# Fifth noble stellation of rhombic triacontahedron

Fifth noble stellation of rhombic triacontahedron | |
---|---|

Rank | 3 |

Type | Noble |

Elements | |

Faces | 30 rectangular-symmetric dodecagrams |

Edges | 60+60+60 |

Vertices | 120 |

Vertex figure | Scalene triangle |

Measures (edge lengths , , ) | |

Edge length ratio | |

Circumradius | |

Number of external pieces | 300 |

Level of complexity | 18 |

Related polytopes | |

Army | Semi-uniform Grid, edge lengths (dipentagon-rectangle), (dipentagon-ditrigon), Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{7-3\sqrt5}{2}}
(ditrigon-rectangle) |

Dual | Fifth noble faceting of icosidodecahedron |

Conjugate | Final stellation of the rhombic triacontahedron |

Convex core | Rhombic triacontahedron |

Abstract & topological properties | |

Flag count | 720 |

Euler characteristic | –30 |

Orientable | Yes |

Genus | 16 |

Properties | |

Symmetry | H_{3}, order 120 |

Flag orbits | 6 |

Convex | No |

Nature | Tame |

History | |

Discovered by | Edmond Hess |

First discovered | 1876 |

The **fifth noble stellation of rhombic triacontahedron** is a noble polyhedron. Its 30 congruent faces are rectangular-symmetric dodecagrams meeting at congruent order-3 vertices. It is a faceting of the same semi-uniform great rhombicosidodecahedron hull as that of the great quasitruncated icosidodecahedron.

The ratio between the shortest and longest edges is 1:**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{\frac{25+10\sqrt5}{3}}}**
≈ 1:3.97327.

## Vertex coordinates[edit | edit source]

A fifth noble stellation of rhombic triacontahedron, centered at the origin, has vertex coordinates given by all permutations of

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\pm\frac12,\,\pm\frac12,\,\pm\frac{2\sqrt5-3}{2}\right),}**

along with all even permutations of:

These are the same coordinates as the great quasitruncated icosidodecahedron.