Seoh Seoxecge frumtraht

{{Rimcraeftniwenglish}}
{{Infobox gesceapu
| nama          = Seoxhyrne
| ymele         = Regular polygon 6 annotated.svg
| gewrit        = Ǣn samaflanc seoxhyrne
| rind          = 6
| sclæfligruna  = {6}
| oferside      = Manigfult
| tweoflaedig   = 120°
}}
Sēo '''seoxecge''' ({{lang-en|hexagon}}) oþþe '''seoxhyrne ġesceap''' is sēo ġesċeap ƿiþ [[seox]] hyrneġen.

In [[geometry]], a '''seoxecge''' (from [[Ancient Greek|Greek]] {{lang|grc|ἕξ}}, {{lang|grc-Latn|hex}}, mēnung "seox", and {{lang|grc|γωνία}}, {{lang|grc-Latn|gonía}}, meaning "corner, angle") is a seoxflanc [[polygon]].<ref>[https://deimel.org/images/plain_cube.gif Cube picture]</ref> Þē total æf þē internal angles æf any [[simple polygon|simple]] (non-self-intersecting) hexagon is 720°.

==Samaflanc seoxhyrne ==
A ''[[samaflanc gesceap|samaflanc]] seoxhyrne'' hæfde sēo [[sclæfligruna]] æf {6} ond [[innyrdigfeald]]s <!--interior angle--> æf 120°.<ref>{{citation|title=Polyhedron Models|first=Magnus J.|last=Wenninger|publisher=Cambridge University Press|year=1974|page=9|isbn=9780521098595|url=https://books.google.com/books?id=N8lX2T-4njIC&pg=PA9|access-date=2015-11-06|archiveurl=https://web.archive.org/web/20160102075753/https://books.google.com/books?id=N8lX2T-4njIC&pg=PA9|archive-date=2016-01-02|url-status=live}}.</ref> Aht cunnen eallswā bēon ācweccaned eallswā a [[sċeortode (gemetrig)|sċeortode]] [[samaflanc þrīhyrne]], t{3}, hwic oðerwiss twā ēgðus æf rindenes.

A samaflanc seoxhyrne is tealde eallswā a seoxhyrne þæt is bā [[samaflanc gesceap|samaflanc]] and [[efnangul gesceap|efnangul]]<!--equiangular-->. Hit is [[twofold gesceap|twofold]], mǣnaneg þæt hit is bā [[hweorfanlic gesceap|hweorfanlic]]<!--cyclic--> (hafs a ymbehringed hring) and [[tangwise gesceap|tangƿise]] (hafs an inwriten hring).

Þē ġecynde wǣg æf þē flacenes īsgelīcs þē hrēodanfex æf þē [[ymbehringed hring]]<!--circumscribed circle--> or [[ymbehringhring]], hwic yamounted<math>\tfrac{2}{\sqrt{3}}</math> tīman þē [[æpþem]] (hrēodanfex æf þē [[inwriten gesceap|inwriten hring]]). Ēall innyrdig [[feald (rīmcræft)|feald]]s 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]] D<sub>6</sub>. 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 þrīhyrne|samaflanc]], and þæt þē samaflanc seoxhyrne cunnen wesan dǣlfæst intō seox samaflanc þrīhyrnes.

Lician [[fēowerecge (gemetrig)|fēowerecge]]s and [[samaflanc]] [[þrīhyrne]]s, samaflanc seoxhyrnes fhit tōgædere þiþout ǣniġ ġeaps tō ''tile þē plane'' (þrī seoxhyrnes meetung aht eall hwyrftstān), and so earon nyttfull fore ācweccaneg [[tessellation]]s. Þē cēlan æf a [[bēohȳf (bēocēpanung)|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 [[samaflanc gesceap|þrīhyrnetorr]], þēah hit is samaflanc.

