Hexagon
Regular hexagon  

A regular hexagon  
Type  Regular polygon 
Edges and vertices  6 
Schläfli symbol  {6}, t{3} 
Coxeter diagram 

Symmetry group  Dihedral (D_{6}), order 2×6 
Internal angle (degrees)  120° 
Dual polygon  Self 
Properties  Convex, cyclic, equilateral, isogonal, isotoxal 
In geometry, a hexagon (from Greek ἕξ hex, "six" and γωνία, gonía, "corner, angle") is a sixsided polygon or 6gon. The total of the internal angles of any simple (nonselfintersecting) hexagon is 720°.
Regular hexagon
A regular hexagon has Schläfli symbol {6}^{[1]} and can also be constructed as a truncated equilateral triangle, t{3}, which alternates two types of edges.
A regular hexagon is defined as a hexagon that is both equilateral and equiangular. It is bicentric, meaning that it is both cyclic (has a circumscribed circle) and tangential (has an inscribed circle).
The common length of the sides equals the radius of the circumscribed circle, which equals times the apothem (radius of the inscribed circle). All internal angles are 120 degrees. A regular hexagon has 6 rotational symmetries (rotational symmetry of order six) and 6 reflection symmetries (six lines of symmetry), making up the dihedral group D_{6}. The longest diagonals of a regular hexagon, connecting diametrically opposite vertices, are twice the length of one side. From this it can be seen that a triangle with a vertex at the center of the regular hexagon and sharing one side with the hexagon is equilateral, and that the regular hexagon can be partitioned into six equilateral triangles.
Like squares and equilateral triangles, regular hexagons fit together without any gaps to tile the plane (three hexagons meeting at every vertex), and so are useful for constructing tessellations. The cells of a beehive honeycomb are hexagonal for this reason and because the shape makes efficient use of space and building materials. The Voronoi diagram of a regular triangular lattice is the honeycomb tessellation of hexagons. It is not usually considered a triambus, although it is equilateral.
Parameters
The maximal diameter (which corresponds to the long diagonal of the hexagon), D, is twice the maximal radius or circumradius, R, which equals the side length, t. The minimal diameter or the diameter of the inscribed circle (separation of parallel sides, flattoflat distance, short diagonal or height when resting on a flat base), d, is twice the minimal radius or inradius, r. The maxima and minima are related by the same factor:
 and, similarly,
The area of a regular hexagon
For any regular polygon, the area can also be expressed in terms of the apothem a and the perimeter p. For the regular hexagon these are given by a = r, and p , so
The regular hexagon fills the fraction of its circumscribed circle.
If a regular hexagon has successive vertices A, B, C, D, E, F and if P is any point on the circumscribing circle between B and C, then PE + PF = PA + PB + PC + PD.
Symmetry
The regular hexagon has Dih_{6} symmetry, order 12. There are 3 dihedral subgroups: Dih_{3}, Dih_{2}, and Dih_{1}, and 4 cyclic subgroups: Z_{6}, Z_{3}, Z_{2}, and Z_{1}.
These symmetries express 9 distinct symmetries of a regular hexagon. John Conway labels these by a letter and group order.^{[2]} r12 is full symmetry, and a1 is no symmetry. d6, a isogonal hexagon constructed by four mirrors can alternate long and short edges, and p6, an isotoxal hexagon constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are duals of each other and have half the symmetry order of the regular hexagon. The i4 forms are regular hexagons flattened or stretched along one symmetry direction. It can be seen as an elongated rhombus, while d2 and p2 can be seen as horizontally and vertically elongated kites. g2 hexagons, with opposite sides parallel are also called hexagonal parallelogons.
Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g6 subgroup has no degrees of freedom but can seen as directed edges.
Example hexagons by symmetry  


Hexagons of symmetry g2, i4, and r12, as parallelogons can tessellate the Euclidean plane by translation. Other hexagon shapes can tile the plane with different orientations.
p6m (*632)  cmm (2*22)  p2 (2222)  p31m (3*3)  pmg (22*)  pg (××)  

r12 
i4 
g2 
d2 
d2 
p2 
a1 
A2 and G2 groups
A2 group roots 
G2 group roots 
The 6 roots of the simple Lie group A2, represented by a Dynkin diagram
The 12 roots of the Exceptional Lie group G2, represented by a Dynkin diagram
Dissection
6cube projection  10 rhomb dissection  

Coxeter states that every zonogon (a 2mgon whose opposite sides are parallel and of equal length) can be dissected into m(m1)/2 parallelograms.^{[3]} In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. This decomposition of a regular hexagon is based on a Petrie polygon projection of a cube, with 3 of 6 square faces. Other parallelogons and projective directions of the cube are dissected within rectangular cuboids.
Dissection of hexagons into 3 rhombs and parallelograms  

