Rhombicosidodecahedron

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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 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
Rhombicosidodecahedron
3.4.5.4
(Vertex figure)
Deltoidalhexecontahedron.jpg
Deltoidal hexecontahedron
(dual polyhedron)
Rhombicosidodecahedron flat.png
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[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:[1]

(±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[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
3-4
Edge
5-4
Face
Square
Face
Triangle
Face
Pentagon
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
Projective
symmetry
[2] [2] [2] [2] [6] [10]
Dual
image
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
Pentagon-centered
Rhombicosidodecahedron stereographic projection triangle.png
Triangle-centered
Rhombicosidodecahedron stereographic projection square.png
Square-centered
Orthographic projection Stereographic projections

Related polyhedra[edit]

Expansion of either a dodecahedron or an icosahedron creates a rhombicosidodecahedron.
Family of uniform icosahedral polyhedra
Symmetry: [5,3], (*532) [5,3]+, (532)
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.png
Uniform polyhedron-53-t0.png Uniform polyhedron-53-t01.png Uniform polyhedron-53-t1.png Uniform polyhedron-53-t12.png Uniform polyhedron-53-t2.png Uniform polyhedron-53-t02.png Uniform polyhedron-53-t012.png Uniform polyhedron-53-s012.png
{5,3} t{5,3} r{5,3} 2t{5,3}=t{3,5} 2r{5,3}={3,5} rr{5,3} tr{5,3} sr{5,3}
Duals to uniform polyhedra
CDel node f1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 5.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 5.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 5.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 5.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 5.pngCDel node fh.pngCDel 3.pngCDel node fh.png
Icosahedron.svg Triakisicosahedron.jpg Rhombictriacontahedron.svg Pentakisdodecahedron.jpg Dodecahedron.svg Deltoidalhexecontahedron.jpg Disdyakistriacontahedron.jpg Pentagonalhexecontahedronccw.jpg
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

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.

Dimensional family of expanded polyhedra and tilings: 3.4.n.4
Symmetry
*n32
[n,3]
Spherical Euclidean Compact hyperbolic Paracompact
*232
[2,3]
D3h
*332
[3,3]
Td
*432
[4,3]
Oh
*532
[5,3]
Ih
*632
[6,3]
P6m
*732
[7,3]
 
*832
[8,3]...
 
*∞32
[∞,3]
 
Expanded
figure
Spherical triangular prism.png
3.4.2.4
Uniform tiling 332-t02.png
3.4.3.4
Uniform tiling 432-t02.png
3.4.4.4
Uniform tiling 532-t02.png
3.4.5.4
Uniform polyhedron-63-t02.png
3.4.6.4
Uniform tiling 73-t02.png
3.4.7.4
Uniform tiling 83-t02.png
3.4.8.4
H2 tiling 23i-5.png
3.4.∞.4
Coxeter
Schläfli
CDel node 1.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node 1.png
rr{2,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
rr{3,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
rr{4,3}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png
rr{5,3}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png
rr{6,3}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node 1.png
rr{7,3}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node 1.png
rr{8,3}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node 1.png
rr{∞,3}
Deltoidal figure Triangular dipyramid.png
V3.4.2.4
Rhombicdodecahedron.jpg
V3.4.3.4
Deltoidalicositetrahedron.jpg
V3.4.4.4
Deltoidalhexecontahedron.jpg
V3.4.5.4
Tiling Dual Semiregular V3-4-6-4 Deltoidal Trihexagonal.svg
V3.4.6.4
Deltoidal triheptagonal til.png
V3.4.7.4
Deltoidal trioctagonal til.png
V3.4.8.4
Deltoidal triapeirogonal til.png
V3.4.∞.4
Coxeter CDel node f1.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node f1.png

Johnson solids[edit]

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

Diminished
J5
Pentagonal cupola.png
76
Diminished rhombicosidodecahedron.png
80
Parabidiminished rhombicosidodecahedron.png
81
Metabidiminished rhombicosidodecahedron.png
83
Tridiminished rhombicosidodecahedron.png
Gyrated and/or diminished
72
Gyrate rhombicosidodecahedron.png
73
Parabigyrate rhombicosidodecahedron.png
74
Metabigyrate rhombicosidodecahedron.png
75
Trigyrate rhombicosidodecahedron.png
77
Paragyrate diminished rhombicosidodecahedron.png
78
Metagyrate diminished rhombicosidodecahedron.png
79
Bigyrate diminished rhombicosidodecahedron.png
82
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
Rhombicosidodecahedron
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

In popular culture[edit]

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.

See also[edit]

Notes[edit]

References[edit]

  • 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]