Periodic and aperiodic tilings of the plane — symmetry groups and beyond
The plane admits exactly 17 wallpaper groups (distinct periodic symmetry groups). The square lattice has p4m symmetry; the hexagonal lattice has p6m. The Cairo tiling (equilateral pentagons, type 4) appears in Islamic architecture. The Pinwheel tiling (Conway & Radin 1994) has tiles in infinitely many orientations yet is aperiodic. The Chair fractal is a self-similar substitution tiling. Click on a tile to highlight its orbit under the symmetry group.