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(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
Wythoff symbol 3 5 | 2
Coxeter diagram CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png
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
Small rhombicosidodecahedron.png
Colored faces
Small rhombicosidodecahedron vertfig.png
(Vertex figure)
Deltoidal hexecontahedron
(dual polyhedron)
Rhombicosidodecahedron flat.png

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[edit]

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[edit]

Cartesian coordinates for the vertices of a rhombicosidodecahedron with edge length 2 centered at the origin are all even permutations of:[1]

(±1, ±1, ±φ3),
(±φ2, ±φ, ±2φ),
(±(2+φ), 0, ±φ2),

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

Orthogonal projections[edit]

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
Image Dodecahedron t02 v.png Dodecahedron t02 e34.png Dodecahedron t02 e45.png Dodecahedron t02 f4.png Dodecahedron t02 A2.png Dodecahedron t02 H3.png
[2] [2] [2] [2] [6] [10]
Dual dodecahedron t02 v.png Dual dodecahedron t02 e34.png Dual dodecahedron t02 e45.png Dual dodecahedron t02 f4.png Dual dodecahedron t02 A2.png Dual dodecahedron t02 H3.png

Spherical tiling[edit]

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.

Uniform tiling 532-t02.png Rhombicosidodecahedron stereographic projection pentagon'.png
Rhombicosidodecahedron stereographic projection triangle.png
Rhombicosidodecahedron stereographic projection square.png
Orthographic projection Stereographic projections

Related polyhedra[edit]

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

Symmetry mutations[edit]

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.

Johnson solids[edit]

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

Pentagonal cupola.png
Diminished rhombicosidodecahedron.png
Parabidiminished rhombicosidodecahedron.png
Metabidiminished rhombicosidodecahedron.png
Tridiminished rhombicosidodecahedron.png
Gyrated and/or diminished
Gyrate rhombicosidodecahedron.png
Parabigyrate rhombicosidodecahedron.png
Metabigyrate rhombicosidodecahedron.png
Trigyrate rhombicosidodecahedron.png
Paragyrate diminished rhombicosidodecahedron.png
Metagyrate diminished rhombicosidodecahedron.png
Bigyrate diminished rhombicosidodecahedron.png
Gyrate bidiminished rhombicosidodecahedron.png

Vertex arrangement[edit]

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.

Small rhombicosidodecahedron.png
Small dodecicosidodecahedron.png
Small dodecicosidodecahedron
Small rhombidodecahedron.png
Small rhombidodecahedron
Small stellated truncated dodecahedron.png
Small stellated truncated dodecahedron
UC36-6 pentagrammic prisms.png
Compound of six pentagrammic prisms
UC37-12 pentagrammic prisms.png
Compound of twelve pentagrammic prisms

Rhombicosidodecahedral graph[edit]

Rhombicosidodecahedral graph
Rhombicosidodecahedral graph.png
5-fold symmetry
Vertices 60
Edges 120
Automorphisms 120
Properties Quartic graph, Hamiltonian, regular

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.[2]

Schlegel diagrams
Rhombicosidodecahedral graph-tricenter.png
3-fold symmetry
Rhombicosidodecahedral graph-squarecenter.png
4-fold symmetry

See also[edit]


  1. ^ Weisstein, Eric W., "Icosahedral group", MathWorld.
  2. ^ Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford University Press, p. 269 


  • 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[edit]