Tiling

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A plane-filling arrangement of plane figures or its generalization to higher dimensions. Formally, a tiling is a collection of disjoint open sets, the closures of which cover the plane. Given a single tile, the so-called first corona is the set of all tiles that have a common boundary point with the tile (including the original tile itself).

Wang's conjecture (1961) stated that if a set of tiles tiled the plane, then they could always be arranged to do so periodically. A periodic tiling of the plane by polygons or space by polyhedra is called a tessellation. The conjecture was refuted in 1966 when R. Berger showed that an aperiodic set of 20426 tiles exists. By 1971, R. Robinson had reduced the number to six and, in 1974, R. Penrose discovered an aperiodic set (when color-matching rules are included) of two tiles: the so-called Penrose tiles. It is not known if there is a single aperiodic tile.

A spiral tiling using a single piece is illustrated on the cover of Grünbaum and Shephard (1986).

The number of tilings possible for convex irregular polygons are given in the following table.

nnameknown tilings
3triangle tilingall
4quadrilateral tilingall
5pentagon tiling14
6hexagon tiling3

There are no tilings for identical convex n-gons for n>=7, although non-identical convex heptagons can tile the plane (Steinhaus 1999, p. 77; Gardner 1984, pp. 248-249).

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