mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a prototile is one of the shapes of a tile in a

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

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 ...

. 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 modu ...

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

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, 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 Tessellation, 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 set of prototiles, aperiodic if copie ...

. In March 2023, four researchers,

Chaim Goodman-Strauss Chaim Goodman-Strauss (born June 22, 1967 in Austin, Texas) is an American mathematician who works in convex geometry, especially aperiodic tiling. He retired from the faculty of the University of Arkansas and currently serves as outreach mathem ...

, David Smith, Joseph Samuel Myers and Craig S. Kaplan, announced the discovery of an aperiodic monohedral prototile (monotile) and a proof that the tile discovered by David Smith is an aperiodic monotile, i.e. a solution to a longstanding open

einstein problem In plane discrete geometry, the einstein problem asks about the existence of a single prototile that by itself forms an aperiodic set of prototiles; that is, a shape that can tessellate space but only in a nonperiodic way. Such a shape is call ...

. In higher dimensions, the problem had been solved earlier: the

Schmitt-Conway-Danzer tile In geometry, the gyrobifastigium is a polyhedron that is constructed by attaching a triangular prism to square face of another one. It is an example of a Johnson solid. It is the only Johnson solid that can tile three-dimensional space. Const ...

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, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...

, and cannot tile space periodically.

References

{{Tessellation Tessellation