Prototile

This form of the Penrose tiling has two prototiles, a fat rhombus (shown blue in the figure) and a thin rhombus (green).

In the mathematical theory of tessellations, a prototile is one of the shapes of a tile in a tessellation.

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. Some of the tiles may be congruent to one or more others. 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.

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

A set of prototiles is said to be aperiodic if every tiling with those prototiles is an aperiodic tiling. The three-dimensional Schmitt-Conway-Danzer tile is the prototile of a monohedral aperiodic tiling of three-dimensional Euclidean space, but it remains open whether there is a monohedral aperiodic prototile for the plane.

References

• Kaplan, Craig S. (2009), Introductory Tiling Theory for Computer Graphics, Synthesis Lectures on Computer Graphics and Animation, Morgan & Claypool Publishers, p. 7, ISBN 9781608450176, http://books.google.com/books?id=OPtQtnNXRMMC&pg=PA7 .
• Cederberg, Judith N. (2001), A Course in Modern Geometries, Undergraduate Texts in Mathematics (2nd ed.), Springer-Verlag, p. 174, ISBN 9780387989723, http://books.google.com/books?id=Fo9tqL99jdMC&pg=PA174 .