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Rectified 5-cell
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<h2 id="wythoff-construction">Wythoff construction</h2> <p>Seen in a <a href="/facts/Configuration_(polytope)/GkelT5uk">configuration matrix</a>, all incidence counts between elements are shown. The diagonal <a href="/facts/F-vector/oVFESPjJ">f-vector</a> numbers are derived through the <a href="/facts/Wythoff_construction/8yTYjp5e">Wythoff construction</a>, dividing the full group order of a subgroup order by removing one mirror at a time.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p> <table><tbody><tr><th>A4</th><th></th><th><a href="/facts/K-face/T2E6BTU4"><i>k</i>-face</a></th><th>fk</th><th>f0</th><th colspan="1">f1</th><th colspan="2">f2</th><th colspan="2">f3</th><th><a href="/facts/Vertex_figure/3u2ySK0C"><i>k</i>-figure</a></th><th>Notes</th></tr><tr><td>A1A2</td><td></td><td>( )</td><th>f0</th><td>10</td><td>6</td><td>3</td><td>6</td><td>3</td><td>2</td><td><a href="/facts/Triangular_prism/Yjp4Kw9M">{3}x{ }</a></td><td>A4/A1A2 = 5!/3!/2 = 10</td></tr><tr><td>A1A1</td><td></td><td>{ }</td><th>f1</th><td>2</td><td>30</td><td>1</td><td>2</td><td>2</td><td>1</td><td><a href="/facts/Isosceles_triangle/rvLe3yqK">{ }v( )</a></td><td>A4/A1A1 = 5!/2/2 = 30</td></tr><tr><td>A2A1</td><td></td><td rowspan="2"><a href="/facts/Triangle/cRXUwsMn">{3}</a></td><th rowspan="2">f2</th><td>3</td><td>3</td><td>10</td><td>*</td><td>2</td><td>0</td><td rowspan="2">{ }</td><td>A4/A2A1 = 5!/3!/2 = 10</td></tr><tr><td>A2</td><td></td><td>3</td><td>3</td><td>*</td><td>20</td><td>1</td><td>1</td><td>A4/A2 = 5!/3! = 20</td></tr><tr><td>A3</td><td></td><td><a href="/facts/Octahedron/G38q92xW">r{3,3}</a></td><th rowspan="2">f3</th><td>6</td><td>12</td><td>4</td><td>4</td><td>5</td><td>*</td><td rowspan="2">( )</td><td rowspan="2">A4/A3 = 5!/4! = 5</td></tr><tr><td>A3</td><td></td><td><a href="/facts/Tetrahedron/7mkSrl6j">{3,3}</a></td><td>4</td><td>6</td><td>0</td><td>4</td><td>*</td><td>5</td></tr></tbody></table> <h2 id="structure">Structure</h2> <p>Together with the simplex and <a href="/facts/24-cell/tfoGnlAD">24-cell</a>, this shape and its <a href="/facts/Dual_polyhedron/rkIHbmAf">dual</a> (a polytope with ten vertices and ten <a href="/facts/Triangular_bipyramid/fU4kIBTV">triangular bipyramid</a> facets) was one of the first 2-simple 2-simplicial 4-polytopes known. This means that all of its two-dimensional faces, and all of the two-dimensional faces of its dual, are triangles. In 1997, Tom Braden found another dual pair of examples, by gluing two rectified 5-cells together; since then, infinitely many 2-simple 2-simplicial polytopes have been constructed.