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 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.
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).
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.
-gons for 
,
although non-identical convex heptagons can tile the plane (Steinhaus 1999, p. 77;
Gardner 1984, pp. 248-249).