{| class="wikitable" width=40%
|+ Forebȳsn
|-
| [[Ymele:Regular Hexagon Inscribed in a Circle.gif|240px]] || [[Ymele:01-Sechseck-Seite-vorgegeben-wiki.svg|263px]]
|-
| Ǣn stepebīstepe ymelestyring æf þē getimbrung æf a samaflanc seoxhyrne usung [[færanfæþm ond streċċaneċġ]], ġiefen bī [[Ewcleod]]'s ''[[Ewcleoden Ādalen|Ādalen]]'', Bōc IU, Trahtnysse 15: þīs is mæglic eallswā 6 <math>=</math> 2 × 3, a afraet æf a anweald æf twā and clæne [[Færmet formesta]]s. || 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 ==
[[Image:Regular hexagon 1.svg|thumb|''R'' = [[Circumradius]]; ''r'' = [[Inradius]]; ''t'' = sīdelǣgum]]

Sēo maximal [[æcgemete#gescap|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:
:<math>\frac{1}{2}d = r = \cos(30^\circ) R = \frac{\sqrt{3}}{2} R = \frac{\sqrt{3}}{2} t</math> &nbsp; and, similarly, <math>d = \frac{\sqrt{3}}{2} D.</math>

Sēo area æf a samaflanc seoxhyrne
:<math>\begin{align}
  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 2.598 R^2 \approx 3.464 r^2\\
    &\approx 0.6495 D^2 \approx 0.866 d^2.
\end{align}</math>

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''<math>{} = 6R = 4r\sqrt{3}</math>, so
:<math>\begin{align}
  A &= \frac{ap}{2} \\
    &= \frac{r \cdot 4r\sqrt{3}}{2} = 2r^2\sqrt{3} \\
    &\approx 3.464 r^2.
\end{align}</math>

Sēo samaflanc seoxhyrne fills þē hlæfdel <math>\tfrac{3\sqrt{3}}{2\pi} \approx 0.8270</math> æf hits [[ymbhrēodscripst hring]]<!--circumscribed circle-->.

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 {{nowrap|PE + PF {{=}} PA + PB + PC + PD}}.

Hit folgode fram sēo reced æf [[ymbhrēodanfex]]<!--circumradius--> to [[onanfex]]<!--inradius--> 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 <math>R</math>, whose distances to þē centroid æf þē regular hexagon and its seox vertices are <math>L</math> and <math>d_i</math> 
respectively, we have<ref name=Mamuka >{{cite journal| last1= Meskhishvili |first1= Mamuka| date=2020|title=Cyclic Averages æf Regular Polygons and Platonic Solids |journal= Communications in Mathematics and Applications|volume=11|pages=335–355|doi= 10.26713/cma.v11i3.1420|doi-broken-date= 31 January 2024|arxiv= 2010.12340|url= https://www.rgnpublications.com/journals/index.php/cma/article/view/1420/1065}}</ref>

:<math> 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), </math>
:<math> 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), </math>
:<math> 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 + 2 R^2 L^2\right). </math>

If <math>d_i</math> are þē distances from þē vertices æf a regular hexagon to any point on its circumcircle, then <ref name= Mamuka />

:<math>\left(\sum_{i=1}^6 d_i^2\right)^2 = 4 \sum_{i=1}^6 d_i^4 .</math>

== Symmetry==
{| class="collapsible collapsed" align=right
! Example hexagons by symmetry
|-
|
{| class=wikitable
|- valign=top
!
! [[File:Hexagon_r12_symmetry.png|60px]]<BR>r12<BR>regular
!
|rowspan=3|
!
! [[File:Hexagon_i4_symmetry.png|60px]]<BR>i4
!
|- valign=top
! [[File:Hexagon_d6_symmetry.png|60px]]<BR>d6<BR>[[isotoxal figure|isotoxal]]
! [[File:Hexagon_g6_symmetry.png|60px]]<BR>g6<BR>directed
! [[File:Hexagon_p6_symmetry.png|60px]]<BR>p6<BR>[[isogonal figure|isogonal]]
! [[File:Hexagon_d3_symmetry.png|60px]]<BR>d2
! [[File:Hexagon_g2_symmetry.png|60px]]<BR>g2<BR>general<BR>[[parallelogon]]
! [[File:Hexagon_p2_symmetry.png|60px]]<BR>p2
|- valign=top
!
! [[File:Hexagon_g3_symmetry.png|60px]]<BR>g3
!
!
! [[File:Hexagon_a1_symmetry.png|60px]]<BR>a1
!
|}
|}

