Tiling

A tiling is a plane-filling arrangement of plane figures or its generalization to higher dimensions. Formally, it 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).

A substitution tiling is constructed by repeatedly replacing tiles according to fixed geometric rules. In a tile inflation rule, tiles are enlarged and replaced by tiles at the original scale (Frank 2008).

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 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. Smith et al. (2024) constructed an aperiodic monotile, the hat polykite, a single polygon that admits aperiodic tilings but no periodic tilings, allowing reflections as well as rotations and translations.

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

The following table summarizes tilings by congruent convex polygons, where is the number of sides. The entry "all" means that every triangle and every convex quadrilateral tiles the plane. For pentagons and hexagons, the numbers count tiling families.

	name	tiling families
3	triangle tiling	all
4	quadrilateral tiling	all
5	pentagon tiling	15
6	hexagon tiling	3

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