Hexagon

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In geometry, a hexagon (from Greek {{#invoke:Lang|lang}}, {{#invoke:Lang|lang}}, meaning "six", and {{#invoke:Lang|lang}}, {{#invoke:Lang|lang}}, meaning "corner, angle") is a six-sided polygon.<ref>Cube picture</ref> The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.

Regular hexagon

A regular hexagon is defined as a hexagon that is both equilateral and 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>Template:Citation.</ref> However, the regular hexagon can also be considered as the cutting off the vertices of an equilateral triangle, which can also be denoted as <math> \mathrm{t}\{3\} </math>.

A regular hexagon 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 or circumcircle, which equals <math>\tfrac{2}{\sqrt{3}}</math> times the apothem (radius of the 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, and that the regular hexagon can be partitioned into six equilateral triangles.

File:Regular hexagon 1.svg

R = Circumradius; r = Inradius; t = side length

The 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:

<math>\frac{1}{2}d = r = \cos(30^\circ) R = \frac{\sqrt{3}}{2} R = \frac{\sqrt{3}}{2} t</math>   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 Template:Nowrap.

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 >Template:Cite journal</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>

Symmetry

File:Hexagon reflections.svg

The six lines of reflection of a regular hexagon, with Dih₆ or r12 symmetry, order 12.

File:Regular hexagon symmetries.svg

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 D₆.<ref>Template:Citation.</ref> There are 16 subgroups. There are 8 up to isomorphism: itself (D₆), 2 dihedral: (D_3, D₂), 4 cyclic: (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.<ref>John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, Template:ISBN (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 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 be seen as directed edges.

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

p6m (*632)	cmm (2*22)	p2 (2222)	p31m (3*3)	pmg (22*)		pg (××)
File:Isohedral tiling p6-13.svg r12	File:Isohedral tiling p6-12.png i4	File:Isohedral tiling p6-7.svg g2	File:Isohedral tiling p6-11.png d2	File:Isohedral tiling p6-10.png d2	File:Isohedral tiling p6-9.svg p2	File:Isohedral tiling p6-1.png a1
Dih6	Dih2	Z2	Dih1			Z1

File:Root system A2.svg A2 group roots Template:Dynkin

File:Root system G2.svg G2 group roots Template:Dynkin2

The 6 roots of the simple Lie group A2, represented by a Dynkin diagram Template:Dynkin, 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 Dynkin diagram Template:Dynkin2 are also in a hexagonal pattern. The two simple roots of two lengths have a 150° angle between them.

Tessellations

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.<ref>Template:Cite book</ref> 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.

Dissection

6-cube projection	12 rhomb dissection
File:6-cube t0 A5.svg	File:6-gon rhombic dissection-size2.svg	File:6-gon rhombic dissection2-size2.svg

Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into Template:Nowrap parallelograms.<ref>Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141</ref> In 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.

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 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 dodecagon, {12}, alternating two types (colors) of edges. An alternated hexagon, h{6}, is an equilateral triangle, {3}. A regular hexagon can be stellated with equilateral triangles on its edges, creating a hexagram. A regular hexagon can be dissected into 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 dodecagon by adding alternating squares and equilateral triangles around it. This pattern repeats within the rhombitrihexagonal tiling.

File:Regular polygon 6 annotated.svg	File:Truncated triangle.svg	File:Regular truncation 3 1000.svg	File:Regular truncation 3 1.5.svg	File:Regular truncation 3 0.55.svg	File:Hexagram.svg	File:Regular polygon 12 annotated.svg	File:Regular polygon 3 annotated.svg
Regular {6}	Truncated t{3} = {6}	Hypertruncated triangles			Stellated Star figure 2{3}	Truncated t{6} = {12}	Alternated h{6} = {3}

File:Crossed-square hexagon.png	File:Medial triambic icosahedron face.svg	File:Great triambic icosahedron face.svg	File:Hexagonal cupola flat.svg	File:Cube petrie polygon sideview.svg	File:3-cube t0.svg	File:3-cube t2.svg	File:5-simplex graph.svg
Crossed hexagon	A concave hexagon	A self-intersecting hexagon (star polygon)	Extended Central {6} in {12}	A skew hexagon, within cube	Dissected {6}	projection octahedron	Complete graph

Self-crossing hexagons

There are six self-crossing hexagons with the vertex arrangement of the regular hexagon:

Self-intersecting hexagons with regular vertices

Dih2			Dih1		Dih3
File:Crossed hexagon1.svg Figure-eight	File:Crossed hexagon2.svg Center-flip	File:Crossed hexagon3.svg Unicursal	File:Crossed hexagon4.svg Fish-tail	File:Crossed hexagon5.svg Double-tail	File:Crossed hexagon6.svg Triple-tail

Hexagonal structures

File:Giant's Causeway (13).JPG

Giant's Causeway closeup

From bees' honeycombs 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.

