Seoxecge

Ēacnung: Forþan þe þām nīwegemyndum þāra rīmcræften ƿēron, sume þisse rēdes synd on Nīƿenglisce.

Seoxhyrne
Ǣn samaflanc seoxhyrne
Rindenes	6
Sclæfligruna	{6}
Oferside	Manigfult
Twēoflaedig horn	120°

Sēo seoxecge (Niwenglisc: hexagon) oþþe seoxhyrne ġesceap is sēo ġesċeap ƿiþ seox hyrneġen.

In geometry, a seoxecge (from Greek ἕξ, hex, mēnung "seox", and γωνία, gonía, meaning "corner, angle") is a seoxflanc polygon. Þē total æf þē internal angles æf any simple (non-self-intersecting) hexagon is 720°.

Samaflanc seoxhyrne

A samaflanc seoxhyrne hæfde sēo sclæfligruna æf {6} ond innyrdigfealds æf 120°. Aht cunnen eallswā bēon ācweccaned eallswā a sċeortode samaflanc þrīhyrne, t{3}, hwic oðerwiss twā ēgðus æf rindenes.

A samaflanc seoxhyrne is tealde eallswā a seoxhyrne þæt is bā samaflanc and efnangul. Hit is twofold, mǣnaneg þæt hit is bā hweorfanlic (hafs a ymbehringed hring) and tangƿise (hafs an inwriten hring).

Þē ġecynde wǣg æf þē flacenes īsgelīcs þē hrēodanfex æf þē ymbehringed hring or ymbehringhring, hwic yamounted${\tfrac {2}{\sqrt {3}}}$ tīman þē æpþem (hrēodanfex æf þē inwriten hring). Ēall innyrdig fealds earon 120 rīminnyrdig. A samaflanc seoxhyrne hafs seox ymbhrǣdlic gemetgemostras (ymbhrǣdlic gemetgemostras ǣf ordre seox) and seox reflection symmetries (seox lines æf symmetry), crǣftung upp þē twēoflaedig flocc D₆. Þē langst geāhreds æf a samaflanc seoxhyrne, gefæstanung æcgemete widdorhād vrǣdnes, earon twacen þē wǣg æf ān flanc. Fram þīs hit cunnen wesan sēon þæt a þrīhyrne ƿīþ a hwyrftstān at þē centrum æf þē samaflanc seoxhyrne and sharung ān flanc ƿīþ þē seoxhyrne is samaflanc, and þæt þē samaflanc seoxhyrne cunnen wesan dǣlfæst intō seox samaflanc þrīhyrnes.

Lician fēowerecges and samaflanc þrīhyrnes, samaflanc seoxhyrnes fhit tōgædere þiþout ǣniġ ġeaps tō tile þē plane (þrī seoxhyrnes meetung aht eall hwyrftstān), and so earon nyttfull fore ācweccaneg tessellations. Þē cēlan æf a bēohȳf huniġcamb earon seoxhyrnelican fore þīs reason and bēacen þæs þē ġesceap crǣften eficient fricu æf ūtstræcan and boldung andweorces. Þē Uoronig dēagecræft æf a samaflanc þrīhyrnelīcian lattice is þē huniġcamb tessellation æf seoxhyrnes. Hit is nāht gestalteg considered a þrīhyrnetorr, þēah hit is samaflanc.

Forebȳsn

Ǣn stepebīstepe ymelestyring æf þē getimbrung æf a samaflanc seoxhyrne usung færanfæþm ond streċċaneċġ, ġiefen bī Ewcleod's Ādalen, Bōc IU, Trahtnysse 15: þīs is mæglic eallswā 6 $=$ 2 × 3, a afraet æf a anweald æf twā and clæne Færmet formestas.	Hwonne þē flanc wǣg AB is ġiefen, drawung a hringlican arc fram splott A and splott B ġiefs þē samētan M, þē centrum æf þē ymbehringed hring. Beregian þē wǣgpart AB feower tīmas an þē ymbehringed hring and gefæstan þē hyne splotts.

