Hexagon

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

Regular hexagon
Regular polygon 6 annotated
A regular hexagon
TypeRegular polygon
Edges and vertices6
Schläfli symbol{6}, t{3}
Coxeter diagramCDel node 1.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node 1.png
Symmetry groupDihedral (D6), order 2×6
Internal angle (degrees)120°
Dual polygonSelf
PropertiesConvex, cyclic, equilateral, isogonal, isotoxal

Regular hexagon

A regular hexagon has Schläfli symbol {1}[1] and can also be constructed as a truncated equilateral triangle, t{3}, which alternates two types of edges.

Regular Hexagon Inscribed in a Circle
A step-by-step animation of the construction of a regular hexagon using compass and straightedge, given by Euclid's Elements, Book IV, Proposition 15: this is possible as 6 2 × 3, a product of a power of two and distinct Fermat primes.
01-Sechseck-Seite-vorgegeben-wiki
When the side length AB is given, then you draw around the point A and around the point B a circular arc. The intersection M is the center of the circumscribed circle. Transfer the line segment AB four times on the circumscribed circle and connect the corner points.

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, which equals times the apothem (radius of the inscribed circle). All internal angles are 120 degrees. A regular hexagon has 6 rotational symmetries (rotational symmetry of order six) and 6 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, although it is equilateral.

Parameters

Regular hexagon 1

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:

    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 circumscribing circle between B and C, then PE + PF = PA + PB + PC + PD.

Symmetry

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

The regular hexagon has Dih6 symmetry, order 12. There are 3 dihedral subgroups: Dih3, Dih2, and Dih1, and 4 cyclic subgroups: Z6, Z3, Z2, and Z1.

These symmetries express 9 distinct symmetries of a regular hexagon. John Conway labels these by a letter and group order.[2] 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 seen as directed edges.

Example hexagons by symmetry
Hexagon r12 symmetry

r12
regular
Hexagon i4 symmetry

i4
Hexagon d6 symmetry

d6
isotoxal
Hexagon g6 symmetry

g6
directed
Hexagon p6 symmetry

p6
isogonal
Hexagon d3 symmetry

d2
Hexagon g2 symmetry

g2
general
parallelogon
Hexagon p2 symmetry

p2
Hexagon g3 symmetry

g3
Hexagon a1 symmetry

a1

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 (××)
Isohedral tiling p6-13
r12
Isohedral tiling p6-12
i4
Isohedral tiling p6-7
g2
Isohedral tiling p6-11
d2
Isohedral tiling p6-10
d2
Isohedral tiling p6-9
p2
Isohedral tiling p6-1
a1

A2 and G2 groups

Root system A2
A2 group roots
Dyn-node n1.pngDyn-3.pngDyn-node n2.png
Root system G2
G2 group roots
Dyn2-nodeg n1.pngDyn2-6a.pngDyn2-node n2.png

The 6 roots of the simple Lie group A2, represented by a Dynkin diagram Dyn-node n1.pngDyn-3.pngDyn-node n2.png, 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 Dyn2-nodeg n1.pngDyn2-6a.pngDyn2-node n2.png are also in a hexagonal pattern. The two simple roots of two lengths have a 150° angle between them.

Dissection

6-cube projection 10 rhomb dissection
6-cube t0 A5 6-gon rhombic dissection-size2 6-gon rhombic dissection2-size2

Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into m(m-1)/2 parallelograms.[3] 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.

Dissection of hexagons into 3 rhombs and parallelograms
2D Rhombs Parallelograms
Hexagon dissection Cube-skew-orthogonal-skew-solid Cuboid diagonal-orthogonal-solid Cuboid skew-orthogonal-solid
Regular {6} Hexagonal parallelogons
3D Square faces Rectangular faces
3-cube graph Cube-skew-orthogonal-skew-frame Cuboid diagonal-orthogonal-frame Cuboid skew-orthogonal-frame
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 3 hexagonal 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 2 types (colors) of edges. An alternated hexagon, h{6}, is a 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 6 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 polygon 6 annotated Truncated triangle Regular truncation 3 1000 Regular truncation 3 1.5 Regular truncation 3 0.55 Hexagram Regular polygon 12 annotated Regular polygon 3 annotated
Regular
{6}
Truncated
t{3} = {6}
Hypertruncated triangles Stellated
Star figure 2{3}
Truncated
t{6} = {12}
Alternated
h{6} = {3}
Medial triambic icosahedron face Great triambic icosahedron face 3-cube t0 Hexagonal cupola flat Cube petrie polygon sideview
A concave hexagon A self-intersecting hexagon (star polygon) Dissected {6} Extended
Central {6} in {12}
A skew hexagon, within cube

