A wallpaper is a mathematical object covering a whole Euclidean plane by repeating a motif indefinitely, in manner that certain isometries keep the drawing unchanged. For each wallpaper there corresponds a group of congruent transformations, with function composition as the group operation. Thus, a wallpaper 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, tessellations, tiles and physical wallpaper.
Example of an Egyptian design with wallpaper group p4m
Example A: Cloth, Tahiti
Example B: Ornamental painting, Nineveh, Assyria
Example C: Painted porcelain, China
In mathematics, a group is a set with an operation that satisfies the following constraints: the operation is associative and has an identity element, and every element of the set has an inverse element.
A periodic wallpaper pattern gives rise to a wallpaper group.