# Octahedron

(Redirected from Oct)
Octahedron
Rank3
TypeRegular
SpaceSpherical
Notation
Bowers style acronymOct
Coxeter diagramo4o3x ()
Schläfli symbol{3,4}
Bracket notation<III>
Elements
Faces8 triangles
Edges12
Vertices6
Vertex figureSquare, edge length 1
Petrie polygons4 skew hexagons
Measures (edge length 1)
Circumradius${\displaystyle \frac{\sqrt2}{2} \approx 0.70711}$
Edge radius${\displaystyle \frac12 = 0.5}$
Inradius${\displaystyle \frac{\sqrt6}{6} \approx 0.40825}$
Volume${\displaystyle \frac{\sqrt2}{3} \approx 0.47140}$
Dihedral angle${\displaystyle \arccos\left(-\frac13\right) \approx 109.47122^\circ}$
Height${\displaystyle \frac{\sqrt6}{3} \approx 0.81650}$
Central density1
Number of external pieces8
Level of complexity1
Related polytopes
ArmyOct
RegimentOct
DualCube
Petrie dualPetrial octahedron
ConjugateNone
Abstract & topological properties
Flag count48
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryB3, order 48
ConvexYes
Net count11
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 also has 3 square pseudofaces. In fact, 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:

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

## 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
{3,3,4} ${\displaystyle 1}$
Cubic honeycomb
{4,3,4} ${\displaystyle \sqrt{2}}$
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.

o4o3o truncations
Name OBSA Schläfli symbol CD diagram Picture
Cube cube {4,3} x4o3o
Truncated cube tic t{4,3} x4x3o
Cuboctahedron co r{4,3} o4x3o
Truncated octahedron toe t{3,4} o4x3x
Octahedron oct {3,4} o4o3x
Small rhombicuboctahedron sirco rr{4,3} x4o3x
Great rhombicuboctahedron girco tr{4,3} x4x3x
Snub cube snic sr{4,3} s4s3s
o3o3o truncations
Name OBSA Schläfli symbol CD diagram Picture
Tetrahedron tet {3,3} x3o3o
Truncated tetrahedron tut t{3,3} x3x3o
Tetratetrahedron = Octahedron oct r{3,3} o3x3o
Truncated tetrahedron tut t{3,3} o3x3x
Tetrahedron tet {3,3} o3o3x
Small rhombitetratetrahedron = Cuboctahedron co rr{3,3} x3o3x
Great rhombitetratetrahedron = Truncated octahedron toe tr{3,3} x3x3x
Snub tetrahedron = Icosahedron ike sr{3,3} s3s3s

The dihedral angle is a supplementary angle to that of the regular tetrahedron. Thus, augmenting one of the faces produces coplanar faces, disqualifying the resulting polyhedron from being a Johnson solid. If two opposite faces are augmented with tetrahedra, the result is a triangular antitegum with 6 identical 60°/120° rhombi for faces.