Hexagonal structures

Giants causeway closeup
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 number of hexagons. This means that honeycombs require less wax to construct and gain lots of 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 Uniform tiling 63-t0 Hexagonal prismatic honeycomb
Parallelogonal Isohedral tiling p6-7 Skew hexagonal prism honeycomb

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 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 ace = bdf.[4]

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.[5]

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.[6]:p. 179

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,[7]

Equilateral triangles on the sides of an arbitrary hexagon

Equilateral in hexagon
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.[8]:Thm. 1

Skew hexagon

Skew polygon in triangular antiprism
A regular skew hexagon seen as edges (black) of a triangular antiprism, symmetry D3d, [2+,6], (2*3), order 12.

A skew hexagon is a skew polygon with 6 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 3-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 cube and octahedron (same as triangular antiprism) have regular skew hexagons as petrie polygons.

Skew hexagons on 3-fold axes
Cube petrie
Cube
Octahedron petrie
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
3-3 duoprism ortho-Dih3
3-3 duoprism
3-3 duopyramid ortho
3-3 duopyramid
5-simplex t0
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[9]:p.184,#286.3 a principal diagonal d1 such that

and a principal diagonal d2 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 (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 CDel node 1.pngCDel 3.pngCDel node 1.pngCDel p.pngCDel node.png and CDel node 1.pngCDel 3.pngCDel node 1.pngCDel p.pngCDel node 1.png.

Hexagons in Archimedean solids
Tetrahedral Octahedral Icosahedral
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node 1.png
Truncated tetrahedron
truncated tetrahedron
Truncated octahedron
truncated octahedron
Great rhombicuboctahedron
truncated cuboctahedron
Truncated icosahedron
truncated icosahedron
Great rhombicosidodecahedron
truncated icosidodecahedron

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

Hexagons in Goldberg polyhedra
Tetrahedral Octahedral Icosahedral
Alternate truncated cube
Chamfered tetrahedron
Truncated rhombic dodecahedron2
Chamfered cube
Truncated rhombic triacontahedron
Chamfered dodecahedron

There are also 9 Johnson solids with regular hexagons:

Johnson solids and near-misses with hexagons
Triangular cupola
triangular cupola
Elongated triangular cupola
elongated triangular cupola
Gyroelongated triangular cupola
gyroelongated triangular cupola
Augmented hexagonal prism
augmented hexagonal prism
Parabiaugmented hexagonal prism
parabiaugmented hexagonal prism
Metabiaugmented hexagonal prism
metabiaugmented hexagonal prism
Triaugmented hexagonal prism
triaugmented hexagonal prism
Augmented truncated tetrahedron
augmented truncated tetrahedron
Triangular hebesphenorotunda
triangular hebesphenorotunda
Truncated triakis tetrahedron
Truncated triakis tetrahedron
Hexpenttri near-miss Johnson solid
Prismoids with hexagons
Hexagonal prism
Hexagonal prism
Hexagonal antiprism
Hexagonal antiprism
Hexagonal pyramid
Hexagonal pyramid
Tilings with regular hexagons
Regular 1-uniform
{6,3}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
r{6,3}
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png
rr{6,3}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png
tr{6,3}
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Uniform tiling 63-t0 Uniform tiling 63-t1 Uniform polyhedron-63-t02 Uniform polyhedron-63-t012
2-uniform tilings
2-uniform 1 2-uniform 10 2-uniform 11 2-uniform 12

Hexagons: natural and human-made

Graphen

The ideal crystalline structure of graphene is a hexagonal grid.

Assembled E-ELT mirror segments undergoing testing

Assembled E-ELT mirror segments

Honey comb

A beehive honeycomb

Carapax

The scutes of a turtle's carapace

Saturn hexagonal north pole feature

North polar hexagonal cloud feature on Saturn, discovered by Voyager 1 and confirmed in 2006 by Cassini [1] [2] [3]

Snowflake 300um LTSEM, 13368

Micrograph of a snowflake

Benzene-aromatic-3D-balls

Benzene, the simplest aromatic compound with hexagonal shape.

Order and Chaos

Hexagonal order of bubbles in a foam.

