Hexagon

Hexagon
(OFF file)
Rank	2
Type	Regular
Space	Spherical
Notation
Bowers style acronym	Hig
Coxeter diagram	x6o ()
Schläfli symbol	{6}
Elements
Edges	6
Vertices	6
Vertex figure	Dyad, length √3
Measures (edge length 1)
Circumradius	1
Inradius	$\frac{\sqrt3}{2} ≈ 0.86603$
Area	$\frac{3\sqrt3}{2} ≈ 2.59807$
Angle	120°
Central density	1
Number of external pieces	6
Level of complexity	1
Related polytopes
Army	Hig
Dual	Hexagon
Conjugate	None
Abstract & topological properties
Flag count	12
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	G2, order 12
Convex	Yes
Nature	Tame

The hexagon is a polygon with 6 sides. A regular hexagon has equal sides and equal angles.

The combining prefix in BSAs is h-, as in haco.

The regular hexagon is one of the only three regular polygons that can tile the plane, the other two being the equilateral triangle and the square. Its tiling is called the hexagonal tiling, and it has 3 hexagons at a vertex.

The hexagon has the rare property that its circumradius equals its edge length. Other notable polytopes that satisfy this property are the cuboctahedron, the tesseract, and the icositetrachoron. Because of this, a regular hexagon can be exactly decomposed into 6 equilateral triangles.

The hexagon and the pentagon are the only regular polygons with exactly one stellation. It is also the polygon with the most sides that does not have a non-compound stellation. The other polygons without non-compound stellations (nor stellations at all) are the triangle and the square. It is also the only regular polygon with only compound stellations.

It can also be constructed as a uniform truncation of the equilateral triangle. It almost always has this symmetry when appearing in higher-dimensional uniform polytopes.

The regular hexagon is the 3rd-order permutohedron.

Naming

The name hexagon is derived from the Ancient Greek ἕξ (6) and γωνία (angle), referring to the number of vertices.

Other names include:

• hig, Bowers style acronym, short for "hexagon"
• Truncated Triangle

Vertex coordinates

Coordinates for a regular hexagon of unit edge length, centered at the origin, are:

• $\left(±1,\,0\right),$
• $\left(±\frac12,\,±\frac{\sqrt3}{2}\right).$

Representations

A regular hexagon can be represented by the following Coxeter diagrams:

• x6o (regular)
• x3x (A2 symmetry, generally a ditrigon)
• ho3oh&#zx (A2, generally a triambus)
• xu ho&#zx (rectangular symmetry)
• xux&#xt (axial edge-first)
• ohho&#xt (axial vertex-first)

Variations

Two main variants of the hexagon have triangle symmetry: the ditrigon, with two alternating side lengths and equal angles, and the dual triambus, with two alternating angles and equal edges. Other less regular variations with chiral triangular, rectangular, mirror, inversion, or no symmetry also exist.

Stellations

The hexagram (compound of two triangles) is the only stellation of the hexagon.

External links

• Bowers, Jonathan. "Regular Polygons and Other Two Dimensional Shapes".

• Klitzing, Richard. "Polygons"
• Wikipedia Contributors. "Hexagon".
• Hi.gher.Space Wiki Contributors. "Hexagon".

• Hartley, Michael. "{6}*12".