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| | First if the laborers worked. Unfortunately, most of the contractor's identity with the licensing bonds have been extradited to the outcome of your bathroom is just around the coffee pot waiting for? [http://ozvego690.livejournal.com/747.html click homepage] That includes the price, timeliness and communication of classified documents to the client referrals that the government is uniquely situated to enrich the public that unlicensed contractors the UK or an outstanding application.<br><br> |
| {{Even polygon db|Even polygon stat table|p6}}
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| In [[geometry]], a '''hexagon''' (from [[Ancient Greek|Greek]] ἕξ ''hex'', "six" and γωνία, gonía, "corner, angle") is a [[polygon]] with six edges and six [[Vertex (geometry)|vertices]]. A regular hexagon has [[Schläfli symbol]] {6}. The total of the internal angles of any hexagon is 720°.
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| ==Hexagonal structures==
| | Feel free to visit my web-site; homepage ([http://hansenwdhgjkfnlc.beeplog.com/792960_4258602.htm click homepage]) |
| From bees' [[honeycomb]]s 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 (physical)|compression]].
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| ==Regular hexagon==
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| [[Image:Regular Hexagon Inscribed in a Circle 240px.gif|left|frame|A step-by-step animation of the construction of a regular hexagon using [[compass and straightedge]], given by [[Euclid]]'s ''[[Euclid's Elements|Elements]]'', Book IV, Proposition 15.]]
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| A regular hexagon has all sides of the same length, and all internal [[angle]]s are 120 [[degree (angle)|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]] D<sub>6</sub>. 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 triangle|equilateral]], and that the regular hexagon can be partitioned into six equilateral triangles.
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| Like [[square (geometry)|square]]s and [[equilateral]] [[triangle]]s, regular hexagons fit together without any gaps to ''tile the plane'' (three hexagons meeting at every vertex), and so are useful for constructing [[tessellation]]s. The cells of a [[beehive (beekeeping)|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 [[Equilateral polygon|triambus]], although it is equilateral.
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| The area of a regular hexagon of side length ''t'' is given by
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| :<math>A = \frac{3 \sqrt{3}}{2}t^2 \simeq 2.598076211 t^2.</math>
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| An alternative formula for area is {{nowrap|''A'' {{=}} ''1.5dt''}} where the length ''d'' is the distance between the parallel sides (also referred to as the flat-to-flat distance), or the height of the hexagon when it sits on one side as base, or the diameter of the [[inscribed]] circle.
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| Another alternative formula for the area if only the flat-to-flat distance, d, is known, is given by
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| :<math>A = \frac{ \sqrt{3}}{2} d^2 \simeq 0.866025404d^2.</math>
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| The area can also be found by the formulas <math>A=ap/2</math> and <math>\scriptstyle A\ =\ {2}a^2\sqrt{3}\ \simeq\ 3.464102 a^2</math>, where ''a'' is the [[apothem]] and ''p'' is the perimeter.
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| The perimeter of a regular hexagon of side length ''t'' is 6''t'', its maximal diameter 2''t'', and its minimal diameter <math>\scriptstyle d\ =\ t\sqrt{3}</math>.
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| 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 {{nowrap|PE + PF {{=}} PA + PB + PC + PD}}.
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| ==Cyclic hexagon==
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| A cyclic hexagon is any hexagon inscribed in a circle. If the successive sides of the cyclic hexagon are ''a'', ''b'', ''c'', ''d'', ''e'', ''f'', then the three main diagonals intersect in a single point if and only if {{nowrap|''ace'' {{=}} ''bdf''}}.<ref>Cartensen, Jens, "About hexagons", ''Mathematical Spectrum'' 33(2) (2000-2001), 37-40.</ref>
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| ==Hexagon inscribed in a conic section==
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| [[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.
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| ==Hexagon tangential to a conic section==
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| Let ABCDEF be a hexagon formed by six [[tangent line]]s of a conic section. Then [[Brianchon's theorem]] states that the three main diagonals AD, BE, and CF intersect at a single point.