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a><a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p> <h2 id="semiregular-polytope">Semiregular polytope</h2> <p>It is one of three <a href="/facts/Semiregular_4-polytopes/Xw9TURVR">semiregular 4-polytopes</a> made of two or more cells which are <a href="/facts/Platonic_solid/jraREXFD">Platonic solids</a>, discovered by <a href="/facts/Thorold_Gosset/3wvHtGPn">Thorold Gosset</a> in his 1900 paper. He called it a <i>tetroctahedric</i> for being made of <a href="/facts/Tetrahedron/7mkSrl6j">tetrahedron</a> and <a href="/facts/Octahedron/G38q92xW">octahedron</a> cells.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> </p><p><a href="/facts/Emanuel_Lodewijk_Elte/ahH8vuy6">E. L. Elte</a> identified it in 1912 as a semiregular polytope, labeling it as tC5. </p> <h2 id="alternate-names">Alternate names</h2> <ul><li>Tetroctahedric (Thorold Gosset)</li> <li>Dispentachoron</li> <li>Rectified 5-cell (<a href="/facts/Norman_Johnson_(mathematician)/bQtL9w56">Norman W. Johnson</a>)</li> <li>Rectified <a href="/facts/Simplex/0FCfNRN8">4-simplex</a></li> <li>Fully truncated 4-simplex</li> <li>Rectified pentachoron (Acronym: rap) (Jonathan Bowers)</li> <li>Ambopentachoron (Neil Sloane & <a href="/facts/John_Horton_Conway/g3PA1zoK">John Horton Conway</a>)</li> <li>(5,2)-<a href="/facts/Hypersimplex/9S1gMgj0">hypersimplex</a> (the convex hull of five-dimensional (0,1)-vectors with exactly two ones)</li></ul> <h2 id="images">Images</h2> <a href="/facts/Orthographic_projection/L3Cv9kYD">orthographic projections</a><table><tbody><tr><th>Ak<a href="/facts/Coxeter_plane/4B80wH8V">Coxeter plane</a></th><th>A4</th><th>A3</th><th>A2</th></tr><tr><th>Graph</th><td></td><td></td><td></td></tr><tr><th><a href="/facts/Dihedral_symmetry/1PUlRh4M">Dihedral symmetry</a></th><td>[5]</td><td>[4]</td><td>[3]</td></tr></tbody></table> <table><tbody><tr><td><a href="/facts/Stereographic_projection/oVyMrWqR">stereographic projection</a>(centered on <a href="/facts/Octahedron/G38q92xW">octahedron</a>)</td><td><a href="/facts/Net_(polytope)/lU8fmGJs">Net (polytope)</a></td></tr><tr><td></td><td><a href="/facts/Tetrahedron/7mkSrl6j">Tetrahedron</a>-centered perspective projection into 3D space, with nearest tetrahedron to the 4D viewpoint rendered in red, and the 4 surrounding octahedra in green. Cells lying on the far side of the polytope have been culled for clarity (although they can be discerned from the edge outlines). The rotation is only of the 3D projection image, in order to show its structure, not a rotation in 4D space.</td></tr></tbody></table> <h2 id="coordinates">Coordinates</h2> <p>The <a href="/facts/Cartesian_coordinate/lkTqvMmB">Cartesian coordinates</a> of the vertices of an origin-centered rectified 5-cell having edge length 2 are: </p> <table><tbody><tr><th colspan="2">Coordinates</th></tr><tr><td> ( 2 5 , 2 6 , 2 3 , 0 ) {\displaystyle \left({\sqrt {\frac {2}{5}}},\ {\frac {2}{\sqrt {6}}},\ {\frac {2}{\sqrt {3}}},\ 0\right)} ( 2 5 , 2 6 , − 1 3 , ± 1 ) {\displaystyle \left({\sqrt {\frac {2}{5}}},\ {\frac {2}{\sqrt {6}}},\ {\frac {-1}{\sqrt {3}}},\ \pm 1\right)} ( 2 5 , − 2 6 , 1 3 , ± 1 ) {\displaystyle \left({\sqrt {\frac {2}{5}}},\ {\frac {-2}{\sqrt {6}}},\ {\frac {1}{\sqrt {3}}},\ \pm 1\right)} ( 2 5 , − 2 6 , − 2 3 , 0 ) {\displaystyle \left({\sqrt {\frac {2}{5}}},\ {\frac {-2}{\sqrt {6}}},\ {\frac {-2}{\sqrt {3}}},\ 0\right)} </td><td> ( − 3 10 , 1 6 , 1 3 , ± 1 ) {\displaystyle \left({\frac {-3}{\sqrt {10}}},\ {\frac {1}{\sqrt {6}}},\ {\frac {1}{\sqrt {3}}},\ \pm 1\right)} ( − 3 10 , 1 6 , − 2 3 , 0 ) {\displaystyle \left({\frac {-3}{\sqrt {10}}},\ {\frac {1}{\sqrt {6}}},\ {\frac {-2}{\sqrt {3}}},\ 0\right)} ( − 3 10 , − 3 2 , 0 , 0 ) {\displaystyle \left({\frac {-3}{\sqrt {10}}},\ -{\sqrt {\frac {3}{2}}},\ 0,\ 0\right)} </td></tr></tbody></table> <p>More simply, the vertices of the <i>rectified 5-cell</i> can be positioned on a <a href="/facts/Hyperplane/2s2ycYYd">hyperplane</a> in 5-space as permutations of (0,0,0,1,1) <i>or</i> (0,0,1,1,1). These construction can be seen as positive <a href="/facts/Orthant/3bkJde4G">orthant</a> facets of the <a href="/facts/Rectified_pentacross/BVp334zE">rectified pentacross</a> or <a href="/facts/Birectified_penteract/81cgLLCG">birectified penteract</a> respectively. </p> <h2 id="related-4-polytopes">Related 4-polytopes</h2> <p>The rectified 5-cell is the <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</a> of the <a href="/facts/5-demicube/zvCeNbU2">5-demicube</a>, and the <a href="/facts/Edge_figure/3u2ySK0C">edge figure</a> of the uniform <a href="/facts/2_21_polytope/nF3nYaQk">221 polytope</a>. </p> <h3>Compound of the rectified 5-cell and its dual</h3> <p>The convex hull the rectified 5-cell and its dual (of the same long radius) is a nonuniform polychoron composed of 30 cells: 10 <a href="/facts/Tetrahedron/7mkSrl6j">tetrahedra</a>, 20 <a href="/facts/Octahedron/G38q92xW">octahedra</a> (as triangular antiprisms), and 20 vertices. Its vertex figure is a <a href="/facts/Triangular_bifrustum/aSE9cekt">triangular bifrustum</a>. </p> <h3>Pentachoron polytopes</h3> <p>The rectified 5-cell is one of 9 <a href="/facts/Uniform_4-polytope/NHxomMxC">Uniform 4-polytopes</a> constructed from the [3,3,3] <a href="/facts/Coxeter_group/bgzHFm1Q">Coxeter group</a>. </p> <table><tbody><tr><th>Name</th><th><a href="/facts/5-cell/Rh6lwDcc">5-cell</a></th><th><a href="/facts/Truncated_5-cell/oqF6jkb7">truncated 5-cell</a></th><th>rectified 5-cell</th><th><a href="/facts/Cantellated_5-cell/pm158bSU">cantellated 5-cell</a></th><th><a href="/facts/Bitruncated_5-cell/oqF6jkb7">bitruncated 5-cell</a></th><th><a href="/facts/Cantitruncated_5-cell/pm158bSU">cantitruncated 5-cell</a></th><th><a href="/facts/Runcinated_5-cell/7A0II7du">runcinated 5-cell</a></th><th><a href="/facts/Runcitruncated_5-cell/7A0II7du">runcitruncated 5-cell</a></th><th><a href="/facts/Omnitruncated_5-cell/7A0II7du">omnitruncated 5-cell</a></th></tr><tr><th><a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläflisymbol</a></th><th>{3,3,3}3r{3,3,3}</th><th>t{3,3,3}3t{3,3,3}</th><th>r{3,3,3}2r{3,3,3}</th><th>rr{3,3,3}r2r{3,3,3}</th><th>2t{3,3,3}</th><th>tr{3,3,3}t2r{3,3,3}</th><th>t0,3{3,3,3}</th><th>t0,1,3{3,3,3}t0,2,3{3,3,3}</th><th>t0,1,2,3{3,3,3}</th></tr><tr><th><a