In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a hexagon (from
Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided
polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two ...
. The total of the internal angles of any
simple
Simple or SIMPLE may refer to:
*Simplicity, the state or quality of being simple
Arts and entertainment
* ''Simple'' (album), by Andy Yorke, 2008, and its title track
* "Simple" (Florida Georgia Line song), 2018
* "Simple", a song by Johnn ...
(non-self-intersecting) hexagon is 720°.
Regular hexagon
A ''
regular hexagon'' has
Schläfli symbol 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
Cycle, cycles, or cyclic may refer to:
Anthropology and social sciences
* Cyclic history, a theory of history
* Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr.
* Social cycle, various cycles in so ...
(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
times the
apothem
The apothem (sometimes abbreviated as apo) of a regular polygon is a line segment from the center to the midpoint of one of its sides. Equivalently, it is the line drawn from the center of the polygon that is perpendicular to one of its sides. ...
(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 incen ...
). All internal
angle
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles ...
s are 120
degrees. A regular hexagon has six
rotational symmetries (''rotational symmetry of order six'') and six
reflection symmetries
In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry.
In 2D t ...
(''six lines of symmetry''), making up the
dihedral group D
6. 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
tessellation
A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of ...
s. 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.
Parameters
The maximal
diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid f ...
(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 pol ...
, ''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 figu ...
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 incen ...
, ''r''. The maxima and minima are related by the same factor:
:
and, similarly,
The area of a regular hexagon
:
For any regular
polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two ...
, the area can also be expressed in terms of the
apothem
The apothem (sometimes abbreviated as apo) of a regular polygon is a line segment from the center to the midpoint of one of its sides. Equivalently, it is the line drawn from the center of the polygon that is perpendicular to one of its sides. ...
''a'' and the perimeter ''p''. For the regular hexagon these are given by ''a'' = ''r'', and ''p''
, so
:
The regular hexagon fills the fraction
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 pol ...
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 incen ...
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
, whose distances to the centroid of the regular hexagon and its six vertices are
and
respectively, we have
:
:
:
If
are the distances from the vertices of a regular hexagon to any point on its circumcircle, then
:
Symmetry
The ''regular hexagon'' has D
6 symmetry. There are 16 subgroups. There are 8 up to isomorphism: itself (D
6), 2 dihedral: (D
3, D
2), 4
cyclic
Cycle, cycles, or cyclic may refer to:
Anthropology and social sciences
* Cyclic history, a theory of history
* Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr.
* Social cycle, various cycles in so ...
: (Z
6, Z
3, Z
2, Z
1) and the trivial (e)
These symmetries express nine distinct symmetries of a regular hexagon.
John Conway 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 ...
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.
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.
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 sym ...
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, 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 . A regular hexagon is a part of the regular
hexagonal tiling, , 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 D
3 symmetry.
A
truncated hexagon, t, is a
dodecagon
In geometry, a dodecagon or 12-gon is any twelve-sided polygon.
Regular dodecagon
A regular dodecagon is a figure with sides of the same length and internal angles of the same size. It has twelve lines of reflective symmetry and rotational s ...
, , alternating two types (colors) of edges. An
alternated hexagon, h, is an
equilateral triangle, . 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
In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. Because the internal angle of the equilater ...
.
A regular hexagon can be extended into a regular
dodecagon
In geometry, a dodecagon or 12-gon is any twelve-sided polygon.
Regular dodecagon
A regular dodecagon is a figure with sides of the same length and internal angles of the same size. It has twelve lines of reflective symmetry and rotational s ...
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, ...
.
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 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.
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? ...
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
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a ...
, 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
Cycle, cycles, or cyclic may refer to:
Anthropology and social sciences
* Cyclic history, a theory of history
* Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr.
* Social cycle, various cycles in so ...
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
An acute triangle (or acute-angled triangle) is a triangle with three acute angles (less than 90°). An obtuse triangle (or obtuse-angled triangle) is a triangle with one obtuse angle (greater than 90°) and two acute angles. Since a triangle's 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
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 ...