Parameters

R = Circumradius; r = Inradius; t = sīdelǣgum

Sēo maximal æcgemete (hƿīc corresponds to þē long diagonal æf þē seoxhyrne), D, is twice þē maximal radius or circumradius, R, hƿīc equals þē sīdelǣgum, t. Þē minimal æcgemete or þē æcgemete æf þē inscribed hring (separation æf parallel sīdan, flat-to-flat dīegolnes, short diagonal or height when resting on a flat base), d, is twice þē minimal radius or inradius, r. Þē maxima and minima are related by þē same factor:

${\frac {1}{2}}d=r=\cos(30^{\circ })R={\frac {\sqrt {3}}{2}}R={\frac {\sqrt {3}}{2}}t$   and, similarly, $d={\frac {\sqrt {3}}{2}}D.$

Sēo area æf a samaflanc seoxhyrne

${\begin{aligned}A&={\frac {3{\sqrt {3}}}{2}}R^{2}=3Rr=2{\sqrt {3}}r^{2}\\[3pt]&={\frac {3{\sqrt {3}}}{8}}D^{2}={\frac {3}{4}}Dd={\frac {\sqrt {3}}{2}}d^{2}\\[3pt]&\approx 2.598R^{2}\approx 3.464r^{2}\\&\approx 0.6495D^{2}\approx 0.866d^{2}.\end{aligned}}$

For ǣniġe samaflanc gesceap, þē area can also be expressed in terms æf þē apothem a and þē perimeter p. For þē samaflanc seoxhyrne þæs are given by a = r, and p${}=6R=4r{\sqrt {3}}$, so

${\begin{aligned}A&={\frac {ap}{2}}\\&={\frac {r\cdot 4r{\sqrt {3}}}{2}}=2r^{2}{\sqrt {3}}\\&\approx 3.464r^{2}.\end{aligned}}$

Sēo samaflanc seoxhyrne fills þē hlæfdel ${\tfrac {3{\sqrt {3}}}{2\pi }}\approx 0.8270$ æf hits ymbhrēodscripst hring.

If a samaflanc seoxhyrne has æfterfylgende vertices A, B, C, D, E, F and if P is ǣniġe ord on þē ymbhrēodhring between B and C, þēos PE + PF = PA + PB + PC + PD.

Hit folgode fram sēo reced æf ymbhrēodanfex to onanfex þæt þē hīeȝþutōwīdþ reced æf a samaflanc seoxhyrne is 1:1.1547005; þæt is, a seoxhyrne with a long diagonal æf 1.0000000 ƿylle hæfde a dīegolnes æf 0.8660254 between parallel sīdan.

Point in plane

For an arbitrary point in þē plane æf a regular hexagon with circumradius $R$, whose distances to þē centroid æf þē regular hexagon and its seox vertices are $L$ and $d_{i}$ respectively, we have

$d_{1}^{2}+d_{4}^{2}=d_{2}^{2}+d_{5}^{2}=d_{3}^{2}+d_{6}^{2}=2\left(R^{2}+L^{2}\right),$

$d_{1}^{2}+d_{3}^{2}+d_{5}^{2}=d_{2}^{2}+d_{4}^{2}+d_{6}^{2}=3\left(R^{2}+L^{2}\right),$

$d_{1}^{4}+d_{3}^{4}+d_{5}^{4}=d_{2}^{4}+d_{4}^{4}+d_{6}^{4}=3\left(\left(R^{2}+L^{2}\right)^{2}+2R^{2}L^{2}\right).$

If $d_{i}$ are þē distances from þē vertices æf a regular hexagon to any point on its circumcircle, then

$\left(\sum _{i=1}^{6}d_{i}^{2}\right)^{2}=4\sum _{i=1}^{6}d_{i}^{4}.$

Symmetry

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

The seox lines æf reflection æf 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 þē middle column are labeled as g for their central gyration orders. Full symmetry æf þē regular form is r12 and no symmetry is labeled a1.

The regular hexagon has D₆ symmetry. There are 16 subgroups. There are 8 up to isomorphism: itself (D₆), 2 dihedral: (D_3, D₂), 4 cyclic: (Z₆, Z₃, Z₂, Z₁) and þē trivial (e)

These symmetries express nine distinct symmetries æf a regular hexagon. John Conway labels these by a letter and group order. r12 is full symmetry, and a1 is no symmetry. p6, an isogonal hexagon constructed by three mirrors can alternate long and short edges, and d6, an isotoxal hexagon constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are duals æf each other and have half þē symmetry order æf þē regular hexagon. Þē 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 æf freedom for irregular forms. Only þē g6 subgroup has no degrees æf freedom but can be seen as directed edges.

