# Icosahedron with inscribed icosidodecahedron

### Model 481

### Description

Icosahedron (black wire) with inscribed (12+20)-plane polygon with 30 vertices (red fiber), congruent to the exemplar in model no.480.

### Additions

Icosahedron The icosahedron is one of the five Platonic solids, see also model 702.

Inscribed (12+20)-plane polygon with 30 vertices
The solid inscribed in the icosahedron is a *icosidodecahedron*. The icosidodecahedron arises from the icosahedron by truncating its vertices. By truncating the vertices to the middle of the edges 12 new hexagons and the previously existing 20 triangles are transformed to 20 smaller triangles.
The icosidodecahedron is circumscribed by

12 pentagons 20 + triangles = 32 faces.

It has 30 vertices and 60 edges. At each vertice two triangles and two pentagons are meeting (3,3,5,5).

The icosidodecahedron is one of 13 Archimedean solids, see also 482.

rhombic triacontahedron
The icosidodecahedron is *polar (dual)* to the *rhombic triacontahedron*, see also model 926. To create this new solid a sphere is inscribed in the icosidodecahedron such that the sphere touches each of the faces in exactly one point. These points of contact create the vertices of the dual solid. By connecting these 32 vertices 30 rhombi are formed. These rhombi are the faces of the rhombic triacontahedron. The number of edges remains the same during the transformation into the dual solid, while the number of vertices and faces is exchanged.

There are 11 Archimedean solids in the collection.

472 | Truncated tetrahedron | inscribed in a tetrahedron |

473 | Truncated octahedron | inscribed in an octahedron |

474 485 |
Cuboctahedron | 474 inscribed in an octahedron 485 inscribed in a cube |

475 | Truncated cube | inscribed in an octahedron |

476 | Rhombicuboctahedron | inscribed in a cube |

478 | Truncated icosahedron | inscribed in an icosahedron |

479 | truncated dodecahedron | inscribed in a dodecahedron |

480 481 |
Icosidodecahedron | 480 inscribed in a dodecahedron 481 inscribed in an icosahedron |

482 | Rhombicosidodecahedron | |

483 | Snub dodecahedron | |

484 | Truncated cube | inscribed in a cube |

The truncated icosidodecahedron and the lost truncated cuboctahedron are not included in the collection.

Showcase of this model is Case number U 7