Uniform 4-polytope

<h2 id="history-of-discovery">History of discovery</h2>
<ul><li>Convex <a href="/facts/Regular_polytope/RYTnEF6Q">Regular polytopes</a>:
<ul><li>1852: <a href="/facts/Ludwig_Schl%25C3%25A4fli/mszJDaFn">Ludwig Schläfli</a> proved in his manuscript <i>Theorie der vielfachen Kontinuität</i> that there are exactly 6 regular polytopes in 4 <a href="/facts/Dimension/0ktmUyac">dimensions</a> and only 3 in 5 or more dimensions.</li></ul></li>
<li><a href="/facts/Schl%25C3%25A4fli-Hess_polychoron/lFf8IH1C">Regular star 4-polytopes</a> (<a href="/facts/Star_polyhedron/XyeJWj3d">star polyhedron</a> cells and/or <a href="/facts/Vertex_figure/3u2ySK0C">vertex figures</a>)
<ul><li>1852: <a href="/facts/Ludwig_Schl%25C3%25A4fli/mszJDaFn">Ludwig Schläfli</a> also found 4 of the 10 regular star 4-polytopes, discounting 6 with cells or vertex figures <a href="/facts/Small_stellated_dodecahedron/lX1zlo6C">{5/2,5}</a> and <a href="/facts/Great_dodecahedron/LWt3Hgg4">{5,5/2}</a>.</li>
<li>1883: <a href="/facts/Edmund_Hess/HJdajw7h">Edmund Hess</a> completed the list of 10 of the nonconvex regular 4-polytopes, in his book (in German) <i>Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder</i> <a href="http://www.hti.umich.edu/cgi/b/bib/bibperm?q1=ABN8623.0001.001">Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder, von dr. Edmund Hess. Mit sechzehn lithographierten tafeln.</a>.</li></ul></li>
<li>Convex <a href="/facts/Semiregular_polytope/Xw9TURVR">semiregular polytopes</a>: (Various definitions before Coxeter's uniform category)
<ul><li>1900: <a href="/facts/Thorold_Gosset/3wvHtGPn">Thorold Gosset</a> enumerated the list of nonprismatic semiregular convex polytopes with regular cells (<a href="/facts/Platonic_solid/jraREXFD">Platonic solids</a>) in his publication <i>On the Regular and Semi-Regular Figures in Space of n Dimensions</i>. In four dimensions, this gives the <a href="/facts/Rectified_5-cell/2UAMuo2P">rectified 5-cell</a>, the <a href="/facts/Rectified_600-cell/S2DeX7LG">rectified 600-cell</a>, and the <a href="/facts/Snub_24-cell/WbEJ3cPQ">snub 24-cell</a>.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a></li>
<li>1910: <a href="/facts/Alicia_Boole_Stott/FoqXBA5T">Alicia Boole Stott</a>, in her publication <i>Geometrical deduction of semiregular from regular polytopes and space fillings</i>, expanded the definition by also allowing <a href="/facts/Archimedean_solid/wYWeHDYT">Archimedean solid</a> and <a href="/facts/Prism_(geometry)/baIRRkCy">prism</a> cells. This construction enumerated 45 semiregular 4-polytopes, corresponding to the nonprismatic forms listed below. The snub 24-cell and <a href="/facts/Grand_antiprism/aohzsR7w">grand antiprism</a> were missing from her list.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a></li>
<li>1911: <a href="/facts/Pieter_Hendrik_Schoute/aWQr1v2I">Pieter Hendrik Schoute</a> published <i>Analytic treatment of the polytopes regularly derived from the regular polytopes</i>, followed Boole-Stott's notations, enumerating the convex uniform polytopes by symmetry based on <a href="/facts/5-cell/Rh6lwDcc">5-cell</a>, <a href="/facts/8-cell/tlqouBoU">8-cell</a>/<a href="/facts/16-cell/X0YAfobw">16-cell</a>, and <a href="/facts/24-cell/tfoGnlAD">24-cell</a>.</li>
<li>1912: <a href="/facts/E._L._Elte/ahH8vuy6">E. L. Elte</a> independently expanded on Gosset's list with the publication <i>The Semiregular Polytopes of the Hyperspaces</i>, polytopes with one or two types of semiregular facets.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a></li></ul></li>
<li>Convex uniform polytopes:
<ul><li>1940: The search was expanded systematically by <a href="/facts/Harold_Scott_MacDonald_Coxeter/D2l5jWGG">H.S.M. Coxeter</a> in his publication <i>Regular and Semi-Regular Polytopes</i>.</li>
<li>Convex uniform 4-polytopes:
<ul><li>1965: The complete list of convex forms was finally enumerated by <a href="/facts/John_Horton_Conway/g3PA1zoK">John Horton Conway</a> and <a href="/facts/Michael_Guy_(computer_scientist)/TjWdLhA3">Michael Guy</a>, in their publication <i>Four-Dimensional Archimedean Polytopes</i>, established by computer analysis, adding only one non-Wythoffian convex 4-polytope, the grand antiprism.</li>