2D  Rhombs  Parallelograms  
Regular {6}  Hexagonal parallelogons  
3D  Square faces  Rectangular faces  
Cube  Rectangular cuboid 
Related polygons and tilings
A regular hexagon has Schläfli symbol {6}. A regular hexagon is a part the regular hexagonal tiling, {6,3}, with 3 hexagonal around each vertex.
A regular hexagon can also be created as a truncated equilateral triangle, with Schläfli symbol t{3}. Seen with two types (colors) of edges, this form only has D_{3} symmetry.
A truncated hexagon, t{6}, is a dodecagon, {12}, alternating 2 types (colors) of edges. An alternated hexagon, h{6}, is a equilateral triangle, {3}. A regular hexagon can be stellated with equilateral triangles on its edges, creating a hexagram. A regular hexagon can be dissected into 6 equilateral triangles by adding a center point. This pattern repeats within the regular triangular tiling.
A regular hexagon can be extended into a regular dodecagon by adding alternating squares and equilateral triangles around it. This pattern repeats within the rhombitrihexagonal tiling.
Regular {6} 
Truncated t{3} = {6} 
Hypertruncated triangles  Stellated Star figure 2{3} 
Truncated t{6} = {12} 
Alternated h{6} = {3} 

A concave hexagon  A selfintersecting hexagon (star polygon)  Dissected {6}  Extended Central {6} in {12} 
A skew hexagon, within cube 

Hexagonal structures
From bees' honeycombs to the Giant's Causeway, hexagonal patterns are prevalent in nature due to their efficiency. In a hexagonal grid each line is as short as it can possibly be if a large area is to be filled with the fewest number of hexagons. This means that honeycombs require less wax to construct and gain lots of strength under compression.
Irregular hexagons with parallel opposite edges are called parallelogons and can also tile the plane by translation. In three dimensions, hexagonal prisms with parallel opposite faces are called parallelohedrons and these can tessellate 3space by translation.
Form  Hexagonal tiling  Hexagonal prismatic honeycomb 

Regular  
Parallelogonal 
Tesselations by hexagons
In addition to the regular hexagon, which determines a unique tessellation of the plane, any irregular hexagon which satisfies the Conway criterion will tile the plane.
Hexagon inscribed in a conic section
Pascal's theorem (also known as the "Hexagrammum Mysticum Theorem") states that if an arbitrary hexagon is inscribed in any conic section, and pairs of opposite sides are extended until they meet, the three intersection points will lie on a straight line, the "Pascal line" of that configuration.
Cyclic hexagon
The Lemoine hexagon is a cyclic hexagon (one inscribed in a circle) with vertices given by the six intersections of the edges of a triangle and the three lines that are parallel to the edges that pass through its symmedian point.
If the successive sides of a cyclic hexagon are a, b, c, d, e, f, then the three main diagonals intersect in a single point if and only if ace = bdf.^{[4]}
If, for each side of a cyclic hexagon, the adjacent sides are extended to their intersection, forming a triangle exterior to the given side, then the segments connecting the circumcenters of opposite triangles are concurrent.^{[5]}
If a hexagon has vertices on the circumcircle of an acute triangle at the six points (including three triangle vertices) where the extended altitudes of the triangle meet the circumcircle, then the area of the hexagon is twice the area of the triangle.^{[6]}^{:p. 179}
Hexagon tangential to a conic section
Let ABCDEF be a hexagon formed by six tangent lines of a conic section. Then Brianchon's theorem states that the three main diagonals AD, BE, and CF intersect at a single point.
In a hexagon that is tangential to a circle and that has consecutive sides a, b, c, d, e, and f,^{[7]}
Equilateral triangles on the sides of an arbitrary hexagon
If an equilateral triangle is constructed externally on each side of any hexagon, then the midpoints of the segments connecting the centroids of opposite triangles form another equilateral triangle.^{[8]}^{:Thm. 1}
Skew hexagon
A skew hexagon is a skew polygon with 6 vertices and edges but not existing on the same plane. The interior of such an hexagon is not generally defined. A skew zigzag hexagon has vertices alternating between two parallel planes.
A regular skew hexagon is vertextransitive with equal edge lengths. In 3dimensions it will be a zigzag skew hexagon and can be seen in the vertices and side edges of a triangular antiprism with the same D_{3d}, [2^{+},6] symmetry, order 12.
The cube and octahedron (same as triangular antiprism) have regular skew hexagons as petrie polygons.
Cube 
Octahedron 
Petrie polygons
The regular skew hexagon is the Petrie polygon for these higher dimensional regular, uniform and dual polyhedra and polytopes, shown in these skew orthogonal projections:
4D  5D  

