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hexagon

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**hexagon**(from Greek á¼•Î¾

*hex*, "six" and Î³Ï‰Î½Î¯Î±,

*gonÃa*, "corner, angle") is a six-sided polygon or 6-gon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720Â°.

## Regular hexagon

A*regular hexagon*has SchlÃ¤fli symbol {1}{{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}}. and can also be constructed as a truncated equilateral triangle, t{3}, which alternates two types of edges.{{Double image|left|Regular Hexagon Inscribed in a Circle.gif|240 |01-Sechseck-Seite-vorgegeben-wiki.svg|263|A step-by-step animation of the construction of a regular hexagon using compass and straightedge, given by Euclid's

*Elements*, Book IV, Proposition 15: this is possible as 6 = 2 Ã— 3, a product of a power of two and distinct Fermat primes.|When the side length {{Overline|AB}} is given, then you draw around the point A and around the point B a circular arc. The intersection M is the center of the circumscribed circle. Transfer the line segment {{Overline|AB}} four times on the circumscribed circle and connect the corner points.|}}A regular hexagon is defined as a hexagon that is both equilateral and equiangular. It is bicentric, meaning that it is both cyclic (has a circumscribed circle) and tangential (has an inscribed circle).The common length of the sides equals the radius of the circumscribed circle, which equals tfrac{2}{sqrt{3}} times the apothem (radius of the inscribed circle). All internal angles are 120 degrees. A regular hexagon has 6 rotational symmetries (

*rotational symmetry of order six*) and 6 reflection symmetries (

*six lines of symmetry*), making up the dihedral group D6. The longest diagonals of a regular hexagon, connecting diametrically opposite vertices, are twice the length of one side. From this it can be seen that a triangle with a vertex at the center of the regular hexagon and sharing one side with the hexagon is equilateral, and that the regular hexagon can be partitioned into six equilateral triangles.Like squares and equilateral triangles, regular hexagons fit together without any gaps to

*tile the plane*(three hexagons meeting at every vertex), and so are useful for constructing tessellations. The cells of a beehive honeycomb are hexagonal for this reason and because the shape makes efficient use of space and building materials. The Voronoi diagram of a regular triangular lattice is the honeycomb tessellation of hexagons. It is not usually considered a triambus, although it is equilateral.{{clear}}

### Parameters

thumb|rightThe maximal 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:

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.

The area of a regular hexagon
begin{align}

A &= frac{3sqrt{3}}{2}R^2 = 3Rr = 2sqrt{3} r^2

&= frac{3sqrt{3}}{8}D^2 = frac{3}{4}Dd = frac{sqrt{3}}{2} d^2

&approx 2.598 R^2 approx 3.464 r^2

&approx 0.6495 D^2 approx 0.866 d^2.

end{align}For any regular polygon, the area can also be expressed in terms of the apothem &= frac{3sqrt{3}}{8}D^2 = frac{3}{4}Dd = frac{sqrt{3}}{2} d^2

&approx 2.598 R^2 approx 3.464 r^2

&approx 0.6495 D^2 approx 0.866 d^2.

*a*and the perimeter

*p*. For the regular hexagon these are given by

*a*=

*r*, and

*p*{} = 6R = 4rsqrt{3}, so

begin{align}

A &= frac{ap}{2}

&= frac{r cdot 4rsqrt{3}}{2} = 2r^2sqrt{3}

&approx 3.464 r^2.

end{align}The regular hexagon fills the fraction tfrac{3sqrt{3}}{2pi} approx 0.8270 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 circumscribing circle between B and C, then {{nowrap|PE + PF {{=}} PA + PB + PC + PD}}.&= frac{r cdot 4rsqrt{3}}{2} = 2r^2sqrt{3}

&approx 3.464 r^2.

