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Great icosahedron
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<h2 id="construction">Construction</h2> <p>The edge length of a great icosahedron is 7 + 3 5 2 {\displaystyle {\frac {7+3{\sqrt {5}}}{2}}} times that of the original icosahedron. </p> <h2 id="images">Images</h2> <table><tbody><tr><th>Transparent model</th><th><a href="/facts/Density_(polytope)/LLQmbxty">Density</a></th><th><a href="/facts/Stellation_diagram/8b5PuPkH">Stellation diagram</a></th><th><a href="/facts/Net_(polyhedron)/lU8fmGJs">Net</a></th></tr><tr><td>A transparent model of the great icosahedron (See also Animation)</td><td>It has a density of 7, as shown in this cross-section.</td><td>It is a <a href="/facts/Stellation/T7NY6CWv">stellation</a> of the icosahedron, counted by Wenninger as model [W41] and the <a href="/facts/List_of_Wenninger_polyhedron_models/hwApnNpt">16th of 17 stellations of the icosahedron</a> and 7th of 59 stellations by <a href="/facts/Donald_Coxeter/D2l5jWGG">Coxeter</a>.</td><td> × 12Net (surface geometry); twelve isosceles pentagrammic pyramids, arranged like the faces of a dodecahedron. Each pyramid folds up like a fan: the dotted lines fold the opposite direction from the solid lines.</td></tr></tbody></table> <a href="/facts/Spherical_tiling/jaglysFR">Spherical tiling</a><table><tbody><tr><td>This polyhedron represents a spherical tiling with a density of 7. (One spherical triangle face is shown above, outlined in blue, filled in yellow)</td></tr></tbody></table> <h2 id="formulas">Formulas</h2> <p>For a great icosahedron with edge length E (the edge of its dodecahedron core), </p><p> Inradius = E ( 3 3 − 15 ) 4 {\displaystyle {\text{Inradius}}={\frac {{\text{E}}(3{\sqrt {3}}-{\sqrt {15}})}{4}}} </p><p> Midradius = E ( 5 − 1 ) 4 {\displaystyle {\text{Midradius}}={\frac {{\text{E}}({\sqrt {5}}-1)}{4}}} </p><p> Circumradius = E 2 ( 5 − 5 ) 4 {\displaystyle {\text{Circumradius}}={\frac {{\text{E}}{\sqrt {2(5-{\sqrt {5}})}}}{4}}} </p><p> Surface Area = 3 3 ( 5 + 4 5 ) E 2 {\displaystyle {\text{Surface Area}}=3{\sqrt {3}}(5+4{\sqrt {5}}){\text{E}}^{2}} </p><p> Volume = 25 + 9 5 4 E 3 {\displaystyle {\text{Volume}}={\tfrac {25+9{\sqrt {5}}}{4}}{\text{E}}^{3}} </p> <h2 id="as-a-snub">As a snub</h2> <p>The <i>great icosahedron</i> can be constructed as a uniform <a href="/facts/Snub_(geometry)/T8opxlLu">snub</a>, with different colored faces and only <a href="/facts/Tetrahedral_symmetry/hIVlf9pu">tetrahedral symmetry</a>: . This construction can be called a <i>retrosnub tetrahedron</i> or <i>retrosnub tetratetrahedron</i>,<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> similar to the <a href="/facts/Snub_tetrahedron/1U93vmXV">snub tetrahedron</a> symmetry of the <a href="/facts/Icosahedron/wWbpnDYp">icosahedron</a>, as a partial faceting of the <a href="/facts/Truncated_octahedron/GA98b4TH">truncated octahedron</a> (or <i><a href="/facts/Omnitruncation/hmbMrRm6">omnitruncated</a> tetrahedron</i>): . It can also be constructed with 2 colors of triangles and <a href="/facts/Pyritohedral_symmetry/hIVlf9pu">pyritohedral symmetry</a> as, or , and is called a <i>retrosnub octahedron</i>. </p> <table><tbody><tr><th>Tetrahedral</th><th>Pyritohedral</th></tr><tr><td></td><td></td></tr><tr><th></th><th></th></tr></tbody></table> <h2 id="related-polyhedra">Related polyhedra</h2> <p>It shares the same <a href="/facts/Vertex_arrangement/DFgCsP0n">vertex arrangement</a> as the regular convex <a href="/facts/Icosahedron/wWbpnDYp">icosahedron</a>. It also shares the same <a href="/facts/Edge_arrangement/DFgCsP0n">edge arrangement</a> as the <a href="/facts/Small_stellated_dodecahedron/lX1zlo6C">small stellated dodecahedron</a>. </p><p>A truncation operation, repeatedly applied to the great icosahedron, produces a sequence of uniform polyhedra. Truncating edges down to points produces the <a href="/facts/Great_icosidodecahedron/6nsz0IF9">great icosidodecahedron</a> as a rectified great icosahedron. The process completes as a birectification, reducing the original faces down to points, and producing the <a href="/facts/Great_stellated_dodecahedron/7efrgSub">great stellated dodecahedron</a>. </p><p>The <a href="/facts/Truncation_(geometry)/DYb589Je">truncated</a> <i>great stellated dodecahedron</i> is a degenerate polyhedron, with 20 triangular faces from the truncated vertices, and 12 (hidden) doubled up