![]() The pattern represented by every finite patch of tiles in a Penrose tiling occurs infinitely many times throughout the tiling. Penrose tilings are self-similar: they may be converted to equivalent Penrose tilings with different sizes of tiles, using processes called inflation and deflation. Roger Penrose in the foyer of the Mitchell Institute for Fundamental Physics and Astronomy, Texas A&M University, standing on a floor with a Penrose tiling Even constrained in this manner, each variation yields infinitely many different Penrose tilings. ![]() This may be done in several different ways, including matching rules, substitution tiling or finite subdivision rules, cut and project schemes, and coverings. The Penrose tilings are obtained by constraining the ways in which these shapes are allowed to fit together in a way that avoids periodic tiling. The original form of Penrose tiling used tiles of four different shapes, but this was later reduced to only two shapes: either two different rhombi, or two different quadrilaterals called kites and darts. There are several different variations of Penrose tilings with different tile shapes. Penrose tilings are named after mathematician and physicist Roger Penrose, who investigated them in the 1970s. ![]() However, despite their lack of translational symmetry, Penrose tilings may have both reflection symmetry and fivefold rotational symmetry. Here, a tiling is a covering of the plane by non-overlapping polygons or other shapes, and a tiling is aperiodic if it does not contain arbitrarily large periodic regions or patches. A Penrose tiling with rhombi exhibiting fivefold symmetryĪ Penrose tiling is an example of an aperiodic tiling. ![]()
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