## Symmetry

File:Hexagon reflections.svg|thumb|160px|left|The six lines of reflection of a regular hexagon, with Dih6 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**.)The

*regular hexagon*has Dih6 symmetry, order 12. There are 3 dihedral subgroups: Dih3, Dih2, and Dih1, and 4 cyclic subgroups: Z6, Z3, Z2, and Z1.These symmetries express 9 distinct symmetries of a regular hexagon. John Conway labels these by a letter and group order.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)

**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 of each other and have half the symmetry order of the regular hexagon. The

**i4**forms are regular hexagons flattened or stretched along one symmetry direction. It can be seen as an elongated rhombus, while

**d2**and

**p2**can be seen as horizontally and vertically elongated kites.

**g2**hexagons, with opposite sides parallel are also called hexagonal parallelogons.Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the

**g6**subgroup has no degrees of freedom but can seen as directed edges.{| class="collapsible collapsed"!Example hexagons by symmetry|{| class=wikitable

!!(File:Hexagon_i4_symmetry.png|60px)i4! |

**g2**,

**i4**, and

**r12**, as parallelogons can tessellate the Euclidean plane by translation. Other hexagon shapes can tile the plane with different orientations.{| class=wikitable!p6m (*632)!cmm (2*22)!p2 (2222)!p31m (3*3)!colspan=2|pmg (22*)!pg (Ã—Ã—)!(File:Isohedral_tiling_p6-13.png|120px)r12!(File:Isohedral_tiling_p6-12.png|120px)i4!(File:Isohedral_tiling_p6-7.png|120px)g2!(File:Isohedral tiling p6-11.png|120px)d2!(File:Isohedral tiling p6-10.png|120px)d2!(File:Isohedral tiling p6-9.png|120px)p2!(File:Isohedral tiling p6-1.png|120px)a1

### A2 and G2 groups {| classwikitable alignright align=center

120px)A2 group roots{{Dynkin | 3|node_n2}} | 120px)G2 group roots{{Dynkin2 | 6a|node_n2}} |

## Dissection{| classwikitable alignright

!6-cube projection!colspan=2|10 rhomb dissection align=center120px) | 140px) | 140px) |

*m*-gon whose opposite sides are parallel and of equal length) can be dissected into

*m*(

*m*-1)/2 parallelograms.Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141In particular this is true for regular polygons 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 parallelogons and projective directions of the cube are dissected within rectangular cuboids.{| class="wikitable collapsible"!colspan=12| Dissection of hexagons into 3 rhombs and parallelograms!rowspan=3|2D!Rhombs!colspan=3|Parallelograms

80px) | 95px) | 120px) | 120px) |

Regular {6} | Hexagonal parallelogons |

95px) | 95px) | 120px) | 120px) |

Cube | Rectangular cuboid |

## Related polygons and tilings

A regular hexagon has SchlÃ¤fli symbol {6}. A regular hexagon is a part of the regular hexagonal tiling, {6,3}, with 3 hexagonal around each vertex.A regular hexagon can also be created as a truncated equilateral triangle, with SchlÃ¤fli symbol t{3}. Seen with two types (colors) of edges, this form only has D3 symmetry.A truncated hexagon, t{6}, is a dodecagon, {12}, alternating 2 types (colors) of edges. An alternated hexagon, h{6}, is a equilateral triangle, {3}. A regular hexagon can be stellated with equilateral triangles on its edges, creating a hexagram. A regular hexagon can be dissected into 6 equilateral triangles by adding a center point. This pattern repeats within the regular triangular tiling.A regular hexagon can be extended into a regular dodecagon by adding alternating squares and equilateral triangles around it. This pattern repeats within the rhombitrihexagonal tiling.{| class=wikitable style="text-align:center;" width=64080px) | missing image! - Truncated triangle.svg>80px]] | 80px) | 80px) | 80px) | Hexagram.svg>80px | 80px) | 80px) |

