![P5-type15-chiral coloring](https://upload.wikimedia.org/wikipedia/commons/c/cb/P5-type15-chiral_coloring.png)
In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a pentagonal tiling is a
tiling of the plane where each individual piece is in the shape of a
pentagon
In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.
A pentagon may be simpl ...
.
A
regular pentagonal
In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.
A pentagon may be simpl ...
tiling on the
Euclidean plane
In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...
is impossible because the
internal angle
In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called an interior angle (or ) if ...
of a
regular pentagon
In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.
A pentagon may be simpl ...
, 108°, is not a divisor of 360°, the angle measure of a whole
turn. However, regular pentagons can tile the
hyperbolic plane
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P'' ...
with
four pentagons around each vertex (
or more) and
sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
with
three pentagons; the latter produces a tiling topologically equivalent to the
dodecahedron
In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...
.
Monohedral convex pentagonal tilings
![Monohedral_pentagonal_tiling_labels](https://upload.wikimedia.org/wikipedia/commons/3/37/Monohedral_pentagonal_tiling_labels.png)
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
The École normale supérieure de Lyon (also known as ENS de Lyon, ENSL or Normale Sup' Lyon) is a French grande école located in the city of Lyon. It is one of the four prestigious écoles normales supérieures in France. The school is ...
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 angle
In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called an interior angle (or ) if ...
s 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
A wallpaper is a mathematical object covering a whole Euclidean plane by repeating a motif indefinitely, in manner that certain isometries keep the drawing unchanged. To a given wallpaper there corresponds a group of such congruent transformatio ...
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)
found the first five types of pentagonal tile. All five can create
isohedral
In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent ...
tilings, meaning that the symmetries of the tiling can take any tile to any other tile (more formally, the
automorphism group
In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
acts transitively
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
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
A wallpaper is a mathematical object covering a whole Euclidean plane by repeating a motif indefinitely, in manner that certain isometries keep the drawing unchanged. To a given wallpaper there corresponds a group of such congruent transformatio ...
symmetry for each tiling is given, with
orbifold notation
In geometry, orbifold notation (or orbifold signature) is a system, invented by the mathematician William Thurston and promoted by John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curvature. The advanta ...
in parentheses. A second lower symmetry group is given if tile
chirality
Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object.
An object or a system is ''chiral'' if it is distinguishable from ...
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.
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).
Types 3, 4, and 5
Kershner (1968) Types 6, 7, 8
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.
James (1975) Type 10
In 1975 Richard E. James III found a ninth type, after reading about Kershner's results in
Martin Gardner
Martin Gardner (October 21, 1914May 22, 2010) was an American popular mathematics and popular science writer with interests also encompassing scientific skepticism, micromagic, philosophy, religion, and literatureespecially the writings of Lewis ...
's "
Mathematical Games
A mathematical game is a game whose rules, strategies, and outcomes are defined by clear mathematical parameters. Often, such games have simple rules and match procedures, such as Tic-tac-toe and Dots and Boxes. Generally, mathematical games ne ...
" column in ''
Scientific American
''Scientific American'', informally abbreviated ''SciAm'' or sometimes ''SA'', is an American popular science magazine. Many famous scientists, including Albert Einstein and Nikola Tesla, have contributed articles to it. In print since 1845, it i ...
'' magazine of July 1975 (reprinted in ). It is indexed as type 10. The tiling is 3-isohedral and non-edge-to-edge.
Rice (1977) Types 9,11,12,13
Marjorie Rice
Marjorie Ruth Rice (née Jeuck) (1923–2017) was an American amateur mathematician most famous for her discoveries of pentagonal tilings in geometry.
Background
Rice was born February 16, 1923, in St. Petersburg, Florida.
Marjorie Rice was a ...
, an amateur mathematician, discovered four new types of
tessellating
A tessellation or tiling is the covering of a surface, often a plane (mathematics), plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to high-dimensional ...
pentagons in 1976 and 1977.
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.
Stein (1985) Type 14
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.
Mann/McLoud/Von Derau (2015) Type 15
University of Washington Bothell
The University of Washington Bothell (UW Bothell) is a branch campus of University of Washington in Bothell, Washington. It was founded in 1989 and is located just northwest of the junction of Interstate 405 and State Route 522, and it shares ...
mathematicians
Casey Mann
Casey Mann is an American mathematician, specializing in discrete and computational geometry, in particular tessellation and knot theory. He is Professor of Mathematics at University of Washington Bothell, and received the PhD at the University ...
,
Jennifer McLoud-Mann
Jennifer McLoud-Mann is an American mathematician known for her 2015 discovery, with Casey Mann and undergraduate student David Von Derau, of the 15th and last class of convex pentagons to tile the plane. She is a professor of mathematics at th ...
, and David Von Derau discovered a 15th monohedral tiling convex pentagon in 2015 using a
computer algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ca ...
