Hexagon
Regular hexagon | |
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Dual polygon | Self |
In geometry, a hexagon (from Greek ἕξ, hex, meaning "six", and γωνία, gonía, meaning "corner, angle") is a six-sided polygon.[1] The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.
Regular hexagon
A regular hexagon has Schläfli symbol {6}[2] 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
The common length of the sides equals the radius of the circumscribed circle or circumcircle, which equals times the
Like
Parameters
The maximal
- and, similarly,
The area of a regular hexagon
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, 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 PE + PF = PA + PB + PC + PD.
It follows from the ratio of
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[3]
If are the distances from the vertices of a regular hexagon to any point on its circumcircle, then [3]
Symmetry
Example hexagons by 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 cyclic: (Z6, Z3, Z2, Z1) and the trivial (e)
These symmetries express nine distinct symmetries of a regular hexagon.
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
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 (××) | |
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r12 |
i4 |
g2 |
d2 |
d2 |
p2 |
a1 |
Dih6 | Dih2 | Z2 | Dih1 | Z1 |
A2 and G2 groups
A2 group roots |
G2 group roots |
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
Dissection
6-cube projection | 12 rhomb dissection | |
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Dissection of hexagons into three rhombs and parallelograms | |||||||||||
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2D | Rhombs | Parallelograms | |||||||||
Regular {6} | Hexagonal parallelogons | ||||||||||
3D | Square faces | Rectangular faces | |||||||||
Cube | Rectangular cuboid |
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.
Regular {6} |
Truncated t{3} = {6} |
Hypertruncated triangles | Stellated Star figure 2{3}
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Truncated t{6} = {12} |
Alternated h{6} = {3} |
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Crossed hexagon |
A concave hexagon | A self-intersecting hexagon (star polygon) | Extended Central {6} in {12} |
A skew hexagon, within cube
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Dissected {6} | projection octahedron |
Complete graph |
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Self-crossing hexagons
There are six self-crossing hexagons with the vertex arrangement of the regular hexagon:
Dih2 | Dih1 | Dih3 | |||
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Figure-eight |
Center-flip |
Unicursal |
Fish-tail |
Double-tail |
Triple-tail |
Hexagonal structures
From bees'
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.
Form | Hexagonal tiling | Hexagonal prismatic honeycomb
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Regular | ||
Parallelogonal |
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
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 ace = bdf.[6]
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.[7]
If a hexagon has vertices on the
Hexagon tangential to a conic section
Let ABCDEF be a hexagon formed by six
In a hexagon that is tangential to a circle and that has consecutive sides a, b, c, d, e, and f,[9]
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.[10]: Thm. 1
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
The cube and octahedron (same as triangular antiprism) have regular skew hexagons as petrie polygons.
Cube |
Octahedron |
Petrie polygons
The regular skew hexagon is the
4D | 5D | |
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3-3 duoprism |
3-3 duopyramid
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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[11]: p.184, #286.3 a principal diagonal d1 such that
and a principal diagonal d2 such that
Polyhedra with hexagons
There is no
Hexagons in Archimedean solids | |||||||||||
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Tetrahedral | Octahedral | Icosahedral | |||||||||
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):
Hexagons in Goldberg polyhedra | |||||||||||
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Tetrahedral | Octahedral | Icosahedral | |||||||||
Chamfered tetrahedron
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Chamfered cube
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Chamfered dodecahedron |
There are also 9 Johnson solids with regular hexagons:
Prismoids with hexagons
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Hexagonal prism |
Hexagonal antiprism |
Hexagonal pyramid |
Tilings with regular hexagons | |||||||||||
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Regular | 1-uniform | ||||||||||
{6,3} |
r{6,3} |
rr{6,3} |
tr{6,3} | ||||||||
2-uniform tilings
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Gallery of natural and artificial hexagons
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The ideal crystalline structure of graphene is a hexagonal grid.
