Nine of the 17 mathematically possible wallpaper groups, built by tiling an asymmetric motif using rotations, reflections and glide reflections. Switch groups to see how rotation centers, mirror lines and glide axes combine to fill the plane.
This tool builds wallpaper groups — the 17 mathematically distinct ways a repeating pattern can symmetrically tile an infinite flat plane. Rather than faking the symmetry visually, each pattern here is constructed exactly from its underlying group operations: an asymmetric motif is duplicated by rotations and reflections about a fixed center to form a small cluster, and that cluster is then repeated by pure translation along a square or hexagonal lattice. This page implements 9 of the 17 groups — p1, p2, pm, pg, cm, p4, p4m, p6 and p6m — covering the full range from no symmetry at all to the maximal 12-fold dihedral symmetry of p6m.
Nine of the 17 wallpaper groups, each built by applying that group's exact rotation, reflection and glide-reflection operations to an off-center motif, then tiling the resulting cluster across a square or hexagonal lattice. The optional overlay reveals the rotation centers, mirror lines and glide axes that generate the pattern.
Pick a wallpaper group and a motif shape, drag the cell-size slider to zoom the tiling in or out, and toggle the symmetry overlay to see the rotation centers and axes that produce each pattern. The stats box updates with a short, technically accurate description of the selected group's operations.
Every wallpaper pattern ever printed on real wallpaper, woven into fabric, or carved into Islamic geometric tilework belongs to exactly one of these 17 groups — there is mathematically no eighteenth possibility for repeating a pattern across a flat plane.
A wallpaper group is a mathematical classification of the symmetries of a pattern that repeats infinitely across a two-dimensional plane. Each group is defined by which combination of translations, rotations, reflections and glide reflections maps the pattern onto itself, without changing its appearance.
The crystallographic restriction theorem limits the rotational symmetries compatible with a repeating lattice to orders 1, 2, 3, 4 and 6, and combining these allowed rotations with the possible reflection and glide-reflection symmetries yields a finite list. Evgraf Fedorov proved in 1891, and George Pólya independently confirmed in 1924, that this exhaustive combinatorial classification produces exactly 17 distinct groups, no more and no fewer.
Both groups have 90° rotation centers at every lattice point on a square lattice. The difference is reflection: p4 contains only rotations, with no mirror symmetry anywhere, while p4m adds four mirror axes through each rotation center, forming the full dihedral symmetry group of order 8 rather than the pure rotation group of order 4.
A glide reflection combines a mirror reflection across a line with a translation by half a lattice period along that same line. It is a distinct symmetry operation from an ordinary mirror reflection or a plain translation — groups like pg contain glide axes but no true mirror lines at all, which is what makes pg visually and mathematically different from pm.
All 17 wallpaper groups can be found in the tile mosaics of the Alhambra palace in Granada, created centuries before the classification theorem existed, and M. C. Escher studied those same tilings to build his own interlocking symmetric prints. Crystallographers also use the wallpaper group classification to describe the symmetry of two-dimensional atomic layers, surface structures and thin films.