Rhombicosidodecahedron

Rhombicosidodecahedron
(Click here for rotating model)
Type	Archimedean solid Uniform polyhedron
Elements	F = 62, E = 120, V = 60 (χ = 2)
Faces by sides	20{3}+30{4}+12{5}
Schläfli symbol	t0,2{5,3}
Wythoff symbol	3 5 | 2
Coxeter–Dynkin
Symmetry	Ih, [5,3], (*532)
Dihedral Angle
References	U27, C30, W14
Properties	Semiregular convex
Colored faces	3.4.5.4 (Vertex figure)
Deltoidal hexecontahedron (dual polyhedron)	Net

In geometry, the rhombicosidodecahedron, or small rhombicosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces.

It has 20 regular triangular faces, 30 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 be called an expanded or cantellated dodecahedron or icosahedron, from truncation operations on either uniform polyhedron.

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 rhombicosidodecahedron. 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 six or twelve 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.

Twelve of the 92 Johnson solids are derived from the rhombicosidodecahedron, four of them by rotation of one or more pentagonal cupolas: the gyrate, parabigyrate, metabigyrate and trigyrate rhombicosidodecahedron. Eight more can be constructed by removing up to three cupolas, sometimes also rotating one or more of the other cupolas.

Cartesian coordinates

Cartesian coordinates for the vertices of a rhombicosidodecahedron with edge length 2 centered at the origin are:

(±1, ±1, ±φ³),

(±φ³, ±1, ±1),

(±1, ±φ³, ±1),

(±φ², ±φ, ±2φ),

(±2φ, ±φ², ±φ),

(±φ, ±2φ, ±φ²),

(±(2+φ), 0, ±φ²),

(±φ², ±(2+φ), 0),

(0, ±φ², ±(2+φ)),

where φ = (1+√5)/2 is the golden ratio (also written τ).

Orthogonal projections

The rhombicosidodecahedron has five special orthogonal projections, centered, on a vertex, on two types of edges, and three types of faces: triangles, squares, and pentagons. The last two correspond to the A₂ and H₂ Coxeter planes.

Orthogonal projections

Centered by	Vertex	Edge 3-4	Edge 5-4	Face Square	Face Triangle	Face Pentagon
Image
Projective symmetry	[2]	[2]	[2]	[2]	[6]	[10]

Related polyhedra

Expansion of a dodecahedron creates a rhombicosidodecahedron and a reverse expansion creates an icosahedron.

The rhombicosidodecahedron shares its vertex arrangement with three nonconvex uniform polyhedra: the small stellated truncated dodecahedron, the small dodecicosidodecahedron (having the triangular and pentagonal faces in common), and the small rhombidodecahedron (having the square faces in common).

It also shares its vertex arrangement with the uniform compounds of six or twelve pentagrammic prisms.

Rhombicosidodecahedron	Small dodecicosidodecahedron	Small rhombidodecahedron
Small stellated truncated dodecahedron	Compound of six pentagrammic prisms	Compound of twelve pentagrammic prisms

See also

• Spinning rhombicosidodecahedron
• Dodecahedron
• Icosahedron
• Icosidodecahedron
• Rhombicuboctahedron
• Truncated icosidodecahedron (great rhombicosidodecahedron)

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)

External links

• Eric W. Weisstein, Small Rhombicosidodecahedron (Archimedean solid) at MathWorld.
• Richard Klitzing, 3D convex uniform polyhedra, x3o5x - srid
• Editable printable net of a Rhombicosidodecahedron with interactive 3D view
• The Uniform Polyhedra
• Virtual Reality Polyhedra The Encyclopedia of Polyhedra
• The Rhombi-Cosi-Dodecahedron Website
• The Rhombicosidodecahedron as a 3D puzzle