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
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 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.

Names

Johannes Kepler in Harmonices Mundi (1618) named this polyhedron a rhombicosidodecahedron, being short for truncated icosidodecahedral rhombus, with icosidodecahedral rhombus being his name for a rhombic triacontahedron.[1][2] There are different truncations of a rhombic triacontahedron into a topological rhombicosidodecahedron: Prominently its rectification (left), the one that creates the uniform solid (center), and the rectification of the dual icosidodecahedron (right), which is the core of the dual compound.

Dimensions

For a rhombicosidodecahedron with edge length a, its surface area and volume are:

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, or 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.

Alternatively, if you expand each of five cubes by moving the faces away from the origin the right amount and rotating each of the five 72° around so they are equidistant from each other, without changing the orientation or size of the faces, and patch the pentagonal and triangular holes in the result, you get a rhombicosidodecahedron. Therefore, it has the same number of squares as five cubes.

Two clusters of faces of the bilunabirotunda, the lunes (each lune featuring two triangles adjacent to opposite sides of one square), can be aligned with a congruent patch of faces on the rhombicosidodecahedron. If two bilunabirotundae are aligned this way on opposite sides of the rhombicosidodecahedron, then a cube can be put between the bilunabirotundae at the very center of the rhombicosidodecahedron.

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 an edge length of 2 centered at the origin are all even permutations of:[3]

(±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+45/2 ≈ 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
Centered by Vertex Edge
3-4
Edge
5-4
Face
Square
Face
Triangle
Face
Pentagon
Solid
Wireframe
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.
A version with golden rectangles is used as vertex element of the construction set Zometool.[4]
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 12 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

Rhombicosidodecahedral graph

Rhombicosidodecahedral graph
Pentagon centered Schlegel diagram
Vertices60
Edges120
Automorphisms120
PropertiesQuartic 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.[5]

Square centered Schlegel diagram

See also

Notes

  1. ^ Ioannis Keppler [i.e., Johannes Kepler] (1619). "Liber II. De Congruentia Figurarum Harmonicarum. XXVIII. Propositio." [Book II. On the Congruence of Harmonic Figures. Proposition XXVIII.]. Harmonices Mundi Libri V [The Harmony of the World in Five Books]. Linz, Austria: Sumptibus Godofredi Tampachii bibl. Francof. excudebat Ioannes Plancus [published by Gottfried Tambach [...] printed by Johann Planck]. p. 64. OCLC 863358134. Unus igitur Trigonicus cum duobus Tetragonicis & uno Pentagonico, minus efficiunt 4 rectis, & congruunt 20 Trigonicum 30 Tetragonis & 12 Pentagonis, in unum Hexacontadyhedron, quod appello Rhombicoſidodecaëdron, ſeu ſectum Rhombum Icoſidododecaëdricum.
  2. ^ Harmonies Of The World by Johannes Kepler, Translated into English with an introduction and notes by E. J. Aiton, A. M. Duncan, "J. V. Field, 1997, ISBN 0-87169-209-0 (page 123)
  3. ^ Weisstein, Eric W. "Icosahedral group". MathWorld.
  4. ^ Weisstein, Eric W. "Zome". MathWorld.
  5. ^ Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford University Press, p. 269

References