[[File:Hexagon reflections.svg|thumb|160px|left|The seox lines æf [[reflection symmetry|reflection]] æf a regular hexagon, with Dih<sub>6</sub> or '''r12''' symmetry, order 12.]]
[[File:Regular hexagon symmetries.svg|thumb|400px|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 D<sub>6</sub> symmetry. There are 16 subgroups. There are 8 up to isomorphism: itself (D<sub>6</sub>), 2 dihedral: (D<sub>3,</sub> D<sub>2</sub>), 4 [[cyclic group|cyclic]]: (Z<sub>6</sub>, Z<sub>3</sub>, Z<sub>2</sub>, Z<sub>1</sub>) and þē trivial (e)

These symmetries express nine distinct symmetries æf a regular hexagon. [[John Horton Conway|John Conway]] labels these by a letter and group order.<ref>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)</ref> '''r12''' is full symmetry, and '''a1''' is no symmetry. '''p6''', an [[isogonal figure|isogonal]] hexagon constructed by three mirrors can alternate long and short edges, and '''d6''', an [[isotoxal figure|isotoxal]] hexagon constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are [[dual polygon|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 [[Elongation (geometry)|elongated]] [[rhombus]], while '''d2''' and '''p2''' can be seen as horizontally and vertically elongated [[Kite (geometry)|kites]]. '''g2''' hexagons, with opposite sides parallel are also called hexagonal [[parallelogon]]s.

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 edge]]s.

Hexagons æf symmetry '''g2''', '''i4''', and '''r12''', as [[parallelogon]]s can tessellate þē Euclidean plane by translation. Other [[Hexagonal tiling#Topologically equivalent tilings|hexagon shapes can tile þē plane]] with different orientations.
{| class=wikitable
!''p''6''m'' (*632)
!''cmm'' (2*22)
!''p''2 (2222)
!''p''31''m'' (3*3)
!colspan=2|''pmg'' (22*)
!''pg'' (××)
|-
![[File:Isohedral_tiling_p6-13.svg|120px]]<BR>[[hexagonal tiling|r12]]
![[File:Isohedral_tiling_p6-12.png|120px]]<BR>i4
![[File:Isohedral_tiling_p6-7.svg|120px]]<BR>g2
![[File:Isohedral tiling p6-11.png|120px]]<BR>d2
![[File:Isohedral tiling p6-10.png|120px]]<BR>d2
![[File:Isohedral tiling p6-9.svg|120px]]<BR>p2
![[File:Isohedral tiling p6-1.png|120px]]<BR>a1
|- valign=top al
!Dih<sub>6</sub>
!Dih<sub>2</sub>
!Z<sub>2</sub>
!colspan=3|Dih<sub>1</sub>
!Z<sub>1</sub>
|}

{{-}}

=== A2 and G2 groups ===
{| class=wikitable align=right style="text-align:center;"
|-
| [[File:Root system A2.svg|120px]]<BR>A2 group roots<BR>{{Dynkin|node_n1|3|node_n2}}
| [[File:Root system G2.svg|120px]]<BR>G2 group roots<BR>{{Dynkin2|nodeg_n1|6a|node_n2}}
|}
The 6 roots æf þē [[simple Lie group]] [[Dynkin diagram#Example: A2|A2]], represented by a [[Dynkin diagram]] {{Dynkin|node_n1|3|node_n2}}, are in a regular hexagonal pattern. Þē two simple roots have a 120° angle between them.

The 12 roots æf þē [[Exceptional Lie group#Exceptional cases|Exceptional Lie group]] [[G2 (mathematics)|G2]], represented by a [[Dynkin diagram]] {{Dynkin2|nodeg_n1|6a|node_n2}} are also in a hexagonal pattern. Þē two simple roots æf two lengths have a 150° angle between them.

