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Bowers style acronymOct
Coxeter diagramo4o3x (CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png)
Schläfli symbol{3,4}
Bracket notation<III>
Faces8 triangles
Vertex figureSquare, edge length 1
Octahedron vertfig.png
Petrie polygons4 skew hexagons
Measures (edge length 1)
Edge radius
Dihedral angle
Central density1
Number of external pieces8
Level of complexity1
Related polytopes
Petrie dualPetrial octahedron
Abstract & topological properties
Flag count48
Euler characteristic2
SymmetryB3, order 48
Net count11

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[edit | edit source]

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

Representations[edit | edit source]

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

In vertex figures[edit | edit source]

Octahedra in vertex figures
Name Picture Schläfli symbol Edge length
Schlegel wireframe 16-cell.png
Cubic honeycomb
Cubic honeycomb.png
Dodecahedral honeycomb
H3 534 CC center.png
Order-4 hexagonal tiling honeycomb
H3 634 FC boundary.png

Variations[edit | edit source]

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

Related polyhedra[edit | edit source]

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
Uniform polyhedron-43-t0.png
Truncated cube tic t{4,3} x4x3o
Uniform polyhedron-43-t01.png
Cuboctahedron co r{4,3} o4x3o
Uniform polyhedron-43-t1.png
Truncated octahedron toe t{3,4} o4x3x
Uniform polyhedron-43-t12.png
Octahedron oct {3,4} o4o3x
Uniform polyhedron-43-t2.png
Small rhombicuboctahedron sirco rr{4,3} x4o3x
Uniform polyhedron-43-t02.png
Great rhombicuboctahedron girco tr{4,3} x4x3x
Uniform polyhedron-43-t012.png
Snub cube snic sr{4,3} s4s3s
Uniform polyhedron-43-s012.png
o3o3o truncations
Name OBSA Schläfli symbol CD diagram Picture
Tetrahedron tet {3,3} x3o3o
Uniform polyhedron-33-t0.png
Truncated tetrahedron tut t{3,3} x3x3o
Uniform polyhedron-33-t01.png
Tetratetrahedron = Octahedron oct r{3,3} o3x3o
Uniform polyhedron-33-t1.png
Truncated tetrahedron tut t{3,3} o3x3x
Uniform polyhedron-33-t12.png
Tetrahedron tet {3,3} o3o3x
Uniform polyhedron-33-t2.png
Small rhombitetratetrahedron = Cuboctahedron co rr{3,3} x3o3x
Uniform polyhedron-33-t02.png
Great rhombitetratetrahedron = Truncated octahedron toe tr{3,3} x3x3x
Uniform polyhedron-33-t012.png
Snub tetrahedron = Icosahedron ike sr{3,3} s3s3s
Uniform polyhedron-33-s012.png

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.

External links[edit | edit source]

  • Klitzing, Richard. "Oct".