# Cube with inscribed rhombicuboctahedron

### Model 476

### Description

Cube with inscribed (6+8+12)-plane polygon with 24 vertices that is polar to the deltoidal icositetrahedron.

### Additions

CubeThe cube is one of the five Platonic solids, see also model 702.

Inscribed (6+8+12)-plane polygon with 24 vertices
The solid inscribed in the cube is a *rhombicuboctahedron*. The rhombicuboctahedron arises from the cube by truncating its vertices and truncating its edges afterwards. By truncating the vertices and edges eight new triangles and 12 squares are created and the previously existing squares are transformed to proportionally smaller squares.
The rhombicuboctahedron is circumscribed by

8 triangles + 12 squares + 6 squares = 26 faces.

It has 4 · 6 = 24 vertices and 48 edges. At each vertice one triangles and three squares are meeting (3,4,4,4).

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

Deltoidal icositetrahedron
The rhombicuboctahedron is *polar (dual)* to the *deltoidal icositetrahedron*. To create this new solid a sphere is inscribed in the rhombicuboctahedron 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 26 vertices 24 deltoids are formed. These deltoids are the faces of the deltoidal icositetrahedron. 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 21

### References

Heesch, H.. Comm.Math.Helv, 6, n=4, Casel 4a.

Hess, E.(1883). *Kugelteilung*, Teubner, mit Figuren, §35, fig. 19.