{{-}}

== Dissection==
{| class=wikitable align=right style="text-align:center;"
! [[6-cubī]] projection
!colspan=2| 12 rhomb dissection
|-
| [[File:6-cube t0 A5.svg|120px]]
| [[File:6-gon rhombic dissection-size2.svg|140px]]
| [[File:6-gon rhombic dissection2-size2.svg|140px]]
|}
[[Coxeter]] states that every [[zonogon]] (a 2''m''-gon whose opposite sides are parallel and æf equal length) can be dissected into {{nowrap|{{frac|1|2}}''m''(''m'' − 1)}} parallelograms.<ref>[[Coxeter]], Mathematical recreations and Essays, Thirteenth edition, p.141</ref> In particular this is true for [[regular polygon]]s 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 [[parallelogon]]s and projective directions æf þē cubs are dissected wiþinnan [[rectangular cuboid]]s.
{| class="wikitable collapsible" style="text-align:center;"
!colspan=12| Dissection æf hexagons into three rhombs and parallelograms
|-
!rowspan=3| 2D
! Rhombs
!colspan=3| Parallelograms
|- valign=top
|[[File:Hexagon_dissection.svg|80px]]
|[[File:Cube-skew-orthogonal-skew-solid.png|95px]]
|[[File:Cuboid_diagonal-orthogonal-solid.png|120px]]
|[[File:Cuboid_skew-orthogonal-solid.png|120px]]
|- valign=top
| Regular {6}
|colspan=3| Hexagonal [[parallelogon]]s
|-
!rowspan=3| 3D
!colspan=2| Square faces
!colspan=2| Rectangular faces
|- valign=top
| [[File:3-cube_graph.svg|95px]]
| [[File:Cube-skew-orthogonal-skew-frame.png|95px]]
| [[File:Cuboid_diagonal-orthogonal-frame.png|120px]]
| [[File:Cuboid_skew-orthogonal-frame.png|120px]]
|- valign=top
|colspan=2| [[Cube]]
|colspan=2| [[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 [[Truncation (geometry)|truncated]] [[equilateral triangle]], with Schläfli symbol t{3}. Seen with two types (colors) æf edges, this form only has D<sub>3</sub> symmetry.

A [[truncation (geometry)|truncated]] hexagon, t{6}, is a [[dodecagon]], {12}, alternating two types (colors) æf edges. An [[Alternation (geometry)|alternated]] hexagon, h{6}, is an [[equilateral triangle]], {3}. A regular hexagon can be [[stellation|stellated]] with equilateral triangles on its edges, creating a [[hexagram]]. A regular hexagon can be dissected into seox [[equilateral triangle]]s 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 [[square]]s and [[equilateral triangle]]s around it. This pattern repeats wiþinnan þē [[rhombitrihexagonal tiling]].

{| class=wikitable style="text-align:center;" width=640
|-
| [[File:Regular polygon 6 annotated.svg|80px]]
| [[Image:Truncated triangle.svg|80px]]
| [[File:Regular truncation 3 1000.svg|80px]]
| [[File:Regular truncation 3 1.5.svg|80px]]
| [[File:Regular truncation 3 0.55.svg|80px]]
| [[Image:Hexagram.svg|80px]]
| [[File:Regular polygon 12 annotated.svg|80px]]
| [[File:Regular polygon 3 annotated.svg|80px]]
|- style="vertical-align:top;"
! Regular<BR>{6}
! Truncated<BR>t{3} = {6}
! colspan=3|Hypertruncated triangles
! Stellated<BR>[[Star figure]] [[Hexagram|2{3}]]
! Truncated<BR>t{6} = [[Dodecagon|{12}]]
! Alternated<BR>h{6} = [[equilateral triangle|{3}]]
|}

{| class=wikitable style="text-align:center;" width=400
|-
|[[File:Crossed-square hexagon.png|80px]]
| [[File:Medial triambic icosahedron face.svg|80px]]
| [[File:Great triambic icosahedron face.svg|80px]]
| [[File:Hexagonal cupola flat.svg|80px]]
| [[File:Cube petrie polygon sideview.svg|80px]]
| [[File:3-cube t0.svg|80px]]
| [[File:3-cube t2.svg|80px]]
| [[File:5-simplex_graph.svg|80px]]
|- style="vertical-align:top;"
! Crossed<BR>hexagon
! A concave hexagon
! A self-intersecting hexagon ([[star polygon]])
! Extended<BR>Central {6} in {12}
! A [[skew regular polygon|skew hexagon]], wiþinnan [[cube]]
! Dissected {6}
! projection<BR>[[octahedron]]
! [[Complete graph]]
|}