. 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 respectively. 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. 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 pentagons tilings
Nonperiodic monohedral pentagonal tilings can also be constructed, like the example below with 6-fold
rotational symmetry
Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which i ...
by Michael Hirschhorn. Angles are A = 140°, B = 60°, C = 160°, D = 80°, E = 100°.
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. 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.
Dual uniform tilings
There are three
isohedral
In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent ...
pentagonal tilings generated as
duals
''Duals'' is a compilation album by the Irish rock band U2. It was released in April 2011 to u2.com subscribers.
Track listing
:* "Where the Streets Have No Name" and "Amazing Grace" are studio mix of U2's performance at the Rose Bowl, Pas ...
of the
uniform tiling
In geometry, a uniform tiling is a tessellation of the plane by regular polygon faces with the restriction of being vertex-transitive.
Uniform tilings can exist in both the Euclidean plane and hyperbolic plane. Uniform tilings are related to the f ...
s, 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 tiling
This table shows the 11 convex uniform tilings (regular and semiregular) of the Euclidean plane, and their dual tilings.
There are three regular and eight semiregular tilings in the plane. The semiregular tilings form new tilings from their dua ...
s. The symmetry of the uniform dual tilings is the same as the uniform tilings. Because the uniform tilings are
isogonal, the duals are
isohedral
In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent ...
.
Dual ''k''-uniform tilings
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:
Pentagonal/hexagonal tessellation
![Pentagonal Tessellation of Hexagons](https://upload.wikimedia.org/wikipedia/commons/b/b2/Pentagonal_Tessellation_of_Hexagons.png)
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:
Non-convex pentagons
![Sphinx tiling pg a](https://upload.wikimedia.org/wikipedia/commons/7/70/Sphinx_tiling_pg_a.svg)
With pentagons that are not required to be
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytope ...
, additional types of tiling are possible. An example is the
sphinx tiling
In geometry, the sphinx tiling is a tessellation of the plane using the "sphinx", a pentagonal hexiamond formed by gluing six equilateral triangles together. The resultant shape is named for its reminiscence to the Great Sphinx at Giza. A sphin ...
, an
aperiodic tiling
An aperiodic tiling is a non-periodic tiling with the additional property that it does not contain arbitrarily large periodic regions or patches. A set of tile-types (or prototiles) is aperiodic if copies of these tiles can form only non- period ...
formed by a pentagonal
rep-tile
In the geometry of tessellations, a rep-tile or reptile is a shape that can be dissected into smaller copies of the same shape. The term was coined as a pun on animal reptiles by recreational mathematician Solomon W. Golomb and popularized by ...
. The sphinx may also tile the plane periodically, by fitting two sphinx tiles together to form a
parallelogram
In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equa ...
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
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each othe ...
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
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...
s into four congruent non-convex pentagons, or
regular hexagons into six congruent non-convex pentagons, and then tile the plane with the resulting unit.
Regular pentagonal tilings in non-Euclidean geometry
A
dodecahedron
In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...
can be considered a regular tiling of 12 pentagons on the surface of a
sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
, with
Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more ...
, having three pentagons around each vertex.
In the
hyperbolic plane
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P'' ...
, there are tilings of regular pentagons, for instance
order-4 pentagonal tiling
In geometry, the order-4 pentagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of . It can also be called a pentapentagonal tiling in a bicolored quasiregular form.
Symmetry
This tiling represents a hyperbolic ...
, , having four pentagons around each vertex. Higher order regular tilings can be constructed on the hyperbolic plane, ending in .
Irregular hyperbolic plane pentagonal tilings
There are an infinite number of dual
uniform tilings in hyperbolic plane
In hyperbolic geometry, a uniform hyperbolic tiling (or regular, quasiregular or semiregular hyperbolic tiling) is an edge-to-edge filling of the hyperbolic plane which has regular polygons as faces and is vertex-transitive ( transitive on its v ...
with isogonal irregular pentagonal faces. They have
s as V3.3.''p''.3.''q''.
![Binary tiling straight](https://upload.wikimedia.org/wikipedia/commons/3/39/Binary_tiling_straight.svg)
The
binary tiling
In geometry, the binary tiling (sometimes called the Böröczky tiling) is a tiling of the hyperbolic plane, resembling a quadtree over the Poincaré half-plane model of the hyperbolic plane. It was first studied mathematically in 1974 by . How ...
can be made into a pentagonal tiling if one replaces the horocyclic edges by line segments.
References
Bibliography
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*
Errata ''Forma'' 25 (1): 49, 2010,
*
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External links
*
Pentagon Tilings15 (monohedral) Tilings with a convex pentagonal tilewith k-isohedral colorings
Code to display the 14th pentagon type tilingCode to display the 15th pentagon type tiling
{{Tessellation
Tessellation