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Rhombicosidodecahedron
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<h2 id="names">Names</h2> <p><a href="/facts/Johannes_Kepler/wk6GTN3C">Johannes Kepler</a> in <i><a href="/facts/Harmonices_Mundi/BDKcKU7Y">Harmonices Mundi</a></i> (1618) named this polyhedron a <i>rhombicosidodecahedron</i>, being short for <i>truncated icosidodecahedral rhombus</i>, with <i>icosidodecahedral rhombus</i> being his name for a <a href="/facts/Rhombic_triacontahedron/HmHWG8JY">rhombic triacontahedron</a>.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a><a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> There are different truncations of a <a href="/facts/Rhombic_triacontahedron/HmHWG8JY">rhombic triacontahedron</a> into a <a href="/facts/Topology/2EqW47cU">topological</a> rhombicosidodecahedron: Prominently its <a href="/facts/Rectification_(geometry)/PkoUdq53">rectification</a> (left), the one that creates the uniform solid (center), and the rectification of the dual <a href="/facts/Icosidodecahedron/P0JluaVA">icosidodecahedron</a> (right), which is the core of the <a href="/facts/Dual_compound/L13pItNN">dual compound</a>. </p> <h2 id="dimensions">Dimensions</h2> <p>For a rhombicosidodecahedron with edge length <i>a</i>, its surface area and volume are: </p> A = ( 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 {\displaystyle {\begin{aligned}A&=\left(30+5{\sqrt {3}}+3{\sqrt {25+10{\sqrt {5}}}}\right)a^{2}&&\approx 59.305\,982\,844\,9a^{2}\\V&={\frac {60+29{\sqrt {5}}}{3}}a^{3}&&\approx 41.615\,323\,782\,5a^{3}\end{aligned}}} <h2 id="geometric-relations">Geometric relations</h2> <p><a href="/facts/Expansion_(geometry)/eMpoRlnm">Expanding</a> an <a href="/facts/Icosidodecahedron/P0JluaVA">icosidodecahedron</a> by moving the faces away from the <a href="/facts/Origin_(mathematics)/Lm26tRCB">origin</a> the right amount, rotating each face so that each triangle vertex continues to touch a pentagon vertex, without changing the size of the faces, and patching the square holes in the result, yields a rhombicosidodecahedron. Therefore, it has the same number of triangles as an <a href="/facts/Icosahedron/wWbpnDYp">icosahedron</a> and the same number of pentagons as a <a href="/facts/Dodecahedron/7r519eCl">dodecahedron</a>, with a square for each edge of either. </p><p>Alternatively, <a href="/facts/Expansion_(geometry)/eMpoRlnm">expanding</a> each of five cubes by moving the faces away from the <a href="/facts/Origin_(mathematics)/Lm26tRCB">origin</a> 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 patching the pentagonal and triangular holes in the result, yields a rhombicosidodecahedron. Therefore, it has the same number of squares as five cubes. </p><p>Two clusters of faces of the <a href="/facts/Bilunabirotunda/Y4HPT5I3">bilunabirotunda</a>, the <i>lunes</i> (each <i>lune</i> 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. </p><p>The rhombicosidodecahedron shares the vertex arrangement with the <a href="/facts/Small_stellated_truncated_dodecahedron/qT9PLHGC">small stellated truncated dodecahedron</a>, and with the uniform compounds of six or twelve <a href="/facts/Pentagrammic_prism/3ofu2hVH">pentagrammic prisms</a>. </p><p>The <a