# Dodecahedron with inscribed truncated dodecahedron

### Model 479

### Description

Dodecahedron (wire) with inscribed (20+12)-plane polygon with 60 vertices (brown), polar figure of the triakis icosahedron.

### Additions

Dodecahedron The dodecahedron is one of the five Platonic solids, see also model 702.

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

20 triangles + 12 decagons = 32 faces.

It has 3 · 20 = 60 vertices and 90 edges. At each vertice one triangle and two decagons are meeting (3,10,10).

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

Triakis icosahedron
The truncated dodecahedron is *polar (dual)* to the *triakis icosahedron*. To create this new solid a sphere is inscribed in the truncated dodecahedron 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 1c3, fig.2..

Hess, E.(1883). *Kugelteilung*, Teubner, mit Figuren, §23, fig. 11.