# Medial rhombic triacontahedron

Medial rhombic triacontahedron | |
---|---|

Rank | 3 |

Type | Uniform dual, Abstractly regular |

Notation | |

Bowers style acronym | Mort |

Coxeter diagram | o5/2m5o () |

Schläfli symbol | |

Elements | |

Faces | 30 rhombi |

Edges | 60 |

Vertices | 12+12 |

Vertex figure | 12 pentagons, 12 pentagrams |

Measures (edge length 1) | |

Inradius | |

Dihedral angle | 120° |

Central density | 3 |

Number of external pieces | 60 |

Related polytopes | |

Dual | Dodecadodecahedron |

Halving | Great dodecahedron |

Conjugate | Medial rhombic triacontahedron |

Convex core | Rhombic triacontahedron |

Abstract & topological properties | |

Flag count | 240 |

Euler characteristic | –6 |

Schläfli type | {4,5} |

Surface | Bring's surface |

Orientable | Yes |

Genus | 4 |

Properties | |

Symmetry | H_{3}, order 120 |

Flag orbits | 2 |

Convex | No |

Nature | Tame |

The **medial rhombic triacontahedron** is a uniform dual polyhedron. It consists of 30 rhombi.

If its dual, the dodecadodecahedron, has an edge length of 1, then the edges of the rhombi will measure . The rhombus faces will have length , and width . The rhombi have two interior angles of , and two of .

## Vertex coordinates[edit | edit source]

A medial rhombic triacontahedron with dual edge length 1 has vertex coordinates given by all even permutations of:

- ,
- .

## Related polytopes[edit | edit source]

This polyhedron is abstractly regular, being a quotient of the order-5 square tiling. As an abstract polytope, it is equivalent to a regular tessellation of Bring's surface. It has a symmetry group of order 240, which is the maximum possible for a tessellation of Bring's surface.

Its realization may also be considered regular if one also counts conjugacies as symmetries.

## External links[edit | edit source]

- Hartley, Michael. "{4,5}*240".
- Klitzing, Richard. "did".
- Wikipedia contributors. "Medial rhombic triacontahedron".
- McCooey, David. "Medial Rhombic Triacontahedron"