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Regular hexagon

A regular hexagon
Edge and vertices 6
Schläfli symbols {6}
Coxeter–Dynkin diagrams
Symmetry group Dihedral (D6)
(with t=edge length)
A = \frac{3 \sqrt{3}}{2}t^2
\simeq 2.598076211 t^2.
Internal angle
Properties convex, cyclic, equilateral, isogonal, isotoxal

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

Regular hexagon

The internal angles of a regular hexagon (where all of the sides are the same) are all 120° and the hexagon has 720 degrees T. It has 6 rotational symmetries and 6 reflection symmetries, making up the dihedral group D6. The longest diagonals of a regular hexagon, connecting diametrically opposite vertices, are twice its sides in length. Like square and equilateral triangle, 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 byA = \frac{3 \sqrt{3}}{2}t^2 \simeq 2.598076211 t^2.

Also, it can be found by the formula A=ap/2, where a is the apothem and p is the perimeter.

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}\,\!.

Related figures

A regular hexagon can also be created as a truncated equilateral triangle, with Schläfli symbol t{3}.

This form only has D3 symmetry.

In this figure, the remaining edges of the original triangle are drawn blue, and new edges from the truncation are red.

The hexagram can be created as a stellation process: extending the 6 edges of a regular hexagon until they meet at 6 new vertices.

A concave hexagon

A self-intersecting hexagon

Petrie polygons

The regular hexagon is the Petrie polygon for four higher dimensional polytopes, shown in these skew orthogonal projections:

Cube (3D)

Octahedron (3D)

16-cell (4D)

5-simplex (5D)

Polyhedra with hexagons

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.

Archimedean solids

truncated tetrahedron

truncated octahedron

truncated icosahedron

truncated cuboctahedron

truncated icosidodecahedron

And 9 Johnson solids:


Hexagonal prism

Hexagonal antiprism

Hexagonal pyramid

Other symmetric polyhedra

Truncated triakis tetrahedron

Truncated rhombic dodecahedron

Truncated rhombic triacontahedron

Regular and uniform tilings with hexagons

The hexagon can form a regular tessellate the plane with a Schläfli symbol {6,3}, having 3 hexagons around every vertex.

A second hexagonal tessellation of the plane can be formed as a truncated triangular tiling, with one of three hexagons colored differently.

A third tessellation of the plane can be formed with three colored hexagons around every vertex.

Trihexagonal tiling

Trihexagonal tiling

Rhombitrihexagonal tiling

Truncated trihexagonal tiling

Hexagons: natural and human-made

Image:Honey_comb.jpg|A beehive honeycombImage:Carapax.svg|The scutes of a turtle's carapaceImage:Saturn_hexagonal_north_pole_feature.jpg|North polar hexagonal cloud feature on Saturn, discovered by Voyager 1 and confirmed in 2006 by Cassini [9442] [9443] [9444]Image:Snowflake 300um LTSEM, 13368.jpg|Micrograph of a snowflakeImage:Hexa-peri-hexabenzocoronene ChemEurJ 2000 1834 commons.jpg|Crystal structure of a molecular hexagon composed of hexagonal aromatic rings reported by Müllen and coworkers in Chem. Eur. J., 2000, 1834-1839.Image:Giants causeway closeup.jpg|Naturally formed basalt columns from Giant's Causewaymarker in Irelandmarker; large masses must cool slowly to form a polygonal fracture patternImage:Fort-Jefferson Dry-Tortugas.jpg|An aerial view of Fort Jefferson in Dry Tortugas National ParkmarkerImage:Jwst front view.jpg|The James Webb Space Telescope mirror is composed of 18 hexagonal segments.Image:France.jpg| Metropolitan Francemarker has a vaguely hexagonal shape. In French, "L'hexagone" sometimes refers to the European mainland of France aka the "metropole" as opposed to the overseas territories such as Corsicamarker, Martiniquemarker or French Guianamarker.

See also

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