  Göttingen Collection of Mathematical Models and Instruments

# Cube with inscribed truncated cube

### Model 484 Category:

### Description

Cube with inscribed (8 + 6) - plane polygon with 8·3 vertices that is polar to the triakis octahedron.

Cube The cube has six squares as faces. It has eight vertices and twelve edges. At each vertex three squares are meeting.

The cube is one of the five Platonic solids. A Platonic solid is a regular, convex polyhedron. The faces of a Platonic solid are congruent, regular polygons. At each vertex the same number of faces are meeting, see also model 411.

The five Platonic solids are Tetrahedron, Hexahedron (Cube), Octahedron, Dodecahedron and Icosahedron.

Inscribed (8+6)-plane polygon with 8 · 3 vertices The solid inscribed in the cube is a truncated cube. The truncated cube arises from the cube by truncating its vertices. By truncating the vertices eight new triangles are created and the previously existing squares are transformed to six octagons. The truncated cube is circumscribed by

8 triangles + 6 octagons = 14 faces.

It has 8·3 = 24 vertices and 36 edges. At each vertice one triangle and two octagons are meeting (3,8,8).

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

Triakis octahedron The truncated cube is polar (dual) to the triakis octahedron. To create this new solid a sphere is inscribed in the truncated 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 14 vertices 24 triangles are formed. These triangles are the faces of the triakis octahedron. 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

Brieskorn, Egbert(1983). Lineare Algebra und analystische Geometrie I, Nachdruck 1985, Vieweg, Braunschweig, p. 19ff.