Hexa-peri-hexabenzocoronene ChemEurJ 2000 1834 commons

Crystal structure of a molecular hexagon composed of hexagonal aromatic rings reported by Müllen and coworkers in Chem. Eur. J., 2000, 1834-1839.

Giants causeway closeup

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

An aerial view of Fort Jefferson in Dry Tortugas National Park

Jwst front view

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

564X573-Carte France geo verte

Metropolitan France has a vaguely hexagonal shape. In French, l'Hexagone refers to the European mainland of France aka the "métropole" as opposed to the overseas territories such as Guadeloupe, Martinique or French Guiana.

Hanksite

Hexagonal Hanksite crystal, one of many hexagonal crystal system minerals

HexagonalBarnKewauneeCountyWisconsinWIS42

Hexagonal barn

Hexaschach

Władysław Gliński's hexagonal chess

Chinese pavilion

Pavilion in the Taiwan Botanical Gardens

See also

References

  1. ^ Wenninger, Magnus J. (1974), Polyhedron Models, Cambridge University Press, p. 9, ISBN 9780521098595.
  2. ^ John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, ISBN 978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278)
  3. ^ Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141
  4. ^ Cartensen, Jens, "About hexagons", Mathematical Spectrum 33(2) (2000–2001), 37–40.
  5. ^ Dergiades, Nikolaos (2014). "Dao's theorem on six circumcenters associated with a cyclic hexagon". Forum Geometricorum. 14: 243–246.
  6. ^ Johnson, Roger A., Advanced Euclidean Geometry, Dover Publications, 2007 (orig. 1960).
  7. ^ Gutierrez, Antonio, "Hexagon, Inscribed Circle, Tangent, Semiperimeter", [4], Accessed 2012-04-17.
  8. ^ Dao Thanh Oai (2015). "Equilateral triangles and Kiepert perspectors in complex numbers". Forum Geometricorum. 15: 105–114.
  9. ^ Inequalities proposed in “Crux Mathematicorum, [5].

External links

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds
1986 Hexagon World Men's Curling Championship

The 1986 Hexagon World Men's Curling Championship, the men's world curling championship, was held from March 31 to April 6 at the CNE Coliseum in Toronto, Ontario, Canada.

1987 Hexagon World Men's Curling Championship

The 1987 Hexagon World Men's Curling Championship was held from March 30 to April 5 at B.C. Place Stadium in Vancouver, British Columbia.

1988 Hexagon World Men's Curling Championship

The 1988 World Men's Curling Championship took place at the Icehalle in Lausanne, Switzerland from April 11–17. The gold medal was won by Team Norway, who also won the Curling competition in the 1988 Olympics in Calgary, Canada. Canada took the silver medal, and Scotland the bronze.

Don Diablo

Don Pepijn Schipper (Dutch pronunciation: [dɔn pɛpɛi̯n sxɪpər]; born 27 February 1980), better known by his stage name Don Diablo, is a Dutch DJ, record producer, musician and singer-songwriter of electronic dance music from Coevorden. He is known for his electronic style of production and vocalizing in most of his songs. He was ranked 7th in the Top 100 DJs – 2018 list by DJ Mag. He was also ranked #1 Producer of the Year by 1001 Tracklists. In 2016 he was ranked the number one Future House Artist of the Year on Beatport.

Hex map

A hex map, hex board, or hex grid is a game board design commonly used in wargames of all scales. The map is subdivided into a hexagonal tiling, small regular hexagons of identical size.

Hexagon (software)

Hexagon is a subdivision modeler owned by DAZ 3D. It was originally developed and published by Eovia and was acquired shortly before the release of version 2.0 by DAZ in 2006. The software drew heavily on Eovia's other modeler, Amapi (it shared the same developers), though it omitted the NURBs and precision measuring tools. The main focus is Subdivision modeling but it includes Spline tools and surface tools. Because of the two omissions, it is not as well suited to product design as Amapi, but is aimed more at illustrative and character modeling with an eye to making it accessible for those new to working in 3D.

Version 2.0 saw the addition of UV unwrapping tools and a selection of 3D painting tools were added, though these are still quite basic and do not include layers. Also added was the facility to paint in displacement - or "3D sculpting", which makes Hexagon particularly well suited for organic modeling. All of this can now be previewed within Hexagon using its new Ambient Occlusion.

Hexagon continues to be developed under DAZ's ownership with version 2.2 released in June 2007, and version 2.5 released in March 2008. In August, 2011 version 2.5.1 was released.

