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Concept
Wallpaper group
Summary
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. To a given wallpaper there corresponds a group of such congruent transformations, with function composition as the group operation. Thus, a wallpaper group (or plane symmetry group or plane crystallographic group) is in 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 and tiles as well as wallpaper.
Any periodic tiling can be seen as a wallpaper. More particularly, we can consider as a wallpaper a tiling by identical tiles edge‐to‐edge, necessarily periodic, and conceive from it a wallpaper by decorating in the same manner every tiling element, and eventually erase partly or entirely the boundaries between these tiles. Conversely, from every wallpaper we can construct such a tiling by identical tiles edge‐to‐edge, which bear each identical ornaments, the identical outlines of these tiles being not necessarily visible on the original wallpaper. Such repeated boundaries delineate a repetitive surface added here in dashed lines.
Such pseudo‐tilings connected to a given wallpaper are in infinite number. For example image 1 shows two models of repetitive squares in two different positions, which have Another repetitive square has an We could indefinitely conceive such repetitive squares larger and larger. An infinity of shapes of repetitive zones are possible for this Pythagorean tiling, in an infinity of positions on this wallpaper. For example in red on the bottom right‐hand corner of image 1, we could glide its repetitive parallelogram in one or another position. In common on the first two images: a repetitive square concentric with each small square tile, their common center being a point symmetry of the wallpaper.
Between identical tiles edge‐to‐edge, an edge is not necessarily a segment of a straight line.
Official source
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