<li>1966 <a href="/facts/Norman_Johnson_(mathematician)/bQtL9w56">Norman Johnson</a> completes his Ph.D. dissertation <i>The Theory of Uniform Polytopes and Honeycombs</i> under advisor Coxeter, completes the basic theory of uniform polytopes for dimensions 4 and higher.</li>
<li>1986 Coxeter published a paper <i>Regular and Semi-Regular Polytopes II</i> which included analysis of the unique <a href="/facts/Snub_24-cell/WbEJ3cPQ">snub 24-cell</a> structure, and the symmetry of the anomalous grand antiprism.</li>
<li>1998<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>-2000: The 4-polytopes were systematically named by Norman Johnson, and given by George Olshevsky's online indexed enumeration (used as a basis for this listing). Johnson named the 4-polytopes as polychora, like polyhedra for 3-polytopes, from the <a href="/facts/Greek_language/pzDd6eIz">Greek</a> roots <i>poly</i> ("many") and <i>choros</i> ("room" or "space").<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> The names of the uniform polychora started with the 6 regular polychora with prefixes based on rings in the Coxeter diagrams; truncation t0,1, cantellation, t0,2, runcination t0,3, with single ringed forms called rectified, and bi, tri-prefixes added when the first ring was on the second or third nodes.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a><a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a></li>
<li>2004: A proof that the Conway-Guy set is complete was published by Marco Möller in his dissertation, <i>Vierdimensionale Archimedische Polytope</i>. Möller reproduced Johnson's naming system in his listing.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a></li>
<li>2008: <i>The Symmetries of Things</i><a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> was published by <a href="/facts/John_Horton_Conway/g3PA1zoK">John H. Conway</a> and contains the first print-published listing of the convex uniform 4-polytopes and higher dimensional polytopes by Coxeter group family, with general <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</a> diagrams for each ringed <a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter diagram</a> permutation—snub, grand antiprism, and duoprisms—which he called proprisms for product prisms. He used his own <i>ijk</i>-ambo naming scheme for the indexed ring permutations beyond truncation and bitruncation, and all of Johnson's names were included in the book index.</li></ul></li></ul></li>
<li>Nonregular uniform star 4-polytopes: (similar to the <a href="/facts/List_of_uniform_polyhedra/robWbdeH">nonconvex uniform polyhedra</a>)
<ul><li>1966: Johnson describes three nonconvex uniform antiprisms in 4-space in his dissertation.<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a></li>
<li>1990-2006: In a collaborative search, up to 2005 a total of 1845 uniform 4-polytopes (convex and nonconvex) had been identified by Jonathan Bowers and George Olshevsky,<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> with an additional four discovered in 2006 for a total of 1849. The count includes the 74 prisms of the 75 <a href="/facts/List_of_uniform_polyhedra/robWbdeH">non-prismatic uniform polyhedra</a> (since that is a finite set – the cubic prism is excluded as it duplicates the tesseract), but not the infinite categories of duoprisms or prisms of antiprisms.<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a></li>
<li>2020-2023: 342 new polychora were found, bringing up the total number of known uniform 4-polytopes to 2191. The list has not been proven complete.<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a><a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a></li></ul></li></ul>
<h2 id="regular-4-polytopes">Regular 4-polytopes</h2>
<p>Regular 4-polytopes are a subset of the uniform 4-polytopes, which satisfy additional requirements. <a href="/facts/List_of_regular_polytopes/Kng25Ymq">Regular 4-polytopes</a> can be expressed with <a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli symbol</a> {<i>p</i>,<i>q</i>,<i>r</i>} have cells of type {<i>p</i>,<i>q</i>}, faces of type {<i>p</i>}, edge figures {<i>r</i>}, and <a href="/facts/Vertex_figure/3u2ySK0C">vertex figures</a> {<i>q</i>,<i>r</i>}.
