Hexagon

For other uses, see Hexagon (disambiguation).

Template:Regular polygon db In geometry, the hexagon (from Greek ἕξ hex, "six" and γωνία, gonía, "corner, angle") is the polygon with six edges and six vertices. The total of the internal angles of any hexagon is 720°.

Hexagonal structures

From bees' honeycombs to the Giant's Causeway, hexagonal patterns are prevalent in nature due to air efficiency. In the hexagonal grid each line is as short as it can possibly be if the 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

File:Regular Hexagon Inscribed in the Circle 240px.gif

A step-by-step animation of the construction of the regular hexagon using compass and straightedge, given by Euclid's Elements, Book IV, Proposition 15.

The regular hexagon has the number of subsymmetries that can be seen by coloring or geometric variations

A regular hexagon is defined as the hexagon that is both equilateral and equiangular. It is bicentric, meaning that it is both cyclic (has the 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 the regular hexagon, connecting diametrically opposite vertices, are twice the length of one side. From this it can be seen that the triangle with the 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 the beehive honeycomb are hexagonal for this reason and because the shape makes efficient use of space and building materials. The Voronoi diagram of the regular triangular lattice is the honeycomb tessellation of hexagons. It is not usually considered the triambus, although it is equilateral.

Parameters

The area of the regular hexagon of side length t is given by

$A={\frac {3{\sqrt {3}}}{2}}t^{2}\simeq 2.598076211t^{2}.$

An alternative formula for the area is

$A={\frac {3}{2}}dt$

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

$A={\frac {\sqrt {3}}{2}}d^{2}\simeq 0.866025404d^{2}.$

The area can also be found by the formulas

$A=ap/2$

and

$A\ =\ {2}a^{2}{\sqrt {3}}\ \simeq \ 3.464102a^{2},$

where a is the apothem and p is the perimeter.

The perimeter of the regular hexagon of side length t is 6t, its maximal diameter 2t, and its minimal diameter $\scriptstyle d\ =\ t{\sqrt {3}}$.

If the 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, an PE + PF = PA + PB + PC + PD.

Related polygons and tilings

A regular hexagon has Schläfli symbol {6}. A regular hexagon is the part the regular hexagonal tiling, {6,3}, with 3 hexagonal around each vertex.

A regular hexagon can also be created as the 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}, aternating 2 types (colors) of edges. An alternated hexagon, h{6} is the equilateral triangle, {3}. A regular hexagon can be stellated with equilateral triangles on its edges, creating the hexagram. A regular hexagon can be dissected into 6 equilateral triangles by adding the center point. This pattern repeats within the regular triangular tiling.

A regular hexagon can be extended into the 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

Tesselations by hexagons

In addition to the regular hexagon, which determines the unique tessellation of the plane, any irregular hexagon which satisfies the Conway criterion will tile the plane.

Hexagon inscribed in the conic section

Pascal's aorem (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 ay meet, the three intersection points will lie on the straight line, the "Pascal line" of that configuration.

Cyclic hexagon

The Lemoine hexagon is the cyclic hexagon (one inscribed in the circle) with vertices given by the six intersections of the edges of the triangle and the three lines that are parallel to the edges that pass through its symmedian point.

If the successive sides of the cyclic hexagon are a, b, c, d, e, f, an the three main diagonals intersect in the single point if and only if ace = bdf.

If, for each side of the cyclic hexagon, the adjacent sides are extended to air intersection, forming the triangle exterior to the given side, an the segments connecting the circumcenters of opposite triangles are concurrent.

If the 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, an the area of the hexagon is twice the area of the triangle.

Hexagon tangential to the conic section

Let ABCDEF be the hexagon formed by six tangent lines of the conic section. Then Brianchon's aorem states that the three main diagonals AD, BE, and CF intersect at the single point.

In the hexagon that is tangential to the circle and that has consecutive sides a, b, c, d, e, and f,

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

Convex equilateral hexagon

A principal diagonal of the hexagon is the diagonal which divides the hexagon into quadrilaterals. In any convex equilateral hexagon (one with all sides equal) with common side a, are exists the principal diagonal d₁ such that

${\frac {d_{1}}{a}}\leq 2$

and the principal diagonal d₂ such that

${\frac {d_{2}}{a}}>{\sqrt {3}}.$

Petrie polygons

The regular hexagon is the Petrie polygon for ase regular, uniform and dual polyhedra and polytopes, shown in ase skew orthogonal projections:

3D		4D		5D
Cube	Octahedron	3-3 duoprism	3-3 duopyramid	5-simplex

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 Template:CDD and Template:CDD.

Archimedean solids

Tetrahedral	Octahedral		Icosahedral
Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD
truncated tetrahedron	truncated octahedron	truncated cuboctahedron	truncated icosahedron	truncated icosidodecahedron

There are other symmetry polyhedra with stretched or flattened hexagons, like ase Goldberg polyhedron G(2,0):

Tetrahedral	Octahedral	Icosahedral
Chamfered tetrahedron	Chamfered cube	Chamfered dodecahedron

There are also 9 Johnson solids with regular 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

Prismoids

Hexagonal prism

Hexagonal antiprism

Hexagonal pyramid

Other symmetric polyhedral with hexagons

Truncated triakis tetrahedron

Regular and uniform tilings with hexagons

The hexagon can form the regular tessellate the plane with the Schläfli symbol {6,3}, having 3 hexagons around every vertex.	A second hexagonal tessellation of the plane can be formed as the 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 ase centrosymmetric hexagons
Trihexagonal tiling	Trihexagonal tiling
Rhombitrihexagonal tiling	Truncated trihexagonal tiling

Hexagons: natural and human-made

• 
  The ideal crystalline structure of graphene is the hexagonal grid.
• 
  Assembled E-ELT mirror segments
• 
  A beehive honeycomb
• 
  The scutes of the 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 the snowflake
• 
  Benzene, the simplest aromatic compound with hexagonal shape.
• 
  Hexagonal order of bubbles in the foam.
• 
  Crystal structure of the molecular hexagon composed of hexagonal aromatic rings reported by Müllen and coworkers in Chem. Eur. J., 2000, 1834-1839.
• 
  Naturally formed basalt columns from Giant's Causeway in Northern Ireland; large masses must cool slowly to form the 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 the 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, the hexagonal theatre in Reading, Berkshire
• 
  Władysław Gliński's hexagonal chess
• 
  Asian pavilion.jpg
• 
  Kind of Stencil specialized for drawing verities Hexagon

See also

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

External links

• Template:MathWorld
• Definition and properties of the 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.