Tessellation

A tiling of regular polygons (in two dimensions), polyhedra (three dimensions), or polytopes ( dimensions) is called a tessellation. Tessellations can be specified using a Schläfli symbol.

The breaking up of self-intersecting polygons into simple polygons is also called tessellation (Woo et al. 1999), or more properly, polygon tessellation.

There are exactly three regular tessellations composed of regular polygons symmetrically tiling the plane.

Tessellations of the plane by two or more convex regular polygons such that the same polygons in the same order surround each polygon vertex are called semiregular tessellations, or sometimes Archimedean tessellations. In the plane, there are eight such tessellations, illustrated above (Ghyka 1977, pp. 76-78; Williams 1979, pp. 37-41; Steinhaus 1999, pp. 78-82; Wells 1991, pp. 226-227).

There are 14 demiregular (or polymorph) tessellations which are orderly compositions of the three regular and eight semiregular tessellations (Critchlow 1970, pp. 62-67; Ghyka 1977, pp. 78-80; Williams 1979, p. 43; Steinhaus 1999, pp. 79 and 81-82).

In three dimensions, a polyhedron which is capable of tessellating space is called a space-filling polyhedron. Examples include the cube, rhombic dodecahedron, and truncated octahedron. There is also a 16-sided space-filler and a convex polyhedron known as the Schmitt-Conway biprism which fills space only aperiodically.

A tessellation of -dimensional polytopes is called a honeycomb.