Hexagon AB

Hexagon AB is a global technology group headquartered in Sweden.

The operation is focused on precision measuring technologies and is divided into three business areas: Geospatial Measuring (Surveying and GPS), Industrial Metrology (Hexagon Metrology) and Technologies. The company markets its products and services under more than 35 different brands worldwide. The Group employs about 18,000 people in 50 countries.

Hexagon's operations encompass hand tools, fixed and portable coordinate measuring machines, GPS systems, construction machine control systems, level meters, laser meters, total stations, sensors for airborne measurement, aftermarket services and software systems, such as PC-DMIS.

Hexagon’s macro products are used within construction and engineering industries, while micro products are used primarily by automotive and aerospace industries, medical industries and design industry. Hexagon's other operations focus on supplying components primarily to the heavy automotive industry as well as key components for industrial robots.

Hexagon held its first annual conference Hexagon 2011 which combined Intergraph®, ERDAS, Leica Geosystems and Hexagon Metrology groups.

Hexagon pool

The Hexagons Pool (Hebrew: בריכת המשושים‎, Breichat HaMeshushim) is a natural pool in the Meshushim Reserve, part of the Yehudiya Forest Nature Reserve, in the central Golan Heights.

The pool, at the bottom of a canyon, is named after the shape of the hexagonal basalt columns that make up its walls. This geological formation was created by the slow cooling of layers of lava flows over a long period. When the lava solidified and cooled, it was split into polygonal shapes due to its contraction.

Hexagonal chess

Hexagonal chess refers to a group of chess variants played on boards composed of hexagon cells. 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 is increased mobility for pieces compared to a standard orthogonal chessboard. (E.g., a rook has six natural directions for movement instead of four.) Three colours are typically used so that no two neighboring cells are the same colour, and a colour-restricted game piece such as the orthodox chess bishop usually comes in sets of three per player in order to maintain the game's balance.

Many different shapes and sizes of hexagon-based boards are used by variants. The nature of the game is also affected by the 30° orientation of the board's cells; the board can be horizontally (Wellisch's, de Vasa's, Brusky's) or vertically (Gliński's, Shafran's, McCooey's) oriented. (E.g., when the sides of hexagonal cells face the players, pawns typically have one straightforward move direction. If a variant's gameboard has cell vertices facing the players, pawns typically have two oblique-forward move directions.) The six-sidedness of the symmetric hexagon gameboard has also resulted in a number of three-player variants.

The first applications of chess on hexagonal boards probably occurred mid-19th century, but two early examples did not include checkmate as the winning objective. More chess-like games for hexagon-based boards started appearing regularly at the beginning of the 20th century. Hexagon-celled gameboards have grown in use for strategy games generally; for example, they are popularly used in modern wargaming.

Hexagonal tiling

In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which three hexagons meet at each vertex. It has Schläfli symbol of {6,3} or t{3,6} (as a truncated triangular tiling).

English mathematician John Conway calls it a hextille.

The internal angle of the hexagon is 120 degrees so three hexagons at a point make a full 360 degrees. It is one of three regular tilings of the plane. The other two are the triangular tiling and the square tiling.

Intergraph

Intergraph Corporation is an American software development and services company. It provides enterprise engineering and geospatially powered software to businesses, governments, and organizations around the world. Intergraph operates through three divisions: Hexagon PPM, Hexagon Safety & Infrastructure, and Hexagon Geospatial. The company’s headquarters is in Huntsville, Alabama, United States.

In 2008, Intergraph was one of the one hundred largest software companies in the world. In July 2010, Intergraph was acquired by Hexagon AB.

IsoHunt

isoHunt was an online torrent files index and repository, where visitors can browse, search, download or upload torrents of various digital content of mostly entertainment nature. The current incarnation arose as a resurrection of the original site that was taken down by legal action from the MPAA; it is not associated in any way with the old staff or owners of the site, and is to be understood as a separate continuation.It originated in 2003 as isohunt.com website for IRC files search and reached over 13.7 million torrents in its database and 20 million peers from indexed torrents. With 7.4 million unique visitors as of May 2006, isoHunt was one of the most popular BitTorrent search engines. Thousands of torrents were added to and deleted from it every day. Users of isoHunt performed over 40 million unique searches per month. On October 19, 2008, isoHunt passed the 1 petabyte mark for torrents indexed globally. The site was the third most popular BitTorrent site as of 2008. According to isoHunt, the total amount of shared content was more than 14.11 petabytes as of June 13, 2012.The site came to an end when the legal battles that isoHunt's founder had been in for years with conglomerates of IP rights holders over allegations of copyright infringing came to a head. A settlement with the MPAA was reached in 2013, stipulating a $110 million reimbursement for damages and the site's closure that followed on October 21, 2013.By the end of October 2013 however, two sites with content presumably mirrored from isohunt.com were reported in media. One of them— isohunt.to became a de facto replacement of the original site.

