# Rhombic triacontahedron

Rhombic triacontahedron | |
---|---|

Rank | 3 |

Type | Uniform dual |

Space | Spherical |

Notation | |

Bowers style acronym | Rhote |

Coxeter diagram | |

Elements | |

Faces | 30 golden rhombi |

Edges | 60 |

Vertices | 12+20 |

Vertex figure | 12 pentagons, 20 triangles |

Measures (edge length 1) | |

Inradius | |

Volume | |

Dihedral angle | 144° |

Central density | 1 |

Related polytopes | |

Army | Rhote |

Regiment | Rhote |

Dual | Icosidodecahedron |

Conjugate | Great rhombic triacontahedron |

Abstract & topological properties | |

Euler characteristic | 2 |

Surface | Sphere |

Orientable | Yes |

Genus | 0 |

Properties | |

Symmetry | H_{3}, order 120 |

Convex | Yes |

Nature | Tame |

The **rhombic triacontahedron**, or **rhote**, is one of the 13 Catalan solids. It has 30 rhombi as faces, with 12 order-5 and 20 order-3 vertices. It is the dual of the uniform icosidodecahedron.

It can also be obtained as the convex hull of a dodecahedron and an icosahedron scaled so that their edges are orthogonal. For this to happen, the icosahedron's edge length must be times that of the dodecahedron's edge length. Each edge of the dodecahedron or icosahedron corresponds to one of the diagonals of the faces.

Each face of this polyhedron is a rhombus with longer diagonal times the shorter diagonal, with acute angle and obtuse angle .

## Vertex coordinates[edit | edit source]

A rhombic triacontahedron of edge length 1 has vertex coordinates given by all permutations of:

Plus all even permutations of:

## Dissection[edit | edit source]

The rhombic triacontahedron can be dissected into 10 acute golden rhombohedra and 10 obtuse golden rhombohedra.^{[1]}^{[2]}^{[3]}

## Related polyhedra[edit | edit source]

The rhombic triacontahedron has many stellations, including 227 fully supported stellations.^{[4]} Some notable stellations of the rhombic triacontahedron include the medial rhombic triacontahedron, great rhombic triacontahedron, rhombihedron, and rhombic hexecontahedron.

## External links[edit | edit source]

- Klitzing, Richard. "rhote".

- Wikipedia Contributors. "Rhombic triacontahedron".
- McCooey, David. "Rhombic Triacontahedron"

## References[edit | edit source]

- ↑ Hart, George. "Dissection of the Rhombic Triacontahedron". Archived from the original on 2012-10-12.
- ↑ Kowalewski, Gehard (1938).
*Der Keplersche Korper und andere Bauspiele*(in German). - ↑ Bardos, Laszlo. "Golden Rhombohedra".
- ↑ Messer, Peter (1995). "Stellations of the Rhombic Triacontahedron and Beyond".
*Structural Topology*(21): 25–46.