whose spatial projection it is — the whole catalog, one wallpaper at a time.

Every spacetime group projects to a wallpaper group: forget the clock, keep every spatial operation. That projection is the symmetry of the long-exposure photograph, and each section here is named by its Conway orbifold — 2222, *632, 3*3 — in the notation of Conway, Burgiel and Goodman-Strauss, The Symmetries of Things.

A spacetime group over that orbifold is the orbifold wearing a clock, and the book's own colour notation says which: a superscript on a feature is the order of the colour permutation that feature induces, so ²2²2²2²2 is 2222 with all four half-turns advancing the animation half a period, and ¹2¹2²2²2 is the same orbifold with only two of them doing so. Each section below describes what happens over one wallpaper group, shows one or two characteristic examples, and links to a page with all of its animation groups at full size. The section counts sum to 275; captions link to the catalog for generators and details.