Wallpaper Groups
All 17 plane symmetry groups — select one to see its tiling pattern
About: There are exactly 17 wallpaper groups — all distinct ways to tile the plane with repeating patterns, classified by their symmetries (translations, rotations, reflections, glide reflections). Proved complete by Fedorov (1891) and Pólya (1924). They appear in Islamic architecture, Escher's art, and crystallography (the 230 space groups extend this to 3D). The groups differ in their highest rotational order: p1 (none), p2 (2-fold), p3/p31m/p3m1 (3-fold), p4/p4m/p4g (4-fold), p6/p6m (6-fold). Only 1, 2, 3, 4, 6-fold rotations tile the plane — the crystallographic restriction theorem.