text: 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. On the top left‑hand corner of image 3, point C  is a vertex of a repetitive pseudo‑rhombus with thick stripes on its whole surface, called pseudo‑rhombus because of a concentric repetitive rhombus  constructed from it by taking out a bit of surface somewhere to append it elsewhere, and keep the area unchanged. By the same process on image 4, a repetitive regular hexagon filled with vertical stripes is constructed from a rhombic repetitive zone  Conversely, from elementary geometric tiles edge‑to‑edge, an artist like M. C. Escher created attractive surfaces many times repeated. On image 2,   the minimum area of a repetitive surface by disregarding colors, each repetitive zone in dashed lines consisting of five pieces in a certain arrangement, to be either a square or a hexagon, like in a proof of the Pythagorean theorem.
  In the present article, a pattern is a repetitive parallelogram of minimal area in a determined position on the wallpaper. Image 1 shows two parallelogram‑shaped patterns — a square is a particular parallelogram —. Image 3 shows rhombic patterns — a rhombus is a particular parallelogram —.
  On this page, all repetitive patterns (of minimal area) are constructed from two translations that generate the group of all translations under which the wallpaper is invariant. With the circle shaped symbol ⵔ of function composition, a pair like  or  generates the group of all translations that transform the Pythagorean tiling into itself.