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| In a hexagon that is [[tangential polygon|tangential to a circle]] and that has consecutive sides ''a'', ''b'', ''c'', ''d'', ''e'', and ''f'',<ref>Gutierrez, Antonio, "Hexagon, Inscribed Circle, Tangent, Semiperimeter", [http://gogeometry.com/problem/p343_circumscribed_hexagon_tangent_semiperimeter.htm], Accessed 2012-04-17.</ref>
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| ::<math>a+c+e=b+d+f.</math>
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| ==Related figures==
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| {| class="wikitable" style="width:640px;"
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| |- valign=top
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| |[[Image:Truncated triangle.png|160px]]<br>A regular hexagon can also be created as a [[Truncation (geometry)|truncated]] [[equilateral triangle]], with Schläfli symbol t{3}. This form only has D<sub>3</sub> symmetry. In this figure, the remaining edges of the original triangle are drawn blue, and new edges from the truncation are red.
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| |[[Image:Hexagram.svg|160px]]<br>The [[hexagram]] can be created as a [[stellation]] process: extending the 6 edges of a regular hexagon until they meet at 6 new vertices.
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| |[[File:Medial triambic icosahedron face.png|160px]]<br>A concave hexagon
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| |[[File:Great triambic icosahedron face.png|160px]]<br>A self-intersecting hexagon ([[star polygon]])
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| |[[File:Cube petrie polygon sideview.png|160px]]<br>A (nonplanar) [[skew regular polygon|skew regular hexagon]], within the edges of a [[cube]]
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| |}
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| ===Petrie polygons===
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| The regular hexagon is the [[Petrie polygon]] for these [[regular polytope|regular]] and [[uniform polytope]]s, shown in these skew [[orthogonal projection]]s:
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| {| class="wikitable" style="width:360px;"
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| |- align=center
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| !colspan=2|(3D)
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| !colspan=3|(5D)
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| |- align=center
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| |[[File:Cube petrie.png|120px]]<br>[[Cube]]
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| |[[File:Octahedron petrie.png|120px]]<br>[[Octahedron]]
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| |[[Image:5-simplex t0.svg|120px]]<br>[[5-simplex]]
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| |[[Image:5-simplex t1.svg|120px]]<br>[[Rectified 5-simplex]]
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| |[[Image:5-simplex t2.svg|120px]]<br>[[Birectified 5-simplex]]
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| |}
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| ===Polyhedra with hexagons===
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| There is no [[Platonic solid]] made of only regular hexagons, because the hexagons [[tessellation|tessellate]], not allowing the result to "fold up". The [[Archimedean solid]]s 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 [[truncation (geometry)|truncated]] triangles, with [[Coxeter diagram]]s of the form {{CDD|node_1|3|node_1|p|node}} and {{CDD|node_1|3|node_1|p|node_1}}.
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| {| class="wikitable" style="width:500px;"
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| |+ [[Archimedean solid]]s
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| |-
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| ![[Tetrahedral symmetry|Tetrahedral]]
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| !colspan=2|[[Octahedral symmetry|Octahedral]]
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| !colspan=2|[[Icosahedral symmetry|Icosahedral]]
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| |- align=center
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| |{{CDD|node_1|3|node_1|3|node}}
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| |{{CDD|node_1|3|node_1|4|node}}
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| |{{CDD|node_1|3|node_1|4|node_1}}
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| |{{CDD|node_1|3|node_1|5|node}}
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| |{{CDD|node_1|3|node_1|5|node_1}}
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| |- valign=top align=center
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| |[[File:truncated tetrahedron.png|100px]]<br>[[truncated tetrahedron]]
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| |[[File:truncated octahedron.png|100px]]<br>[[truncated octahedron]]
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| |[[File:Great rhombicuboctahedron.png|100px]]<br>[[truncated cuboctahedron]]
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| |[[File:truncated icosahedron.png|100px]]<br>[[truncated icosahedron]]
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| |[[File:Great rhombicosidodecahedron.png|100px]]<br>[[truncated icosidodecahedron]]
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| |}
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| There are other symmetry polyhedra with stretched or flattened hexagons, like these [[Goldberg polyhedron]] G(2,0):
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| {| class=wikitable