href="/facts/Zometool/bx6DlUR2">Zometool</a> kits for making <a href="/facts/Geodesic_dome/XmHEPKmK">geodesic domes</a> and other <a href="/facts/Polyhedron/UC0p9FTF">polyhedra</a> 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 <a href="/facts/Golden_rectangle/Y7YQsbYh">golden rectangles</a>. </p><p>Twelve of the 92 <a href="/facts/Johnson_solid/8SR7CyNz">Johnson solids</a> are derived from the rhombicosidodecahedron, four of them by rotation of one or more <a href="/facts/Pentagonal_cupola/LfcF0Wjf">pentagonal cupolae</a>: the <a href="/facts/Gyrate_rhombicosidodecahedron/axpKMvyx">gyrate</a>, <a href="/facts/Parabigyrate_rhombicosidodecahedron/xBbqGcqJ">parabigyrate</a>, <a href="/facts/Metabigyrate_rhombicosidodecahedron/Cy9RWlEE">metabigyrate</a>, and <a href="/facts/Trigyrate_rhombicosidodecahedron/MEBDS0Cf">trigyrate rhombicosidodecahedron</a>. Eight more can be constructed by removing up to three cupolae, sometimes also rotating one or more of the other cupolae. </p> <h2 id="cartesian-coordinates">Cartesian coordinates</h2> <p><a href="/facts/Cartesian_coordinates/lkTqvMmB">Cartesian coordinates</a> for the vertices of a rhombicosidodecahedron with an edge length of 2 centered at the origin are all <a href="/facts/Even_permutation/ludhCMJz">even permutations</a> of:<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> (±1, ±1, ±<i>φ</i>3), (±<i>φ</i>2, ±<i>φ</i>, ±2<i>φ</i>), (±(2+<i>φ</i>), 0, ±<i>φ</i>2), <p>where <i>φ</i> = 1 + √5/2 is the <a href="/facts/Golden_ratio/anloSRmD">golden ratio</a>. Therefore, the <a href="/facts/Circumradius/RafYT4am">circumradius</a> of this rhombicosidodecahedron is the common distance of these points from the origin, namely √<i>φ</i>6+2 = √<i>8φ+7</i> for edge length 2. For unit edge length, R must be halved, giving </p> <i>R</i> = √8<i>φ</i>+7/2 = √11+4√5/2 ≈ 2.233. <h2 id="orthogonal-projections">Orthogonal projections</h2> <p>The <i>rhombicosidodecahedron</i> has six special <a href="/facts/Orthogonal_projection/sElXGkxD">orthogonal projections</a>, 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 <a href="/facts/Coxeter_plane/4B80wH8V">Coxeter planes</a>. </p> Orthogonal projections<table><tbody><tr><th>Centered by</th><th>Vertex</th><th>Edge3-4</th><th>Edge5-4</th><th>FaceSquare</th><th>FaceTriangle</th><th>FacePentagon</th></tr><tr><th>Solid</th><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>Wireframe</th><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>Projectivesymmetry</th><td>[2]</td><td>[2]</td><td>[2]</td><td>[2]</td><td>[6]</td><td>[10]</td></tr><tr><th>Dualimage</th><td></td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table> <h2 id="spherical-tiling">Spherical tiling</h2> <p>The rhombicosidodecahedron can also be represented as a <a href="/facts/Spherical_tiling/jaglysFR">spherical tiling</a>, and projected onto the plane via a <a href="/facts/Stereographic_projection/oVyMrWqR">stereographic projection</a>. This projection is <a href="/facts/Conformal_map/cEpQnMad">conformal</a>, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane. </p> <table><tbody><tr><td></td><td><a href="/facts/Pentagon/xoq5oJeQ">Pentagon</a>-centered</td><td><a href="/facts/Triangle/cRXUwsMn">Triangle</a>-centered</td><td><a href="/facts/Square/JHJeMvmL">Square</a>-centered</td></tr><tr><th><a