Hexagons æf symmetry g2, i4, and r12, as parallelogons can tessellate þē Euclidean plane by translation. Other hexagon shapes can tile þē plane with different orientations.

p6m (*632)	cmm (2*22)	p2 (2222)	p31m (3*3)	pmg (22*)		pg (××)
r12	i4	g2	d2	d2	p2	a1
Dih6	Dih2	Z2	Dih1			Z1

A2 and G2 groups

A2 group roots Bysen:Dynkin

G2 group roots Bysen:Dynkin2

The 6 roots æf þē simple Lie group A2, represented by a Dynkin diagram Bysen:Dynkin, are in a regular hexagonal pattern. Þē two simple roots have a 120° angle between them.

The 12 roots æf þē Exceptional Lie group G2, represented by a Dynkin diagram Bysen:Dynkin2 are also in a hexagonal pattern. Þē two simple roots æf two lengths have a 150° angle between them.

Dissection

6-cubī projection	12 rhomb dissection

Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and æf equal length) can be dissected into Bysen:Fracm(m − 1) parallelograms. In particular this is true for regular polygons with evenly many sides, in which case þē parallelograms are all rhombi. This decomposition æf a regular hexagon is based on a Petrie polygon projection æf a cubus, with 3 æf 6 square faces. Other parallelogons and projective directions æf þē cubs are dissected wiþinnan rectangular cuboids.

Dissection æf hexagons into three 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 æf þē regular hexagonal tiling, {6,3}, with three hexagonal faces 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) æf edges, this form only has D₃ symmetry.

A truncated hexagon, t{6}, is a dodecagon, {12}, alternating two types (colors) æf edges. An alternated hexagon, h{6}, is an 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 seox equilateral triangles by adding a center point. This pattern repeats wiþinnan þē 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 wiþinnan þē rhombitrihexagonal tiling.

Regular {6}	Truncated t{3} = {6}	Hypertruncated triangles			Stellated Star figure 2{3}	Truncated t{6} = {12}	Alternated h{6} = {3}

Crossed hexagon	A concave hexagon	A self-intersecting hexagon (star polygon)	Extended Central {6} in {12}	A skew hexagon, wiþinnan cube	Dissected {6}	projection octahedron	Complete graph

Self-crossing hexagons

There are seox self-crossing hexagons with þē vertex arrangement æf þē regular hexagon:

Self-intersecting hexagons with regular vertices

Dih2			Dih1		Dih3
Figure-eight	Center-flip	Unicursal	Fish-tail	Double-tail	Triple-tail

Hexagonal structures

Giant's Causeway closeup

From bees' honeycombs to þē 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 þē fewest hexagons. This means that honeycombs require less wax to construct and gain much strength under compression.

Irregular hexagons with parallel opposite edges are called parallelogons and can also tile þē 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

Heafodgewrit: Hexagonal tiling

In addition to þē regular hexagon, which determines a unique tessellation æf þē plane, any irregular hexagon which satisfies þē Conwēg criterion will tile þē plane.

Hexagon inscribed in a conic section

Pascal's theorem (also known as þē "Hexagrammum Mysticum Theorem") states that if an arbitrary hexagon is inscribed in any conic section, and pairs æf opposite sides are extended until they meet, þē three intersection points will lie on a straight line, þē "Pascal line" æf that configuration.

Cyclic hexagon

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

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

If, for each side æf a cyclic hexagon, þē adjacent sides are extended to their intersection, forming a triangle exterior to þē given side, then þē segments connecting þē circumcenters æf opposite triangles are concurrent.

If a hexagon has vertices on þē circumcircle æf an acute triangle at þē seox points (including three triangle vertices) where þē extended altitudes æf þē triangle meet þē circumcircle, then þē area æf þē hexagon is twice þē area æf þē triangle.Bysen:Rp

Hexagon tangential to a conic section

Let ABCDEF be a hexagon formed by seox tangent lines æf a conic section. Then Brianchon's theorem states that þē 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,

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

Equilateral triangles on þē sides æf an arbitrary hexagon

Equilateral triangles on þē sides æf an arbitrary hexagon

If an equilateral triangle is constructed externally on each side æf any hexagon, then þē midpoints æf þē segments connecting þē centroids æf opposite triangles form another equilateral triangle.Bysen:Rp

