# Octahedron with inscribed snub cube

### Model 475

### Description

Octahedron with inscribed (6+8+24)-plane polygon with 24 vertices (green), polar figure of the pentagon icositetrahedron.

### Additions

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

Inscribed (6+8+24)-plane polygon with 24 vertices
The solid inscribed in the octahedron is a *snub cube*.
The snub cube has

6 squares + 8 triangles + 24 triangles = 38 faces.

It has 24 vertices and 60 edges. At each vertice four triangles and one squares are meeting (3,3,3,3,4).

There is also a mirror image of the snub cube.

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

Pentagonal icositetrahedron
The snub cube is *polar (dual)* to the *pentagonal icositetrahedron*. To create this new solid a sphere is inscribed in the snub cube 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 38 vertices 24 pentagons are formed. These pentagons are the faces of the pentagonal 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==5, Case 2a.

Hess, E.(1883). *Kugelteilung*, Teubner, mit Figuren, §42, fig. 24.