Spherical polyhedron

<h2 id="history">History</h2>
<p>During the 10th Century, the Islamic scholar <a href="/facts/Ab%25C5%25AB_al-Waf%25C4%2581%2527_B%25C5%25ABzj%25C4%2581n%25C4%25AB/ynklB2fP">Abū al-Wafā' Būzjānī</a> (Abu'l Wafa) studied spherical polyhedra as part of a work on the geometry needed by craftspeople and architects.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>
</p><p>The work of <a href="/facts/Buckminster_Fuller/fLvx87hc">Buckminster Fuller</a> on <a href="/facts/Geodesic_dome/XmHEPKmK">geodesic domes</a> in the mid 20th century triggered a boom in the study of spherical polyhedra.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> At roughly the same time, <a href="/facts/Harold_Scott_MacDonald_Coxeter/D2l5jWGG">Coxeter</a> used them to enumerate all but one of the <a href="/facts/Uniform_polyhedron/PHnprqRC">uniform polyhedra</a>, through the construction of kaleidoscopes (<a href="/facts/Wythoff_construction/8yTYjp5e">Wythoff construction</a>).<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
</p>
<h2 id="examples">Examples</h2>
<p>All <a href="/facts/Regular_polyhedron/HJ6oGyug">regular polyhedra</a>, <a href="/facts/Semiregular_polyhedron/fHygpjHu">semiregular polyhedra</a>, and their duals can be projected onto the sphere as tilings:
</p>
<table><tbody><tr><th><a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläflisymbol</a></th><th>{p,q}</th><th>t{p,q}</th><th>r{p,q}</th><th>t{q,p}</th><th>{q,p}</th><th>rr{p,q}</th><th>tr{p,q}</th><th>sr{p,q}</th></tr><tr><th><a href="/facts/Vertex_configuration/wWaBS6t0">Vertexconfig.</a></th><th>pq</th><th>q.2p.2p</th><th>p.q.p.q</th><th>p.2q.2q</th><th>qp</th><th>q.4.p.4</th><th>4.2q.2p</th><th>3.3.q.3.p</th></tr><tr><th rowspan="2">Tetrahedralsymmetry(3 3 2)</th><td rowspan="2"><a href="/facts/Tetrahedron/7mkSrl6j">33</a></td><td><a href="/facts/Truncated_tetrahedron/K9ItvmUK">3.6.6</a></td><td><a href="/facts/Octahedron/G38q92xW">3.3.3.3</a></td><td><a href="/facts/Truncated_tetrahedron/K9ItvmUK">3.6.6</a></td><td rowspan="2"><a href="/facts/Tetrahedron/7mkSrl6j">33</a></td><td><a href="/facts/Cuboctahedron/hD7p0Mem">3.4.3.4</a></td><td><a href="/facts/Truncated_octahedron/GA98b4TH">4.6.6</a></td><td><a href="/facts/Icosahedron/wWbpnDYp">3.3.3.3.3</a></td></tr><tr><td><a href="/facts/Triakis_tetrahedron/qz6NNDMU">V3.6.6</a></td><td><a href="/facts/Cube/65lUaAqN">V3.3.3.3</a></td><td><a href="/facts/Triakis_tetrahedron/qz6NNDMU">V3.6.6</a></td><td><a href="/facts/Rhombic_dodecahedron/X3npAlmr">V3.4.3.4</a></td><td><a href="/facts/Tetrakis_hexahedron/RjfM6K49">V4.6.6</a></td><td><a href="/facts/Dodecahedron/7r519eCl">V3.3.3.3.3</a></td></tr><tr><th rowspan="2">Octahedralsymmetry(4 3 2)</th><td rowspan="2"><a href="/facts/Cube/65lUaAqN">43</a></td><td><a href="/facts/Truncated_cube/XVAUG4fD">3.8.8</a></td><td><a href="/facts/Cuboctahedron/hD7p0Mem">3.4.3.4</a></td><td><a href="/facts/Truncated_octahedron/GA98b4TH">4.6.6</a></td><td