# Penrose tiling

**A Penrose tiling is a covering of the plane by non-overlapping polygons or other shapes. Despite their lack of translational symmetry, Penrose tilings may have both reflection symmetry and fivefold rotational symmetry. The study of these tilings has been important in the understanding of physical materials that also form quasicrystals. Penrosetilings have also been applied in architecture and decoration, as in the floor tiling shown.**

### About Penrose tiling in brief

A Penrose tiling is a covering of the plane by non-overlapping polygons or other shapes. Aperiodic means that shifting any tiling with these shapes by any finite distance, without rotation, cannot produce the same tiling. Despite their lack of translational symmetry, Penrose tilings may have both reflection symmetry and fivefold rotational symmetry. The study of these tilings has been important in the understanding of physical materials that also form quasicrystals. Penrosetilings have also been applied in architecture and decoration, as in the floor tiling shown. The first Pen rose tiling was created by Raphael Robinson in 1971. The original form of Pen rose tilings 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. The pattern represented by every finite patch of tiles in a Pen Rose tiling occurs infinitely many times throughout the tiling, as a physical structure. The subject of aperiodic tilings received new interest in the 1960s when logician Hao Wang noted connections between decision problems and tilings. In particular, he introduced tilings by square plates with colored edges, now known as Wang dominoes or tiles. Wang’s student Robert Berger proved that the Domino Problem was undecidable in his 1964 thesis, but the latter did not appear in his 1968 monograph.

The matching required in a tiling by Wanginoes can easily be achieved by modifying the edges of the edges so that they can fit together as prescribed by the jigsaw puzzle pieces. In 1971, Raphael Robinson used this technique to obtain an an undecidability of a set of six prototiles to obtain a PenRose tiling of just six protOTiles. A tiling that has no periods is non-periodic. The tiles in the square tiling have only one shape, and it is common for other tilings to have only a finite number of shapes. That is, each tile in the tiler must be congruent to one of these prototile. The tiling or tile the plane is said to be a periodic if it has periods that shift the tiled in two different directions, and in this case its tilings are also called a Periodic Tiling. The most familiar tilings, such as covering a floor with squares meeting edge-to-edge, are examples of periodic tilings and are called Periodic Tile Tiling (PTT) A Periodic tile is called a periodic tiling when it is periodic when it hasperiodic tiles that tile the same surface in different directions. A Periodical Tiling is also known as a Tiling of the Plane (TPT) and can be found on the Wikipedia page. It is a type of tiling where the pattern of tiles can be seen by looking at the shape and size of each tile.

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This page is based on the article Penrose tiling published in Wikipedia (as of Jan. 04, 2021) and was automatically summarized using artificial intelligence.