href="/facts/Coxeter%25E2%2580%2593Dynkin_diagram/Ik6LmXJB">Coxeterdiagram</a></th><th></th><th></th><th></th><th></th><th></th><th></th><th></th><th></th><th></th></tr><tr><th><a href="/facts/Schlegel_diagram/qwsDv1H8">Schlegeldiagram</a></th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>A4<a href="/facts/Coxeter_plane/4B80wH8V">Coxeter plane</a>Graph</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>A3 Coxeter planeGraph</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>A2 Coxeter planeGraph</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table> <h3>Semiregular polytopes</h3> <p>The rectified 5-cell is second in a dimensional series of <a href="/facts/Uniform_k21_polytope/vv3lWMYd">semiregular polytopes</a>. Each progressive <a href="/facts/Uniform_polytope/nFcNAzLy">uniform polytope</a> is constructed as the <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</a> of the previous polytope. <a href="/facts/Thorold_Gosset/3wvHtGPn">Thorold Gosset</a> identified this series in 1900 as containing all <a href="/facts/Regular_polytope/RYTnEF6Q">regular polytope</a> facets, containing all <a href="/facts/Simplex/0FCfNRN8">simplexes</a> and <a href="/facts/Orthoplex/erk0byF6">orthoplexes</a> (<a href="/facts/Tetrahedron/7mkSrl6j">tetrahedrons</a> and <a href="/facts/Octahedron/G38q92xW">octahedrons</a> in the case of the rectified 5-cell). The <a href="/facts/Coxeter_symbol/PfHvVPBg">Coxeter symbol</a> for the rectified 5-cell is 021. </p> <table><tbody><tr><th colspan="12"><a href="/facts/Uniform_k_21_polytope/vv3lWMYd">k21 figures</a> in n dimensions</th></tr><tr><th>Space</th><th colspan="6">Finite</th><th>Euclidean</th><th>Hyperbolic</th></tr><tr><th><a href="/facts/En_(Lie_algebra)/NUSj3xlQ">En</a></th><th><a href="/facts/Three-dimensional_space/KAqKWUbS">3</a></th><th><a href="/facts/Four-dimensional_space/vFQd0M7T">4</a></th><th><a href="/facts/Five-dimensional_space/bcrNVzhj">5</a></th><th><a href="/facts/Six-dimensional_space/Lh23NhTa">6</a></th><th><a href="/facts/Seven-dimensional_space/Cw6WuFgL">7</a></th><th><a href="/facts/Eight-dimensional_space/KFwQyKy3">8</a></th><th><a href="/facts/Nine-dimensional_space/a3RGcxrx">9</a></th><th><a href="/facts/Ten-dimensional_space/HUJFIPYH">10</a></th></tr><tr><th><a href="/facts/Coxeter_group/bgzHFm1Q">Coxetergroup</a></th><td>E3=A2A1</td><td>E4=A4</td><td>E5=D5</td><td><a href="/facts/E6_(mathematics)/U9YEMtWj">E6</a></td><td><a href="/facts/E7_(mathematics)/KjFez9PW">E7</a></td><td><a href="/facts/E8_(mathematics)/NEaKT71H">E8</a></td><td>E9 = E ~ 8 {\displaystyle {\tilde {E}}_{8}} = E8+</td><td>E10 = T ¯ 8 {\displaystyle {\bar {T}}_{8}} = E8++</td></tr><tr><th><a href="/facts/Coxeter%25E2%2580%2593Dynkin_diagram/Ik6LmXJB">Coxeterdiagram</a></th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th><a