Rhombicosidodecahedron

The rhombicosidodecahedron, or small rhombicosidodecahedron, is an Archimedean solid. It has 20 regular triangular faces, 30 regular square faces, 12 regular pentagonal faces, 60 vertices and 120 edges.

The name "rhombicosidodecahedron" refers to the fact that the 30 square faces lie in the same planes as the 30 faces of the rhombic triacontahedron which is dual to the icosidodecahedron.

It can also called a "cantellated dodecahedron" or a "cantellated icosahedron" from truncation operations of the uniform polyhedron.

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Geometric relations

If you blow up an icosahedron by moving the faces away from the origin the right amount, without changing the orientation or size of the faces, and do the same to its dual dodecahedron, and patch the square holes in the result, you get a rhombicosadodecahedron. Therefore, it has the same number of triangles as an icosahedron and the same number of pentagons as a dodecahedron, with a square for each edge of either.

The rhombicosidodecahedron shares the vertex arrangement with the small stellated truncated dodecahedron, and with the uniform compounds of 6 or 12 pentagrammic prisms.

The Zometool kits for making geodesic domes and other polyhedra use slotted balls as connectors. The balls are "expanded" small rhombicosidodecahedra, with the squares replaced by rectangles. The expansion is chosen so that the resulting rectangles are golden rectangles.

Cartesian coordinates

Cartesian coordinates for the vertices of a rhombicosidodecahedron centered at the origin are: (±1, ±1, ±τ³),: (±τ³, ±1, ±1),: (±1, ±τ³, ±1),: (±τ², ±τ, ±2τ),: (±2τ, ±τ², ±τ),: (±τ, ±2τ, ±τ²),: (±(2+τ), 0, ±τ²),: (±τ², ±(2+τ), 0),: (0, ±τ², ±(2+τ)),where τ = (1+√5)/2 is the golden ratio (also written φ).

Vertex arrangement

The rhombicosidodecahedron shares its vertex arrangement with 3 nonconvex uniform polyhedrons:

ee also

*
*dodecahedron
*icosahedron
*icosidodecahedron
*rhombicuboctahedron
*truncated icosidodecahedron (great rhombicosidodecahedron)

References

* (Section 3-9)

External links

*
* [http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra]
* [http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] The Encyclopedia of Polyhedra
* [http://www.rhombicosidodec.rack111.com/index.html The Rhombi-Cosi-Dodecahedron Website]