# Excavated expanded cuboctahedron

Excavated expanded cuboctahedron | |
---|---|

Rank | 3 |

Type | Quasi-convex Stewart toroid |

Elements | |

Faces | 8 triangles, 6+24+48 squares |

Edges | 24+24+24+24+24+48 |

Vertices | 6+8+24+24 |

Measures (edge length 1) | |

Volume | |

Surface area | |

Dihedral angles | 4-4 (rhombic prism lacing, narrow): |

4-4 (rhombic prism lacing, wide): | |

3-4: | |

4-4 (6-fold square to 24-fold square): | |

4-4 (trip lacings): 60° | |

Related polytopes | |

Convex hull | Expanded cuboctahedron |

Abstract & topological properties | |

Flag count | 672 |

Euler characteristic | –20 |

Orientable | Yes |

Genus | 11 |

Properties | |

Symmetry | B_{3}, order 48 |

Flag orbits | 14 |

Convex | No |

Nature | Tame |

History | |

Discovered by | Alex Doskey |

First discovered | 2004 |

The **excavated expanded cuboctahedron** is a quasi-convex Stewart toroid. It can be obtained by excavating 12 unit-edge-length rhombic prisms from an expanded cuboctahedron (blending their rhombus faces such that the prisms lie *inside* the shape), then excavating a rhombic dodecahedron from the center (blending it with the remaining rhombus faces of the prisms and opening up that space). This removes all of the rhombic faces, so the toroid is regular-faced. Its convex hull is an equilateral expanded cuboctahedron, with 30 squares (divided into two sets of 6 and 24), 12 rhombi, and 8 triangles.

It can also be obtained by outer-blending six square pyramids, eight tetrahedra, and twenty-four triangular prisms together.

## Vertex coordinates[edit | edit source]

An excavated expanded cuboctahedron of edge length 1 has vertex coordinates given by all permutations of

- ,
- ,
- ,
- .

## Related polyhedra[edit | edit source]

If the square pyramids and tetrahedra are partially-expanded into their respective cupolae (square and triangular), an excavated truncated rhombicuboctahedron will be formed.

If the triangular prisms are removed (and the pyramids brought together), a dissection of the cuboctahedron will be formed. It is a part of the tetrahedral-octahedral honeycomb.

## External links[edit | edit source]

- Wikipedia contributors. "Dissection of the expanded cuboctahedron".
- Doskey, Alex. "Prism Expansions".
- Webb. Robert. "Prism-Expanded Dissected Cuboctahedron".