80px) | 80px) | 80px) | 80px) | 80px) |

## Hexagonal structures

File:Giants causeway closeup.jpg -170px) | 170px) |

170px) | 240px) |

## Tesselations by hexagons

In addition to the regular hexagon, which determines a unique tessellation of the plane, any irregular hexagon which satisfies the Conway criterion will tile the plane.## Hexagon inscribed in a conic section

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

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

*b*,

*c*,

*d*,

*e*,

*f*, then the three main diagonals intersect in a single point if and only if {{nowrap|

*ace*{{=}}

*bdf*}}.Cartensen, Jens, "About hexagons",

*Mathematical Spectrum*33(2) (2000â€“2001), 37â€“40.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.JOURNAL, Dergiades, Nikolaos, Dao's theorem on six circumcenters associated with a cyclic hexagon, Forum Geometricorum, 14, 2014, 243-246,weblink 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.Johnson, Roger A.,

*Advanced Euclidean Geometry*, Dover Publications, 2007 (orig. 1960).{{rp|p. 179}}

## Hexagon tangential to a conic section

Let ABCDEF be a hexagon formed by six tangent lines of a conic section. Then Brianchon's theorem states that the three main diagonals AD, BE, and CF intersect at a single point.In a hexagon that is tangential to a circle and that has consecutive sides*a*,

*b*,

*c*,

*d*,

*e*, and

*f*,Gutierrez, Antonio, "Hexagon, Inscribed Circle, Tangent, Semiperimeter", weblink, Accessed 2012-04-17.

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

## 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 centroids of opposite triangles form another equilateral triangle.JOURNAL, Dao Thanh Oai, 2015, Equilateral triangles and Kiepert perspectors in complex numbers, Forum Geometricorum, 15, 105-114,weblink {{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 antiprismtriangular antiprismA**skew hexagon**is a skew polygon with 6 vertices and edges but not existing on the same plane. The interior of such an hexagon is not generally defined. A

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

**regular skew hexagon**is vertex-transitive with equal edge lengths. In 3-dimensions it will be a zig-zag skew hexagon and can be seen in the vertices and side edges of a triangular antiprism with the same D3d, [2+,6] symmetry, order 12.The cube and octahedron (same as triangular antiprism) have regular skew hexagons as petrie polygons.{| class="wikitable"|+Skew hexagons on 3-fold axes align=center

100px)Cube | 100px)Octahedron |

### Petrie polygons

The regular skew hexagon is the Petrie polygon for these higher dimensional regular, uniform and dual polyhedra and polytopes, shown in these skew orthogonal projections:{| class="wikitable" style="width:360px;" align=center!colspan=2|4D!5D100px)3-3 duoprism | 100px)3-3 duopyramid | Image:5-simplex t0.svg>100px5-simplex |

## Convex equilateral hexagon

A*principal diagonal*of a hexagon is a diagonal which divides the hexagon into quadrilaterals. In any convex equilateral hexagon (one with all sides equal) with common side

*a*, there exists

*Inequalities proposed in â€œCrux Mathematicorumâ€*, weblink.{{rp|p.184,#286.3}} a principal diagonal

*d*1 such that

frac{d_1}{a} leq 2

and a principal diagonal *d*2 such that

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

### Polyhedra with hexagons

There is no Platonic solid made of only regular hexagons, because the hexagons tessellate, not allowing the result to "fold up". The Archimedean solids with some hexagonal faces are the truncated tetrahedron, truncated octahedron, truncated icosahedron (of soccer ball and fullerene fame), truncated cuboctahedron and the truncated icosidodecahedron. These hexagons can be considered truncated triangles, with Coxeter diagrams of the form {{CDD|node_1|3|node_1|p|node}} and {{CDD|node_1|3|node_1|p|node_1}}.{| class="wikitable collapsible collapsed"!colspan=12|Hexagons in Archimedean solids!Tetrahedral!colspan=2|Octahedral!colspan=2|Icosahedralnode_1 | node_1 | node}} | node_1 | node_1 | node}} | node_1 | node_1 | node_1}} | node_1 | node_1 | node}} | node_1 | node_1 | node_1}} |