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

Hexagonal prism tessellations

Form	Hexagonal tiling	Hexagonal prismatic honeycomb
Regular	File:Uniform tiling 63-t0.svg	File:Hexagonal prismatic honeycomb.png
Parallelogonal	File:Isohedral tiling p6-7.svg	File:Skew hexagonal prism honeycomb.png

Tesselations by hexagons

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} 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 Template:Nowrap.<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.<ref>Template:Cite journal</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>Template:Rp

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,<ref>Gutierrez, Antonio, "Hexagon, Inscribed Circle, Tangent, Semiperimeter", [1] Template:Webarchive, 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

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.<ref>Template:Cite journal</ref>Template:Rp

Skew hexagon

File:Skew polygon in triangular antiprism.png

A regular skew hexagon seen as edges (black) of a triangular antiprism, symmetry D_3d, [2⁺,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_3d, [2⁺,6] symmetry, order 12.

The cube and octahedron (same as triangular antiprism) have regular skew hexagons as petrie polygons.

Skew hexagons on 3-fold axes

File:Cube petrie.png Cube

File:Octahedron petrie.png 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:

4D		5D
File:3-3 duoprism ortho-Dih3.png 3-3 duoprism	File:3-3 duopyramid ortho.png 3-3 duopyramid	File:5-simplex t0.svg 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<ref name="Crux">Inequalities proposed in "Crux Mathematicorum", [2] Template:Webarchive.</ref>Template:Rp a principal diagonal d₁ such that

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

and a principal diagonal d₂ 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 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 Template:CDD and Template:CDD.

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

There are also 9 Johnson solids with regular hexagons:

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

• Graphen.jpg

  The ideal crystalline structure of graphene is a hexagonal grid.

• Assembled E-ELT mirror segments undergoing testing.jpg

  Assembled E-ELT mirror segments

• Honey comb.jpg

  A beehive honeycomb

• Carapax.svg

  The scutes of a turtle's carapace

• PIA20513 - Basking in Light.jpg

  Saturn's hexagon, a hexagonal cloud pattern around the north pole of the planet

• Snowflake 300um LTSEM, 13368.jpg

  Micrograph of a snowflake

• Benzene-aromatic-3D-balls.png

  Benzene, the simplest aromatic compound with hexagonal shape.

• Order and Chaos.tif

  Hexagonal order of bubbles in a foam.

• Hexa-peri-hexabenzocoronene ChemEurJ 2000 1834 commons.jpg

  Crystal structure of a molecular hexagon composed of hexagonal aromatic rings.

• 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

• Fort-Jefferson Dry-Tortugas.jpg

  An aerial view of Fort Jefferson in Dry Tortugas National Park

• Jwst front view.jpg

  The James Webb Space Telescope mirror is composed of 18 hexagonal segments.

• 564X573-Carte France geo verte.png

  lang}} refers to Metropolitan France for its vaguely hexagonal shape.

• Hanksite.JPG

  Hexagonal Hanksite crystal, one of many hexagonal crystal system minerals

• HexagonalBarnKewauneeCountyWisconsinWIS42.jpg

  Hexagonal barn

• Reading the Hexagon Theatre.jpg

  The Hexagon, a hexagonal theatre in Reading, Berkshire

• Hexaschach.jpg

  Władysław Gliński's hexagonal chess

• Chinese pavilion.jpg

  Pavilion in the Taiwan Botanical Gardens

• 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: six-sided star within a regular hexagon
• Unicursal hexagram: single path, six-sided star, within a hexagon
• Honeycomb theorem
• Havannah: abstract board game played on a six-sided hexagonal grid
• Central place theory

External links

• {{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web

|_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:Hexagon%7CHexagon.html}} |title = Hexagon |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}

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

Template:Polytopes

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