Octahedron with inscribed snub cube

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.