rowspan="2"><a href="/facts/Octahedron/G38q92xW">34</a></td><td><a href="/facts/Rhombicuboctahedron/TmxUmEnl">3.4.4.4</a></td><td><a href="/facts/Truncated_cuboctahedron/nd2ltxcm">4.6.8</a></td><td><a href="/facts/Snub_cube/pbnr40gk">3.3.3.3.4</a></td></tr><tr><td><a href="/facts/Triakis_octahedron/mJ3s18DL">V3.8.8</a></td><td><a href="/facts/Rhombic_dodecahedron/X3npAlmr">V3.4.3.4</a></td><td><a href="/facts/Tetrakis_hexahedron/RjfM6K49">V4.6.6</a></td><td><a href="/facts/Deltoidal_icositetrahedron/bu6NBY9X">V3.4.4.4</a></td><td><a href="/facts/Disdyakis_dodecahedron/wwAbxHWL">V4.6.8</a></td><td><a href="/facts/Pentagonal_icositetrahedron/2v6EVUPX">V3.3.3.3.4</a></td></tr><tr><th rowspan="2">Icosahedralsymmetry(5 3 2)</th><td rowspan="2"><a href="/facts/Dodecahedron/7r519eCl">53</a></td><td><a href="/facts/Truncated_dodecahedron/PVTAsUWQ">3.10.10</a></td><td><a href="/facts/Icosidodecahedron/P0JluaVA">3.5.3.5</a></td><td><a href="/facts/Truncated_icosahedron/LR8ugaY9">5.6.6</a></td><td rowspan="2"><a href="/facts/Icosahedron/wWbpnDYp">35</a></td><td><a href="/facts/Rhombicosidodecahedron/AaDckLGP">3.4.5.4</a></td><td><a href="/facts/Truncated_icosidodecahedron/d5v9VpFY">4.6.10</a></td><td><a href="/facts/Snub_dodecahedron/oouEPVxD">3.3.3.3.5</a></td></tr><tr><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/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><tr><th>Dihedralexample(p=6)(2 2 6)</th><td><a href="/facts/Dihedron/YkNgovlv">62</a></td><td><a href="/facts/Dihedron/YkNgovlv">2.12.12</a></td><td><a href="/facts/Dihedron/YkNgovlv">2.6.2.6</a></td><td><a href="/facts/Hexagonal_prism/oxYc0nF2">6.4.4</a></td><td><a href="/facts/Hosohedron/t2dyV0bW">26</a></td><td><a href="/facts/Hexagonal_prism/oxYc0nF2">2.4.6.4</a></td><td><a href="/facts/Dodecagonal_prism/50AfF1sz">4.4.12</a></td><td><a href="/facts/Hexagonal_antiprism/bt1W4cs6">3.3.3.6</a></td></tr></tbody></table>

<table><tbody><tr><th><i>n</i></th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th><th>...</th></tr><tr><th><i>n</i>-<a href="/facts/Prism_(geometry)/baIRRkCy">Prism</a>(2 2 p)</th><td></td><td></td><td></td><td></td><td></td><td></td><td>...</td></tr><tr><th><i>n</i>-<a href="/facts/Bipyramid/MkzrCA0W">Bipyramid</a>(2 2 p)</th><td></td><td></td><td></td><td></td><td></td><td></td><td>...</td></tr><tr><th><i>n</i>-<a href="/facts/Antiprism/FgP5jIVd">Antiprism</a></th><td></td><td></td><td></td><td></td><td></td><td></td><td>...</td></tr><tr><th><i>n</i>-<a href="/facts/Trapezohedron/8QpjlnXe">Trapezohedron</a></th><td></td><td></td><td></td><td></td><td></td><td></td><td>...</td></tr></tbody></table>
<h2 id="improper-cases">Improper cases</h2>
<p>Spherical tilings allow cases that polyhedra do not, namely <a href="/facts/Hosohedra/t2dyV0bW">hosohedra</a>: figures as {2,n}, and <a href="/facts/Dihedra/YkNgovlv">dihedra</a>: figures as {n,2}. Generally, regular hosohedra and regular dihedra are used.