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'',
:
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
Skew may refer to:
In mathematics
* Skew lines, neither parallel nor intersecting.
* Skew normal distribution, a probability distribution
* Skew field or division ring
* Skew-Hermitian matrix
* Skew lattice
* Skew polygon, whose vertices do ...
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 fa ...
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 D
3d,
+,6">+,6symmetry, order 12.
The
cube 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
:
and a principal diagonal ''d''
2 such that
:
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
In geometry, the truncated icosahedron is an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares ...
(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 fa ...
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 solids with regular hexagons:
Gallery of natural and artificial hexagons
Image:Graphen.jpg, The ideal crystalline structure of graphene
Graphene () is an allotrope of carbon consisting of a Single-layer materials, single layer of atoms arranged in a hexagonal lattice nanostructure. 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
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 m ...
, 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
Benzene is an organic chemical compound with the molecular formula C6H6. The benzene molecule is composed of six carbon atoms joined in a planar ring with one hydrogen atom attached to each. Because it contains only carbon and hydrogen atoms ...
, the simplest aromatic compound
Aromatic compounds, also known as "mono- and polycyclic aromatic hydrocarbons", are organic compounds containing one or more aromatic rings. The parent member of aromatic compounds is benzene. The word "aromatic" originates from the past groupin ...
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
Basalt (; ) is an aphanitic (fine-grained) extrusive igneous rock formed from the rapid cooling of low-viscosity lava rich in magnesium and iron (mafic lava) exposed at or very near the surface of a rocky planet or moon. More than 90 ...
columns from Giant's Causeway in Northern Ireland
Northern Ireland ( ga, Tuaisceart Éireann ; sco, label=Ulster Scots dialect, Ulster-Scots, Norlin Airlann) is a part of the United Kingdom, situated in the north-east of the island of Ireland, that is #Descriptions, variously described as ...
; 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
Dry Tortugas National Park is a national park located about west of Key West in the Gulf of Mexico. The park preserves Fort Jefferson and the seven Dry Tortugas islands, the westernmost and most isolated of the Florida Keys. The archipelago ...
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, ''l'Hexagone'' refers to Metropolitan France for its vaguely hexagonal shape.
Image:Hanksite.JPG, Hexagonal Hanksite crystal, one of many hexagonal crystal system
In crystallography, the hexagonal crystal family is one of the six crystal families, which includes two crystal systems (hexagonal and trigonal) and two lattice systems (hexagonal and rhombohedral). While commonly confused, the trigonal crysta ...
minerals
File:HexagonalBarnKewauneeCountyWisconsinWIS42.jpg, Hexagonal barn
Image:Reading the Hexagon Theatre.jpg, The Hexagon, a hexagonal theatre
Theatre or theater is a collaborative form of performing art that uses live performers, usually actors or actresses, to present the experience of a real or imagined event before a live audience in a specific place, often a stage. The perfor ...
in Reading, Berkshire
Reading ( ) is a town and borough in Berkshire, southeast England. Located in the Thames Valley at the confluence of the rivers Thames and Kennet, the Great Western Main Line railway and the M4 motorway serve the town. Reading is east ...
Image:Hexaschach.jpg, Władysław Gliński's hexagonal chess
Image:Chinese pavilion.jpg, Pavilion in the Taiwan
Taiwan, officially the Republic of China (ROC), is a country in East Asia, at the junction of the East and South China Seas in the northwestern Pacific Ocean, with the People's Republic of China (PRC) to the northwest, Japan to the no ...
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: 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 one-to-one fashion, often (but not always) by means of an involution operation: if the dual of is , then the ...
and tessellates
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
*
Hexagonal crystal system
In crystallography, the hexagonal crystal family is one of the six crystal families, which includes two crystal systems (hexagonal and trigonal) and two lattice systems (hexagonal and rhombohedral). While commonly confused, the trigonal crysta ...
*
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 conjecture
*
Havannah: abstract board game played on a six-sided hexagonal grid
References
External links
*
Definition and properties of a hexagonwith interactive animation an
An Introduction to Hexagonal Geometryo
Hexneta 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