href="/facts/Orthographic_projection/L3Cv9kYD">Orthographic projection</a></th><th colspan="3"><a href="/facts/Stereographic_projection/oVyMrWqR">Stereographic projections</a></th></tr></tbody></table> <h2 id="related-polyhedra">Related polyhedra</h2> <table><tbody><tr><th colspan="8">Family of uniform icosahedral polyhedra</th></tr><tr><th colspan="7"><a href="/facts/List_of_spherical_symmetry_groups/3X3YcHoQ">Symmetry</a>: <a href="/facts/Icosahedral_symmetry/pDlwAy23">[5,3]</a>, (*532)</th><th>[5,3]+, (532)</th></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th></th><th></th><th></th><th></th><th></th><th></th><th></th><th></th></tr><tr><td><a href="/facts/Dodecahedron/7r519eCl">{5,3}</a></td><td><a href="/facts/Truncated_dodecahedron/PVTAsUWQ">t{5,3}</a></td><td><a href="/facts/Icosidodecahedron/P0JluaVA">r{5,3}</a></td><td><a href="/facts/Truncated_icosahedron/LR8ugaY9">t{3,5}</a></td><td><a href="/facts/Icosahedron/wWbpnDYp">{3,5}</a></td><td>rr{5,3}</td><td><a href="/facts/Truncated_icosidodecahedron/d5v9VpFY">tr{5,3}</a></td><td><a href="/facts/Snub_dodecahedron/oouEPVxD">sr{5,3}</a></td></tr><tr><th colspan="8">Duals to uniform polyhedra</th></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td><a href="/facts/Icosahedron/wWbpnDYp">V5.5.5</a></td><td><a href="/facts/Triakis_icosahedron/VoVKELcj">V3.10.10</a></td><td><a href="/facts/Rhombic_triacontahedron/HmHWG8JY">V3.5.3.5</a></td><td><a href="/facts/Pentakis_dodecahedron/TksT30Qt">V5.6.6</a></td><td><a href="/facts/Dodecahedron/7r519eCl">V3.3.3.3.3</a></td><td><a href="/facts/Deltoidal_hexecontahedron/IWvmY471">V3.4.5.4</a></td><td><a href="/facts/Disdyakis_triacontahedron/WkV7L9Pg">V4.6.10</a></td><td><a href="/facts/Pentagonal_hexecontahedron/uXUsI7L5">V3.3.3.3.5</a></td></tr></tbody></table> <h3>Symmetry mutations</h3> <p>This polyhedron is topologically related as a part of a sequence of <a href="/facts/Cantellation_(geometry)/kS24Tde3">cantellated</a> polyhedra with vertex figure (3.4.n.4), which continues as tilings of the <a href="/facts/Hyperbolic_space/kI7tnwdk">hyperbolic plane</a>. These <a href="/facts/Vertex-transitive/5nC8PdBm">vertex-transitive</a> figures have (*n32) reflectional <a href="/facts/Orbifold_notation/HCbBoa2J">symmetry</a>. </p> <table><tbody><tr><th colspan="9">*<i>n</i>32 symmetry mutation of expanded tilings: 3.4.<i>n</i>.4</th></tr><tr><th rowspan="2">Symmetry*<i>n</i>32[n,3]</th><th colspan="4"><a href="/facts/List_of_spherical_symmetry_groups/3X3YcHoQ">Spherical</a></th><th><a href="/facts/List_of_planar_symmetry_groups/3buKYIft">Euclid.</a></th><th colspan="2">Compact hyperb.</th><th>Paracomp.</th></tr><tr><th>*232[2,3]</th><th>*332[3,3]</th><th>*432[4,3]</th><th>*532[5,3]</th><th>*632[6,3]</th><th>*732[7,3]</th><th>*832[8,3]...</th><th>*∞32[∞,3]</th></tr><tr><th>Figure</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th><a href="/facts/Vertex_configuration/wWaBS6t0">Config.</a></th><th><a href="/facts/Triangular_prism/Yjp4Kw9M">3.4.2.4</a></th><th><a href="/facts/Cuboctahedron/hD7p0Mem">3.4.3.4</a></th><th><a href="/facts/Rhombicuboctahedron/TmxUmEnl">3.4.4.4</a></th><th>3.4.5.4</th><th><a href="/facts/Rhombitrihexagonal_tiling/RJPMacIY">3.4.6.4</a></th><th><a href="/facts/Rhombitriheptagonal_tiling/VSK9wJcN">3.4.7.4</a></th><th><a href="/facts/Rhombitrioctagonal_tiling/xULl9VnV">3.4.8.4</a></th><th><a href="/facts/Rhombitriapeirogonal_tiling/1Wy0rv3j">3.4.