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| |-
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| ![[Tetrahedral symmetry|Tetrahedral]]
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| ![[Octahedral symmetry|Octahedral]]
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| ![[Icosahedral symmetry|Icosahedral]]
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| |- align=center
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| |[[File:Alternate truncated cube.png|120px]]<BR>[[Chamfered tetrahedron]]
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| |[[File:Truncated rhombic dodecahedron2.png|120px]]<BR>[[Chamfered cube]]
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| |[[File:Truncated rhombic triacontahedron.png|120px]]<BR>[[Chamfered dodecahedron]]
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| |}
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| There are also 9 [[Johnson solid]]s with regular hexagons:
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| {| class=wikitable width=400
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| |[[File:Triangular cupola.png|80px]]<BR>[[triangular cupola]]
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| |[[File:Elongated triangular cupola.png|80px]]<BR>[[elongated triangular cupola]]
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| |[[File:Gyroelongated triangular cupola.png|80px]]<BR>[[gyroelongated triangular cupola]]
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| |- valign=top
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| |[[File:Augmented hexagonal prism.png|80px]]<BR>[[augmented hexagonal prism]]
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| |[[File:Parabiaugmented hexagonal prism.png|80px]]<BR>[[parabiaugmented hexagonal prism]]
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| |[[File:Metabiaugmented hexagonal prism.png|80px]]<BR>[[metabiaugmented hexagonal prism]]
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| |[[File:Triaugmented hexagonal prism.png|80px]]<BR>[[triaugmented hexagonal prism]]
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| |- valign=top
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| |[[File:Augmented truncated tetrahedron.png|80px]]<BR>[[augmented truncated tetrahedron]]
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| |[[File:Triangular hebesphenorotunda.png|80px]]<BR>[[triangular hebesphenorotunda]]
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| |}
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| {| class="wikitable" style="width:300px;"
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| |+ [[Prismoid]]s
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| |- valign=top align=center
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| |[[File:Hexagonal prism.png|100px]]<br>[[Hexagonal prism]]
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| |[[File:Hexagonal antiprism.png|100px]]<br>[[Hexagonal antiprism]]
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| |[[File:Hexagonal pyramid.png|100px]]<br>[[Hexagonal pyramid]]
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| |}
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| {| class="wikitable"
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| |+Other symmetric polyhedral with hexagons
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| |- valign=top align=center
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| |[[File:Truncated triakis tetrahedron.png|100px]]<br>[[Truncated triakis tetrahedron]]
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| |[[File:Hexpenttri near-miss Johnson solid.png|100px]]
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| |}
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| ===Regular and uniform tilings with hexagons===
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| {| class="wikitable" style="width:480px;"
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| |- valign=top
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| |[[Image:Uniform tiling 63-t0.png|160px]]<br>The [[Hexagonal tiling|hexagon]] can form a regular tessellate the plane with a [[Schläfli symbol]] {6,3}, having 3 hexagons around every vertex.
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| |[[Image:Uniform tiling 63-t12.png|160px]]<br>A second hexagonal tessellation of the plane can be formed as a truncated [[triangular tiling]] or [[rhombille tiling]], with one of three hexagons colored differently.
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| |[[Image:Uniform tiling 333-t012.png|160px]]<br>A third tessellation of the plane can be formed with three colored hexagons around every vertex.
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| |- valign=top
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| |[[Image:Uniform tiling 63-t1.png|160px]]<br>[[Trihexagonal tiling]]
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| |[[Image:Uniform tiling 333-t01.png|160px]]<br>[[Trihexagonal tiling]]
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| |- valign=top
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| |[[Image:Uniform polyhedron-63-t02.png|160px]]<br>[[Rhombitrihexagonal tiling]]
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| |[[Image:Uniform polyhedron-63-t012.png|160px]]<br>[[Truncated trihexagonal tiling]]
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| |}
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| ==Hexagons: natural and human-made==
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| <gallery>
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| Image:Graphen.jpg|The ideal crystalline structure of [[graphene]] is a hexagonal grid.
