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Inmathematics, aprototile is one of the shapes of a tile in atessellation.[1]
A tessellation of the plane or of any other space is a cover of the space byclosed shapes, called tiles, that havedisjointinteriors. Some of the tiles may becongruent to one or more others. IfS is the set of tiles in a tessellation, a setR of shapes is called a set of prototiles if no two shapes inR are congruent to each other, and every tile inS is congruent to one of the shapes inR.[2]
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 samecardinality, so the number of prototiles is well defined. A tessellation is said to bemonohedral if it has exactly one prototile.
A set of prototiles is said to be aperiodic if every tiling with those prototiles is anaperiodic tiling. In March 2023, four researchers,Chaim Goodman-Strauss,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 openeinstein problem.[3][4]
In higher dimensions, the problem had been solved earlier: theSchmitt-Conway-Danzer tile is the prototile of a monohedral aperiodic tiling of three-dimensionalEuclidean space, and cannot tile space periodically.
