View source for Hexagon

{{Short description|Shape with six sides}}
{{For|the crystal system|Hexagonal crystal family}}
{{Other uses}}
{{Redirect|Hexagonal|the FIFA World Cup qualifying tournament in North America|Hexagonal (CONCACAF)}}
{{Regular polygon db|Regular polygon stat table|p6}}

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

==Regular hexagon==
A regular hexagon is defined as a hexagon that is both [[equilateral polygon|equilateral]] and [[equiangular polygon|equiangular]]. In other words, a hexagon is said to be regular if the edges are all equal in length, and each of its [[internal angle]] is equal to 120°. The [[Schläfli symbol]] denotes this polygon as <math> \{6\} </math>.<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> However, the regular hexagon can also be considered as the [[Truncation (geometry)|cutting off the vertices]] of an [[equilateral triangle]], which can also be denoted as <math> \mathrm{t}\{3\} </math>.

A regular hexagon is [[bicentric polygon|bicentric]], meaning that it is both [[cyclic polygon|cyclic]] (has a circumscribed circle) and [[tangential polygon|tangential]] (has an inscribed circle). The common length of the sides equals the radius of the [[circumscribed circle]] or [[circumcircle]], which equals <math>\tfrac{2}{\sqrt{3}}</math> times the [[apothem]] (radius of the [[inscribed figure|inscribed circle]]).

=== Measurement ===
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 triangle|equilateral]], and that the regular hexagon can be partitioned into six equilateral triangles.

[[Image:Regular hexagon 1.svg|thumb|''R'' = [[Circumradius]]; ''r'' = [[Inradius]]; ''t'' = side length]]

The maximal [[diameter#Polygons|diameter]] (which corresponds to the long [[diagonal]] of the hexagon), ''D'', is twice the maximal radius or [[circumradius]], ''R'', which equals the side length, ''t''. The minimal diameter or the diameter of the [[inscribed]] circle (separation of parallel sides, flat-to-flat distance, short diagonal or height when resting on a flat base), ''d'', is twice the minimal radius or [[inradius]], ''r''. The maxima and minima are related by the 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>

The area of a regular hexagon
:<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 any regular [[polygon]], the area can also be expressed in terms of the [[apothem]] ''a'' and the perimeter ''p''. For the regular hexagon these 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>

The regular hexagon fills the fraction <math>\tfrac{3\sqrt{3}}{2\pi} \approx 0.8270</math> of its [[circumscribed circle]].

If a regular hexagon has successive vertices A, B, C, D, E, F and if P is any point on the circumcircle between B and C, then {{nowrap|PE + PF {{=}} PA + PB + PC + PD}}.

It follows from the ratio of [[circumradius]] to [[inradius]] that the height-to-width ratio of a regular hexagon is 1:1.1547005; that is, a hexagon with a long [[diagonal]] of 1.0000000 will have a distance of 0.8660254 or cos(30°) between parallel sides.

=== Point in plane ===
For an arbitrary point in the plane of a regular hexagon with circumradius <math>R</math>, whose distances to the centroid of the regular hexagon and its six 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 of 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= 1 November 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 the distances from the vertices of 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>

=== Construction ===
{{multiple image
| align = center
| image1 = Regular Hexagon Inscribed in a Circle.gif
| width1 = 240 
| alt1 = 
| caption1 = A step-by-step animation of the construction of a regular hexagon using [[compass and straightedge]], given by [[Euclid]]'s ''[[Euclid's Elements|Elements]]'', Book IV, Proposition 15: this is possible as 6 <math>=</math> 2 × 3, a product of a power of two and distinct [[Fermat prime]]s.
| image2 = 01-Sechseck-Seite-vorgegeben-wiki.svg
| width2 = 263
| alt2 = 
| caption2 = When the side length {{Overline|AB}} is given, drawing a circular arc from point A and point B gives the [[intersection]] M, the center of the [[circumscribed circle]]. Transfer the [[line segment]] {{Overline|AB}} four times on the circumscribed circle and connect the corner points.
| footer = 
}}

=== Symmetry ===
[[File:Hexagon reflections.svg|thumb|160px|The six lines of [[reflection symmetry|reflection]] of 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 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'''.]]

A regular hexagon has six [[rotational symmetries]] (''rotational symmetry of order six'') and six [[reflection symmetries]] (''six lines of symmetry''), making up the [[dihedral group]] D<sub>6</sub>.<ref>{{citation
 | last1 = Johnston | first1 = Bernard L.
 | last2 = Richman | first2 = Fred
 | year = 1997
 | publisher = CRC Press
 | title = Numbers and Symmetry: An Introduction to Algebra
 | url = https://books.google.com/books?id=koUfrlgsmUcC&pg=PA92
 | page = 92
| isbn = 978-0-8493-0301-2
 }}.</ref> 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 the trivial (e)

These symmetries express nine distinct symmetries of 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]] 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 [[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 of freedom for irregular forms. Only the '''g6''' subgroup has no degrees of freedom but can be seen as [[directed edge]]s.

Hexagons of symmetry '''g2''', '''i4''', and '''r12''', as [[parallelogon]]s can tessellate the Euclidean plane by translation. Other [[Hexagonal tiling#Topologically equivalent tilings|hexagon shapes can tile the 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>
|}

{| class="wikitable skin-invert-image" 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 of the [[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. The two simple roots have a 120° angle between them.

The 12 roots of the [[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. The two simple roots of two lengths have a 150° angle between them.