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AssembledE-ELTmirror segments
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A beehive honeycomb
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The scutes of a turtle's carapace
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Saturn's hexagon, a hexagonal cloud pattern around the north pole of the planet
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Micrograph of a snowflake
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Benzene, the simplest aromatic compound with hexagonal shape.
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Hexagonal order of bubbles in a foam.
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Crystal structure of a molecular hexagon composed of hexagonal aromatic rings.
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Naturally formed basalt columns from Giant's Causeway in Northern Ireland; large masses must cool slowly to form a polygonal fracture pattern
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An aerial view of Fort Jefferson in Dry Tortugas National Park
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The James Webb Space Telescope mirror is composed of 18 hexagonal segments.
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In French, l'Hexagone refers to Metropolitan France for its vaguely hexagonal shape.
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Hexagonalhexagonal crystal systemminerals
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Hexagonal barn
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Władysław Gliński's hexagonal chess
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Pavilion in the Taiwan Botanical Gardens
See also
- self-dual and tessellates Euclidean space
- Hexagonal crystal system
- Hexagonal number
- regular tilingof 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
- ^ Cube picture
- ISBN 9780521098595, archivedfrom the original on 2016-01-02, retrieved 2015-11-06.
- ^ doi:10.26713/cma.v11i3.1420 (inactive 31 January 2024).)
{{cite journal}}
: CS1 maint: DOI inactive as of January 2024 (link - ISBN 978-1-56881-220-5(Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278)
- Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141
- ^ Cartensen, Jens, "About hexagons", Mathematical Spectrum 33(2) (2000–2001), 37–40.
- ^ Dergiades, Nikolaos (2014). "Dao's theorem on six circumcenters associated with a cyclic hexagon". Forum Geometricorum. 14: 243–246. Archived from the original on 2014-12-05. Retrieved 2014-11-17.
- ^ Johnson, Roger A., Advanced Euclidean Geometry, Dover Publications, 2007 (orig. 1960).
- ^ Gutierrez, Antonio, "Hexagon, Inscribed Circle, Tangent, Semiperimeter", [1] Archived 2012-05-11 at the Wayback Machine, Accessed 2012-04-17.
- ^ Dao Thanh Oai (2015). "Equilateral triangles and Kiepert perspectors in complex numbers". Forum Geometricorum. 15: 105–114. Archived from the original on 2015-07-05. Retrieved 2015-04-12.
- ^ Inequalities proposed in "Crux Mathematicorum", [2] Archived 2017-08-30 at the Wayback Machine.
External links
- 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.
- Hexagons are the Bestagons on .
Family | An | Bn | I2(p) / Dn | E6 / E7 / E8 / F4 / G2 | Hn
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Regular polygon | Triangle | Square | p-gon | Hexagon | Pentagon | |||||||
Uniform polyhedron | Tetrahedron | Octahedron • Cube | Demicube | Dodecahedron • Icosahedron | ||||||||
Uniform polychoron
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Pentachoron | 16-cell • Tesseract | Demitesseract | 24-cell | 120-cell • 600-cell | |||||||
Uniform 5-polytope | 5-simplex | 5-orthoplex • 5-cube | 5-demicube | |||||||||
Uniform 6-polytope | 6-simplex | 6-orthoplex • 6-cube | 6-demicube | 122 • 221 | ||||||||
Uniform 7-polytope | 7-simplex | 7-orthoplex • 7-cube | 7-demicube | 132 • 231 • 321 | ||||||||
Uniform 8-polytope | 8-simplex | 8-orthoplex • 8-cube | 8-demicube | 142 • 241 • 421 | ||||||||
Uniform 9-polytope | 9-simplex | 9-orthoplex • 9-cube | 9-demicube | |||||||||
Uniform 10-polytope | 10-simplex | 10-orthoplex • 10-cube | 10-demicube | |||||||||
Uniform n-polytope | n-simplex | n-orthoplex • n-cube | n-demicube | 1k2 • 2k1 • k21 | n-pentagonal polytope | |||||||
Topics: List of regular polytopes and compounds
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