# Stella octangula

Stella octangula | |
---|---|

Rank | 3 |

Type | Regular |

Space | Spherical |

Notation | |

Bowers style acronym | So |

Coxeter diagram | β4o3o () |

Elements | |

Components | 2 tetrahedra |

Faces | 8 triangles |

Edges | 12 |

Vertices | 8 |

Vertex figure | Equilateral triangle, edge length 1 |

Measures (edge length 1) | |

Circumradius | |

Inradius | |

Volume | |

Dihedral angle | |

Central density | 2 |

Number of external pieces | 24 |

Level of complexity | 3 |

Related polytopes | |

Army | Cube, edge length |

Regiment | So |

Dual | Stella octangula |

Conjugate | None |

Convex core | Octahedron |

Abstract & topological properties | |

Flag count | 48 |

Schläfli type | {3,3} |

Orientable | Yes |

Properties | |

Symmetry | B_{3}, order 48 |

Convex | No |

Nature | Tame |

The **stella octangula**, **stellated octahedron**, **so**, or **compound of two tetrahedra** is a regular polyhedron compound. It's made out of 8 triangles, 3 joining at each vertex. It can be constructed by taking a tetrahedron and overlaying it with its central inversion. Alternatively, it can be created by stellating the octahedron.

It can also be considered an antiprism based on the compound of two digons {4/2}.

Its quotient prismatic equivalent is the hexadecachoron, which is four-dimensional.

## Gallery[edit | edit source]

## Representations[edit | edit source]

- (full symmetry)
- (β2β4o)
- (β2β2β)
- xo3oo3ox

## Vertex coordinates[edit | edit source]

The vertices of a stella octangula of edge length 1 can be given by:

These arise from the fact that a tetrahedron can be constructed as the alternation of the cube. Taking even changes of sign and odd changes of sign reconstructs the two component tetrahedra.

Alternate coordinates can be derived from those of the triangle, by considering the tetrahedron as a triangular pyramid:

- ,
- ,
- .

These are more complicated, but generalize to two-simplex compounds of any dimension.

## External links[edit | edit source]

- Bowers, Jonathan. "Polyhedron Category C1: Compound Regulars" (#1).

- Klitzing, Richard. "so".

- Wikipedia Contributors. "Stellated octahedron".