=== Self-crossing hexagons===
There are seox [[Star polygon|self-crossing hexagons]] with þē [[vertex arrangement]] æf þē regular hexagon:
{| class=wikitable style="width:400px; text-align:center;"
|+ Self-intersecting hexagons with regular vertices
!colspan=3| Dih<sub>2</sub>
!colspan=2| Dih<sub>1</sub>
! Dih<sub>3</sub>
|- valign=top
| [[File:Crossed hexagon1.svg|100px]]<BR>Figure-eight
| [[File:Crossed hexagon2.svg|100px]]<BR>Center-flip
| [[File:Crossed hexagon3.svg|100px]]<BR>[[Unicursal hexagram|Unicursal]]
| [[File:Crossed hexagon4.svg|100px]]<BR>Fish-tail
| [[File:Crossed hexagon5.svg|100px]]<BR>Double-tail
| [[File:Crossed hexagon6.svg|100px]]<BR>Triple-tail
|}

==Hexagonal structures==
[[File:Giant's Causeway (13).JPG|thumb|Giant's Causeway closeup]]
From bees' [[honeycomb]]s 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 (physics)|compression]].

Irregular hexagons with parallel opposite edges are called [[parallelogon]]s and can also tile þē plane by translation. In three dimensions, [[hexagonal prism]]s with parallel opposite faces are called [[parallelohedron]]s and these can tessellate 3-space by translation.
{| class=wikitable style="text-align:center;"
|+ Hexagonal prism tessellations
! Form
! [[Hexagonal tiling]]
! [[Hexagonal prismatic honeycomb]]
|-
! Regular
| [[File:Uniform tiling 63-t0.svg|170px]]
| [[File:Hexagonal prismatic honeycomb.png|170px]]
|-
! Parallelogonal
| [[File:Isohedral tiling p6-7.png|170px]]
| [[File:Skew hexagonal prism honeycomb.png|240px]]
|}

==Tesselations by hexagons==
{{main|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 [[extended side|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 polygon|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 {{nowrap|''ace'' {{=}} ''bdf''}}.<ref>Cartensen, Jens, "About hexagons", ''Mathematical Spectrum'' 33(2) (2000–2001), 37–40.</ref>

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 lines|concurrent]].<ref>{{cite journal|author=Dergiades, Nikolaos|title=Dao's theorem on seox circumcenters associated with a cyclic hexagon|journal=[[Forum Geometricorum]]|volume=14|date=2014|pages=243–246|url=http://forumgeom.fau.edu/FG2014volume14/FG201424index.html|access-date=2014-11-17|archive-url=https://web.archive.org/web/20141205210609/http://forumgeom.fau.edu/FG2014volume14/FG201424index.html|archive-date=2014-12-05|url-status=live|archivedate=2014-12-05|archiveurl=https://web.archive.org/web/20141205210609/http://forumgeom.fau.edu/FG2014volume14/FG201424index.html}}</ref>

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.<ref name=Johnson>Johnson, Roger A., ''Advanced Euclidean Geometry'', Dover Publications, 2007 (orig. 1960).</ref>{{rp|p. 179}}

==Hexagon tangential to a conic section==
Let ABCDEF be a hexagon formed by seox [[tangent line]]s 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 polygon|tangential to a circle]] and that has consecutive sides ''a'', ''b'', ''c'', ''d'', ''e'', and ''f'',<ref>Gutierrez, Antonio, "Hexagon, Inscribed Circle, Tangent, Semiperimeter", [http://gogeometry.com/problem/p343_circumscribed_hexagon_tangent_semiperimeter.htm] {{Webarchive|url=https://web.archive.org/web/20120511025055/http://gogeometry.com/problem/p343_circumscribed_hexagon_tangent_semiperimeter.htm|date=2012-05-11}}, Accessed 2012-04-17.</ref>