Isotoxal figure

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 edges, there is a translation, rotation and/or reflection that will move one edge to the other, while leaving the region occupied by the object unchanged.

The term isotoxal is derived from the Greek τοξον meaning arc.

KH-9 Hexagon

KH-9 (BYEMAN codename HEXAGON), commonly known as Big Bird or Keyhole-9, was a series of photographic reconnaissance satellites launched by the United States between 1971 and 1986. Of twenty launch attempts by the National Reconnaissance Office, all but one were successful. Photographic film aboard the KH-9 was sent back to Earth in recoverable film return capsules for processing and interpretation. The best ground resolution achieved by the main cameras was better than 0.6 metres (2 ft 0 in).They are also officially known as the Broad Coverage Photo Reconnaissance satellites (Code 467), built by Lockheed Corporation for the National Reconnaissance Office.The KH-9 was declassified in September 2011 and an example was put on public display for a single day on September 17, 2011 in the parking lot of the Steven F. Udvar-Hazy Center of the National Air and Space Museum.On January 26, 2012 the National Museum of the United States Air Force put a KH-9 on public display along with its predecessors the KH-7 and KH-8.

List of Qualcomm Snapdragon systems-on-chip

This is a list of Qualcomm Snapdragon chips. Snapdragon is a family of mobile system on a chip (SoC) made by Qualcomm for use in smartphones, tablets, and smartbook devices.

Pascal's theorem

In projective geometry, Pascal's theorem (also known as the hexagrammum mysticum theorem) states that if six arbitrary points are chosen on a conic (which may be an ellipse, parabola or hyperbola in an appropriate affine plane) and joined by line segments in any order to form a hexagon, then the three pairs of opposite sides of the hexagon (extended if necessary) meet at three points which lie on a straight line, called the Pascal line of the hexagon. It is named after Blaise Pascal.

The theorem is also valid in the Euclidean plane, but the statement needs to be adjusted to deal with the special cases when opposite sides are parallel.

Quiz! Hexagon II

Quiz! Hexagon II (クイズ!ヘキサゴンII, Kuizu! Hekisagon Tsū) was a Japanese quiz variety show on Fuji Television, airing Wednesdays from 19:00-19:57 Japan Standard Time. The show began airing on October 19, 2005, ending on September 28, 2011 with 247 episodes aired; its predecessor, Quiz! Hexagon - This Evening is a Quiz Parade!! (クイズ!ヘキサゴン 今夜はクイズパレード!!, Kuizu! Hekisagon Kon'ya wa Kuizu Parēdo!!) aired from June 5 through October 12, 2005.

Saturn's hexagon

Saturn's hexagon is a persisting hexagonal cloud pattern around the north pole of Saturn, located at about 78°N.

The sides of the hexagon are about 14,500 km (9,000 mi) long, which is more than the diameter of Earth (about 12,700 km (7,900 mi)). The hexagon may be a bit greater than 29,000 km (18,000 mi) wide, may be 300 km (190 mi) high, and may be a jet stream made of atmospheric gases moving at 320 km/h (200 mph). It rotates with a period of 10h 39m 24s, the same period as Saturn's radio emissions from its interior. The hexagon does not shift in longitude like other clouds in the visible atmosphere.Saturn's hexagon was discovered during the Voyager mission in 1981 and was later revisited by Cassini-Huygens in 2006. During the Cassini mission, the hexagon changed from a mostly blue color to more of a golden color. Saturn's south pole does not have a hexagon, according to Hubble observations; however, it does have a vortex, and there is also a vortex inside the northern hexagon. Multiple hypotheses for the hexagonal cloud pattern have been developed.

The Hexagon

The Hexagon is a multi-purpose theatre and arts venue in Reading, Berkshire, England. Built in 1977 in the shape of an elongated hexagon, the theatre is operated by Reading Borough Council under the name "Reading Arts and Venues" along with South Street Arts Centre and Reading's concert hall.

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