Rhombicosidodecahedron

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Small rhombicosidodecahedron
(Click here for rotating model)
Type	Archimedean solid
Elements	F=62, E=120, V=60 (χ=2)
Faces by sides	20{3}+30{4}+12{5}
Schläfli symbol	$r\begin{Bmatrix} 3 \\ 5 \end{Bmatrix}$
Wythoff symbol	3 5 \| 2
Coxeter-Dynkin
Symmetry	I_h
References	U₂₇, C₃₀, W₁₄
Properties	Semiregular convex
Colored faces	3.4.5.4 (Vertex figure)
Deltoidal hexecontahedron (dual polyhedron)	Net

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.

1 Geometric relations
2 Cartesian coordinates
3 See also
4 References
5 External links

[edit] 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.

Kids at St. Germaine School in Bethel Park, PA built the worlds largest rhombicosidodecahedron, which stood at over 9 feet tall. The workers included, but are certainly not limited to (in alphabetical order) Bridget Bielich, Callie Hanua, Glenn Huetter, Carl Mitchell, Anthony "TJ" Serafini, and Leanna Talarico. Bielich's grandfather, Walter, donated the cardboard for the project. The shape stood on its own, although for several weeks it sat in a cold convent, haunted by dead nuns and supported by chairs. It was painted blue, purple, and pink. Most of the guys in the class did not like that it was purple and pink, because they are insecure and, sometimes, a little whiney. The building of the shape caused great unity among the 8th grade class, and it also took up a lot of math class time. The record-breaking construction was written about in the Post-Gazette.

[edit] 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.

[edit] See also

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

[edit] References

Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. (Section 3-9)