Rhombicuboctahedron

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In geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular and eighteen square faces. There are 24 identical vertices, with one triangle and three squares meeting at each. Note that six of the squares only share vertices with the triangles while the other twelve share an edge. The polyhedron has octahedral symmetry, like the cube and octahedron. Its dual is called the deltoidal icositetrahedron or trapezoidal icositetrahedron, although its faces are not really true trapezoids.

The name rhombicuboctahedron refers to the fact that 12 of the square faces lie in the same planes as the 12 faces of the rhombic dodecahedron which is dual to the cuboctahedron. Great rhombicuboctahedron is an alternative name for a truncated cuboctahedron, whose faces are parallel to those of the (small) rhombicuboctahedron.

It can also be called a cantellated cube or a cantellated octahedron from truncation operations of the uniform polyhedron.

If the original rhombicuboctahedron has unit edge length, its dual strombic icositetrahedron has edge lengths

Contents

Area and volume

The area A and the volume V of the rhombicuboctahedron of edge length a are:

Cartesian coordinates

Cartesian coordinates for the vertices of a rhombicuboctahedron centred at the origin, with edge length 2 units, are all permutations of

Geometric relations

There are three pairs of parallel planes that each intersect the rhombicuboctahedron through eight edges in the form of a regular octagon. The rhombicuboctahedron may be divided along any of these to obtain an octagonal prism with regular faces and two additional polyhedra called square cupolae, which count among the Johnson solids. These can be reassembled to give a new solid called the pseudorhombicuboctahedron (or elongated square gyrobicupola) with the symmetry of a square antiprism. In this the vertices are all locally the same as those of a rhombicuboctahedron, with one triangle and three squares meeting at each, but are not all identical with respect to the entire polyhedron, since some are closer to the symmetry axis than others.

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