</p>
Family of regular hosohedra · *<i>n</i>22 symmetry mutations of regular hosohedral tilings: <i>nn</i><table><tbody><tr><th>Space</th><th colspan="5">Spherical</th><th></th><th>Euclidean</th></tr><tr><th>Tiling  name</th><th><a href="/facts/Henagonal_hosohedron/gx05DD40">Henagonal  hosohedron</a></th><th><a href="/facts/Digonal_hosohedron/t2dyV0bW">Digonal  hosohedron</a></th><th><a href="/facts/Trigonal_hosohedron/t2dyV0bW">Trigonal  hosohedron</a></th><th><a href="/facts/Square_hosohedron/t2dyV0bW">Square  hosohedron</a></th><th><a href="/facts/Pentagonal_hosohedron/t2dyV0bW">Pentagonal  hosohedron</a></th><th>...</th><th><a href="/facts/Apeirogonal_hosohedron/nT5IPbED">Apeirogonal  hosohedron</a></th></tr><tr><th>Tiling  image</th><td></td><td></td><td></td><td></td><td></td><td>...</td><td></td></tr><tr><th><a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli  symbol</a></th><td>{2,1}</td><td>{2,2}</td><td>{2,3}</td><td>{2,4}</td><td>{2,5}</td><td>...</td><td>{2,∞}</td></tr><tr><th><a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter  diagram</a></th><td></td><td></td><td></td><td></td><td></td><td>...</td><td></td></tr><tr><th>Faces and edges</th><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>...</td><td>∞</td></tr><tr><th>Vertices</th><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>...</td><td>2</td></tr><tr><th><a href="/facts/Vertex_configuration/wWaBS6t0">Vertex  config.</a></th><td>2</td><td>2.2</td><td>23</td><td>24</td><td>25</td><td>...</td><td>2∞</td></tr></tbody></table>
Family of regular dihedra · *<i>n</i>22 symmetry mutations of regular dihedral tilings: <i>nn</i><table><tbody><tr><th>Space</th><th colspan="5">Spherical</th><th></th><th>Euclidean</th></tr><tr><th>Tiling  name</th><th><a href="/facts/Monogonal_dihedron/gx05DD40">Monogonal  dihedron</a></th><th><a href="/facts/Digonal_dihedron/YkNgovlv">Digonal  dihedron</a></th><th><a href="/facts/Trigonal_dihedron/YkNgovlv">Trigonal  dihedron</a></th><th><a href="/facts/Square_dihedron/t2dyV0bW">Square  dihedron</a></th><th><a href="/facts/Pentagonal_dihedron/YkNgovlv">Pentagonal  dihedron</a></th><th>...</th><th><a href="/facts/Apeirogonal_dihedron/LpbWIQN9">Apeirogonal  dihedron</a></th></tr><tr><th>Tiling  image</th><td></td><td></td><td></td><td></td><td></td><td>...</td><td></td></tr><tr><th><a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli  symbol</a></th><td>{1,2}</td><td>{2,2}</td><td>{3,2}</td><td>{4,2}</td><td>{5,2}</td><td>...</td><td>{∞,2}</td></tr><tr><th><a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter  diagram</a></th><td></td><td></td><td></td><td></td><td></td><td>...</td><td></td></tr><tr><th>Faces</th><td>2 <a href="/facts/Monogon/gx05DD40">{1}</a></td><td>2 <a href="/facts/Digon/vur5IQ5d">{2}</a></td><td>2 <a href="/facts/Equilateral_triangle/MJomsYDS">{3}</a></td><td>2 <a href="/facts/Square/JHJeMvmL">{4}</a></td><td>2 <a href="/facts/Pentagon/jeHEARHH">{5}</a></td><td>...</td><td>2 <a href="/facts/Apeirogon/r0V9hleb">{∞}</a></td></tr><tr><th>Edges and  vertices</th><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>...</td><td>∞</td></tr><tr><th><a href="/facts/Vertex_configuration/wWaBS6t0">Vertex  config.</a></th><td>1.1</td><td>2.2</td><td>3.3</td><td>4.4</td><td>5.5</td><td>...</td><td>∞.∞</td></tr></tbody></table>
<h2 id="relation-to-tilings-of-the-projective-plane">Relation to tilings of the projective plane</h2>
<p>Spherical polyhedra having at least one <a href="/facts/Inversive_geometry/VqFmbfDI">inversive symmetry</a> are related to <a href="/facts/Projective_polyhedra/dJGo1hGD">projective polyhedra</a><a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> (tessellations of the <a href="/facts/Real_projective_plane/gGVd0ctY">real projective plane</a>) – just as the sphere has a 2-to-1 <a href="/facts/Covering_map/bDRGasl7">covering map</a> of the projective plane, projective polyhedra correspond under 2-fold cover to spherical polyhedra that are symmetric under <a href="/facts/Reflection_through_the_origin/UzswAm8N">reflection through the origin</a>.
