# Octahedron with inscribed truncated octahedron

### Model 473

### Description

Octahedron (wire) with inscribed (6+8)-plane polygon with 6·4 vertices (Kepler's cuboctahedron, brown), polar to the tetrakis hexahedron.

### Additions

OctahedronThe octahedron is one of the five Platonic solids, see also model 702.

Inscribed (6+8)-plane polygon with 6·4 vertices
The solid inscribed in the octahedron is a *truncated octahedron*. The truncated octahedron arises from the octahedron by truncating its vertices. By truncating the vertices six new squares are created and the previously existing triangles are transformed to eigt hexagons.
The truncated octahedron is circumscribed by

6 squares + 8 hexagons = 14 faces.

It has 6 · 4 = 24 vertices and 36 edges. At each vertice one triangle and two hexagons are meeting (4,6,6).

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

Tetrakis hexahedron
The truncated octahedron is *polar (dual)* to the *tetrakis hexahedron*. To create this new solid a sphere is inscribed in the truncated octahedron 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 14 vertices 24 triangles are formed. These triangles are the faces of the triakis octahedron. 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, Case 2c with fig. 5.

Hess, E.(1883). *Kugelteilung*, Teubner, mit Figuren, §20, Fig.8.