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 dualdodecahedron, 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 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.
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
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 H2Coxeter 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.
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
^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. OCLC863358134. 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.
^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, ISBN0-87169-209-0 (page 123)
Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN0-486-23729-X. (Section 3-9)
Cromwell, P. (1997). Polyhedra. United Kingdom: Cambridge. pp. 79–86 Archimedean solids. ISBN0-521-55432-2.
The Big Bang Theory Series 8 Episode 2 - The Junior Professor Solution: features this solid as the answer to an impromptu science quiz the main four characters have in Leonard and Sheldon's apartment, and is also illustrated in Chuck Lorre's Vanity Card #461 at the end of that episode.