# Octahedral pyramid

Octahedral pyramid | |
---|---|

Rank | 4 |

Type | Segmentotope |

Space | Spherical |

Notation | |

Bowers style acronym | Octpy |

Coxeter diagram | oo4oo3ox&#x |

Elements | |

Cells | 8 tetrahedra, 1 octahedron |

Faces | 8+12 triangles |

Edges | 6+12 |

Vertices | 1+6 |

Vertex figures | 1 octahedron, edge length 1 |

6 square pyramids, edge length 1 | |

Measures (edge length 1) | |

Circumradius | |

Hypervolume | |

Dichoral angles | Tet–3–tet: 120° |

Tet–3–oct: 60° | |

Heights | Point atop oct: |

Trig atop gyro tet: | |

Central density | 1 |

Related polytopes | |

Army | Octpy |

Regiment | Octpy |

Dual | Cubic pyramid |

Conjugate | None |

Abstract & topological properties | |

Euler characteristic | 0 |

Orientable | Yes |

Properties | |

Symmetry | B_{3}×I, order 48 |

Convex | Yes |

Nature | Tame |

The **octahedral pyramid**, or **octpy**, is a Blind polytope and CRF segmentochoron (designated K-4.3 on Richard Klitzing's list). It has 8 regular tetrahedra and 1 regular octahedron as cells. As the name suggests, it is a pyramid based on the octahedron.

Two octahedral pyramids can be attached at their bases to form a regular hexadecachoron. An octahedral pyramid can be further cut in half to produce two square scalenes.

It is part of an infinite family of Blind polytopes known as the orthoplecial pyramids, which generalize the square pyramid to higher dimensions.

Apart from being a point atop octahedron, it has an alternate segmentochoron representation as a triangle atop gyro tetrahedron seen as a triangular pyramid.

## Vertex coordinates[edit | edit source]

The vertices of an octahedral pyramid of edge length 1 are given by:

with all permutations of the first 3 coordinates of:

## Representations[edit | edit source]

An octahedral pyramid has the following Coxeter diagrams:

- oo4oo3ox&#x (full symmetry)
- oo3ox3oo&#x (base is in A
_{3}symmetry, tetratetrahedral pyramid) - oxo3oox&#x (base is in A
_{2}symmetry only, triangular antiprismatic pyramid)

## Segmentochoron display[edit | edit source]

Point atop octahedron

## External links[edit | edit source]

- Klitzing, Richard. "octpy".

- Wikipedia Contributors. "Octahedral pyramid".
- Hi.gher.Space Wiki Contributors. "Octahedral pyramid".