| 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.
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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 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.
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 for the vertices of a rhombicosidodecahedron with edge length 2 centered at the origin are:[1]
where φ = (1+√5)/2 is the golden ratio (also written τ).
The rhombicosidodecahedron has five special orthogonal projections, centered, on a vertex, on two types of edges, and two types of faces: triangles and pentagons. The last two correspond to the A2 and H2 Coxeter planes.
| Centered by | Vertex | Edge 3-4 |
Edge 5-4 |
Face Triangle |
Face Pentagon |
|---|---|---|---|---|---|
| Image | |||||
| Projective symmetry |
[2] | [2] | [2] | [6] | [10] |
The rhombicosidodecahedron shares its vertex arrangement with 3 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 6 or 12 pentagrammic prisms.
Rhombicosidodecahedron |
Small dodecicosidodecahedron |
Small rhombidodecahedron |
Small stellated truncated dodecahedron |
Compound of six pentagrammic prisms |
Compound of twelve pentagrammic prisms |
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