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In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.


Regular hexagon

A '' regular hexagon'' has
Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more ...
and can also be constructed as a truncated equilateral triangle, t, 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 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 \tfrac times the apothem (radius of the
inscribed circle In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...
). All internal angles 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 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 In geometry, an equilateral polygon is a polygon which has all sides of the same length. Except in the triangle case, an equilateral polygon does not need to also be equiangular polygon, equiangular (have all angles equal), but if it does then it ...
, although it is equilateral.


Parameters

The maximal diameter (which corresponds to the long diagonal of the hexagon), ''D'', is twice the maximal radius or
circumradius In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...
, ''R'', which equals the side length, ''t''. The minimal diameter or the diameter of the
inscribed {{unreferenced, date=August 2012 An inscribed triangle of a circle In geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. To say that "figure F is inscribed in figur ...
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 In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...
, ''r''. The maxima and minima are related by the same factor: :\fracd = r = \cos(30^\circ) R = \frac R = \frac t   and, similarly, d = \frac D. The area of a regular hexagon :\begin A &= \fracR^2 = 3Rr = 2\sqrt r^2 \\ pt &= \fracD^2 = \fracDd = \frac d^2 \\ pt &\approx 2.598 R^2 \approx 3.464 r^2\\ &\approx 0.6495 D^2 \approx 0.866 d^2. \end 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'' = 6R = 4r\sqrt, so :\begin A &= \frac \\ &= \frac = 2r^2\sqrt \\ &\approx 3.464 r^2. \end The regular hexagon fills the fraction \tfrac \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
circumradius In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...
to
inradius In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...
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 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(R^2 + L^2\right), : 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), : 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). If d_i are the distances from the vertices of a regular hexagon to any point on its circumcircle, then :\left(\sum_^6 d_i^2\right)^2 = 4 \sum_^6 d_i^4 .


Symmetry

The ''regular hexagon'' has D6 symmetry. There are 16 subgroups. There are 8 up to isomorphism: itself (D6), 2 dihedral: (D3, D2), 4 cyclic: (Z6, Z3, Z2, Z1) and the trivial (e) These symmetries express nine distinct symmetries of a regular hexagon.
John Conway John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches o ...
labels these by a letter and group order. 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 In geometry, a polytope (for example, a polygon or a polyhedron) or a tiling is isotoxal () or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given two ...
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
parallelogon In geometry, a parallelogon is a polygon with parallel opposite sides (hence the name) that can tile a plane by translation (rotation is not permitted). Parallelogons have an even number of sides and opposite sides that are equal in length. A l ...
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 seen as directed edges. Hexagons of symmetry g2, i4, and r12, as
parallelogon In geometry, a parallelogon is a polygon with parallel opposite sides (hence the name) that can tile a plane by translation (rotation is not permitted). Parallelogons have an even number of sides and opposite sides that are equal in length. A l ...
s can tessellate the Euclidean plane by translation. Other hexagon shapes can tile the plane with different orientations.


A2 and G2 groups

The 6 roots of the simple Lie group A2, represented by a Dynkin diagram , are in a regular hexagonal pattern. The two simple roots have a 120° angle between them. The 12 roots of the
Exceptional Lie group In mathematics, a simple Lie group is a connected non-abelian Lie group ''G'' which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symm ...
G2, represented by a Dynkin diagram are also in a hexagonal pattern. The two simple roots of two lengths have a 150° angle between them.


Dissection

Coxeter states that every zonogon (a 2''m''-gon whose opposite sides are parallel and of equal length) can be dissected into parallelograms. 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 In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only r ...
, with 3 of 6 square faces. Other
parallelogon In geometry, a parallelogon is a polygon with parallel opposite sides (hence the name) that can tile a plane by translation (rotation is not permitted). Parallelogons have an even number of sides and opposite sides that are equal in length. A l ...
s and projective directions of the cube are dissected within rectangular cuboids.


Related polygons and tilings

A regular hexagon has
Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more ...
. A regular hexagon is a part of the regular
hexagonal tiling In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of or (as a truncated triangular tiling). English mathemat ...
, , with three hexagonal faces around each vertex. A regular hexagon can also be created as a truncated equilateral triangle, with Schläfli symbol t. Seen with two types (colors) of edges, this form only has D3 symmetry. A truncated hexagon, t, is a dodecagon, , alternating two types (colors) of edges. An alternated hexagon, h, is an equilateral triangle, . A regular hexagon can be
stellated In geometry, stellation is the process of extending a polygon in two dimensions, polyhedron in three dimensions, or, in general, a polytope in ''n'' dimensions to form a new figure. Starting with an original figure, the process extends specific el ...
with equilateral triangles on its edges, creating a
hexagram , can be seen as a compound composed of an upwards (blue here) and downwards (pink) facing equilateral triangle, with their intersection as a regular hexagon (in green). A hexagram ( Greek language, Greek) or sexagram (Latin) is a six-pointed ...
. 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 In geometry, the rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. There are one triangle, two squares, and one hexagon on each vertex. It has Schläfli symbol of rr. John Conway calls it a rhombihexadeltille.Conway, 2 ...
.


Self-crossing hexagons

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


Hexagonal structures

From bees' honeycombs to the Giant's Causeway, hexagonal patterns are prevalent in nature due to their efficiency. In a
hexagonal grid In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of or (as a truncated triangular tiling). English mathematic ...
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
parallelogon In geometry, a parallelogon is a polygon with parallel opposite sides (hence the name) that can tile a plane by translation (rotation is not permitted). Parallelogons have an even number of sides and opposite sides that are equal in length. A l ...
s 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.