</p><p>The existence of a regular 4-polytope {<i>p</i>,<i>q</i>,<i>r</i>} is constrained by the existence of the regular polyhedra {<i>p</i>,<i>q</i>} which becomes cells, and {<i>q</i>,<i>r</i>} which becomes the <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</a>.
</p><p>Existence as a finite 4-polytope is dependent upon an inequality:<a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a>
</p>

sin
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              π
              p
            
          
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        sin
        ⁡
        
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              π
              r
            
          
          )
        
        >
        cos
        ⁡
        
          (
          
            
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        .
      
    
    {\displaystyle \sin \left({\frac {\pi }{p}}\right)\sin \left({\frac {\pi }{r}}\right)>\cos \left({\frac {\pi }{q}}\right).}

<p>The 16 <a href="/facts/List_of_regular_polytopes/Kng25Ymq">regular 4-polytopes</a>, with the property that all cells, faces, edges, and vertices are congruent:
</p>
<ul><li>6 <a href="/facts/Regular_convex_4-polytope/lFf8IH1C">regular convex 4-polytopes</a>: <a href="/facts/5-cell/Rh6lwDcc">5-cell</a> {3,3,3}, <a href="/facts/8-cell/tlqouBoU">8-cell</a> {4,3,3}, <a href="/facts/16-cell/X0YAfobw">16-cell</a> {3,3,4}, <a href="/facts/24-cell/tfoGnlAD">24-cell</a> {3,4,3}, <a href="/facts/120-cell/W4nroBjJ">120-cell</a> {5,3,3}, and <a href="/facts/600-cell/tNBtDUe9">600-cell</a> {3,3,5}.</li>
<li>10 <a href="/facts/Regular_star_4-polytope/lFf8IH1C">regular star 4-polytopes</a>: <a href="/facts/Icosahedral_120-cell/zqVMqIt8">icosahedral 120-cell</a> {3,5,5/2}, <a href="/facts/Small_stellated_120-cell/v58V84tU">small stellated 120-cell</a> {5/2,5,3}, <a href="/facts/Great_120-cell/DWoWl2GJ">great 120-cell</a> {5,5/2,5}, <a href="/facts/Grand_120-cell/PkhQyduS">grand 120-cell</a> {5,3,5/2}, <a href="/facts/Great_stellated_120-cell/5rEn9UcI">great stellated 120-cell</a> {5/2,3,5}, <a href="/facts/Grand_stellated_120-cell/bRqLFA8B">grand stellated 120-cell</a> {5/2,5,5/2}, <a href="/facts/Great_grand_120-cell/8MBEpfnm">great grand 120-cell</a> {5,5/2,3}, <a href="/facts/Great_icosahedral_120-cell/MPbGSeyI">great icosahedral 120-cell</a> {3,5/2,5}, <a href="/facts/Grand_600-cell/LqttGCQw">grand 600-cell</a> {3,3,5/2}, and <a href="/facts/Great_grand_stellated_120-cell/IIMNuXc0">great grand stellated 120-cell</a> {5/2,3,3}.</li></ul>
<h2 id="convex-uniform-4-polytopes">Convex uniform 4-polytopes</h2>
<h3>Symmetry of uniform 4-polytopes in four dimensions</h3>
