Wallpaper group

A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art, especially in textiles, tiles, and wallpaper.

The simplest wallpaper group, Group p1, applies when there is no symmetry beyond simple translation of a pattern in two dimensions. The following patterns have more forms of symmetry, including some rotational and reflectional symmetries:

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  Example A: Cloth, Tahiti
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  Example B: Ornamental painting, Nineveh, Assyria
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  Example C: Painted porcelain, China

Examples A and B have the same wallpaper group; it is called p4m in the IUCr notation and *442 in the orbifold notation. Example C has a different wallpaper group, called p4g or 4*2 . The fact that A and B have the same wallpaper group means that they have the same symmetries, regardless of the designs' superficial details; whereas C has a different set of symmetries.

The number of symmetry groups depends on the number of dimensions in the patterns. Wallpaper groups apply to the two-dimensional case, intermediate in complexity between the simpler frieze groups and the three-dimensional space groups.

A proof that there are only 17 distinct groups of such planar symmetries was first carried out by Evgraf Fedorov in 1891^[1] and then derived independently by George Pólya in 1924.^[2] The proof that the list of wallpaper groups is complete came only after the much harder case of space groups had been done. The seventeen wallpaper groups are listed below; see § The seventeen groups.