href="/facts/Coxeter_notation/ArUw6goe">Symmetry</a></th><td>[3−1,2,1]</td><td>[30,2,1]</td><td>[31,2,1]</td><td>[32,2,1]</td><td>[33,2,1]</td><td>[34,2,1]</td><td>[35,2,1]</td><td>[36,2,1]</td></tr><tr><th><a href="/facts/Group_order/Cl3tKGFg">Order</a></th><td>12</td><td>120</td><td>1,920</td><td>51,840</td><td>2,903,040</td><td>696,729,600</td><td colspan="2">∞</td></tr><tr><th>Graph</th><td></td><td></td><td></td><td></td><td></td><td></td><td>-</td><td>-</td></tr><tr><th>Name</th><td><a href="/facts/Triangular_prism/Yjp4Kw9M">−121</a></td><td>021</td><td><a href="/facts/Demipenteract/zvCeNbU2">121</a></td><td><a href="/facts/2_21_polytope/nF3nYaQk">221</a></td><td><a href="/facts/3_21_polytope/AEUzwA1A">321</a></td><td><a href="/facts/4_21_polytope/QFGglCkk">421</a></td><td><a href="/facts/5_21_honeycomb/f971VRma">521</a></td><td><a href="/facts/6_21_honeycomb/m7yEza2C">621</a></td></tr></tbody></table> <h3>Isotopic polytopes</h3> Isotopic uniform truncated simplices<table><tbody><tr><th>Dim.</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th><th>8</th></tr><tr><th>Name<a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter</a></th><td><a href="/facts/Hexagon/DPzuCzWE">Hexagon</a> = t{3} = {6}</td><td><a href="/facts/Octahedron/G38q92xW">Octahedron</a> = r{3,3} = {31,1} = {3,4} { 3 3 } {\displaystyle \left\{{\begin{array}{l}3\\3\end{array}}\right\}} </td><td>Decachoron<a href="/facts/Bitruncated_5-cell/oqF6jkb7">2t{33}</a></td><td>Dodecateron<a href="/facts/Birectified_5-simplex/H5kx0rFW">2r{34}</a> = {32,2} { 3 , 3 3 , 3 } {\displaystyle \left\{{\begin{array}{l}3,3\\3,3\end{array}}\right\}} </td><td>Tetradecapeton<a href="/facts/Tritruncated_6-simplex/YhGJGUuI">3t{35}</a></td><td>Hexadecaexon<a href="/facts/Trirectified_7-simplex/kwsH3Rdg">3r{36}</a> = {33,3} { 3 , 3 , 3 3 , 3 , 3 } {\displaystyle \left\{{\begin{array}{l}3,3,3\\3,3,3\end{array}}\right\}} </td><td>Octadecazetton<a href="/facts/Quadritruncated_8-simplex/aQ2TjRuJ">4t{37}</a></td></tr><tr><th>Images</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>Vertex figure</th><td>( )∨( )</td><td><a href="/facts/Square/JHJeMvmL">{ }×{ }</a></td><td><a href="/facts/Tetragonal_disphenoid/7rU7jc7U">{ }∨{ }</a></td><td><a href="/facts/3-3_duoprism/B1dhsgrg">{3}×{3}</a></td><td><a href="/facts/5-simplex/KIfBFfhl">{3}∨{3}</a></td><td>{3,3}×{3,3}</td><td><a href="/facts/6-simplex/mInfSIeH">{3,3}∨{3,3}</a></td></tr><tr><th>Facets</th><td></td><td><a href="/facts/Triangle/cRXUwsMn">{3}</a> </td><td><a href="/facts/Truncated_tetrahedron/K9ItvmUK">t{3,3}</a> </td><td>r{3,3,3} </td><td><a href="/facts/Bitruncated_5-simplex/GFrpum4c">2t{3,3,3,3}</a> </td><td><a href="/facts/Birectified_6-simplex/b3R5YeFN">2r{3,3,3,3,3}</a> </td><td><a href="/facts/Tritruncated_7-simplex/aw2XKIxf">3t{3,3,3,3,3,3}</a> </td></tr><tr><th>Asintersectingdual<a href="/facts/Simplex/0FCfNRN8">simplexes</a></th><td> ∩ </td><td> ∩ </td><td> ∩ </td><td> ∩ </td><td> ∩ </td><td> ∩ </td><td> ∩ </td></tr></tbody></table> <h2 id="notes">Notes</h2> <ul><li><a href="/facts/Thorold_Gosset/3wvHtGPn">T. Gosset</a>: <i>On the Regular and Semi-Regular Figures in Space of n Dimensions</i>, <a href="/facts/Messenger_of_Mathematics/anb7QjuX">Messenger of Mathematics</a>, Macmillan, 1900</li> <li><a href="/facts/John_Horton_Conway/g3PA1zoK">J.H. Conway</a> and <a href="/facts/Michael_Guy_(computer_scientist)/TjWdLhA3">M.J.T. Guy</a>: <i>Four-Dimensional Archimedean Polytopes</i>, Proceedings of the Colloquium on Convexity at Copenhagen, page 38 and 39, 1965</li> <li><a href="/facts/Harold_Scott_MacDonald_Coxeter/D2l5jWGG">H.S.M. Coxeter</a>: <ul><li>H.S.M. Coxeter, <i>Regular Polytopes</i>, 3rd Edition, Dover New York, 1973</li> <li>Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-471-01003-6 <a href="http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html">[1]</a> <ul><li>(Paper 22) H.S.M. Coxeter, <i>Regular and Semi Regular Polytopes I</i>, [Math. Zeit. 46 (1940) 380–407, MR 2,10]</li> <li>(Paper 23) H.S.M. Coxeter, <i>Regular and Semi-Regular Polytopes II</i>, [Math. Zeit. 188 (1985) 559-591]</li> <li>(Paper 24) H.S.M. Coxeter, <i>Regular and Semi-Regular Polytopes III</i>, [Math. Zeit. 200 (1988) 3-45]</li></ul></li></ul></li> <li><a href="/facts/Norman_Johnson_(mathematician)/bQtL9w56">Norman Johnson</a> <i>Uniform Polytopes</i>, Manuscript (1991) <ul><li>N.W. Johnson: <i>The Theory of Uniform Polytopes and Honeycombs</i>, Ph.D. (1966)</li></ul></li> <li><a href="/facts/John_Horton_Conway/g3PA1zoK">John H. Conway</a>, Heidi Burgiel, <a href="/facts/Chaim_Goodman-Strauss/4IKWbx8w">Chaim Goodman-Strauss</a>, <i>The Symmetries of Things</i> 2008, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-56881-220-5 (Chapter 26)</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="http://www.polytope.de/nr03.html">Rectified 5-cell</a> - data and images <ul><li><a href="https://web.archive.org/web/20070204075028/members.aol.com/Polycell/section1.html">1. Convex uniform polychora based on the pentachoron - Model 2</a>, George Olshevsky.</li></ul></li> <li>Klitzing, Richard. <a href="https://bendwavy.org/klitzing/dimensions/polychora.htm">"4D uniform polytopes (polychora) x3o3o3o - rap"</a>.</li></ul> <table><tbody><tr><th colspan="13"><ul><li>v</li><li>t</li><li>e</li></ul>Fundamental convex <a href="/facts/Regular_polytope/RYTnEF6Q">regular</a> and <a href="/facts/Uniform_polytope/nFcNAzLy">uniform polytopes</a> in dimensions 2–10</th></tr><tr><th><a href="/facts/Coxeter_group/bgzHFm1Q">Family</a></th><td><a href="/facts/Simple_Lie_group/LkHTNYU9"><i>A</i><i>n</i></a></td><td><a href="/facts/Simple_Lie_group/LkHTNYU9"><i>B</i><i>n</i></a></td><td><i>I</i>2(p) / <a href="/facts/Simple_Lie_group/LkHTNYU9"><i>D</i><i>n</i></a></td><td><a href="/facts/E6_(mathematics)/U9YEMtWj"><i>E</i>6</a> / <a href="/facts/E7_(mathematics)/KjFez9PW"><i>E</i>7</a> / <a href="/facts/E8_(mathematics)/NEaKT71H"><i>E</i>8</a> / <a