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Rhombicosidodecahedron :
| 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{53}
|
| t0,2{5,3} | |
| Wythoff symbol | 3 5 | 2 |
| Coxeter diagram |
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| 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 |
In geometry, the 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.
Dimensions:
For a rhombicosidodecahedron with edge length a, its surface area and volume are:
A=(30+53+325+105)a2≈59.3059828449a2V=60+2953a3≈41.6153237825a3
Cartesian coordinates:
Cartesian coordinates for the vertices of a rhombicosidodecahedron with an edge length of 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. Therefore, the circumradius of this rhombicosidodecahedron is the common distance of these points from the origin, namely √φ6+2 = √8φ+7 for edge length 2. For unit edge length, R must be halved, giving
R = √8φ+7/2 = √11+4√5/2 ≈ 2.233.
Orthogonal projections:
Orthogonal projections in Geometria (1543) by Augustin Hirschvogel
The rhombicosidodecahedron has six 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
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| Centered by | Vertex | Edge 3-4 | Edge 5-4 | Face Square | Face Triangle | Face Pentagon |
|---|---|---|---|---|---|---|
| Wireframe | ||||||
| Projective symmetry | [2] | [2] | [2] | [2] | [6] | [10] |
| Dual image |
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Rhombicosidodecahedral graph:
| Rhombicosidodecahedral graph | |
|---|---|
Pentagon centered Schlegel diagram | |
| Vertices | 60 |
| Edges | 120 |
| Automorphisms | 120 |
| Properties | Quartic graph, Hamiltonian, regular |
| Table of graphs and parameters |
Square centered Schlegel diagram
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 |
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