Göttingen Collection of Mathematical Models and Instruments

Octahedron with inscribed snub cube

Model 475

Octahedron with inscribed snub cube
B II 180


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


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
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
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


Heesch, H.. Comm.Math.Helv, 6, n==5, Case 2a.

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