# Icosahedron with inscribed truncated icosahedron

### Model 478

### Description

Icosahedron (wire) with inscribed (12+20)-plane polygon with 60 vertices (brown), polar to the pentakis dodecahedron.

### Additions

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

Inscribed (12+20)-plane polygon with 60 vertices
The solid inscribed in the icosahedron is a *truncated icosahedron*. The truncated icosahedron arises from the icosahedron by truncating its vertices. By truncating the vertices 12 new pentagons and the previously existing 20 triangles are transformed to 20 hexagons.
The truncated icosahedron is circumscribed by

12 pentagons + 20 hexagons = 32 faces.

It has 5 · 12 = 60 vertices and 90 edges. At each vertice one pentagon and two hexagons are meeting (5,6,6).

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

Pentakis dodecahedron
The truncated icosahedron is *polar (dual)* to the *pentakis dodecahedron*. To create this new solid a sphere is inscribed in the truncated icosahedron 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 60 triangles are formed. These deltoids are the faces of the pentakis dodecahedron. 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 20

### References

Heesch, H.. Comm.Math.Helv, 6, n=3, Case 3c, fig.9..

Hess, E.(1883). *Kugelteilung*, Teubner, mit Figuren, §21, fig. 9.