:<math>a + c + e = b + d + f.</math>

==Equilateral triangles on þē sides æf an arbitrary hexagon==

[[File:Equilateral in hexagon.svg|thumb|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 [[centroid]]s æf opposite triangles form another equilateral triangle.<ref>{{cite journal|author=Dao Thanh Oai|date=2015|title=Equilateral triangles and Kiepert perspectors in complex numbers|journal=Forum Geometricorum|volume=15|pages=105–114|url=http://forumgeom.fau.edu/FG2015volume15/FG201509index.html|access-date=2015-04-12|archive-url=https://web.archive.org/web/20150705033424/http://forumgeom.fau.edu/FG2015volume15/FG201509index.html|archive-date=2015-07-05|url-status=live|archivedate=2015-07-05|archiveurl=https://web.archive.org/web/20150705033424/http://forumgeom.fau.edu/FG2015volume15/FG201509index.html}}</ref>{{rp|Thm. 1}}
{{-}}

== Skew hexagon==
[[File:Skew polygon in triangular antiprism.png|160px|thumb|A regular skew hexagon seen as edges (black) æf a [[triangular antiprism]], symmetry D<sub>3d</sub>, [2<sup>+</sup>,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<sub>3d</sub>, [2<sup>+</sup>,6] symmetry, order 12.

The [[cube]] and [[octahedron]] (same as triangular antiprism) have regular skew hexagons as petrie polygons.
{| class="wikitable" style="text-align:center;"
|+ Skew hexagons on 3-fold axes
|-
| [[File:Cube petrie.png|100px]]<br>[[Cube]]
| [[File:Octahedron petrie.png|100px]]<br>[[Octahedron]]
|}

===Petrie polygons===
The regular skew hexagon is þē [[Petrig polygon]] for these higher dimensional [[regular polytope|regular]], uniform and dual polyhedra and polytopes, shown in these skew [[orthogonal projection]]s:

{| class="wikitable" style="width:360px; text-align:center;"
|-
!colspan=2| 4D
! 5D
|- valign=top
| [[File:3-3 duoprism ortho-Dih3.png|100px]]<BR>[[3-3 duoprism]]
| [[File:3-3 duopyramid ortho.png|100px]]<BR>[[3-3 duopyramid]]
| [[Image:5-simplex t0.svg|100px]]<br>[[5-simplex]]
|}

==Convex equilateral hexagon==

A ''principal diagonal'' æf a hexagon is a diagonal which divides þē hexagon into quadrilaterals. In any convex [[equilateral polygon|equilateral]] hexagon (one with all sides equal) with common side ''a'', there exists<ref name="Crux">''Inequalities proposed in "[[Crux Mathematicorum]]"'', [http://www.imomath.com/othercomp/Journ/ineq.pdf] {{Webarchive|url=https://web.archive.org/web/20170830032311/http://imomath.com/othercomp/Journ/ineq.pdf|date=2017-08-30}}.</ref>{{rp|p.184,#286.3}} a principal diagonal ''d''<sub>1</sub> such that

:<math>\frac{d_1}{a} \leq 2</math>

and a principal diagonal ''d''<sub>2</sub> such that

:<math>\frac{d_2}{a} > \sqrt{3}.</math>

===Polyhedra with hexagons===
There is no [[Platonic solid]] made æf only regular hexagons, because þē hexagons [[tessellation|tessellate]], not allowing þē result to "fold up". Þē [[Archimedean solid]]s 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 [[truncation (geometry)|truncated]] triangles, with [[Coxeter diagram]]s æf þē form {{CDD|node_1|3|node_1|p|node}} and {{CDD|node_1|3|node_1|p|node_1}}.

