Göttingen Collection of Mathematical Models and Instruments

# Octahedron with inscribed cuboctahedron

### Model 474

 Category: B II 179

### Description

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).

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 474485 Cuboctahedron 474 inscribed in an octahedron485 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 480481 Icosidodecahedron 480 inscribed in a dodecahedron481 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

### References

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

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