100px)truncated tetrahedron | 100px)truncated octahedron | 100px)truncated cuboctahedron | 100px)truncated icosahedron | 100px)truncated icosidodecahedron |

120px)Chamfered tetrahedron | 120px)Chamfered cube | 120px)Chamfered dodecahedron |

80px)triangular cupola | 80px)elongated triangular cupola | 80px)gyroelongated triangular cupola |

80px)augmented hexagonal prism | 80px)parabiaugmented hexagonal prism | 80px)metabiaugmented hexagonal prism | 80px)triaugmented hexagonal prism |

80px)augmented truncated tetrahedron | 80px)triangular hebesphenorotunda | 80px)Truncated triakis tetrahedron | 80px) |

100px)Hexagonal prism | 100px)Hexagonal antiprism | 100px)Hexagonal pyramid |

hexagonal tiling>{6,3}{{CDD | 6 | 3|node}} | Trihexagonal tiling>r{6,3}{{CDD | 6 | 3|node}} | Rhombitrihexagonal tiling>rr{6,3}{{CDD | 6 | 3|node_1}} | Truncated trihexagonal tiling>tr{6,3}{{CDD | 6 | 3|node_1}} |

Image:Uniform tiling 63-t0.png>120px | Image:Uniform tiling 63-t1.png>120px | Image:Uniform polyhedron-63-t02.png>120px | Image:Uniform polyhedron-63-t012.png>120px |

2-uniform tilings |

120px) | 120px) | 120px) | 120px) |

## Hexagons: natural and human-made

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 segmentsImage:Honey comb.jpg|A beehive honeycombImage:Carapax.svg|The scutes of a turtle's carapaceImage:Saturn hexagonal north pole feature.jpg|North polar hexagonal cloud feature on Saturn, discovered by Voyager 1 and confirmed in 2006 by Cassini weblink weblink weblinkImage:Snowflake 300um LTSEM, 13368.jpg|Micrograph of a snowflakeFile: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 molecular hexagon composed of hexagonal aromatic rings reported by MÃ¼llen and coworkers in Chem. Eur. J., 2000, 1834-1839.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 patternImage:Fort-Jefferson Dry-Tortugas.jpg|An aerial view of Fort Jefferson in Dry Tortugas National ParkImage:Jwst front view.jpg|The James Webb Space Telescope mirror is composed of 18 hexagonal segments.File:564X573-Carte France geo verte.png|Metropolitan France has a vaguely hexagonal shape. In French,*l'Hexagone*refers to the European mainland of France aka the "mÃ©tropole" as opposed to the overseas territories such as Guadeloupe, Martinique or French Guiana.Image:Hanksite.JPG|Hexagonal Hanksite crystal, one of many hexagonal crystal system mineralsFile:HexagonalBarnKewauneeCountyWisconsinWIS42.jpg|Hexagonal barnImage:Reading the Hexagon Theatre.jpg|The Hexagon, a hexagonal theatre in Reading, BerkshireImage:Hexaschach.jpg|WÅ‚adysÅ‚aw GliÅ„ski's hexagonal chessImage:Chinese pavilion.jpg|Pavilion in the Taiwan Botanical GardensImage:Mustosen talon ikkuna 1870 1.jpg|Hexagonal window

## See also

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

## References

{{reflist|30em}}## External links

{{wiktionary}}- {{MathWorld|title=Hexagon|urlname=Hexagon}}

- Definition and properties of a hexagon with interactive animation and construction with compass and straightedge.
- Cymatics â€“ Hexagonal shapes occurring within water sound images{{Dead link|date=May 2010}}
- Cassini Images Bizarre Hexagon on Saturn
- Saturn's Strange Hexagon
- A hexagonal feature around Saturn's North Pole
- "Bizarre Hexagon Spotted on Saturn" â€“ from Space.com (27 March 2007)

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