Pentagonal tiling

In geometry, a pentagonal tiling is a tiling of the plane where each individual piece is in the shape of a pentagon. A regular pentagonal tiling on the Euclidean plane is impossible because the internal angle of a regular pentagon, 108°, is not a divisor of 360°, the angle measure of a whole turn. However, regular pentagons can tile the hyperbolic plane with four pentagons around each vertex (or more) and sphere with three pentagons; the latter produces a tiling that is topologically equivalent to the dodecahedron. Fifteen types of convex pentagons are known to tile the plane monohedrally (i.e. with one type of tile). The most recent one was discovered in 2015. This list has been shown to be complete by (result subject to peer-review). showed that there are only eight edge-to-edge convex types, a result obtained independently by . Michaël Rao of the École normale supérieure de Lyon claimed in May 2017 to have found the proof that there are in fact no convex pentagons that tile beyond these 15 types. As of 11 July 2017, the first half of Rao's proof had been independently verified (computer code available) by Thomas Hales, a professor of mathematics at the University of Pittsburgh. As of December 2017, the proof was not yet fully peer-reviewed. Each enumerated tiling family contains pentagons that belong to no other type; however, some individual pentagons may belong to multiple types. In addition, some of the pentagons in the known tiling types also permit alternative tiling patterns beyond the standard tiling exhibited by all members of its type. The sides of length a, b, c, d, e are directly clockwise from the angles at vertices A, B, C, D, E respectively. (Thus, A, B, C, D, E are opposite to d, e, a, b, c respectively.) Many of these monohedral tile types have degrees of freedom. These freedoms include variations of internal angles and edge lengths. In the limit, edges may have lengths that approach zero or angles that approach 180°. Types 1, 2, 4, 5, 6, 7, 8, 9, and 13 allow parametric possibilities with nonconvex prototiles.