An example pentagonal tile with angle labels A,B,C,D, and E, and edge length labels a,b,c,d, and e
Fifteen types of convex pentagons are known to tile the plane monohedrally (i.e. with one type of tile).[2] The most recent one was discovered in 2015. This list has been shown to be complete by Rao (2017) (result subject to peer-review). Bagina (2011) showed that there are only eight edge-to-edge convex types, a result obtained independently by Sugimoto (2012).
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.[3] As of 11 July 2017, the first half of Rao's proof had been independently verified (computer code available[4]) by Thomas Hales, a professor of mathematics at the University of Pittsburgh.[5] 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 tiles
1
2
3
4
5
B + C = 180° A + D + E = 360°
c = e B + D = 180°
a = b, d = c + e A = C = D = 120°
b = c, d = e B = D = 90°
a = b, d = e A = 60°, D = 120°
6
7
8
9
10
a = d = e, b = c B + D = 180°, 2B = E
b = c = d = e B + 2E = 2C + D = 360°
b = c = d = e 2B + C = D + 2E = 360°
b = c = d = e 2A + C = D + 2E = 360°
a = b = c + e A = 90°, B + E = 180° B + 2C = 360°
11
12
13
14
15
2a + c = d = e A = 90°, C + E = 180° 2B + C = 360°
2a = d = c + e A = 90°, C + E = 180° 2B + C = 360°
d = 2a = 2e B = E = 90° 2A + D = 360°
2a = 2c = d = e A = 90°, B ≈ 145.34°, C ≈ 69.32° D ≈ 124.66°, E ≈ 110.68° (2B + C = 360°, C + E = 180°)
a = c = e, b = 2a A = 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.
Reinhardt (1918)
Reinhardt (1918) 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 groupacts 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.[6] 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.[7]
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.
Type 1
There are many tiling topologies that contain type 1 pentagons. Five example topologies are given below.
d = c + e A = 90°, 2B + C = 360° C + D = 180°, B + E = 270°
Type 2
These type 2 examples are isohedral. The second is an edge-to-edge variation. They both have pgg (22×) symmetry. If mirror image tiles (yellow and green) are considered distinct, the symmetry is p2 (2222).
a = b = c, d = e A = 60°, B = 120°, C = 90° D = 120°, E = 150°
Kershner (1968) Types 6, 7, 8
Kershner (1968) found three more types of pentagonal tile, bringing the total to eight. He claimed incorrectly that this was the complete list of pentagons that can tile the plane.
These examples are 2-isohedral and edge-to-edge. Types 7 and 8 have chiral pairs of tiles, which are colored as pairs in yellow-green and the other as two shades of blue. The pgg symmetry is reduced to p2 when chiral pairs are considered distinct.
Type 6
Type 6 (Also type 5)
Type 7
Type 8
p2 (2222)
pgg (22×)
pgg (22×)
p2 (2222)
p2 (2222)
a = d = e, b = c B + D = 180°, 2B = E
a = d = e, b = c, B = 60° A = C = D = E = 120°
b = c = d = e B + 2E = 2C + D = 360°
b = c = d = e 2B + C = D + 2E = 360°
4-tile primitive unit
4-tile primitive unit
8-tile primitive unit
8-tile primitive unit
James (1975) Type 10
In 1975 Richard E. James III found a ninth type, after reading about Kershner's results in Martin Gardner's "Mathematical Games" column in Scientific American magazine of July 1975 (reprinted in Gardner (1988)).[8] It is indexed as type 10. The tiling is 3-isohedral and non-edge-to-edge.
All four tilings are 2-isohedral. The chiral pairs of tiles are colored in yellow and green for one isohedral set, and two shades of blue for the other set. The pgg symmetry is reduced to p2 when the chiral pairs are considered distinct.
The tiling by type 9 tiles is edge-to-edge, but the others are not.
Each primitive unit contains eight tiles.
