# Octahedron

Octahedron | |
---|---|

Rank | 3 |

Type | Regular |

Space | Spherical |

Notation | |

Bowers style acronym | Oct |

Coxeter diagram | o4o3x () |

Schläfli symbol | {3,4} |

Bracket notation | <III> |

Conway notation | O |

Stewart notation | S_{3} |

Elements | |

Faces | 8 triangles |

Edges | 12 |

Vertices | 6 |

Vertex figure | Square, edge length 1 |

Petrie polygons | 4 skew hexagons |

Holes | 3 squares |

Measures (edge length 1) | |

Circumradius | |

Edge radius | |

Inradius | |

Volume | |

Dihedral angle | |

Height | |

Central density | 1 |

Number of external pieces | 8 |

Level of complexity | 1 |

Related polytopes | |

Army | Oct |

Regiment | Oct |

Dual | Cube |

Petrie dual | Petrial octahedron |

Skewing | Skew octahedron |

Conjugate | None |

Abstract & topological properties | |

Flag count | 48 |

Euler characteristic | 2 |

Surface | Sphere |

Orientable | Yes |

Genus | 0 |

Properties | |

Symmetry | B_{3}, order 48 |

Convex | Yes |

Net count | 11 |

Nature | Tame |

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:

- o4o3x () (regular)
- o3x3o () (A3 symmetry, tetratetrahedron)
- s2s3s () (generally a triangular antiprism)
- s2s6o () (similar to above, as alternated hexagonal prism)
- xo3ox&#x (A2 axial, generally a triangular antipodium)
- oxo4ooo&#xt (BC2 axial, generally a square bipyramid)
- oxo oxo&#xt (generally a rectangular bipyramid)
- xox oqo&#xt (A1×A1 axial, edge-first)
- oxox&#xr (single symmetry axis only)
- qo ox4oo&#xt (BC2 prism symmetry square bipyramid)
- qo ox ox&#xt (brick symmetry rectangle bipyramid)
- qoo oqo ooq&#zx (brick symmetry, rhombic bipyramid)

## In vertex figures[edit | edit source]

Name | Picture | Schläfli symbol | Edge length |
---|---|---|---|

Hexadecachoron | {3,3,4} | ||

Cubic honeycomb | {4,3,4} | ||

Dodecahedral honeycomb | {5,3,4} | ||

Order-4 hexagonal tiling honeycomb | {6,3,4} |

## Variations[edit | edit source]

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

- Tetratetrahedron - 2 sets of 4 triangles - just a coloring with no true variants in measures
- Triangular antiprism - 2 equilateral bases, 6 isosceles sides, vertex transitive
- Triangular antipodium - as above with 2 different sized bases and 2 sets of 3 isosceles sides
- Square tegum - 8 isosceles triangles, square prism symmetry
- Rectangular tegum - 2 sets of 4 isosceles triangles
- Rhombic tegum - 8 scalene triangles, digonal prism symmetry
- Digonal scalenohedron - 8 scalene triangles, digonal antiprism symmetry

## 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:

- Small icosicosahedron (5)
- Snub octahedron (4)
- Inner disnub octahedron (8, with rotational freedom)
- Hexagrammic disnub octahedron (8)
- Outer disnub octahedron (8, with rotational freedom)
- Inner disnub tetrahedron (4, with rotational freedom)
- Hexagrammic disnub tetrahedron (4)
- Outer disnub tetrahedron (4, with rotational freedom)
- Snub icosahedron (10)
- Great snub icosahedron (10)
- Outer disnub icosahedron (20, with rotational freedom)
- Inner disnub icosahedron (20, with rotational freedom)
- Great disnub icosahedron (20, with rotational freedom)
- Disnub icosahedron (20)

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

The octahedron has one stellation, the stella octangula.

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 () |

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.

## External links[edit | edit source]

- Bowers, Jonathan. "Polyhedron Category 1: Regulars" (#3).

- Bowers, Jonathan. "Batch 1: Oct and Co Facetings" (#1 under oct).

- Klitzing, Richard. "Oct".

- Quickfur. "The Octahedron".

- Wikipedia Contributors. "Octahedron".
- McCooey, David. "Octahedron"

- Hi.gher.Space Wiki Contributors. "Octahedron".

- Hartley, Michael. "{3,4}*48".