<p class="note">Main article: <a href="/facts/Point_groups_in_four_dimensions/FtoaKNqe">Point groups in four dimensions</a></p>
Orthogonal subgroups<table><tbody><tr><td>The 24 mirrors of F4 can be decomposed into 2 orthogonal D4 groups:<ol><li> =  (12 mirrors)</li><li> =  (12 mirrors)</li></ol></td></tr><tr><td>The 10 mirrors of B3×A1 can be decomposed into orthogonal groups, 4A1 and D3:<ol><li> =  (3+1 mirrors)</li><li> =  (6 mirrors)</li></ol></td></tr></tbody></table>
<p>There are 5 fundamental mirror symmetry <a href="/facts/Point_group/41NSqUN6">point group</a> families in 4-dimensions: A4 = , B4 = , D4 = , F4 = , H4 = .<a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a> There are also 3 prismatic groups A3A1 = , B3A1 = , H3A1 = , and duoprismatic groups: I2(p)×I2(q) = . Each group defined by a <a href="/facts/Goursat_tetrahedron/kDgl1J5z">Goursat tetrahedron</a> <a href="/facts/Fundamental_domain/e68KDYWu">fundamental domain</a> bounded by mirror planes.
</p><p>Each reflective uniform 4-polytope can be constructed in one or more reflective point group in 4 dimensions by a <a href="/facts/Wythoff_construction/8yTYjp5e">Wythoff construction</a>, represented by rings around permutations of nodes in a <a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter diagram</a>. Mirror <a href="/facts/Hyperplane/2s2ycYYd">hyperplanes</a> can be grouped, as seen by colored nodes, separated by even-branches. Symmetry groups of the form [a,b,a], have an extended symmetry, [[a,b,a]], doubling the symmetry order. This includes [3,3,3], [3,4,3], and [<i>p</i>,2,<i>p</i>]. Uniform polytopes in these group with symmetric rings contain this extended symmetry.
</p><p>If all mirrors of a given color are unringed (inactive) in a given uniform polytope, it will have a lower symmetry construction by removing all of the inactive mirrors. If all the nodes of a given color are ringed (active), an <a href="/facts/Alternation_(geometry)/H2nVs4xj">alternation</a> operation can generate a new 4-polytope with chiral symmetry, shown as "empty" circled nodes", but the geometry is not generally adjustable to create uniform solutions.
</p>
<table><tbody><tr><th><a href="/facts/Coxeter_group/bgzHFm1Q">Weylgroup</a></th><th><a href="/facts/John_Horton_Conway/g3PA1zoK">Conway</a>Quaternion</th><th>Abstractstructure</th><th><a href="/facts/Group_order/Cl3tKGFg">Order</a></th><th colspan="2"><a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeterdiagram</a></th><th><a href="/facts/Coxeter_notation/ArUw6goe">Coxeternotation</a></th><th><a href="/facts/Coxeter_notation/ArUw6goe">Commutatorsubgroup</a></th><th><a href="/facts/Coxeter_number/4B80wH8V">Coxeternumber</a>(h)</th><th colspan="4"><a href="/facts/Kaleidoscope/PTR2SCDm">Mirrors</a><i>m</i>=2<i>h</i></th></tr><tr><th