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# Hexagon  ( Constructible Polygons )

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In , a hexagon (from ἕξ, hex, meaning "six", and γωνία, gonía, meaning "corner, angle") is a six-sided or 6-gon. The total of the internal angles of any (non-self-intersecting) hexagon is 720°.

Regular hexagon
A hexagon has Schläfli symbol {6}. and can also be constructed as a truncated equilateral triangle, t{3}, which alternates two types of edges.

A regular hexagon is defined as a hexagon that is both equilateral and equiangular. It is bicentric, meaning that it is both (has a circumscribed circle) and tangential (has an inscribed circle).

The common length of the sides equals the radius of the circumscribed circle or , which equals $\tfrac\left\{2\right\}\left\{\sqrt\left\{3\right\}\right\}$ times the (radius of the ). All internal are 120 degrees. A regular hexagon has six rotational symmetries ( rotational symmetry of order six) and six reflection symmetries ( six lines of symmetry), making up the 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 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 , regular hexagons fit together without any gaps to tile the plane (three hexagons meeting at every vertex), and so are useful for constructing . The cells of a beehive are hexagonal for this reason and because the shape makes efficient use of space and building materials. The of a regular triangular lattice is the honeycomb tessellation of hexagons. It is not usually considered a triambus, although it is equilateral.

Parameters
The maximal diameter (which corresponds to the long of the hexagon), D, is twice the maximal radius or , R, which equals the side length, t. The minimal diameter or the diameter of the 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 , r. The maxima and minima are related by the same factor:
$\frac\left\{1\right\}\left\{2\right\}d = r = \cos\left(30^\circ\right) R = \frac\left\{\sqrt\left\{3\right\}\right\}\left\{2\right\} R = \frac\left\{\sqrt\left\{3\right\}\right\}\left\{2\right\} t$   and, similarly, $d = \frac\left\{\sqrt\left\{3\right\}\right\}\left\{2\right\} D.$

The area of a regular hexagon

\begin\left\{align\right\}
 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}

For any regular , the area can also be expressed in terms of the a and the perimeter p. For the regular hexagon these are given by a = r, and p$\left\{\right\} = 6R = 4r\sqrt\left\{3\right\}$, so

\begin\left\{align\right\}
 A &= \frac{ap}{2} \\
&= \frac{r \cdot 4r\sqrt{3}}{2} = 2r^2\sqrt{3} \\
&\approx 3.464 r^2.

\end{align}

The regular hexagon fills the fraction $\tfrac\left\{3\sqrt\left\{3\right\}\right\}\left\{2\pi\right\} \approx 0.8270$ 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 circumcircle between B and C, then .

It follows from the ratio of to that the height-to-width ratio of a regular hexagon is 1:1.1547005; that is, a hexagon with a long of 1.0000000 will have a distance of 0.8660254 between parallel sides.

Point in plane
For an arbitrary point in the plane of a regular hexagon with circumradius $R$, whose distances to the centroid of the regular hexagon and its six vertices are $L$ and $d_i$ respectively, we have

$d_1^2 + d_4^2 = d_2^2 + d_5^2 = d_3^2+ d_6^2= 2\left\left(R^2 + L^2\right\right),$
$d_1^2 + d_3^2+ d_5^2 = d_2^2 + d_4^2+ d_6^2 = 3\left\left(R^2 + L^2\right\right),$
$d_1^4 + d_3^4+ d_5^4 = d_2^4 + d_4^4+ d_6^4 = 3\left\left(\left\left(R^2 + L^2\right\right)^2 + 2 R^2 L^2\right\right).$

If $d_i$ are the distances from the vertices of a regular hexagon to any point on its circumcircle, then

$\left\left(\sum_\left\{i=1\right\}^6 d_i^2\right\right)^2 = 4 \sum_\left\{i=1\right\}^6 d_i^4 .$

Symmetry
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The regular hexagon has D6 symmetry. There are 16 subgroups. There are 8 up to isomorphism: itself (D6), 2 dihedral: (D3, D2), 4 : (Z6, Z3, Z2, Z1) and the trivial (e)

These symmetries express nine 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, (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 hexagon constructed by three mirrors can alternate long and short edges, and d6, an hexagon constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are 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 , while d2 and p2 can be seen as horizontally and vertically elongated kites. g2 hexagons, with opposite sides parallel are also called hexagonal .

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 .

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

A2 and G2 groups

A2 group roots

G2 group roots
The 6 roots of the simple Lie group , represented by a , are in a regular hexagonal pattern. The two simple roots have a 120° angle between them.

The 12 roots of the Exceptional Lie group G2, represented by a are also in a hexagonal pattern. The two simple roots of two lengths have a 150° angle between them.

