# Pentagon tiling

The 14 known pentagon tilings

In geometry, a pentagon tiling is a tiling of the plane by pentagons. A regular pentagonal tiling on the Euclidean plane is impossible because the internal angle of a regular pentagon, 3π/5, is not a divisor of 2π. Fourteen types of monohedral tiling by convex pentagons are known as of 2012, and it is not known whether this list is complete.

## History

Karl Reinhardt (1918) found the 5 pentagon tilings that are "tile transitive", meaning that the symmetries of the tiling can take any tile to any other tile (more formally, the automorphism group acts transitively on the tiles). R. B. Kershner (1968) found 3 more tilings, all of which are not tile transitive; he claimed incorrectly that this was the complete list of pentagons that can tile the plane. Richard E. James III found a 9th tiling in 1975, after reading about Kershner's results in Martin Gardner's Mathematical Games column of July 1975 (reprinted in (Gardner 1988)); James' tiling is monohedral (it only uses one type of tile) but not tile transitive. Schattschneider (1978) described how Marjorie Rice, an amateur mathematician, discovered four new types of tessellating pentagons in 1976 and 1977. Schattschneider (1985) described a 14th tiling found by Rolf Stein in 1985. O. Bagina (2011) (in Russian) proved 8 edge-to-edge convex types, a result obtained independently by T. Sugimoto in 2012.

## Non-convex pentagons

With pentagons that are not required to be convex, additional types of tiling are possible. An example is the sphinx tiling, an aperiodic tiling formed by a pentagonal rep-tile (Godrèche 1989). The sphinx may also tile the plane periodically, by fitting two sphinx tiles together to form a parallelogram and then tiling the plane by translates of this parallelogram, a pattern that can also be used for certain other shapes of non-convex pentagons.

## Dual uniform tilings

There are 3 isohedral pentagonal tilings generated as duals of the uniform tilings:

## Regular pentagonal tilings in non-Euclidean geometry

A dodecahedron can be considered a regular tiling of 12 pentagons on the surface of a sphere, with Schläfli symbol {5,3}, having 3 pentagons around each vertex.

In the hyperbolic plane, there are tilings of regular pentagons, for instance order-4 pentagonal tiling, with Schläfli symbol {5,4}, having 4 pentagons around reach vertex. Higher order regular tilings {5,n} can be constructed on the hyperbolic plane, ending in {5,∞}.

Sphere Hyperbolic plane

{5,3}

{5,4}

{5,5}

{5,6}

{5,7}

{5,8}
...{5,∞}

## Irregular hyperbolic plane pentagonal tilings

There are an infinite number of dual uniform tilings in hyperbolic plane with isogonal irregular pentagonal faces. They have face configurations as V3.3.p.3.q.

Order p-q floret pentagonal tiling
7-3 8-3 9-3 ... 5-4 6-4 7-4 ... 5-5

V3.3.3.3.7
V3.3.3.3.8 V3.3.3.3.9 ...
V3.3.4.3.5
V3.3.4.3.6 V3.3.4.3.7 ... V3.3.5.3.5 ...