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}
Conway notation	eD or aaD
Schläfli symbols	rr{5,3} or $r{\begin{Bmatrix}5\\3\end{Bmatrix}}$
	t0,2{5,3}
Wythoff symbol	3 5 | 2
Coxeter diagram
Symmetry group	Ih, H3, [5,3], (*532), order 120
Rotation group	I, [5,3]+, (532), order 60
Dihedral angle	3-4: 159°05′41″ (159.09°) 4-5: 148°16′57″ (148.28°)
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 expand 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" 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 cupolae: the gyrate, parabigyrate, metabigyrate and trigyrate rhombicosidodecahedron. Eight more can be constructed by removing up to three cupolae, sometimes also rotating one or more of the other cupolae.

Cartesian coordinates

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

(±1, ±1, ±ϕ³),

(±ϕ², ±ϕ, ±2ϕ),

(±(2+ϕ), 0, ±ϕ²),

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

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]
Dual image

Spherical tiling

The rhombicosidodecahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.

	Pentagon-centered	Triangle-centered	Square-centered
Orthographic projection	Stereographic projections

Related polyhedra

Expansion of either a dodecahedron or an icosahedron creates a rhombicosidodecahedron.

Family of uniform icosahedral polyhedra
Symmetry: [5,3], (*532)							[5,3]+, (532)
{5,3}	t{5,3}	r{5,3}	t{3,5}	{3,5}	rr{5,3}	tr{5,3}	sr{5,3}
Duals to uniform polyhedra
V5.5.5	V3.10.10	V3.5.3.5	V5.6.6	V3.3.3.3.3	V3.4.5.4	V4.6.10	V3.3.3.3.5

Symmetry mutations

This polyhedron is topologically related as a part of a sequence of cantellated polyhedra with vertex figure (3.4.n.4), which continues as tilings of the hyperbolic plane. These vertex-transitive figures have (*n32) reflectional symmetry.

*n32 symmetry mutation of expanded tilings: 3.4.n.4
Symmetry *n32 [n,3]	Spherical				Euclid.	Compact hyperb.		Paracomp.
	*232 [2,3]	*332 [3,3]	*432 [4,3]	*532 [5,3]	*632 [6,3]	*732 [7,3]	*832 [8,3]...	*∞32 [∞,3]
Figure
Config.	3.4.2.4	3.4.3.4	3.4.4.4	3.4.5.4	3.4.6.4	3.4.7.4	3.4.8.4	3.4.∞.4

Johnson solids

There are 13 related Johnson solids, 5 by diminishment, and 8 including gyrations:

Diminished

J5

76

80

81

83

Gyrated and/or diminished

72	73	74	75
77	78	79	82

Vertex arrangement

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

In popular culture

The rhombicosidodecahedron is mentioned in Episode 2 of Season 8 of The Big Bang Theory as Sheldon, Howard, Raj and Leonard are quizzing one another on various scientific subjects. Chuck Lorre also included an image of the rhombicosidodecahedron on vanity plate #461 at the end of the episode.

Rhombicosidodecahedral graph

Rhombicosidodecahedral graph
5-fold symmetry
Vertices	60
Edges	120
Automorphisms	120
Properties	Quartic graph, Hamiltonian, regular
Table of graphs and parameters

In the mathematical field of graph theory, a rhombicosidodecahedral graph is the graph of vertices and edges of the rhombicosidodecahedron, one of the Archimedean solids. It has 60 vertices and 120 edges, and is a quartic graph Archimedean graph.

Schlegel diagrams

3-fold symmetry

4-fold symmetry

See also

• Truncated 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)
• Cromwell, P. (1997). Polyhedra. United Kingdom: Cambridge. pp. 79-86 Archimedean solids. ISBN 0-521-55432-2.

External links

• Weisstein, Eric W., "Small Rhombicosidodecahedron" ("Archimedean solid") at MathWorld.
  • Weisstein, Eric W. "Small rhombicosidodecahedron graph". MathWorld.
• Klitzing, Richard. "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