Type 9
Type 11
Type 12
Type 13
pgg (22×)
p2 (2222)
b=c=d=e 2A+C=D+2E=360°
2a+c=d=e A=90°, 2B+C=360° C+E=180°
2a=d=c+e A=90°, 2B+C=360° C+E=180°
d=2a=2e B=E=90°, 2A+D=360°
8-tile primitive unit
8-tile primitive unit
8-tile primitive unit
8-tile primitive unit
Stein (1985) Type 14
A 14th convex pentagon type was found by Rolf Stein in 1985.[10]
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 Bobtuse with . Other relations can easily be deduced.
The primitive units contain six tiles respectively. It has p2 (2222) symmetry.
University of Washington Bothell mathematicians Casey Mann, Jennifer McLoud-Mann, and David Von Derau discovered a 15th monohedral tiling convex pentagon in 2015 using a computer algorithm.[11][12] It is 3-isohedral and non-edge-to-edge, drawn with 6 colors, 2 shades of 3 colors, representing chiral pairs of the three isohedral positions. The pgg symmetry is reduced to p2 when the chiral pairs are considered distinct. It has completely determined tiles, with no degrees of freedom. The primitive units contain twelve tiles. It has pgg (22×) symmetry, and p2 (2222) if chiral pairs are considered distinct.
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.[5] 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.
Nonperiodic monohedral pentagonal tilings
Nonperiodic monohedral pentagonal tilings can also be constructed, like the example below with 6-fold rotational symmetry by Michael Hirschhorn. Angles are A = 140°, B = 60°, C = 160°, D = 80°, E = 100°.[13][14]
In 2016 it could be shown by Bernhard Klaassen that every discrete rotational symmetry type can be represented by a monohedral pentagonal tiling from the same class of pentagons.[15] Examples for 5-fold and 7-fold symmetry are shown below. Such tilings are possible for any type of n-fold rotational symmetry with n>2.
5-fold rotational symmetry in a monohedral pentagonal tiling
7-fold rotational symmetry in a monohedral pentagonal tiling
Dual uniform tilings
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.
The k-uniform tilings with valence-5 vertices also have pentagonal dual tilings, containing the same three shaped pentagons as the semiregular duals above, but contain a mixture of pentagonal types. A k-uniform tiling has a k-isohedral dual tiling and are represented by different colors and shades of colors below.
For example these 2, 3, 4, and 5-uniform duals are all pentagonal:[18][19]
2-isohedral
3-isohedral
p4g (4*2)
pgg (22×)
p2 (2222)
p6 (*632)
4-isohedral
5-isohedral
pgg (22×)
p2 (2222)
p6m (*632)
5-isohedral
pgg (22×)
p2 (2222)
Pentagonal/hexagonal tessellation
Pentagonal subdivisions of a hexagon
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).
Non-convex pentagons
Periodic tiling by the sphinx
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.[20] The sphinx may also tile the plane periodically, by fitting two sphinx tiles together to form a parallelogram and then tiling the plane by translation of this parallelogram,[20] a pattern that can be extended to any non-convex pentagon that has two consecutive angles adding to 2π.
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.[21] 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.
In non-Euclidean geometry
Spherical tiling
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.
One may also consider a degenerate tiling by two hemispheres, with the great circle between them subdivided into five equal arcs, as a pentagonal tiling with Schläfli symbol {5,2}.
Regular hyperbolic tilings
In the hyperbolic plane, one can construct regular pentagons that have any interior angle for . The resulting pentagons tile the plane regularly, with pentagons around each vertex. For instance, the order-4 pentagonal tiling, {5,4}, has four right-angled pentagons around each vertex. A limiting case is the infinite-order pentagonal tiling {5,∞} produced by ideal regular pentagons. These pentagons have ideal points as their vertices, with angle equal to zero.
Polygonal hyperbolic binary tiling with 60-120-60-120-120-degree pentagons
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.[22]
Related Research Articles
In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or abstract polyhedra, but not all can also be constructed as geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron.
In Euclidean geometry, a kite is a quadrilateral with reflection symmetry across a diagonal. Because of this symmetry, a kite has two equal angles and two pairs of adjacent equal-length sides. Kites are also known as deltoids, but the word deltoid may also refer to a deltoid curve, an unrelated geometric object sometimes studied in connection with quadrilaterals. A kite may also be called a dart, particularly if it is not convex.
