# Octahedron with inscribed cuboctahedron

### Model 474

### Description

Octahedron (black) with inscribed cuboctahedron (6+8-plane polygon with 12 vertices), red. Polar figure of the the rhombus dodecahedron.

### Additions

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

Inscribed (6+8)-plane polygon with 12 vertices
The solid inscribed in the octahedron is a *cuboctahedron*. The cuboctahedron arises from the octahedron by truncating its vertices. By truncating the vertices to the middle of the edges six new squares are created and the previously existing triangles are transformed to smaller triangles.
The cuboctahedron is circumscribed by

6 squares + 8 triangles = 14 faces.

It has 12 vertices and 24 edges. At each vertice two triangles and two squares are meeting (3,3,4,4).

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

Rhombic dodecahedron
The cuboctahedron is *polar (dual)* to the *rhombic dodecahedron*, see also model 925. To create this new solid a sphere is inscribed in the cuboctahedron 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 12 rhombi are formed. These rhombis are the faces of the rhombic 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=4, Case 3a1 with fig. 12.

Hess, E.(1883). *Kugelteilung*, Teubner, mit Figuren, §32, figure 17.