{| class="wikitable collapsible collapsed" style="text-align:center;"
!colspan=12|Hexagons in [[Archimedean solid]]s
|-
! [[Tetrahedral symmetry|Tetrahedral]]
!colspan=2| [[Octahedral symmetry|Octahedral]]
!colspan=2| [[Icosahedral symmetry|Icosahedral]]
|-
| {{CDD|node_1|3|node_1|3|node}}
| {{CDD|node_1|3|node_1|4|node}}
| {{CDD|node_1|3|node_1|4|node_1}}
| {{CDD|node_1|3|node_1|5|node}}
| {{CDD|node_1|3|node_1|5|node_1}}
|- valign=top
| [[File:truncated tetrahedron.png|100px]]<br>[[truncated tetrahedron]]
| [[File:truncated octahedron.png|100px]]<br>[[truncated octahedron]]
| [[File:Great rhombicuboctahedron.png|100px]]<br>[[truncated cuboctahedron]]
| [[File:truncated icosahedron.png|100px]]<br>[[truncated icosahedron]]
| [[File:Great rhombicosidodecahedron.png|100px]]<br>[[truncated icosidodecahedron]]
|}

There are other symmetry polyhedra with stretched or flattened hexagons, like these [[Goldberg polyhedron]] G(2,0):
{| class="wikitable collapsible collapsed" style="text-align:center;"
! colspan=12 | Hexagons in Goldberg polyhedra
|-
! [[Tetrahedral symmetry|Tetrahedral]]
! [[Octahedral symmetry|Octahedral]]
! [[Icosahedral symmetry|Icosahedral]]
|-
| [[File:Alternate truncated cube.png|120px]]<BR>[[Chamfered tetrahedron]]
| [[File:Truncated rhombic dodecahedron2.png|120px]]<BR>[[Chamfered cube]]
| [[File:Truncated rhombic triacontahedron.png|120px]]<BR>[[Chamfered dodecahedron]]
|}

There are also 9 [[Johnson solid]]s with regular hexagons:
{| class="wikitable collapsible collapsed" style="width:400px; text-align:center;"
!colspan=12| Johnson solids with hexagons
|- valign=top
| [[File:Triangular cupola.png|80px]]<BR>[[triangular cupola]]
| [[File:Elongated triangular cupola.png|80px]]<BR>[[elongated triangular cupola]]
| [[File:Gyroelongated triangular cupola.png|80px]]<BR>[[gyroelongated triangular cupola]]
|- valign=top
| [[File:Augmented hexagonal prism.png|80px]]<BR>[[augmented hexagonal prism]]
| [[File:Parabiaugmented hexagonal prism.png|80px]]<BR>[[parabiaugmented hexagonal prism]]
| [[File:Metabiaugmented hexagonal prism.png|80px]]<BR>[[metabiaugmented hexagonal prism]]
|- valign=top
| [[File:Triaugmented hexagonal prism.png|80px]]<BR>[[triaugmented hexagonal prism]]
| [[File:Augmented truncated tetrahedron.png|80px]]<BR>[[augmented truncated tetrahedron]]
| [[File:Triangular hebesphenorotunda.png|80px]]<BR>[[triangular hebesphenorotunda]]
|}

{| class="wikitable collapsible collapsed" style="text-align:center;"
!colspan=12| [[Prismoid]]s with hexagons
|- valign=top
| [[File:Hexagonal prism.png|100px]]<br>[[Hexagonal prism]]
| [[File:Hexagonal antiprism.png|100px]]<br>[[Hexagonal antiprism]]
| [[File:Hexagonal pyramid.png|100px]]<br>[[Hexagonal pyramid]]
|}

{| class="wikitable collapsible collapsed" style="width:480px;"
!colspan=12| Tilings with regular hexagons
|-
! Regular
!colspan=3| 1-uniform
|- style="text-align:center;"
|[[hexagonal tiling|{6,3}]]<BR>{{CDD|node_1|6|node|3|node}}
|[[Trihexagonal tiling|r{6,3}]]<BR>{{CDD|node|6|node_1|3|node}}
|[[Rhombitrihexagonal tiling|rr{6,3}]]<BR>{{CDD|node_1|6|node|3|node_1}}
|[[Truncated trihexagonal tiling|tr{6,3}]]<BR>{{CDD|node_1|6|node_1|3|node_1}}
|-
|[[Image:Uniform tiling 63-t0.svg|120px]]
|[[Image:Uniform tiling 63-t1.png|120px]]
|[[Image:Uniform polyhedron-63-t02.png|120px]]
|[[Image:Uniform polyhedron-63-t012.png|120px]]
|- style="text-align:center;"
|colspan=4|[[2-uniform tiling]]s
|-
|[[File:2-uniform 1.png|120px]]
|[[File:2-uniform 10.png|120px]]
|[[File:2-uniform 11.png|120px]]
|[[File:2-uniform 12.png|120px]]
|}

