# Rhombicosidodecahedron

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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: (&plusmn;1, &plusmn;1, &plusmn;&tau;3),: (&plusmn;&tau;3, &plusmn;1, &plusmn;1),: (&plusmn;1, &plusmn;&tau;3, &plusmn;1),: (&plusmn;&tau;2, &plusmn;&tau;, &plusmn;2&tau;),: (&plusmn;2&tau;, &plusmn;&tau;2, &plusmn;&tau;),: (&plusmn;&tau;, &plusmn;2&tau;, &plusmn;&tau;2),: (&plusmn;(2+&tau;), 0, &plusmn;&tau;2),: (&plusmn;&tau;2, &plusmn;(2+&tau;), 0),: (0, &plusmn;&tau;2, &plusmn;(2+&tau;)),where &tau; = (1+&radic;5)/2 is the golden ratio (also written &phi;).

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)

*
* [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]

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### Look at other dictionaries:

• rhombicosidodecahedron — noun An Archimedean solid with 20 regular triangular faces, 30 regular square faces, 12 regular pentagonal faces, 60 vertices and 120 edges …   Wiktionary

• rhombicosidodecahedron — An Archimedean solid composed of sixty two faces …   Grandiloquent dictionary

• Nonconvex great rhombicosidodecahedron — Type Uniform star polyhedron Elements F = 62, E = 120 V = 60 (χ = 2) Faces by sides 20{3}+30{4}+12{5/2} …   Wikipedia

• Metabidiminished rhombicosidodecahedron — Type Johnson J80 J81 J82 Fac …   Wikipedia

• Small complex rhombicosidodecahedron — Type Uniform star polyhedron Elements F = 62, E = 120 (60x2) V = 20 (χ = 38) Faces by sides 20{3}+12{5/2}+30{4} …   Wikipedia

• Diminished rhombicosidodecahedron — Type Johnson J75 J76 J77 Faces …   Wikipedia

• Parabidiminished rhombicosidodecahedron — Infobox Polyhedron with net Polyhedron Type=Johnson J79 J80 J81 Face List=10 triangles 2x10 squares 10 pentagons 2 decagons Edge Count=90 Vertex Count=50 Symmetry Group= D 5d Vertex List=20(4.5.10) 10+20(3.4.5.4) Dual= Property List=convex Net In …   Wikipedia

• Tridiminished rhombicosidodecahedron — Infobox Polyhedron with net Polyhedron Type=Johnson J82 J83 J84 Face List=2+3 triangles 3x3+6 squares 3x3 pentagons 3 decagons Edge Count=75 Vertex Count=45 Symmetry Group= C 3v Vertex List=5x6(4.5.10) 3x3+6(3.4.5.4) Dual= Property List=convex… …   Wikipedia

• Trigyrate rhombicosidodecahedron — Infobox Polyhedron with net Polyhedron Type=Johnson J74 J75 J76 Face List=2+2x3+2x6 triangles 4x3+3.6 squares 4x3 pentagons Edge Count=120 Vertex Count=60 Symmetry Group= C 3v Vertex List=5x6(3.42.5) 4x3+3x6(3.4.5.4) Dual= Property List=convex… …   Wikipedia

• Metabigyrate rhombicosidodecahedron — Type Johnson J73 J74 J75 Faces …   Wikipedia