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 In the mathematical theory of tessellations, the Conway criterion, named for the English mathematician John Horton Conway, is a sufficient rule for when a prototile will tile the plane. It consists of the following requirements:Will It Tile? Try ...
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 In geometry, the Lemoine hexagon is a cyclic hexagon 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. There are two definitions of th ...
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 . 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. 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).


Hexagon tangential to a conic section

Let ABCDEF be a hexagon formed by six tangent lines of a conic section. Then
Brianchon's theorem In geometry, Brianchon's theorem is a theorem stating that when a hexagon is circumscribed around a conic section, its principal diagonals (those connecting opposite vertices) meet in a single point. It is named after Charles Julien Brianchon (1 ...
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'', :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 centroids of opposite triangles form another equilateral triangle.


Skew hexagon

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 In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in ...
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 In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...
with the same D3d, +,6symmetry, order 12. The
cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only r ...
and octahedron (same as triangular antiprism) have regular skew hexagons as petrie polygons.


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:


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"''

.
a principal diagonal ''d''1 such that :\frac \leq 2 and a principal diagonal ''d''2 such that :\frac > \sqrt.


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 solid In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids (which are composed ...
s with some hexagonal faces are the truncated tetrahedron,
truncated octahedron In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular hexagon, hexagons and 6 Squa ...
, truncated icosahedron (of soccer ball and fullerene fame),
truncated cuboctahedron In geometry, the truncated cuboctahedron is an Archimedean solid, named by Kepler as a truncation of a cuboctahedron. It has 12 square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices, and 72 edges. Since each of its fac ...
and the
truncated icosidodecahedron In geometry, a truncated icosidodecahedron, rhombitruncated icosidodecahedron,Wenninger Model Number 16 great rhombicosidodecahedron,Williams (Section 3-9, p. 94)Cromwell (p. 82) omnitruncated dodecahedron or omnitruncated icosahedronNorman Wooda ...
. These hexagons can be considered truncated triangles, with Coxeter diagrams of the form and . There are other symmetry polyhedra with stretched or flattened hexagons, like these Goldberg polyhedron G(2,0): There are also 9
Johnson solid In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that isohedral, each face must be the same polygon, or that the same polygons join around each Vertex (geometry), ver ...
s with regular hexagons:


Gallery of natural and artificial hexagons

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 A carapace is a Dorsum (biology), dorsal (upper) section of the exoskeleton or shell in a number of animal groups, including arthropods, such as crustaceans and arachnids, as well as vertebrates, such as turtles and tortoises. In turtles and tor ...
Image:PIA20513 - Basking in Light.jpg,
Saturn's hexagon Saturn's hexagon is a persistent approximately hexagonal cloud pattern around the north pole of the planet Saturn, located at about 78°N. The sides of the hexagon are about long, which is about longer than the diameter of Earth. The 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 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 The James Webb Space Telescope (JWST) is a space telescope which conducts infrared astronomy. As the largest optical telescope in space, its high resolution and sensitivity allow it to view objects too old, distant, or faint for the Hubble Spa ...
mirror is composed of 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 Hanksite is a sulfate mineral, distinguished as one of only a handful that contain both carbonate and sulfate ion groups. It has the chemical formula Sodium, Na22Potassium, K(Sulfate, SO4)9(Carbonate, CO3)2Chlorine, Cl. Occurrence It was first de ...
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 Hexagonal chess is a group of chess variants played on boards composed of hexagon . The best known is Gliński's variant, played on a symmetric 91-cell hexagonal board. Since each hexagonal cell not on a board edge has six neighbor cells, there ...
Image:Chinese pavilion.jpg, Pavilion in the Taiwan Botanical Gardens Image:Mustosen talon ikkuna 1870 1.jpg,
Hexagonal window A hexagonal window (also Melnikov's or honeycomb window) is a hexagon-shaped window, resembling a bee cell or crystal lattice of graphite. The window can be vertically or horizontally oriented, openable or fixed. It can also be regular or elongately ...


See also

*
24-cell In geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called C24, or the icositetrachoron, octaplex (short for "octahedral complex"), icosatetrahedroid, oct ...
: a
four-dimensional A four-dimensional space (4D) is a mathematical extension of the concept of three-dimensional or 3D space. Three-dimensional space is the simplest possible abstraction of the observation that one only needs three numbers, called ''dimensions'', ...
figure which, like the hexagon, has orthoplex facets, is
self-dual In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a Injective function, one-to-one fashion, often (but not always) by means of an Involution (mathematics), involutio ...
and tessellates Euclidean space * Hexagonal crystal system * Hexagonal number *
Hexagonal tiling In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of or (as a truncated triangular tiling). English mathemat ...
: a regular tiling of hexagons in a plane *
Hexagram , can be seen as a compound composed of an upwards (blue here) and downwards (pink) facing equilateral triangle, with their intersection as a regular hexagon (in green). A hexagram ( Greek language, Greek) or sexagram (Latin) is a six-pointed ...
: six-sided star within a regular hexagon * Unicursal hexagram: single path, six-sided star, within a hexagon * Honeycomb conjecture * Havannah: abstract board game played on a six-sided hexagonal grid


References


External links

*
Definition and properties of a hexagon
with interactive animation an


An Introduction to Hexagonal Geometry
o
Hexnet
a website devoted to hexagon mathematics. * – an animated internet video about hexagons by CGP Grey. {{Polytopes 6 (number) Constructible polygons Polygons by the number of sides Elementary shapes