# Icosidodecahedron

Icosidodecahedron | |
---|---|

Rank | 3 |

Type | Uniform |

Space | Spherical |

Notation | |

Bowers style acronym | Id |

Coxeter diagram | o5x3o () |

Elements | |

Faces | 20 triangles, 12 pentagons |

Edges | 60 |

Vertices | 30 |

Vertex figure | Rectangle, edge lengths 1 and (1+√5)/2 |

Measures (edge length 1) | |

Circumradius | |

Volume | |

Dihedral angle | |

Central density | 1 |

Number of external pieces | 32 |

Level of complexity | 2 |

Related polytopes | |

Army | Id |

Regiment | Id |

Dual | Rhombic triacontahedron |

Conjugate | Great icosidodecahedron |

Abstract & topological properties | |

Flag count | 240 |

Euler characteristic | 2 |

Surface | Sphere |

Orientable | Yes |

Genus | 0 |

Properties | |

Symmetry | H_{3}, order 120 |

Convex | Yes |

Nature | Tame |

The **icosidodecahedron**, or **id**, is a quasiregular polyhedron and one of the 13 Archimedean solids. It consists of 20 equilateral triangles and 12 pentagons, with two of each joining at a vertex. It can be derived as a rectified dodecahedron or icosahedron.

## Vertex coordinates[edit | edit source]

An icosidodecahedron of side length 1 has vertex coordinates given by all permutations of

and even permutations of

The first set of vertices corresponds to a scaled octahedron which can be inscribed into the icosidodecahedron.

## Representations[edit | edit source]

An icosidodecahedron can be represented by the following Coxeter diagrams:

- o5x3o (full symmetry)
- xoxfo5ofxox&#xt (H2 axial, as pentagonal gyrobirotunda, triangle-first)
- oxFfofx3xfofFxo&#xt (A2 axial)
- VooFxf oVofFx ooVxfF&#zx (A1×A1×A1 symmetry)
- oxFf(oV)fFxo ofxF(Vo)Fxfo&#xt (A1×A1 axial)

## Related polyhedra[edit | edit source]

The icosidodecahedron is the colonel of a three-member regiment that also includes the small icosihemidodecahedron and the small dodecahemidodecahedron.

The icosidodecahedron can be split along an equatorial decagonal section to produce two pentagonal rotundas. Since the bases of these rotundas are in opposite orientations, an icosidodecahedron can be called the **pentagonal gyrobirotunda**. If one rotunda is rotated 36°, so that triangles connect to triangles and pentagons connect to pentagons, the result is the pentagonal orthobirotunda. If a decagonal prism is inserted between the two halves of the icosidodecahedron, the result is the elongated pentagonal gyrobirotunda.

Name | OBSA | Schläfli symbol | CD diagram | Picture |
---|---|---|---|---|

Dodecahedron | doe | {5,3} | x5o3o | |

Truncated dodecahedron | tid | t{5,3} | x5x3o | |

Icosidodecahedron | id | r{5,3} | o5x3o | |

Truncated icosahedron | ti | t{3,5} | o5x3x | |

Icosahedron | ike | {3,5} | o5o3x | |

Small rhombicosidodecahedron | srid | rr{5,3} | x5o3x | |

Great rhombicosidodecahedron | grid | tr{5,3} | x5x3x | |

Snub dodecahedron | snid | sr{5,3} | s5s3s |

## External links[edit | edit source]

- Bowers, Jonathan. "Polyhedron Category 3: Quasiregulars" (#24).

- Bowers, Jonathan. "Batch 3: Id, Did, and Gid Facetings" (#1 under id).

- Klitzing, Richard. "Id".

- Quickfur. "The Icosidodecahedron".

- Wikipedia Contributors. "Icosidodecahedron".
- McCooey, David. "Icosidodecahedron"

- Hi.gher.Space Wiki Contributors. "Rhodomesohedron".