# Inverted snub dodecadodecahedron

Inverted snub dodecadodecahedron | |
---|---|

Rank | 3 |

Type | Uniform |

Notation | |

Bowers style acronym | Isdid |

Coxeter diagram | s5/3s5s () |

Elements | |

Faces | 60 triangles, 12 pentagons, 12 pentagrams |

Edges | 60+60+30 |

Vertices | 60 |

Vertex figure | Irregular pentagon, edge lengths 1, 1, (√5–1)/2, 1, (1+√5)/2 |

Measures (edge length 1) | |

Circumradius | ≈ 0.85163 |

Volume | ≈ 4.61431 |

Dihedral angles | 3–3: ≈ 130.49074° |

5–3: ≈ 68.64088° | |

5/2–3: ≈ 11.12448° | |

Central density | 9 |

Number of external pieces | 372 |

Level of complexity | 39 |

Related polytopes | |

Army | Non-uniform snid |

Regiment | Isdid |

Dual | Medial inverted pentagonal hexecontahedron |

Conjugate | Snub dodecadodecahedron |

Convex core | Dodecahedron |

Abstract & topological properties | |

Flag count | 600 |

Euler characteristic | –6 |

Orientable | Yes |

Properties | |

Symmetry | H_{3}+, order 60 |

Chiral | Yes |

Convex | No |

Nature | Tame |

The **inverted snub dodecadodecahedron** or **isdid**, is a uniform polyhedron. It consists of 60 snub triangles, 12 pentagrams, and 12 pentagons. Three triangles, 1 pentagon, and one pentagram meeting at each vertex. It can be constructed by alternation of the quasitruncated dodecadodecahedron and then setting all edge lengths to be equal.

This polyhedron has tiny holes near the pentagrams' "notched" regions, as well as ambiguously-tunneled "arches" opposite the notches. These make the inverted snub dodecadodecahedron the only uniform polyhedron to have holes not caused by filling method (e.g. the great dirhombicosidodecahedron's numerous, very obvious holes when binary-filled).

## Measures[edit | edit source]

The circumradius *R* ≈ 0.85163 of the inverted snub dodecadodecahedron with unit edge length is the smallest positive real root of:

Its volume *V* ≈ 4.61431 is given by the smallest positive real root of:

These same polynomials define the circumradius and volume of the snub dodecadodecahedron.

## Related polyhedra[edit | edit source]

The inverted disnub dodecadodecahedron is a uniform polyhedron compound composed of the 2 opposite chiral forms of the inverted snub dodecadodecahedron.

## External links[edit | edit source]

- Bowers, Jonathan. "Polyhedron Category 6: Snubs" (#69).

- Klitzing, Richard. "isdid".
- Wikipedia contributors. "Inverted snub dodecadodecahedron".
- McCooey, David. "Inverted Snub Dodecadodecahedron"