href="/facts/F4_(mathematics)/30H8CQ8v"><i>F</i>4</a> / <a href="/facts/G2_(mathematics)/Bka4pKjQ"><i>G</i>2</a></td><td><a href="/facts/H4_(mathematics)/bgzHFm1Q">Hn</a></td></tr><tr><th><a href="/facts/Regular_polygon/ws89Nhpt">Regular polygon</a></th><td><a href="/facts/Equilateral_triangle/MJomsYDS">Triangle</a></td><td><a href="/facts/Square/JHJeMvmL">Square</a></td><td><a href="/facts/Regular_polygon/ws89Nhpt">p-gon</a></td><td><a href="/facts/Hexagon/DPzuCzWE">Hexagon</a></td><td><a href="/facts/Pentagon/jeHEARHH">Pentagon</a></td></tr><tr><th><a href="/facts/Uniform_polyhedron/PHnprqRC">Uniform polyhedron</a></th><td><a href="/facts/Tetrahedron/7mkSrl6j">Tetrahedron</a></td><td><a href="/facts/Octahedron/G38q92xW">Octahedron</a> • <a href="/facts/Cube/65lUaAqN">Cube</a></td><td><a href="/facts/Tetrahedron/7mkSrl6j">Demicube</a></td><td></td><td><a href="/facts/Regular_dodecahedron/xClYqhcp">Dodecahedron</a> • <a href="/facts/Regular_icosahedron/1U93vmXV">Icosahedron</a></td></tr><tr><th><a href="/facts/Uniform_polychoron/NHxomMxC">Uniform polychoron</a></th><td><a href="/facts/5-cell/Rh6lwDcc">Pentachoron</a></td><td><a href="/facts/16-cell/X0YAfobw">16-cell</a> • <a href="/facts/Tesseract/tlqouBoU">Tesseract</a></td><td><a href="/facts/16-cell/X0YAfobw">Demitesseract</a></td><td><a href="/facts/24-cell/tfoGnlAD">24-cell</a></td><td><a href="/facts/120-cell/W4nroBjJ">120-cell</a> • <a href="/facts/600-cell/tNBtDUe9">600-cell</a></td></tr><tr><th><a href="/facts/Uniform_5-polytope/mqxsoqwY">Uniform 5-polytope</a></th><td><a href="/facts/5-simplex/KIfBFfhl">5-simplex</a></td><td><a href="/facts/5-orthoplex/vU1TvdQ1">5-orthoplex</a> • <a href="/facts/5-cube/XABdhjof">5-cube</a></td><td><a href="/facts/5-demicube/zvCeNbU2">5-demicube</a></td><td></td><td></td></tr><tr><th><a href="/facts/Uniform_6-polytope/pxqFTw2F">Uniform 6-polytope</a></th><td><a href="/facts/6-simplex/mInfSIeH">6-simplex</a></td><td><a href="/facts/6-orthoplex/eGV6o0ol">6-orthoplex</a> • <a href="/facts/6-cube/xN74tdD0">6-cube</a></td><td><a href="/facts/6-demicube/1oTufnFp">6-demicube</a></td><td><a href="/facts/1_22_polytope/7XXjWGQ8">122</a> • <a href="/facts/2_21_polytope/nF3nYaQk">221</a></td><td></td></tr><tr><th><a href="/facts/Uniform_7-polytope/suCJSvTN">Uniform 7-polytope</a></th><td><a href="/facts/7-simplex/6DwtPplH">7-simplex</a></td><td><a href="/facts/7-orthoplex/mIJa3rQp">7-orthoplex</a> • <a href="/facts/7-cube/4af1W7tq">7-cube</a></td><td><a href="/facts/7-demicube/fGVUTvTI">7-demicube</a></td><td><a href="/facts/1_32_polytope/um16Pd2V">132</a> • <a href="/facts/2_31_polytope/zkjkKJ6X">231</a> • <a href="/facts/3_21_polytope/AEUzwA1A">321</a></td><td></td></tr><tr><th><a href="/facts/Uniform_8-polytope/M93v3h3c">Uniform 8-polytope</a></th><td><a href="/facts/8-simplex/QUhcnmk2">8-simplex</a></td><td><a href="/facts/8-orthoplex/dXFkXtPn">8-orthoplex</a> • <a href="/facts/8-cube/vEpKH3le">8-cube</a></td><td><a