Skew hexagon

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

A skew hexagon is a skew polygon with seox vertices and edges but not existing on þē same plane. Þē interior æf such a 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 three dimensions it will be a zig-zag skew hexagon and can be seen in þē vertices and side edges æf a triangular antiprism with þē 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 þē Petrig 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 æf a hexagon is a diagonal which divides þē hexagon into quadrilaterals. In any convex equilateral hexagon (one with all sides equal) with common side a, there existsBysen:Rp 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 æf only regular hexagons, because þē hexagons tessellate, not allowing þē result to "fold up". Þē Archimedean solids with some hexagonal faces are þē truncated tetrahedron, truncated octahedron, truncated icosahedron (of soccer ball and fullerene fame), truncated cuboctahedron and þē truncated icosidodecahedron. These hexagons can be considered truncated triangles, with Coxeter diagrams æf þē form Bysen:CDD and Bysen:CDD.

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

There are also 9 Johnson solids with regular hexagons:

Tilings with regular hexagons
Regular	1-uniform
{6,3} Bysen:CDD	r{6,3} Bysen:CDD	rr{6,3} Bysen:CDD	tr{6,3} Bysen:CDD
2-uniform tilings

Gallery of natural and artificial hexagons

• 
  The ideal crystalline structure æf graphene is a hexagonal grid.
• 
  Assembled E-ELT mirror segments
• 
  A beehive honeycomb
• 
  The scutes æf a turtle's carapace
• 
  Saturn's hexagon, a hexagonal cloud pattern around þē north pole æf þē planet
• 
  Micrograph æf a snowflake
• 
  Benzene, þē simplest aromatic compound with hexagonal shape.
• 
  Hexagonal order æf bubbles in a foam.
• 
  Crystal structure æf a molecular hexagon composed æf hexagonal aromatic rings.
• 
  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 æf Fort Jefferson in Dry Tortugas National Park
• 
  The James Webb Space Telescope mirror is composed æf 18 hexagonal segments.
• 
  In French, l'Hexagone refers to Metropolitan France for its vaguely hexagonal shape.
• 
  Hexagonal Hanksite crystal, one æf many hexagonal crystal system minerals
• 
  Hexagonal barn
• 
  The Hexagon, a hexagonal theatre in Reading, Berkshire
• 
  Władysław Gliński's hexagonal chess
• 
  Pavilion in þē Taiwan Botanical Gardens
• 
  Hexagonal window

See also

• 24-cell: a four-dimensional figure which, like þē hexagon, has orthoplex facets, is self-dual and tessellates Euclidean space
• Hexagonal crystal system
• Hexagonal number
• Hexagonal tiling: a regular tiling æf hexagons in a plane
• Hexagram: seoxflanc steorra wiþinnan a regular hexagon
• Unicursal hexagram: single path, seoxflanc steorra, wiþinnan a hexagon
• Honeycomb conjecture
• Havannah: abstract board game played on a seoxflanc hexagonal grid

Fruman

Gesceapu
Þrīhyrn	Cēplere • Ġetælhyrne • Samaflanc • Sċeorthyrne • Riht • Twāflanc • Ƿilnunȝ
Fēowerecge	Antiparallelogram • Bicentric • Crossed • Cyclic • Equidiagonal • Ex-tangential • Harmonic • Isosceles trapezoid • Kite • Orthodiagonal • Parallelogram • Langfēowerecge • Right kite • Right trapezoid • Rhombus • Fēowertynefeald • Tangential • Tangential trapezoid • Ungewunelic
By flanccounten (1–10)	Ānecge (1) • Twāecge (2) • Þrīecge (3) • Fēowerecge (4) • Fifecge (5) • Seoxecge (6) • Seofonecge (7) • Eahtaecge (8) • Nigonecge (9) • Tīenecge (10)
By flanccounten (11–20)	Endleofanecge (11) • Tƿelfecge (12) • Þrēotīeneecge (13) • Fēowertīeneecge (14) • Fīftīeneecge (15) • Seoxtīeneecge (16) • Seofontīeneecge (17) • Eahtatīeneecge (18) • Nigontīeneecge (19) • Tƿēntiġecge (20)
Steorraġesceapu	Fīfhyrnedesteorra • Pentagram • Hexagram • Heptagram • Octagram • Enneagram • Decagram • Hendecagram • Dodecagram

External links

• Bysen:MathWorld

• Definition and properties of a hexagon with interactive animation and construction with compass and straightedge.
• An Introduction to Hexagonal Geometry on Hexnet a website devoted to hexagon mathematics.
• Bysen:YouTube – an animated internet video about hexagons by CGP Grey.