∞.4</a></th></tr></tbody></table> <h3>Johnson solids</h3> <p>There are 12 related <a href="/facts/Johnson_solid/8SR7CyNz">Johnson solids</a>, 5 by diminishment, and 8 including gyrations: </p> Diminished<table><tbody><tr><td><a href="/facts/Pentagonal_cupola/LfcF0Wjf">J5</a></td><td><a href="/facts/Diminished_rhombicosidodecahedron/hAECv7Ht">76</a></td><td><a href="/facts/Parabidiminished_rhombicosidodecahedron/tBBvbtQd">80</a></td><td><a href="/facts/Metabidiminished_rhombicosidodecahedron/VmcoPTUH">81</a></td><td><a href="/facts/Tridiminished_rhombicosidodecahedron/cMEGkzlE">83</a></td></tr></tbody></table> Gyrated and/or diminished<table><tbody><tr><td><a href="/facts/Gyrate_rhombicosidodecahedron/axpKMvyx">72</a></td><td><a href="/facts/Parabigyrate_rhombicosidodecahedron/xBbqGcqJ">73</a></td><td><a href="/facts/Metabigyrate_rhombicosidodecahedron/Cy9RWlEE">74</a></td><td><a href="/facts/Trigyrate_rhombicosidodecahedron/MEBDS0Cf">75</a></td></tr><tr><td><a href="/facts/Paragyrate_diminished_rhombicosidodecahedron/PLjHv31u">77</a></td><td><a href="/facts/Metagyrate_diminished_rhombicosidodecahedron/LhWVdH88">78</a></td><td><a href="/facts/Bigyrate_diminished_rhombicosidodecahedron/3CAw70Fn">79</a></td><td><a href="/facts/Gyrate_bidiminished_rhombicosidodecahedron/XelV8fxk">82</a></td></tr></tbody></table> <h3>Vertex arrangement</h3> <p>The rhombicosidodecahedron shares its <a href="/facts/Vertex_arrangement/DFgCsP0n">vertex arrangement</a> with three <a href="/facts/Nonconvex_uniform_polyhedra/rSv9XpHQ">nonconvex uniform polyhedra</a>: the <a href="/facts/Small_stellated_truncated_dodecahedron/qT9PLHGC">small stellated truncated dodecahedron</a>, the <a href="/facts/Small_dodecicosidodecahedron/AQ4YIlL5">small dodecicosidodecahedron</a> (having the triangular and pentagonal faces in common), and the <a href="/facts/Small_rhombidodecahedron/Uq7u4yGu">small rhombidodecahedron</a> (having the square faces in common). </p><p>It also shares its vertex arrangement with the <a href="/facts/Polyhedron_compound/L13pItNN">uniform compounds</a> of <a href="/facts/Compound_of_six_pentagrammic_prisms/pMh7vyGW">six</a> or <a href="/facts/Compound_of_twelve_pentagrammic_prisms/LvvCVwbM">twelve pentagrammic prisms</a>. </p> <table><tbody><tr><td>Rhombicosidodecahedron</td><td><a href="/facts/Small_dodecicosidodecahedron/AQ4YIlL5">Small dodecicosidodecahedron</a></td><td><a href="/facts/Small_rhombidodecahedron/Uq7u4yGu">Small rhombidodecahedron</a></td></tr><tr><td><a href="/facts/Small_stellated_truncated_dodecahedron/qT9PLHGC">Small stellated truncated dodecahedron</a></td><td><a href="/facts/Compound_of_six_pentagrammic_prisms/pMh7vyGW">Compound of six pentagrammic prisms</a></td><td><a href="/facts/Compound_of_twelve_pentagrammic_prisms/LvvCVwbM">Compound of twelve pentagrammic prisms</a></td></tr></tbody></table> <h2 id="rhombicosidodecahedral-graph">Rhombicosidodecahedral graph</h2> <p>In the <a href="/facts/Mathematics/pxTouaz4">mathematical</a> field of <a href="/facts/Graph_theory/VVPCd4TX">graph theory</a>, a rhombicosidodecahedral graph is the <a href="/facts/1-skeleton/EAt2F7tC">graph of vertices and edges</a> of the rhombicosidodecahedron, one of the <a href="/facts/Archimedean_solid/wYWeHDYT">Archimedean solids</a>. It has 60 <a href="/facts/Vertex_(graph_theory)/WhlWmL3o">vertices</a> and 120 edges, and is a <a href="/facts/Quartic_graph/vcvX5azv">quartic graph</a> <a href="/facts/Archimedean_graph/V3eecXxk">Archimedean graph</a>.