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Wallpaper group
Classification of a two-dimensional repetitive pattern / From Wikipedia, the free encyclopedia
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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.
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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:
- Example A: Cloth, Tahiti
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.