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Hexagon

Related subjects: Mathematics

Background Information

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Regular hexagon
Hexagon.svg
A regular hexagon, {6}
Edges and vertices 6
Schläfli symbols {6}
t{3}
Coxeter–Dynkin diagrams CDW ring.svgCDW 6.pngCDW dot.svg
CDW ring.svgCDW 3.pngCDW ring.svg
Symmetry group Dihedral (D6)
Area
(with t=edge length)		 A = \frac{3 \sqrt{3}}{2}t^2
\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}.

Regular hexagon

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 t\,\! is given by A = \frac{3 \sqrt{3}}{2}t^2 \simeq 2.598076211 t^2.

The perimeter of a regular hexagon of side length t\,\! is, of course, 6t\,\!, its maximal diameter 2t\,\!, and its minimal diameter 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.

Hexagons: natural and human-made

  • A beehive honeycomb

  • The scutes of a turtle's carapace

  • North polar hexagonal cloud feature on Saturn, discovered by Voyager 1 and confirmed in 2006 by Cassini

  • Micrograph of a snowflake

  • Hexa-peri-hexabenzocoronene ChemEurJ 2000 1834.jpg

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

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

  • An aerial view of Fort Jefferson in Dry Tortugas National Park

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

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

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