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Truncated_rhombicosidodecahedron/nW5z4pv3">Truncated rhombicosidodecahedron</a></li></ul> <h2 id="notes">Notes</h2> <ul><li><a href="/facts/Robert_Williams_(geometer)/1FJDMSJ2">Williams, Robert</a> (1979). <i>The Geometrical Foundation of Natural Structure: A Source Book of Design</i>. Dover Publications, Inc. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-486-23729-X. (Section 3-9)</li> <li>Cromwell, P. (1997). <i>Polyhedra</i>. United Kingdom: Cambridge. pp. 79–86 <i>Archimedean solids</i>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-55432-2.</li> <li><i><a href="/facts/The_Big_Bang_Theory/xxPyPod6">The Big Bang Theory</a> Series 8 Episode 2 - The Junior Professor Solution</i>: 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 <a href="/facts/Chuck_Lorre/7fKLIx8M">Chuck Lorre</a>'s Vanity Card #461 at the end of that episode.</li></ul> <h2 id="external-links">External links</h2> Look up <i>rhombicosidodecahedron</i> in Wiktionary, the free dictionary. <ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a>, "<a href="https://mathworld.wolfram.com/SmallRhombicosidodecahedron.html">Small Rhombicosidodecahedron</a>" ("<a href="http://mathworld.wolfram.com/ArchimedeanSolid.html">Archimedean solid</a>") at <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>. <ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/SmallRhombicosidodecahedralGraph.html">"Small rhombicosidodecahedron graph"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li></ul></li> <li>Klitzing, Richard. <a href="https://bendwavy.org/klitzing/dimensions/polyhedra.htm">"3D convex uniform polyhedra x3o5x - srid"</a>.</li> <li><a href="http://www.dr-mikes-math-games-for-kids.com/polyhedral-nets.html?net=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&name=Rhombicosidodecahedron#applet">Editable printable net of a Rhombicosidodecahedron with interactive 3D view</a></li> <li><a href="http://www.mathconsult.ch/showroom/unipoly/">The Uniform Polyhedra</a></li> <li><a href="http://www.georgehart.com/virtual-polyhedra/vp.html">Virtual Reality Polyhedra</a> The Encyclopedia of Polyhedra</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>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. <a href="/wiki/Johannes_Kepler" target="_blank">/wiki/Johannes_Kepler</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>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) <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Weisstein, Eric W. "Icosahedral group". MathWorld. <a href="/wiki/Eric_W._Weisstein" target="_blank">/wiki/Eric_W._Weisstein</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford University Press, p. 269 <a href="/wiki/Oxford_University_Press" target="_blank">/wiki/Oxford_University_Press</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> </ol>