==Gallery of natural and artificial hexagons==
<gallery mode="packed">
Image:Graphen.jpg|The ideal crystalline structure æf [[graphene]] is a hexagonal grid.
Image:Assembled E-ELT mirror segments undergoing testing.jpg|Assembled [[E-ELT]] mirror segments
Image:Honey comb.jpg|A beehive [[honeycomb]]
Image:Carapax.svg|The scutes æf a turtle's [[carapace]]
Image:PIA20513 - Basking in Light.jpg|[[Saturn's hexagon]], a hexagonal cloud pattern around þē north pole æf þē planet
Image:Snowflake 300um LTSEM, 13368.jpg|Micrograph æf a snowflake
File:Benzene-aromatic-3D-balls.png|[[Benzene]], þē simplest [[aromatic compound]] with hexagonal shape.
File:Order and Chaos.tif|Hexagonal order æf bubbles in a foam.
Image:Hexa-peri-hexabenzocoronene ChemEurJ 2000 1834 commons.jpg|Crystal structure æf a [[Hexabenzocoronene|molecular hexagon]] composed æf hexagonal aromatic rings.
Image:Giants causeway closeup.jpg|Naturally formed [[basalt]] columns from [[Giant's Causeway]] in [[Northern Ireland]]; large masses must cool slowly to form a polygonal fracture pattern
Image:Fort-Jefferson Dry-Tortugas.jpg|An aerial view æf Fort Jefferson in [[Dry Tortugas National Park]]
Image:Jwst front view.jpg|The [[James Webb Space Telescope]] mirror is composed æf 18 hexagonal segments.
File:564X573-Carte France geo verte.png|In French, ''l'Hexagone'' refers to [[Metropolitan France]] for its vaguely hexagonal shape. 
Image:Hanksite.JPG|Hexagonal [[Hanksite]] crystal, one æf many [[hexagonal crystal system]] minerals
File:HexagonalBarnKewauneeCountyWisconsinWIS42.jpg|Hexagonal barn
Image:Reading the Hexagon Theatre.jpg|[[The Hexagon]], a hexagonal [[theatre]] in [[Reading, Berkshire]]
Image:Hexaschach.jpg|Władysław Gliński's [[hexagonal chess]]
Image:Chinese pavilion.jpg|Pavilion in þē [[Taiwan]] Botanical Gardens
Image:Mustosen talon ikkuna 1870 1.jpg|[[Hexagonal window]]
</gallery>

==See also==
* [[24-cell]]: a [[four-dimensional space|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 (board game)|Havannah]]: abstract board game played on a seoxflanc hexagonal grid

== Fruman ==
{{reflist}}

{{Gesceapu}}
[[Flocc:Ġesċeapu]]

==External links==
{{wiktionary}}
*{{MathWorld|title=Hexagon|urlname=Hexagon}}

*[http://www.mathopenref.com/hexagon.html Definition and properties of a hexagon] with interactive animation and [http://www.mathopenref.com/consthexagon.html construction with compass and straightedge].
*[https://hexnet.org/content/hexagonal-geometry An Introduction to Hexagonal Geometry] on [https://web.archive.org/web/19980204100717/http://www.hexnet.org/ Hexnet] a website devoted to hexagon mathematics.
*{{YouTube|thOifuHs6eY|Hexagons are the Bestagons}} – an [[animation|animated]] [[internet video]] about hexagons by [[CGP Grey]].
<br />

{{Center|{{Polytopes}} }}
{{Polygons}}

[[Category:6 (number)]]
[[Category:Constructible polygons]]
[[Category:Polygons by the number of sides]]
[[Category:Elementary shapes]]