Prototile

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

[1] 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.

If S is the set of tiles in a tessellation, a set R of shapes is called a set of prototiles if no two shapes in R are congruent to each other, and every tile in S is congruent to one of the shapes in R.[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.

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 open einstein problem.

[3][4] In higher dimensions, the problem had been solved earlier: the Schmitt-Conway-Danzer tile is the prototile of a monohedral aperiodic tiling of three-dimensional Euclidean space, and cannot tile space periodically.