</p><p>The best-known examples of projective polyhedra are the regular projective polyhedra, the quotients of the <a href="/facts/Centrally_symmetric/UzswAm8N">centrally symmetric</a> <a href="/facts/Platonic_solid/jraREXFD">Platonic solids</a>, as well as two infinite classes of even <a href="/facts/Dihedra/YkNgovlv">dihedra</a> and <a href="/facts/Hosohedra/t2dyV0bW">hosohedra</a>:<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>
</p>
<ul><li><a href="/facts/Hemicube_(geometry)/l3BnMxDt">Hemi-cube</a>, {4,3}/2</li>
<li><a href="/facts/Hemi-octahedron/HbRkd20f">Hemi-octahedron</a>, {3,4}/2</li>
<li><a href="/facts/Hemi-dodecahedron/44VmMjGY">Hemi-dodecahedron</a>, {5,3}/2</li>
<li><a href="/facts/Hemi-icosahedron/1PDWTJ1M">Hemi-icosahedron</a>, {3,5}/2</li>
<li>Hemi-dihedron, {2p,2}/2, p≥1</li>
<li>Hemi-hosohedron, {2,2p}/2, p≥1</li></ul>
<h2 id="see-also">See also</h2>

Wikimedia Commons has media related to Spherical polyhedra.

<ul><li><a href="/facts/Spherical_geometry/WGLQb1Ys">Spherical geometry</a></li>
<li><a href="/facts/Spherical_trigonometry/3G3Fhk2h">Spherical trigonometry</a></li>
<li><a href="/facts/Polyhedron/UC0p9FTF">Polyhedron</a></li>
<li><a href="/facts/Projective_polyhedron/dJGo1hGD">Projective polyhedron</a></li>
<li><a href="/facts/Toroidal_polyhedron/g6SS84k2">Toroidal polyhedron</a></li>
<li><a href="/facts/Conway_polyhedron_notation/MVmYMqID">Conway polyhedron notation</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Sarhangi, Reza (September 2008). "Illustrating Abu al-Wafā' Būzjānī: Flat images, spherical constructions". Iranian Studies. 41 (4): 511–523. doi:10.1080/00210860802246184. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Popko, Edward S. (2012). Divided Spheres: Geodesics and the Orderly Subdivision of the Sphere. CRC Press. p. xix. ISBN 978-1-4665-0430-1. Buckminster Fuller's invention of the geodesic dome was the biggest stimulus for spherical subdivision research and development. <a href="978-1-4665-0430-1" target="_blank">978-1-4665-0430-1</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Coxeter, H.S.M.; Longuet-Higgins, M.S.; Miller, J.C.P. (1954). "Uniform polyhedra". Phil. Trans. 246 A (916): 401–50. JSTOR 91532. <a href="/wiki/Harold_Scott_MacDonald_Coxeter" target="_blank">/wiki/Harold_Scott_MacDonald_Coxeter</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>McMullen, Peter; Schulte, Egon (2002). "6C. Projective Regular Polytopes". Abstract Regular Polytopes. Cambridge University Press. pp. 162–5. ISBN 0-521-81496-0. <a href="0-521-81496-0" target="_blank">0-521-81496-0</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Coxeter, H.S.M. (1969). "§21.3 Regular maps'". Introduction to Geometry (2nd ed.). Wiley. pp. 386–8. ISBN 978-0-471-50458-0. MR 0123930. <a href="978-0-471-50458-0" target="_blank">978-0-471-50458-0</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
</ol>

Spherical polyhedron open-in-new

Spherical polyhedron