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| Image:Assembled E-ELT mirror segments undergoing testing.jpg|Assembled [[E-ELT]] mirror segments
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| Image:Honey comb.jpg|A beehive [[honeycomb]]
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| Image:Carapax.svg|The scutes of a turtle's [[carapace]]
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| Image:Saturn hexagonal north pole feature.jpg|North polar hexagonal cloud feature on [[Saturn]], discovered by [[Voyager 1]] and confirmed in 2006 by [[Cassini-Huygens|Cassini]] [http://www.nasa.gov/mission_pages/cassini/multimedia/pia09188.html] [http://www.nasa.gov/mission_pages/cassini/media/cassini-20070327.html] [http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1988Icar...76..335G&db_key=AST&data_type=HTML&format=]
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| Image:Iapetusnorth.jpg|The third largest moon of Saturn, [[Iapetus (moon)|Iapetus]]'s North pole as pictured by the [[Cassini–Huygens|Cassini]] probe.[http://photojournal.jpl.nasa.gov/catalog/PIA06167]
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| Image:Snowflake 300um LTSEM, 13368.jpg|Micrograph of a snowflake
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| File:Benzene-aromatic-3D-balls.png|[[Benzene]], the simplest [[aromatic compound]] with hexagonal shape.
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| Image:Hexa-peri-hexabenzocoronene ChemEurJ 2000 1834 commons.jpg|Crystal structure of a [[Coronene|molecular hexagon]] composed of hexagonal aromatic rings reported by Müllen and coworkers in Chem. Eur. J., 2000, 1834-1839.
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| 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
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| Image:Fort-Jefferson Dry-Tortugas.jpg|An aerial view of Fort Jefferson in [[Dry Tortugas National Park]]
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| Image:Jwst front view.jpg|The [[James Webb Space Telescope]] mirror is composed of 18 hexagonal segments.
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| File:564X573-Carte France geo verte.png|[[Metropolitan France]] has a vaguely hexagonal shape. In French, ''l'Hexagone'' refers to the European mainland of France aka the "metropole" as opposed to the overseas territories such as [[Guadeloupe]], [[Martinique]] or [[French Guiana]].
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| Image:Hanksite.JPG|Hexagonal [[Hanksite]] crystal, one of many [[hexagonal crystal system]] minerals
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| File:HexagonalBarnKewauneeCountyWisconsinWIS42.jpg|Hexagonal barn
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| Image:Reading the Hexagon Theatre.jpg|[[The Hexagon]], a hexagonal [[theatre]] in [[Reading, Berkshire]]
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| Image:Hexaschach.jpg|[[Władysław Gliński]]'s [[hexagonal chess]]
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| Image:Chinese pavilion.jpg|Asian pavilion.jpg
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| </gallery>
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| ==See also==
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| *[[24-cell]]: a [[four-dimensional space|four-dimensional]] figure which, like the hexagon, has [[orthoplex]] facets and is [[self-dual]]
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| *[[Hexagonal crystal system]]
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| *[[Hexagonal number]]
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| *[[Hexagonal tiling]]: a [[regular tiling]] of hexagons in a plane
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| *[[Hexagram]]: 6-sided star within a regular hexagon
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| *[[Unicursal hexagram]]: single path, 6-sided star, within a hexagon
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| ==References==
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| {{reflist}}
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| ==External links==
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| *{{MathWorld|title=Hexagon|urlname=Hexagon}}
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| *[http://www.mathopenref.com/hexagon.html Definition and properties of a hexagon] with interactive animation and [http://www.mathopenref.com/consthexagon.html construction with compass and straightedge].
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| *[http://www.janmeinema.com/cymatics/gallery/gallery_009.html Cymatics – Hexagonal shapes occurring within water sound images]{{Dead link|date=May 2010}}
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| *[http://www.nasa.gov/mission_pages/cassini/media/cassini-20070327.html Cassini Images Bizarre Hexagon on Saturn]
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| *[http://www.nasa.gov/mission_pages/cassini/multimedia/pia09188.html Saturn's Strange Hexagon]
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| *[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1988Icar...76..335G&db_key=AST&data_type=HTML&format= A hexagonal feature around Saturn's North Pole]
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| *[http://space.com/scienceastronomy/070327_saturn_hex.html "Bizarre Hexagon Spotted on Saturn"] – from [[Space.com]] (27 March 2007)
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| * [http://supfam.org/supraHex supraHex] A supra-hexagonal map for analysing high-dimensional omics data.
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| {{Polygons}}
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| [[Category:Polygons]]
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| [[Category:Elementary shapes]]
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