Cube with inscribed cuboctahedron
Model 485

Category: 
Description
Cube with inscribed (6 + 8)  plane polygon with 4·3 vertices. Polar figure of the the rhombus dodecahedron.
Additions
Cube The cube is one of the five Platonic solids, see also model 702.
Inscribed (6+8)plane polygon with 4 · 3 vertices The solid inscribed in the cube is a cuboctahedron. The cuboctahedron arises from the cube by truncating its vertices. By truncating the vertices to the middle of the edges eight new triangles are created and the previously existing squares are transformed to six smaller squares. The cuboctahedron is circumscribed by
6 squares + 8 triangles = 14 faces.
It has 4·3 = 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 rhombi 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 
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.
Showcase of this model is Case number 20