A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of geometries.
In geometry, a polytope or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in the same or reverse order, and with the same angles between corresponding faces.
Euclidean plane tilings by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler in his Harmonices Mundi.
In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of {6,3} or t{3,6} .
In geometry, the snub hexagonal tiling is a semiregular tiling of the Euclidean plane. There are four triangles and one hexagon on each vertex. It has Schläfli symbol sr{3,6}. The snub tetrahexagonal tiling is a related hyperbolic tiling with Schläfli symbol sr{4,6}.
In geometry, the elongated triangular tiling is a semiregular tiling of the Euclidean plane. There are three triangles and two squares on each vertex. It is named as a triangular tiling elongated by rows of squares, and given Schläfli symbol {3,6}:e.
In geometry, the rhombille tiling, also known as tumbling blocks, reversible cubes, or the dice lattice, is a tessellation of identical 60° rhombi on the Euclidean plane. Each rhombus has two 60° and two 120° angles; rhombi with this shape are sometimes also called diamonds. Sets of three rhombi meet at their 120° angles, and sets of six rhombi meet at their 60° angles.
In geometry, a Cairo pentagonal tiling is a tessellation of the Euclidean plane by congruent convex pentagons, formed by overlaying two tessellations of the plane by hexagons and named for its use as a paving design in Cairo. It is also called MacMahon's net after Percy Alexander MacMahon, who depicted it in his 1921 publication New Mathematical Pastimes. John Horton Conway called it a 4-fold pentille.
In geometry, a vertex configuration is a shorthand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex. For uniform polyhedra there is only one vertex type and therefore the vertex configuration fully defines the polyhedron.
In geometry, a tessellation of dimension 2 or higher, or a polytope of dimension 3 or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent but must be transitive, i.e. must lie within the same symmetry orbit. In other words, for any two faces A and B, there must be a symmetry of the entire figure by translations, rotations, and/or reflections that maps A onto B. For this reason, convex isohedral polyhedra are the shapes that will make fair dice.
In geometry, a polytope or a tiling is isotoxal or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given two edges, there is a translation, rotation, and/or reflection that will move one edge to the other while leaving the region occupied by the object unchanged.
In geometry, a shape is said to be anisohedral if it admits a tiling, but no such tiling is isohedral (tile-transitive); that is, in any tiling by that shape there are two tiles that are not equivalent under any symmetry of the tiling. A tiling by an anisohedral tile is referred to as an anisohedral tiling.
A Pythagorean tiling or two squares tessellation is a tiling of a Euclidean plane by squares of two different sizes, in which each square touches four squares of the other size on its four sides. Many proofs of the Pythagorean theorem are based on it, explaining its name. It is commonly used as a pattern for floor tiles. When used for this, it is also known as a hopscotch pattern or pinwheel pattern, but it should not be confused with the mathematical pinwheel tiling, an unrelated pattern.
In geometry, a plesiohedron is a special kind of space-filling polyhedron, defined as the Voronoi cell of a symmetric Delone set. Three-dimensional Euclidean space can be completely filled by copies of any one of these shapes, with no overlaps. The resulting honeycomb will have symmetries that take any copy of the plesiohedron to any other copy.
In geometry, a planigon is a convex polygon that can fill the plane with only copies of itself. In the Euclidean plane there are 3 regular planigons; equilateral triangle, squares, and regular hexagons; and 8 semiregular planigons; and 4 demiregular planigons which can tile the plane only with other planigons.
References
↑ Chung, Ping Ngai; Fernandez, Miguel A.; Li, Yifei; Mara, Michael; Morgan, Frank; Plata, Isamar Rosa; Shah, Nirlee; Vieira, Luis Sordo; Wikner, Elena (2012-05-01), "Isoperimetric Pentagonal Tilings", Notices of the American Mathematical Society, 59 (5): 632, doi:10.1090/noti838, ISSN0002-9920
1 2 3 Reinhardt 1918, pp.77–81 (caution: there is at least one obvious mistake within this paper, i.e. γ+δ angle sum needs to equal π, not 2π for the first two tiling types defined on page 77)
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