Hexagon

Regular hexagon
A regular hexagon
Type	Regular polygon
Edges and vertices	6
Schläfli symbol	{6}, t{3}
Coxeter diagram
Symmetry group	Dihedral (D6), 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 6-sided polygon or 6-gon. The total of the internal angles of any 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 $\tfrac{2\sqrt{3}}{3}$ 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 D₆. 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 area of a regular hexagon of side length t is given by

$A = \frac{3 \sqrt{3}}{2}t^2 \simeq 2.598 t^2.$

An alternative formula for the area is

$A=\frac{3}{2}d \cdot t$

where the length d is the distance between the parallel sides (also referred to as the flat-to-flat 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 flat-to-flat distance, d, is known, is given by

$A = \frac{ \sqrt{3}}{2} d^2 \simeq 0.866d^2.$

The area can also be found by the formulas

$A=ap/2$

and

$A\ =\ {2}a^2\sqrt{3}\ \simeq\ 3.464 a^2,$

where a is the apothem and p is the perimeter.

The regular hexagon fills the fraction $\tfrac{3\sqrt{3}}{2\pi}\approx 0.8270$ of its circumscribed circle.

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.

Symmetry

The six lines of reflection of a regular hexagon, with Dih₆ or r12 symmetry, order 12.

The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars) Cyclic symmetries in the middle column are labeled as g for their central gyration orders. Full symmetry of the regular form is r12 and no symmetry is labeled a1.

The regular hexagon has Dih₆ symmetry, order 12. There are 3 dihedral subgroups: Dih₃, Dih₂, and Dih₁, and 4 cyclic subgroups: Z₆, Z₃, Z₂, and Z₁.

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
r12 regular i4 d6 isotoxal g6 directed p6 isogonal d2 g2 general parallelogon p2 g3 a1

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

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 D₃ symmetry.

A truncated hexagon, t{6}, is an 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 self-intersecting hexagon (star polygon)	Dissected {6}	Extended Central {6} in {12}	A skew hexagon, within cube

Hexagonal structures

Giants causeway closeup

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 3-space by translation.

Hexagonal prism tessellations

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.^[3]

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.^[4]

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.^{[5]: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,^[6]

$a+c+e=b+d+f.$

Equilateral triangles on the sides of an arbitrary hexagon

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.^{[7]:Thm. 1}

Skew hexagon

A regular skew hexagon seen as edges of a triangular antiprism, symmetry D_3d, [2⁺,6], (2*3), order 12.

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 zig-zag hexagon has vertices alternating between two parallel planes.

A regular skew hexagon is vertex-transitive with equal edge lengths. In 3-dimensions it will be a zig-zag 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.

Skew hexagons on 3-fold axes

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
3-3 duoprism	3-3 duopyramid	5-simplex

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^{[8]:p.184,#286.3} a principal diagonal d₁ such that

$\frac{d_1}{a} \leq 2$

and a principal diagonal d₂ such that

$\frac{d_2}{a} > \sqrt{3}.$

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 .

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 near-misses 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 1-uniform {6,3} r{6,3} rr{6,3} tr{6,3} 2-uniform tilings

Hexagons: natural and human-made

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  The ideal crystalline structure of graphene is a hexagonal grid.

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  Assembled E-ELT mirror segments

• 

  A beehive honeycomb

• 

  The scutes of a turtle's carapace

• 

  North polar hexagonal cloud feature on Saturn, discovered by Voyager 1 and confirmed in 2006 by Cassini [1] [2] [3]

• 

  Micrograph of a snowflake

• 

  Benzene, the simplest aromatic compound with hexagonal shape.

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  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, 1834-1839.

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  Naturally formed basalt columns from Giant's Causeway in Northern Ireland; large masses must cool slowly to form a polygonal fracture pattern

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  An aerial view of Fort Jefferson in Dry Tortugas National Park

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  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.

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  Hexagonal Hanksite crystal, one of many hexagonal crystal system minerals

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  Hexagonal barn

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  The Hexagon, a hexagonal theatre in Reading, Berkshire

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  Władysław Gliński's hexagonal chess

• 

  Asian pavilion.jpg

See also

• 24-cell: a four-dimensional figure which, like the hexagon, has orthoplex facets, is self-dual and tessellates Euclidean space
• Hexagonal crystal system
• Hexagonal number
• Hexagonal tiling: a regular tiling of hexagons in a plane
• Hexagram: 6-sided star within a regular hexagon
• Unicursal hexagram: single path, 6-sided star, within a hexagon

References

1. Wenninger, Magnus J. (1974), Polyhedron Models, Cambridge University Press, p. 9, ISBN 9780521098595.
2. John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, ISBN 978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278)
3. Cartensen, Jens, "About hexagons", Mathematical Spectrum 33(2) (2000–2001), 37–40.
4. Nikolaos Dergiades, "Dao's theorem on six circumcenters associated with a cyclic hexagon", Forum Geometricorum 14, 2014, 243--246. http://forumgeom.fau.edu/FG2014volume14/FG201424index.html
5. Johnson, Roger A., Advanced Euclidean Geometry, Dover Publications, 2007 (orig. 1960).
6. Gutierrez, Antonio, "Hexagon, Inscribed Circle, Tangent, Semiperimeter", , Accessed 2012-04-17.
7. Dao Thanh Oai (2015), "Equilateral triangles and Kiepert perspectors in complex numbers", Forum Geometricorum 15, 105--114. http://forumgeom.fau.edu/FG2015volume15/FG201509index.html
8. Inequalities proposed in “Crux Mathematicorum”, .

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 supra-hexagonal map for analysing high-dimensional omics data.