^ Harmonies Of The World by Johannes Kepler, Translated into English with an introduction and notes by E.Rhombicosidodecahedral graph Rhombicosidodecahedral graph It also shares its vertex arrangement with the uniform compounds of six or twelve pentagrammic prisms. 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). There are 12 related Johnson solids, 5 by diminishment, and 8 including gyrations: * n32 symmetry mutation of expanded tilings: 3.4. These vertex-transitive figures have (*n32) reflectional symmetry. 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. A version with golden rectangles is used as vertex element of the construction set Zometool. Related polyhedra Expansion of either a dodecahedron or an icosahedron creates a rhombicosidodecahedron. Straight lines on the sphere are projected as circular arcs on the plane. This projection is conformal, preserving angles but not areas or lengths. The rhombicosidodecahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. The last two correspond to the A 2 and H 2 Coxeter planes. 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. Orthogonal projections in Geometria (1543) by Augustin Hirschvogel For unit edge length, R must be halved, giving Therefore, the circumradius of this rhombicosidodecahedron is the common distance of these points from the origin, namely √ φ 6+2 = √ 8φ+7 for edge length 2. Where φ = 1 + √ 5 / 2 is the golden ratio. Eight more can be constructed by removing up to three cupolae, sometimes also rotating one or more of the other cupolae.Ĭartesian coordinates for the vertices of a rhombicosidodecahedron with an edge length of 2 centered at the origin are all even permutations of: (☑, ☑, ± φ 3), (± φ 2, ± φ, ☒ φ), (±(2+ φ), 0, ± φ 2), 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. The expansion is chosen so that the resulting rectangles are golden rectangles. The balls are "expanded" rhombicosidodecahedra, with the squares replaced by rectangles. The Zometool kits for making geodesic domes and other polyhedra use slotted balls as connectors. The rhombicosidodecahedron shares the vertex arrangement with the small stellated truncated dodecahedron, and with the uniform compounds of six or twelve pentagrammic prisms. 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. 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. Therefore, it has the same number of squares as five cubes. 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.Īlternatively, 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. 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. 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.įor a rhombicosidodecahedron with edge length a, its surface area and volume are:Ī = ( 30 + 5 3 + 3 25 + 10 5 ) a 2 ≈ 59.305 982 844 9 a 2 V = 60 + 29 5 3 a 3 ≈ 41.615 323 782 5 a 3 Geometric relations 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.
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