colspan="13">Irreducible</th></tr><tr><th>A4</th><td>+1/60[I×I].21</td><td><a href="/facts/Symmetric_group/RUksK5l5">S5</a></td><td>120</td><td></td><td></td><td>[3,3,3]</td><td>[3,3,3]+</td><td>5</td><td></td><td>10</td><td></td><td></td></tr><tr><th>D4</th><td>±1/3[T×T].2</td><td>1/2.2S4</td><td>192</td><td></td><td></td><td>[31,1,1]</td><td rowspan="2">[31,1,1]+</td><td>6</td><td></td><td>12</td><td></td><td></td></tr><tr><th>B4</th><td>±1/6[O×O].2</td><td>2S4 = S2≀S4</td><td>384</td><td></td><td></td><td>[4,3,3]</td><td>8</td><td>4</td><td>12</td><td></td><td></td></tr><tr><th>F4</th><td>±1/2[O×O].23</td><td>3.2S4</td><td>1152</td><td></td><td></td><td>[3,4,3]</td><td>[3+,4,3+]</td><td>12</td><td>12</td><td>12</td><td></td><td></td></tr><tr><th>H4</th><td>±[I×I].2</td><td>2.(A5×A5).2</td><td>14400</td><td></td><td></td><td>[5,3,3]</td><td>[5,3,3]+</td><td>30</td><td></td><td>60</td><td></td><td></td></tr><tr><th colspan="13">Prismatic groups</th></tr><tr><th>A3A1</th><td>+1/24[O×O].23</td><td>S4×D1</td><td>48</td><td></td><td></td><td>[3,3,2] = [3,3]×[ ]</td><td rowspan="2">[3,3]+</td><td>-</td><td></td><td>6</td><td>1</td><td></td></tr><tr><th>B3A1</th><td>±1/24[O×O].2</td><td>S4×D1</td><td>96</td><td></td><td></td><td>[4,3,2] = [4,3]×[ ]</td><td>-</td><td>3</td><td>6</td><td>1</td><td></td></tr><tr><th>H3A1</th><td>±1/60[I×I].2</td><td>A5×D1</td><td>240</td><td></td><td></td><td>[5,3,2] = [5,3]×[ ]</td><td>[5,3]+</td><td>-</td><td></td><td>15</td><td>1</td><td></td></tr><tr><th colspan="13">Duoprismatic groups (Use 2p,2q for even integers)</th></tr><tr><th>I2(<i>p</i>)I2(<i>q</i>)</th><td>±1/2[D2<i>p</i>×D2<i>q</i>]</td><td>D<i>p</i>×D<i>q</i></td><td>4<i>pq</i></td><td></td><td></td><td>[<i>p</i>,2,<i>q</i>] = [<i>p</i>]×[<i>q</i>]</td><td rowspan="3">[<i>p</i>+,2,<i>q</i>+]</td><td>-</td><td></td><td><i>p</i> </td><td><i>q</i> </td><td></td></tr><tr><th>I2(<i>2p</i>)I2(<i>q</i>)</th><td>±1/2[D4<i>p</i>×D2<i>q</i>]</td><td>D2<i>p</i>×D<i>q</i></td><td>8<i>pq</i></td><td></td><td></td><td>[2<i>p</i>,2,<i>q</i>] = [2<i>p</i>]×[<i>q</i>]</td><td>-</td><td><i>p</i> </td><td><i>p</i> </td><td><i>q</i> </td><td></td></tr><tr><th>I2(<i>2p</i>)I2(<i>2q</i>)</th><td>±1/2[D4<i>p</i>×D4<i>q</i>]</td><td>D<i>2p</i>×D<i>2q</i></td><td>16<i>pq</i></td><td></td><td></td><td>[2<i>p</i>,2,2<i>q</i>] = [2<i>p</i>]×[2<i>q</i>]</td><td>-</td><td><i>p</i> </td><td><i>p</i> </td><td><i>q</i> </td><td><i>q</i> </td></tr></tbody></table>
<h3>Enumeration</h3>
<p>There are 64 convex uniform 4-polytopes, including the 6 regular convex 4-polytopes, and excluding the infinite sets of the <a href="/facts/Duoprism/D73YMRth">duoprisms</a> and the <a href="/facts/Uniform_antiprismatic_prism/thA0prtm">antiprismatic prisms</a>.
