In the mathematical theory of

tessellation A tessellation or tiling is the covering of a surface, often a plane (mathematics), plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to high-dimensional ...

s, a prototile is one of the shapes of a tile in a tessellation.

Definition

A tessellation of the plane or of any other space is a cover of the space by closed shapes, called tiles, that have disjoint

interiors ''Interiors'' is a 1978 American drama film written and directed by Woody Allen. It stars Kristin Griffith, Mary Beth Hurt, Richard Jordan, Diane Keaton, E. G. Marshall, Geraldine Page, Maureen Stapleton, and Sam Waterston. Allen's first ful ...

. Some of the tiles may be

congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In mod ...

to one or more others. If is the set of tiles in a tessellation, a set of shapes is called a set of prototiles if no two shapes in are congruent to each other, and every tile in is congruent to one of the shapes in . It is possible to choose many different sets of prototiles for a tiling: translating or rotating any one of the prototiles produces another valid set of prototiles. However, every set of prototiles has the same

cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...

, so the number of prototiles is well defined. A tessellation is said to be monohedral if it has exactly one prototile.

Aperiodicity

A set of prototiles is said to be aperiodic if every tiling with those prototiles is an

aperiodic tiling An aperiodic tiling is a non-periodic tiling with the additional property that it does not contain arbitrarily large periodic regions or patches. A set of tile-types (or prototiles) is aperiodic if copies of these tiles can form only non- period ...

. It is unknown whether there exists a single two-dimensional shape (called an ''

einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...

''). that forms the prototile of an aperiodic tiling, but not of any periodic tiling. That is, the existence of a single-tile (monohedral) aperiodic prototile set is an open problem. The

Socolar–Taylor tile The Socolar–Taylor tile is a single non-connected tile which is aperiodic on the Euclidean plane, meaning that it admits only non-periodic tilings of the plane (due to the Sierpinski's triangle-like tiling that occurs), with rotations and ref ...

forms two-dimensional aperiodic tilings, but is defined by combinatorial matching conditions rather than purely by its shape. In higher dimensions, the problem is solved: the

Schmitt-Conway-Danzer tile In geometry, the gyrobifastigium is the 26th Johnson solid (). It can be constructed by joining two face-regular triangular prisms along corresponding square faces, giving a quarter-turn to one prism. It is the only Johnson solid that can tile ...

is the prototile of a monohedral aperiodic tiling of three-dimensional

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...

, and cannot tile space periodically.

References

{{Tessellation Tessellation