# Compound of twenty octahedra with rotational freedom

(Redirected from Outer disnub icosahedron)

Compound of twenty octahedra with rotational freedom | |
---|---|

Rank | 3 |

Type | Uniform |

Notation | |

Bowers style acronym | Addasi |

Elements | |

Components | 20 octahedra |

Faces | 40+120 triangles |

Edges | 120+120 |

Vertices | 120 |

Vertex figure | Square, edge length 1 |

Measures (edge length 1) | |

Circumradius | |

Inradius | |

Volume | |

Dihedral angle | |

Central density | 20 |

Related polytopes | |

Army | Semi-uniform Grid |

Regiment | Addasi |

Dual | Compound of twenty cubes with rotational freedom |

Conjugate | Compound of twenty octahedra with rotational freedom |

Abstract & topological properties | |

Schläfli type | {3,4} |

Orientable | Yes |

Properties | |

Symmetry | H_{3}, order 120 |

Convex | No |

Nature | Tame |

The **altered disnub icosahedron** (OBSA: **addasi**) or **compound of twenty octahedra with rotational freedom** is a uniform polyhedron compound. It consists of 40+120 triangles, with 4 triangles joining at each vertex.

## Vertex coordinates[edit | edit source]

The vertices of an altered disnub icosahedron of edge length 1 and rotation angle θ are given by all even permutations of:

- ,
- ,
- ,
- ,
- .

## Rotational variants[edit | edit source]

This compound has rotational freedom, represented by an angle θ . The 20 octahedra can each be thought of as triangular antiprisms, and can be rotated in pairs going opposite directions while still maintaining vertex-transitivity. This compound goes through the following named phases as θ increases:

*θ*= 0°: Double-cover of the snub icosahedron- : General phase sometimes known as the
**outer disnub icosahedron**(OBSA:**oddasi**).*θ*= (?): Special case known as the**hexagrammic disnub icosahedron**(OBSA:**hiddasi**) where the snub triangles form hexagrams.

- : Vertices coincide by pairs to form the disnub icosahedron
- : General phase sometimes known as the
**inner disnub icosahedron**(OBSA:**iddasi**) - : Octahedra coincide by four, forming a quadruple-cover of the small icosicosahedron
- : General phase known as the
**great disnub icosahedron**(OBSA:**giddasi**) *θ*= 60°: Double-cover of the great snub icosahedron

## External links[edit | edit source]

- Bowers, Jonathan. "Polyhedron Category C9: Octahedral Continuums" (#63, 63b, 64, 65).

- Klitzing, Richard. "addasi".
- Wikipedia contributors. "Compound of twenty octahedra with rotational freedom".