href="/facts/8-demicube/sWAomIhw">8-demicube</a></td><td><a href="/facts/1_42_polytope/DTgYDddB">142</a> • <a href="/facts/2_41_polytope/R9z0wNUz">241</a> • <a href="/facts/4_21_polytope/QFGglCkk">421</a></td><td></td></tr><tr><th><a href="/facts/Uniform_9-polytope/UdFL9ns2">Uniform 9-polytope</a></th><td><a href="/facts/9-simplex/dHV7qhxN">9-simplex</a></td><td><a href="/facts/9-orthoplex/D7LjUcAh">9-orthoplex</a> • <a href="/facts/9-cube/RNgCPzoM">9-cube</a></td><td><a href="/facts/9-demicube/dkTpfqss">9-demicube</a></td><td></td><td></td></tr><tr><th><a href="/facts/Uniform_10-polytope/Qc5NglV8">Uniform 10-polytope</a></th><td><a href="/facts/10-simplex/oeft7Qjn">10-simplex</a></td><td><a href="/facts/10-orthoplex/x6ysTC9U">10-orthoplex</a> • <a href="/facts/10-cube/mmmyBfJJ">10-cube</a></td><td><a href="/facts/10-demicube/TskxfLy9">10-demicube</a></td><td></td><td></td></tr><tr><th>Uniform <i>n</i>-<a href="/facts/Polytope/a76nXrL0">polytope</a></th><td><i>n</i>-<a href="/facts/Simplex/0FCfNRN8">simplex</a></td><td><i>n</i>-<a href="/facts/Cross-polytope/erk0byF6">orthoplex</a> • <i>n</i>-<a href="/facts/Hypercube/q2mRPbIz">cube</a></td><td><i>n</i>-<a href="/facts/Demihypercube/ackHKsEa">demicube</a></td><td><a href="/facts/Uniform_1_k2_polytope/eqTstjgl">1k2</a> • <a href="/facts/Uniform_2_k1_polytope/pQv1I7AC">2k1</a> • <a href="/facts/Uniform_k_21_polytope/vv3lWMYd">k21</a></td><td><i>n</i>-<a href="/facts/Pentagonal_polytope/4uxePdKt">pentagonal polytope</a></td></tr><tr><th colspan="13">Topics: <a href="/facts/Polytope_families/Gun0RgmR">Polytope families</a> • <a href="/facts/Regular_polytope/RYTnEF6Q">Regular polytope</a> • <a href="/facts/List_of_regular_polytopes_and_compounds/Kng25Ymq">List of regular polytopes and compounds</a></th></tr></tbody></table> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Conway, 2008 <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Klitzing, Richard. "o3x4o3o - rap". <a href="https://bendwavy.org/klitzing/dimensions/../incmats/rap.htm" target="_blank">https://bendwavy.org/klitzing/dimensions/../incmats/rap.htm</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Eppstein, David; Kuperberg, Greg; Ziegler, Günter M. (2003), "Fat 4-polytopes and fatter 3-spheres", in Bezdek, Andras (ed.), Discrete Geometry: In honor of W. Kuperberg's 60th birthday, Pure and Applied Mathematics, vol. 253, pp. 239–265, arXiv:math.CO/0204007. <a href="/wiki/David_Eppstein" target="_blank">/wiki/David_Eppstein</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Paffenholz, Andreas; Ziegler, Günter M. (2004), "The Et-construction for lattices, spheres and polytopes", Discrete & Computational Geometry, 32 (4): 601–621, arXiv:math.MG/0304492, doi:10.1007/s00454-004-1140-4, MR 2096750, S2CID 7603863. <a href="/wiki/G%C3%BCnter_M._Ziegler" target="_blank">/wiki/G%C3%BCnter_M._Ziegler</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Gosset, 1900 <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> </ol>