Dissection
states that every (a 2 m-gon whose opposite sides are parallel and of equal length) can be dissected into parallelograms., Mathematical recreations and Essays, Thirteenth edition, p.141 In particular this is true for with evenly many sides, in which case the parallelograms are all rhombi. This decomposition of a regular hexagon is based on a projection of a , with 3 of 6 square faces. Other and projective directions of the cube are dissected within rectangular cuboids.
Regular {6}Hexagonal
Rectangular cuboid

Related polygons and tilings
A regular hexagon has Schläfli symbol {6}. A regular hexagon is a part of the regular , {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) of edges, this form only has D3 symmetry.

A truncated hexagon, t{6}, is a , {12}, alternating two types (colors) of edges. An alternated hexagon, h{6}, is an equilateral triangle, {3}. A regular hexagon can be with equilateral triangles on its edges, creating a . A regular hexagon can be dissected into six equilateral triangles by adding a center point. This pattern repeats within the regular triangular tiling.

A regular hexagon can be extended into a regular by adding alternating and equilateral triangles around it. This pattern repeats within the rhombitrihexagonal tiling.

Self-crossing hexagons
There are six with the vertex arrangement of the regular hexagon:
 + Self-intersecting hexagons with regular vertices !colspan=3 Dih2 !colspan=2 Dih1 ! Dih3 Figure-eight Center-flip Unicursal Fish-tail Double-tail Triple-tail

Hexagonal structures
From bees' to the Giant's Causeway, hexagonal patterns are prevalent in nature due to their efficiency. In a 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 to construct and gain much strength under compression.

Irregular hexagons with parallel opposite edges are called and can also tile the plane by translation. In three dimensions, with parallel opposite faces are called and these can tessellate 3-space by translation.

 + Hexagonal prism tessellations ! Form ! Hexagonal tiling ! Hexagonal prismatic honeycomb

Tesselations by hexagons
In addition to the regular hexagon, which determines a unique tessellation of the plane, any irregular hexagon which satisfies the 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 , and pairs of opposite until they meet, the three intersection points will lie on a straight line, the "Pascal line" of that configuration.

Cyclic hexagon
The is a 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 .

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 .Cartensen, Jens, "About hexagons", Mathematical Spectrum 33(2) (2000–2001), 37–40.

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 .

If a hexagon has vertices on the of an 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).

Hexagon tangential to a conic section
Let ABCDEF be a hexagon formed by six 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", [1] , Accessed 2012-04-17.

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

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 of opposite triangles form another equilateral triangle.

Skew hexagon
A skew hexagon is a with six 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 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 D3d, 2+,6 symmetry, order 12.

The and (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 the for these higher dimensional , uniform and dual polyhedra and polytopes, shown in these skew orthogonal projections:

3-3 duoprism

3-3 duopyramid

5-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 "Crux Mathematicorum", [2] . a principal diagonal d1 such that

$\frac\left\{d_1\right\}\left\{a\right\} \leq 2$

and a principal diagonal d2 such that

$\frac\left\{d_2\right\}\left\{a\right\} > \sqrt\left\{3\right\}.$

Polyhedra with hexagons
There is no made of only regular hexagons, because the hexagons , not allowing the result to "fold up". The Archimedean solids with some hexagonal faces are the truncated tetrahedron, truncated octahedron, truncated icosahedron (of and fame), truncated cuboctahedron and the truncated icosidodecahedron. These hexagons can be considered truncated triangles, with of the form and .

truncated tetrahedron

truncated octahedron

truncated cuboctahedron

truncated icosahedron

truncated icosidodecahedron

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

Chamfered tetrahedron

Chamfered dodecahedron

There are also 9 with regular hexagons:

triangular cupola

elongated triangular cupola

gyroelongated triangular cupola

augmented hexagonal prism

parabiaugmented hexagonal prism

metabiaugmented hexagonal prism

triaugmented hexagonal prism

augmented truncated tetrahedron

triangular hebesphenorotunda

Truncated triakis tetrahedron

Hexagonal antiprism

Hexagonal pyramid

r{6,3}
rr{6,3}
tr{6,3}
2-uniform tilings

Gallery of natural and artificial hexagons
Image:Graphen.jpg|The ideal crystalline structure of is a hexagonal grid. Image:Assembled E-ELT mirror segments undergoing testing.jpg|Assembled mirror segments Image:Honey comb.jpg|A beehive Image:Carapax.svg|The scutes of a turtle's 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|, 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. Image:Giants causeway closeup.jpg|Naturally formed columns from Giant's Causeway in ; 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|Metropolitan France has a vaguely hexagonal shape. In French, l'Hexagone refers to the European mainland of France. Image:Hanksite.JPG|Hexagonal crystal, one of many hexagonal crystal system minerals File:HexagonalBarnKewauneeCountyWisconsinWIS42.jpg|Hexagonal barn Image:Reading the Hexagon Theatre.jpg|, a hexagonal in Reading, Berkshire Image:Hexaschach.jpg|Władysław Gliński's Image:Chinese pavilion.jpg|Pavilion in the Botanical Gardens Image:Mustosen talon ikkuna 1870 1.jpg|

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