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Great icosahedron
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<table><tbody><tr><th colspan="2">Great icosahedron</th></tr><tr><td colspan="2"></td></tr><tr><td>Type</td><td><a href="/facts/Kepler%E2%80%93Poinsot_polyhedron/mzUUgBYS">Kepler–Poinsot polyhedron</a></td></tr><tr><td><a href="/facts/Stellation/T7NY6CWv">Stellation</a> core</td><td><a href="/facts/List_of_Wenninger_polyhedron_models/hwApnNpt">icosahedron</a></td></tr><tr><td><a href="/facts/Euler_characteristic/JCjzWUr2">Elements</a></td><td><i>F</i> = 20, <i>E</i> = 30<i>V</i> = 12 (χ = 2)</td></tr><tr><td>Faces by sides</td><td>20{3}</td></tr><tr><td><a href="/facts/Schl%C3%A4fli_symbol/jNQR77UU">Schläfli symbol</a></td><td>{3,5⁄2}</td></tr><tr><td><a href="/facts/Face_configuration/wWaBS6t0">Face configuration</a></td><td>V(53)/2</td></tr><tr><td><a href="/facts/Wythoff_symbol/ECD0Wl5x">Wythoff symbol</a></td><td>5⁄2 | 2 3</td></tr><tr><td><a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter diagram</a></td><td></td></tr><tr><td><a href="/facts/List_of_spherical_symmetry_groups/3X3YcHoQ">Symmetry group</a></td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23">Ih</a>, H3, [5,3], (*532)</td></tr><tr><td><a href="/facts/List_of_uniform_polyhedra/robWbdeH">References</a></td><td><a href="/facts/Uniform_polyhedron/PHnprqRC">U</a>53, <a href="/facts/Harold_Scott_MacDonald_Coxeter/D2l5jWGG">C</a>69, <a href="/facts/List_of_Wenninger_polyhedron_models/hwApnNpt">W</a>41</td></tr><tr><td>Properties</td><td><a href="/facts/Regular_polyhedron/HJ6oGyug">Regular</a> <a href="/facts/Nonconvex_polyhedron/XyeJWj3d">nonconvex</a> <a href="/facts/Deltahedron/qDjkasx9">deltahedron</a></td></tr><tr><td>(35)/2(<a href="/facts/Vertex_figure/3u2ySK0C">Vertex figure</a>)</td><td><a href="/facts/Great_stellated_dodecahedron/7efrgSub">Great stellated dodecahedron</a>(<a href="/facts/Dual_polyhedron/rkIHbmAf">dual polyhedron</a>)</td></tr></tbody></table> <p>In <a href="/facts/Geometry/wTQf5ffQ">geometry</a>, the great icosahedron is one of four <a href="/facts/Kepler%E2%80%93Poinsot_polyhedra/mzUUgBYS">Kepler–Poinsot polyhedra</a> (<a href="/facts/Nonconvex/2pRHcRgC">nonconvex</a> <a href="/facts/List_of_regular_polytopes/Kng25Ymq">regular polyhedra</a>), with <a href="/facts/Schl%C3%A4fli_symbol/jNQR77UU">Schläfli symbol</a> {3,5⁄2} and <a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter-Dynkin diagram</a> of . It is composed of 20 intersecting triangular faces, having five triangles meeting at each vertex in a <a href="/facts/Pentagram/cN0ke8tp">pentagrammic</a> sequence. </p><p>The great icosahedron can be constructed analogously to the pentagram, its two-dimensional analogue, via the extension of the (<i>n</i>–1)-dimensional <a href="/facts/Simplex/0FCfNRN8">simplex</a> faces of the core n-polytope (equilateral triangles for the great icosahedron, and <a href="/facts/Line_segment/V8E0M4qk">line segments</a> for the pentagram) until the figure regains regular faces. The <a href="/facts/Grand_600-cell/LqttGCQw">grand 600-cell</a> can be seen as its four-dimensional analogue using the same process. </p>