Pentagonal tiling

A tiling of the plane by pentagons

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.) + 15 monohedral pentagonal tiles12345B + C = 180°A + D + E = 360°c = eB + D = 180°a = b, d = c + eA = C = D = 120°b = c, d = eB = D = 90°a = b, d = eA = 60°, D = 120°678910a = d = e, b = cB + D = 180°, 2B = Eb = c = d = eB + 2E = 2C + D = 360°b = c = d = e2B + C = D + 2E = 360°b = c = d = e2A + C = D + 2E = 360°a = b = c + eA = 90°, B + E = 180°B + 2C = 360°11121314152a + c = d = eA = 90°, C + E = 180°2B + C = 360°2a = d = c + eA = 90°, C + E = 180°2B + C = 360°d = 2a = 2eB = E = 90°2A + D = 360°2a = 2c = d = eA = 90°, B ≈ 145.34°, C ≈ 69.32°D ≈ 124.66°, E ≈ 110.68°(2B + C = 360°, C + E = 180°)a = c = e, b = 2aA = 150°, B = 60°, C = 135°D = 105°, E = 90°

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. Periodic tilings are characterised by their wallpaper group symmetry, for example p2 (2222) is defined by four 2 fold gyration points. This nomenclature is used in the diagrams below, where the tiles are also colored by their k isohedral positions within the symmetry. A primitive unit is a section of the tiling that generates the whole tiling using only translations, and is as small as possible.

found the first five types of pentagonal tile. All five can create isohedral tilings, 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). B. Grünbaum and G. C. Shephard have shown that there are exactly twenty four distinct "types" of isohedral tilings of the plane by pentagons according to their classification scheme. All use Reinhardt's tiles, usually with additional conditions necessary for the tiling. There are two tilings by all type 2 tiles, and one by all of each of the other four types. Fifteen of the other eighteen tilings are by special cases of type 1 tiles. Nine of the twenty four tilings are edge to edge. There are also 2 isohedral tilings by special cases of type 1, type 2, and type 4 tiles, and 3 isohedral tilings, all edge to edge, by special cases of type 1 tiles. There is no upper bound on k for k isohedral tilings by certain tiles that are both type 1 and type 2, and hence neither on the number of tiles in a primitive unit. The wallpaper group symmetry for each tiling is given, with orbifold notation in parentheses. A second lower symmetry group is given if tile chirality exists, where mirror images are considered distinct. These are shown as yellow and green tiles in those cases.

A 14th convex pentagon type was found by Rolf Stein in 1985. The tiling is 3 isohedral and non edge to edge. It has completely determined tiles, with no degrees of freedom. The exact proportions are specified by and angle B obtuse with Other relations can easily be deduced. The primitive units contain six tiles respectively. It has p2 (2222) symmetry. + Type 142a=2c=d=eA=90°, B≈145.34°, C≈69.32°,D≈124.66°, E≈110.68°(2B+C=360°, C+E=180°).6 tile primitive unit

In July 2017 Michaël Rao completed a computer assisted proof showing that there are no other types of convex pentagons that can tile the plane. The complete list of convex polygons that can tile the plane includes the above 15 pentagons, three types of hexagons, and all quadrilaterals and triangles. A consequence of this proof is that no convex polygon exists that tiles the plane only aperiodically, since all of the above types allow for a periodic tiling.

There are three isohedral pentagonal tilings generated as duals of the uniform tilings, those with 5 valence vertices. They represent special higher symmetry cases of the 15 monohedral tilings above. Uniform tilings and their duals are all edge to edge. These dual tilings are also called Laves tilings. The symmetry of the uniform dual tilings is the same as the uniform tilings. Because the uniform tilings are isogonal, the duals are isohedral. cmm (2*22)p4g (4*2)p6 (632)Prismatic pentagonal tilingInstance of type 1Floret pentagonal tilingInstance of types 1, 5 and 6120°, 120°, 120°, 90°, 90° V3.3.3.4.4120°, 120°, 90°, 120°, 90°V3.3.4.3.4120°, 120°, 120°, 120°, 60°V3.3.3.3.6

Pentagons have a peculiar relationship with hexagons. As demonstrated graphically below, some types of hexagons can be subdivided into pentagons. For example, a regular hexagon bisects into two type 1 pentagons. Subdivision of convex hexagons is also possible with three (type 3), four (type 4) and nine (type 3) pentagons. By extension of this relation, a plane can be tessellated by a single pentagonal prototile shape in ways that generate hexagonal overlays. For example:
Planar tessellation by a single pentagonal prototile (type 1) with overlays of regular hexagons (each comprising 2 pentagons). Planar tessellation by a single pentagonal prototile (type 3) with overlays of regular hexagons (each comprising 3 pentagons). Planar tessellation by a single pentagonal prototile (type 4) with overlays of semiregular hexagons (each comprising 4 pentagons). Planar tessellation by a single pentagonal prototile (type 3) with overlays of two sizes of regular hexagons (comprising 3 and 9 pentagons respectively).

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. 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 be extended to any non convex pentagon that has two consecutive angles adding to 2. It is possible to divide an equilateral triangle into three congruent non convex pentagons, meeting at the center of the triangle, and to tile the plane with the resulting three pentagon unit. A similar method can be used to subdivide squares into four congruent non convex pentagons, or regular hexagons into six congruent non convex pentagons, and then tile the plane with the resulting unit.

A dodecahedron can be considered a regular tiling of 12 pentagons on the surface of a sphere, with Schläfli symbol {5,3}, having three pentagons around each vertex. In the hyperbolic plane, there are tilings of regular pentagons, for instance order 4 pentagonal tiling, {5,4}, having four pentagons around each vertex. Higher order regular tilings {5,n} can be constructed on the hyperbolic plane, ending in {5,∞}. SphereHyperbolic plane{5,2}{5,3}{5,4}{5,5}{5,6}{5,7}{5,8} {5,∞}

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 tiling7 38 39 3 5 46 47 4 5 5V3.3.3.3.7V3.3.3.3.8V3.3.3.3.9 V3.3.4.3.5V3.3.4.3.6V3.3.4.3.7 V3.3.5.3.5

A version of the binary tiling, with its tiles bounded by hyperbolic line segments rather than arcs of horocycles, forms pentagonal tilings that must be non periodic, in the sense that their symmetry groups can be one dimensional but not two dimensional.