Göttingen Collection of Mathematical Models and Instruments

Octahedron with inscribed cuboctahedron

Model 474

B II 179


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


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
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 20


Heesch, H.. Comm.Math.Helv, 6, n=4, Case 3a1 with fig. 12.

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