Hexagon

From Wikipedia, the free encyclopedia

For other uses, see Hexagon (disambiguation).

Regular hexagon
Image:Hexagon.svg A regular hexagon, {6}
Edges and vertices	6
Schläfli symbols	{6} t{3}
Coxeter–Dynkin diagrams	Image:CDW ring.svg Image:CDW 6.png Image:CDW dot.svg Image:CDW ring.svg Image:CDW 3.png Image:CDW ring.svg
Symmetry group	Dihedral (D₆)
Area (with t=edge length)	Failed to parse (Missing texvc executable; please see math/README to configure.): A = \frac{3 \sqrt{3}}{2}t^2 Failed to parse (Missing texvc executable; please see math/README to configure.): \simeq 2.598076211 t^2.
Internal angle (degrees)	120°

In geometry, a hexagon is a polygon with six edges and six vertices. A regular hexagon has Schläfli symbol {6}.

1 Regular hexagon
2 Hexagons: in nature and by humankind
3 See also
4 External links

[edit] Regular hexagon

Image:HexagonConstructionAni.gif

A regular hexagon is constructible with compass and straightedge. The following is a step-by-step animated method of this, given by Euclid's Elements, Book IV, Proposition 15.

The internal angles of a regular hexagon (one where all sides and all angles are equal) are all 120° and the hexagon has 720 degrees. It has 6 lines of symmetry. 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.

The area of a regular hexagon of side length Failed to parse (Missing texvc executable; please see math/README to configure.): t\,\!

is given by

Failed to parse (Missing texvc executable; please see math/README to configure.): A = \frac{3 \sqrt{3}}{2}t^2 \simeq 2.598076211 t^2.

The perimeter of a regular hexagon of side length Failed to parse (Missing texvc executable; please see math/README to configure.): t\,\!

is, of course, Failed to parse (Missing texvc executable; please see math/README to configure.): 6t\,\!

, its maximal diameter Failed to parse (Missing texvc executable; please see math/README to configure.): 2t\,\! , and its minimal diameter Failed to parse (Missing texvc executable; please see math/README to configure.): t\sqrt{3}\,\! .

There is no platonic solid made of regular hexagons. 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.

[edit] Hexagons: in nature and by humankind

Honey comb.jpg

A beehive honeycomb

Carapax.svg

The scutes of a turtle's carapace

Saturn hexagonal north pole feature.jpg

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

Snowflake 300um LTSEM, 13368.jpg

Micrograph of a snowflake

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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.jpg

Naturally formed basalt columns from Giant's Causeway in Ireland; large masses must cool slowly to form a polygonal fracture pattern

Fort-Jefferson Dry-Tortugas.jpg

An aerial view of Fort Jefferson in Dry Tortugas National Park

Jwst front view.jpg

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

France.jpg

France has a vaguely hexagonal shape. In French, "L'hexagone" sometimes refers to the country.