pentagonal faces ({10/2}) as truncations of the original pentagram faces, the latter forming two <a href="/facts/Great_dodecahedron/LWt3Hgg4">great dodecahedra</a> inscribed within and sharing the edges of the icosahedron. </p> <table><tbody><tr><th>Name</th><th><a href="/facts/Great_stellated_dodecahedron/7efrgSub">Greatstellateddodecahedron</a></th><th><a href="/facts/Truncated_great_stellated_dodecahedron/c4zVP1dT">Truncated great stellated dodecahedron</a></th><th><a href="/facts/Great_icosidodecahedron/6nsz0IF9">Greaticosidodecahedron</a></th><th><a href="/facts/Truncated_great_icosahedron/Dakffsbg">Truncatedgreaticosahedron</a></th><th>Greaticosahedron</th></tr><tr><th><a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter-Dynkindiagram</a></th><td></td><td></td><td></td><td></td><td></td></tr><tr><th>Picture</th><td></td><td></td><td></td><td></td><td></td></tr></tbody></table> <ul><li><a href="/facts/Magnus_Wenninger/2QeSYKu2">Wenninger, Magnus</a> (1974). <i>Polyhedron Models</i>. <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-09859-9.</li> <li><a href="/facts/Harold_Scott_MacDonald_Coxeter/D2l5jWGG">Coxeter, Harold Scott MacDonald</a>; Du Val, P.; Flather, H. T.; Petrie, J. F. (1999). <i>The fifty-nine icosahedra</i> (3rd ed.). Tarquin. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-899618-32-3. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0676126">0676126</a>. (1st Edn University of Toronto (1938))</li> <li><a href="/facts/Harold_Scott_MacDonald_Coxeter/D2l5jWGG">H.S.M. Coxeter</a>, <i><a href="/facts/Regular_Polytopes_(book)/jomuRNha">Regular Polytopes</a></i>, (3rd edition, 1973), Dover edition, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-486-61480-8, 3.6 6.2 <i>Stellating the Platonic solids</i>, pp. 96–104</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a>, "<a href="https://mathworld.wolfram.com/GreatIcosahedron.html">Great icosahedron</a>" ("<a href="http://mathworld.wolfram.com/UniformPolyhedron.html">Uniform polyhedron</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/IcosahedronStellations.html">"Fifteen stellations of the icosahedron"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li></ul></li> <li><a href="http://gratrix.net/polyhedra/uniform/summary">Uniform polyhedra and duals</a></li></ul> <table><tbody><tr><td colspan="10">Notable <a href="/facts/The_Fifty-Nine_Icosahedra/syuwcVfg">stellations of the icosahedron</a></td></tr><tr><td><a href="/facts/Platonic_solid/jraREXFD">Regular</a></td><td colspan="3"><a href="/facts/Dual_uniform_polyhedron/eSEanSNt">Uniform duals</a></td><td colspan="3"><a href="/facts/Compound_polyhedron/L13pItNN">Regular compounds</a></td><td><a href="/facts/Kepler%E2%80%93Poinsot_polyhedron/mzUUgBYS">Regular star</a></td><td colspan="2">Others</td></tr><tr><th><a href="/facts/Regular_icosahedron/1U93vmXV">(Convex) icosahedron</a></th><th><a href="/facts/Small_triambic_icosahedron/1LU1h5dU">Small triambic icosahedron</a></th><th><a href="/facts/Medial_triambic_icosahedron/F4txMCXY">Medial triambic icosahedron</a></th><th><a href="/facts/Great_triambic_icosahedron/F4txMCXY">Great triambic icosahedron</a></th><th><a href="/facts/Compound_of_five_octahedra/8M825cd5">Compound of five octahedra</a></th><th><a href="/facts/Compound_of_five_tetrahedra/KKFAnLxh">Compound of five tetrahedra</a></th><th><a href="/facts/Compound_of_ten_tetrahedra/z9LgGAPL">Compound of ten tetrahedra</a></th><th>Great icosahedron</th><th><a href="/facts/Excavated_dodecahedron/lf3pYPTs">Excavated dodecahedron</a></th><th><a href="/facts/Final_stellation_of_the_icosahedron/UMKwLhvr">Final stellation</a></th></tr><tr><td></td><td></td><td colspan="2"></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td colspan="2"></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td colspan="10">The stellation process on the icosahedron creates a number of related <a href="/facts/Polyhedron/UC0p9FTF">polyhedra</a> and <a href="/facts/Compound_polyhedron/L13pItNN">compounds</a> with <a href="/facts/Icosahedral_symmetry/pDlwAy23">icosahedral symmetry</a>.</td></tr></tbody></table> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Klitzing, Richard. "uniform polyhedra Great icosahedron". <a href="https://bendwavy.org/klitzing/dimensions/../incmats/gike.htm" target="_blank">https://bendwavy.org/klitzing/dimensions/../incmats/gike.htm</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> </ol>