# Octahedron

Octahedron Rank3
TypeRegular
SpaceSpherical
Bowers style acronymOct
Info
Coxeter diagramo4o3x (     )
Schläfli symbol{3,4}
Bracket notation<III>
SymmetryBC3, order 48
ArmyOct
RegimentOct
Elements
Vertex figureSquare, edge length 1
Faces8 triangles
Edges12
Vertices6
Measures (edge length 1)
Circumradius$\frac{\sqrt2}{2} ≈ 0.70711$ Edge radius$\frac12 = 0.5$ Inradius$\frac{\sqrt6}{6} ≈ 0.40825$ Volume$\frac{\sqrt2}{3} ≈ 0.47140$ Dihedral angle$\arccos\left(-\frac13\right) ≈ 109.47122°$ Height$\frac{\sqrt6}{3} ≈ 0.81650$ Central density1
Euler characteristic2
Number of pieces8
Level of complexity1
Related polytopes
DualCube
ConjugateOctahedron
Properties
ConvexYes
OrientableYes
NatureTame

The octahedron, or oct, is one of the five Platonic solids. It consists of 8 equilateral triangles, joined 4 to a square vertex. It is the 3 dimensional orthoplex.

It can be built by joining two square pyramids by their square face, which makes it the square tegum.

It can also be constructed by rectifying the tetrahedron.

It is also the uniform triangular antiprism, and is a segmentohedron in this form.

It occurs as cells in one regular polychoron, namely the icositetrachoron.

## Vertex coordinates

An octahedron of side length 1 has vertex coordinates given by all permutations of:

• $\left(±\frac{\sqrt2}{2},\,0,\,0\right).$ ## Representations

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

## In vertex figures

Octahedra in vertex figures
Name Picture Schläfli symbol Edge length
Hexadecachoron {3,3,4} $1$ Cubic honeycomb {4,3,4} $\sqrt{2}$ Dodecahedral honeycomb {5,3,4}
Order-4 hexagonal tiling honeycomb {6,3,4}

## Variations

Other variants of the octahedron exist, using 8 triangular faces with 6 4-fold vertices. Some of these include:

## Related polyhedra

The octahedron is the colonel of a two-member regiment that also includes the tetrahemihexahedron.

The octahedron is the regular-faced square bipyramid. If a cube, seen as a square prism, is inserted between the two haves, the result is an elongated square bipyramid.

A number of uniform polyhedron compounds are composed of octahedra, all but one of them featured octahedra in triangular antiprism symmetry:

There is also an infinite family of prismatic octahedron compounds, the antiprisms of compounds of triangles:

The octahedron has one stellation, the stella octangula.