# Hexagon

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Hexagon
Rank2
TypeRegular
Notation
Bowers style acronymHig
Coxeter diagramx6o ()
Schläfli symbol{6}
Elements
Edges6
Vertices6
Vertex figureDyad, length 3
Measures (edge length 1)
Circumradius1
Inradius${\displaystyle {\frac {\sqrt {3}}{2}}\approx 0.86603}$
Area${\displaystyle {\frac {3{\sqrt {3}}}{2}}\approx 2.59807}$
Angle120°
Central density1
Number of external pieces6
Level of complexity1
Related polytopes
ArmyHig
DualHexagon
ConjugateNone
Abstract & topological properties
Flag count12
Euler characteristic0
OrientableYes
Properties
SymmetryG2, order 12
Flag orbits1
ConvexYes
NatureTame

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

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 (as well as all expanded simplices), 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"
• 6-gon
• Truncated triangle

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

## Vertex coordinates

Coordinates for a regular hexagon of unit edge length, centered at the origin, can be given as:

• ${\displaystyle \left(\pm 1,\,0\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}}\right)}$.

Integer coordinates for the regular hexagon can be given in 3-dimensional space as all permutations of:

• ${\displaystyle \pm \left(2,-1,-1\right)}$.

The result is a regular hexagon with edge length ${\displaystyle {\sqrt {6}}}$.

Excluding skew polygons, the hexagon is one of three regular polygons which can be expressed with rational coordinates. The others being the square and triangle.

## Representations

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

## In vertex figures

The regular unit hexagon appears as the vertex figure of the triangular tiling. Various non-regular hexagons appear as vertex figures of some snub polyhedra.

## 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.

A non-compound, self-intersecting hexagon may be called a unicursal hexagram.

### Other skew hexagons

The hexagon is one of five regular hexagons in Euclidean space, the other four being skew:

Hexagons in Euclidean space
Name Extended Schläfli symbol Dimensions
hexagon ${\displaystyle \left\{{\dfrac {6}{1}}\right\}}$ 2
hexagonal-triangular coil ${\displaystyle \left\{{\dfrac {6}{1,2}}\right\}}$ 4
skew hexagon ${\displaystyle \left\{{\dfrac {6}{1,3}}\right\}}$ 3
skew triangle ${\displaystyle \left\{{\dfrac {6}{2,3}}\right\}}$ 3
skew hexagonal-triangular coil ${\displaystyle \left\{{\dfrac {6}{1,2,3}}\right\}}$ 5

## Stellations

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