Hexagon
Regular hexagon  

A regular hexagon


Type  Regular polygon 
Edges and vertices  6 
Schläfli symbol  {6} 
Coxeter diagram  
Symmetry group  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 polygon with six edges and six vertices. A regular hexagon has Schläfli symbol {6}. The total of the internal angles of any hexagon is 720°.
Contents
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.
Regular hexagon
A regular hexagon has all sides of the same length, and that length is also the radius of the circumscribed 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.
The area of a regular hexagon of side length t is given by
 A = \frac{3 \sqrt{3}}{2}t^2 \simeq 2.598076211 t^2.
An alternative formula for area is A = 1.5dt where the length d is the distance between the parallel sides (also referred to as the flattoflat distance), or the height of the hexagon when it sits on one side as base, or the diameter of the inscribed circle.
Another alternative formula for the area if only the flattoflat distance, d, is known, is given by
 A = \frac{ \sqrt{3}}{2} d^2 \simeq 0.866025404d^2.
The area can also be found by the formulas A=ap/2 and \scriptstyle A\ =\ {2}a^2\sqrt{3}\ \simeq\ 3.464102 a^2, where a is the apothem and p is the perimeter.
The perimeter of a regular hexagon of side length t is 6t, its maximal diameter 2t, and its minimal diameter \scriptstyle d\ =\ t\sqrt{3}.
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.
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.^{[1]}
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.^{[2]}
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,^{[3]}
 a+c+e=b+d+f.
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^{[4]}^{:p.184,#286.3} a principal diagonal d_{1} such that
 \frac{d_1}{a} \leq 2
and a principal diagonal d_{2} such that
 \frac{d_2}{a} > \sqrt{3}.
Related figures
A regular hexagon can also be created as a truncated equilateral triangle, with Schläfli symbol t{3}. This form only has D_{3} symmetry. In this figure, the remaining edges of the original triangle are drawn blue, and new edges from the truncation are red. 
The hexagram can be created as a stellation process: extending the 6 edges of a regular hexagon until they meet at 6 new vertices. 
A concave hexagon 
A selfintersecting hexagon (star polygon) 
A (nonplanar) skew regular hexagon, within the edges of a cube 
Petrie polygons
The regular hexagon is the Petrie polygon for these regular and uniform polytopes, shown in these skew orthogonal projections:
(3D)  (4D)  (5D)  

Cube 
Octahedron 
33 duopyramid 
33 duoprism 
5simplex 
Rectified 5simplex 
Birectified 5simplex 
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 and .
Tetrahedral  Octahedral  Icosahedral  

truncated tetrahedron 
truncated octahedron 
truncated cuboctahedron 
truncated icosahedron 
truncated icosidodecahedron 
Tetrahedral  Octahedral  Icosahedral 

Chamfered tetrahedron 
Chamfered cube 
Chamfered dodecahedron 
Hexagonal prism 
Hexagonal antiprism 
Hexagonal pyramid 
Truncated triakis tetrahedron 
Regular and uniform tilings with hexagons
The hexagon can form a regular tessellate the plane with a Schläfli symbol {6,3}, having 3 hexagons around every vertex. 
A second hexagonal tessellation of the plane can be formed as a truncated triangular tiling or rhombille tiling, with one of three hexagons colored differently. 
A third tessellation of the plane can be formed with three colored hexagons around every vertex. 
The hexagonal tiling can be distorted, like these centrosymmetric hexagons 
Trihexagonal tiling 
Trihexagonal tiling 

Rhombitrihexagonal tiling 
Truncated trihexagonal tiling 
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 "metropole" 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

Asian pavilion.jpg
See also
 24cell: a fourdimensional figure which, like the hexagon, has orthoplex facets and is selfdual
 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
References
 ^ Cartensen, Jens, "About hexagons", Mathematical Spectrum 33(2) (2000–2001), 37–40.
 ^ Nikolaos Dergiades, "Dao's theorem on six circumcenters associated with a cyclic hexagon", Forum Geometricorum 14, 2014, 243246. http://forumgeom.fau.edu/FG2014volume14/FG201424index.html
 ^ Gutierrez, Antonio, "Hexagon, Inscribed Circle, Tangent, Semiperimeter", [4], Accessed 20120417.
 ^ Inequalities proposed in “Crux Mathematicorum”, [5].
External links
 Weisstein, Eric W., "Hexagon", MathWorld.
 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)
 supraHex A suprahexagonal map for analysing highdimensional omics data.