33 duoprism 
33 duopyramid 
5simplex 
Convex equilateral hexagon
A principal diagonal of a hexagon is a diagonal which divides the hexagon into quadrilaterals. In any convex equilateral hexagon (one with all sides equal) with common side a, there exists^{[9]}^{:p.184,#286.3} a principal diagonal d_{1} such that
and a principal diagonal d_{2} such that
Polyhedra with hexagons
There is no Platonic solid made of only regular hexagons, because the hexagons tessellate, not allowing the result to "fold up". The Archimedean solids with some hexagonal faces are the truncated tetrahedron, truncated octahedron, truncated icosahedron (of soccer ball and fullerene fame), truncated cuboctahedron and the truncated icosidodecahedron. These hexagons can be considered truncated triangles, with Coxeter diagrams of the form
Hexagons in Archimedean solids  

Tetrahedral  Octahedral  Icosahedral  
truncated tetrahedron 
truncated octahedron 
truncated cuboctahedron 
truncated icosahedron 
truncated icosidodecahedron 
There are other symmetry polyhedra with stretched or flattened hexagons, like these Goldberg polyhedron G(2,0):
Hexagons in Goldberg polyhedra  

Tetrahedral  Octahedral  Icosahedral  
Chamfered tetrahedron 
Chamfered cube 
Chamfered dodecahedron 
There are also 9 Johnson solids with regular hexagons:
Johnson solids and nearmisses with hexagons  

triangular cupola 
elongated triangular cupola 
gyroelongated triangular cupola  
augmented hexagonal prism 
parabiaugmented hexagonal prism 
metabiaugmented hexagonal prism 
triaugmented hexagonal prism  
augmented truncated tetrahedron 
triangular hebesphenorotunda 
Truncated triakis tetrahedron 
Prismoids with hexagons  

Hexagonal prism 
Hexagonal antiprism 
Hexagonal pyramid 
Tilings with regular hexagons  

Regular  1uniform  
{6,3} 
r{6,3} 
rr{6,3} 
tr{6,3}  
2uniform tilings  
Hexagons: natural and humanmade
 The ideal crystalline structure of graphene is a hexagonal grid.
 Assembled EELT mirror segments
 A beehive honeycomb
 The scutes of a turtle's carapace
 Micrograph of a snowflake
 Benzene, the simplest aromatic compound with hexagonal shape.
 Hexagonal order of bubbles in a foam.
 Crystal structure of a molecular hexagon composed of hexagonal aromatic rings reported by Müllen and coworkers in Chem. Eur. J., 2000, 18341839.
 Naturally formed basalt columns from Giant's Causeway in Northern Ireland; large masses must cool slowly to form a polygonal fracture pattern
 An aerial view of Fort Jefferson in Dry Tortugas National Park
 The James Webb Space Telescope mirror is composed of 18 hexagonal segments.
 Metropolitan France has a vaguely hexagonal shape. In French, l'Hexagone refers to the European mainland of France aka the "métropole" as opposed to the overseas territories such as Guadeloupe, Martinique or French Guiana.
 Hexagonal Hanksite crystal, one of many hexagonal crystal system minerals
 Hexagonal barn
 The Hexagon, a hexagonal theatre in Reading, Berkshire
 Władysław Gliński's hexagonal chess
 Pavilion in the Taiwan Botanical Gardens
 Hexagonal window
See also
 24cell: a fourdimensional figure which, like the hexagon, has orthoplex facets, is selfdual and tessellates Euclidean space
 Hexagonal crystal system
 Hexagonal number
 Hexagonal tiling: a regular tiling of hexagons in a plane
 Hexagram: 6sided star within a regular hexagon
 Unicursal hexagram: single path, 6sided star, within a hexagon
 Honeycomb conjecture
References
 ↑ Wenninger, Magnus J. (1974), Polyhedron Models, Cambridge University Press, p. 9, ISBN 9780521098595 .
 ↑ John H. Conway, Heidi Burgiel, Chaim GoodmanStrauss, (2008) The Symmetries of Things, ISBN 9781568812205 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275278)
 ↑ Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141
 ↑ Cartensen, Jens, "About hexagons", Mathematical Spectrum 33(2) (2000–2001), 37–40.
 ↑ Dergiades, Nikolaos (2014). "Dao's theorem on six circumcenters associated with a cyclic hexagon". Forum Geometricorum. 14: 243–246.
 ↑ Johnson, Roger A., Advanced Euclidean Geometry, Dover Publications, 2007 (orig. 1960).
 ↑ Gutierrez, Antonio, "Hexagon, Inscribed Circle, Tangent, Semiperimeter", , Accessed 20120417.
 ↑ Dao Thanh Oai (2015). "Equilateral triangles and Kiepert perspectors in complex numbers". Forum Geometricorum. 15: 105–114.
 ↑ Inequalities proposed in “Crux Mathematicorum”, .
External links
Look up hexagon in Wiktionary, the free dictionary. 
 Definition and properties of a hexagon with interactive animation and construction with compass and straightedge.
 Cymatics – Hexagonal shapes occurring within water sound images
 Cassini Images Bizarre Hexagon on Saturn
 Saturn's Strange Hexagon
 A hexagonal feature around Saturn's North Pole
 "Bizarre Hexagon Spotted on Saturn" – from Space.com (27 March 2007)