Rhombicosidodecahedron

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Template:Semireg polyhedra db 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.

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

Image:Rhombicosidodecahedron flat.png

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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.

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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 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.

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