=== Tessellations ===
Like [[square (geometry)|square]]s and [[equilateral triangle]]s, regular hexagons fit together without any gaps to ''tile the plane'' (three hexagons meeting at every vertex), and so are useful for constructing [[tessellation]]s.<ref>{{cite book
 | first = Maciej | last = Dunajski
 | year = 2022
 | publisher = Oxford University Press
 | title = Geometry: A Very Short Introduction
 | url = https://books.google.com/books?id=zyRXEAAAQBAJ&pg=PA26
 | page = 26
 | isbn = 978-0-19-968368-0
}}</ref> The cells of a [[beehive (beekeeping)|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.

== Dissection==
{| class="wikitable skin-invert-image" align=right style="text-align:center;"
! [[6-cube]] 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 of 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 the parallelograms are all rhombi. This decomposition of a regular hexagon is based on a [[Petrie polygon]] projection of a [[cube]], with 3 of 6 square faces. Other [[parallelogon]]s and projective directions of the cube are dissected within [[rectangular cuboid]]s.
{| class="wikitable skin-invert-image collapsible" style="text-align:center;"
!colspan=12| Dissection of 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 of the 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) of 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) of 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 six [[equilateral triangle]]s 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 [[square]]s and [[equilateral triangle]]s around it. This pattern repeats within the [[rhombitrihexagonal tiling]].

{| class="wikitable skin-invert-image" 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 skin-invert-image" 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]], within [[cube]]
! Dissected {6}
! projection<BR>[[octahedron]]
! [[Complete graph]]
|}

=== Self-crossing hexagons===
There are six [[Star polygon|self-crossing hexagons]] with the [[vertex arrangement]] of the regular hexagon:
{| class="wikitable skin-invert-image" 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 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 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 the 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.svg|170px]]
| [[File:Skew hexagonal prism honeycomb.png|240px]]
|}

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

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 lines|concurrent]].<ref>{{cite journal|author=Dergiades, Nikolaos|title=Dao's theorem on six 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}}</ref>

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.<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 six [[tangent line]]s 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 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 the sides of an arbitrary hexagon==

[[File:Equilateral in hexagon.svg|thumb|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 [[centroid]]s of 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}}</ref>{{rp|Thm. 1}}
{{-}}

== Skew hexagon==
[[File:Skew polygon in triangular antiprism.png|160px|thumb|A regular skew hexagon seen as edges (black) of a [[triangular antiprism]], symmetry D<sub>3d</sub>, [2<sup>+</sup>,6], (2*3), order 12.]]
A '''skew hexagon''' is a [[skew polygon]] with six vertices and edges but not existing on the same plane. The interior of 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 the vertices and side edges of a [[triangular antiprism]] with the 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 the [[Petrie polygon]] for these higher dimensional [[regular polytope|regular]], uniform and dual polyhedra and polytopes, shown in these skew [[orthogonal projection]]s:

{| class="wikitable skin-invert-image" 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'' of a hexagon is a diagonal which divides the 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 of only regular hexagons, because the hexagons [[tessellation|tessellate]], not allowing the result to "fold up". The [[Archimedean solid]]s 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 [[truncation (geometry)|truncated]] triangles, with [[Coxeter diagram]]s of the 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]]
|}

==Hexagon versus Sexagon==
The debate over whether hexagons should be referred to as "sexagons" has its roots in the etymology of the term. The prefix "hex-" originates from the Greek word "hex," meaning six, while "sex-" comes from the Latin "sex," also signifying six. Some linguists and mathematicians argue that since many English mathematical terms derive from Latin, the use of "sexagon" would align with this tradition. Historical discussions date back to the 19th century, when mathematicians began to standardize terminology in geometry. However, the term "hexagon" has prevailed in common usage and academic literature, solidifying its place in mathematical terminology despite the historical argument for "sexagon." The consensus remains that "hexagon" is the appropriate term, reflecting its Greek origins and established usage in mathematics. (see [[Numeral_prefix#Occurrences]]).

==Gallery of natural and artificial hexagons==
<gallery mode="packed">
Image:Graphen.jpg|The ideal crystalline structure of [[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 of a turtle's [[carapace]]
Image:PIA20513 - Basking in Light.jpg|[[Saturn's hexagon]], a hexagonal cloud pattern around the north pole of the planet
Image:Snowflake 300um LTSEM, 13368.jpg|Micrograph of a snowflake
File:Benzene-aromatic-3D-balls.png|[[Benzene]], the simplest [[aromatic compound]] with hexagonal shape.
File:Order and Chaos.tif|Hexagonal order of bubbles in a foam.
Image:Hexa-peri-hexabenzocoronene ChemEurJ 2000 1834 commons.jpg|Crystal structure of a [[Hexabenzocoronene|molecular hexagon]] composed of 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 of Fort Jefferson in [[Dry Tortugas National Park]]
Image:Jwst front view.jpg|The [[James Webb Space Telescope]] mirror is composed of 18 hexagonal segments.
File:564X573-Carte France geo verte.png|In French, {{Lang|fr|l'Hexagone}} refers to [[Metropolitan France]] for its vaguely hexagonal shape. 
Image:Hanksite.JPG|Hexagonal [[Hanksite]] crystal, one of 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 the [[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 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]]: six-sided star within a regular hexagon
* [[Unicursal hexagram]]: single path, six-sided star, within a hexagon
* [[Honeycomb theorem]]
* [[Havannah (board game)|Havannah]]: abstract board game played on a six-sided hexagonal grid
* [[Central place theory]]

==References==
{{reflist|30em}}

==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]]