</p>
<ul><li>5 are polyhedral prisms based on the <a href="/facts/Platonic_solid/jraREXFD">Platonic solids</a> (1 overlap with regular since a cubic hyperprism is a <a href="/facts/Tesseract/tlqouBoU">tesseract</a>)</li>
<li>13 are polyhedral prisms based on the <a href="/facts/Archimedean_solid/wYWeHDYT">Archimedean solids</a></li>
<li>9 are in the self-dual regular A4 [3,3,3] group (<a href="/facts/5-cell/Rh6lwDcc">5-cell</a>) family.</li>
<li>9 are in the self-dual regular F4 [3,4,3] group (<a href="/facts/24-cell/tfoGnlAD">24-cell</a>) family. (Excluding snub 24-cell)</li>
<li>15 are in the regular B4 [3,3,4] group (<a href="/facts/Tesseract/tlqouBoU">tesseract</a>/<a href="/facts/16-cell/X0YAfobw">16-cell</a>) family (3 overlap with 24-cell family)</li>
<li>15 are in the regular H4 [3,3,5] group (<a href="/facts/120-cell/W4nroBjJ">120-cell</a>/<a href="/facts/600-cell/tNBtDUe9">600-cell</a>) family.</li>
<li>1 special snub form in the [3,4,3] group (<a href="/facts/24-cell/tfoGnlAD">24-cell</a>) family.</li>
<li>1 special non-Wythoffian 4-polytope, the grand antiprism.</li>
<li>TOTAL: 68 − 4 = 64</li></ul>
<p>These 64 uniform 4-polytopes are indexed below by George Olshevsky. Repeated symmetry forms are indexed in brackets.
</p><p>In addition to the 64 above, there are 2 infinite prismatic sets that generate all of the remaining convex forms:
</p>
<ul><li>Set of <a href="/facts/Uniform_antiprismatic_hyperprism/thA0prtm">uniform antiprismatic prisms</a> - sr{<i>p</i>,2}×{ } - Polyhedral prisms of two <a href="/facts/Antiprisms/FgP5jIVd">antiprisms</a>.</li>
<li>Set of uniform <a href="/facts/Duoprism/D73YMRth">duoprisms</a> - {<i>p</i>}×{<i>q</i>} - A <a href="/facts/Cartesian_product/BfqBWzdL">Cartesian product</a> of two polygons.</li></ul>
<h3>The A4 family</h3>
<p class="note">Further information: <a href="/facts/A4_polytope/CIr2PM4I">A4 polytope</a></p>
<p>The 5-cell has <a href="/facts/Pentachoric_symmetry/FtoaKNqe"><i>diploid pentachoric</i> [3,3,3] symmetry</a>,<a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a> of <a href="/facts/Symmetry_order/tmtq9goq">order</a> 120, isomorphic to the permutations of five elements, because all pairs of vertices are related in the same way.
</p><p>Facets (cells) are given, grouped in their Coxeter diagram locations by removing specified nodes.
</p>
[3,3,3] uniform polytopes<table><tbody><tr><th rowspan="2">#</th><th rowspan="2">Name Bowers name (and acronym)</th><th rowspan="2"><a href="/facts/Vertex_figure/3u2ySK0C">Vertexfigure</a></th><th rowspan="2"><a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter diagram</a>and <a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli</a>symbols</th><th colspan="4">Cell counts by location</th><th colspan="4">Element counts</th></tr><tr><th>Pos. 3(5)</th><th>Pos. 2(10)</th><th>Pos. 1(10)</th><th>Pos. 0(5)</th><th>Cells</th><th>Faces</th><th>Edges</th><th>Vertices</th></tr><tr><th>1</th><td><a href="/facts/5-cell/Rh6lwDcc">5-cell</a>Pentachoron<a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a> (pen)</td><td></td><td>{3,3,3}</td><td>(4)<a href="/facts/Tetrahedron/7mkSrl6j">(3.3.3)</a></td><td></td><td></td><td></td><td>5</td><td>10</td><td>10</td><td>5</td></tr><tr><th>2</th><td><a href="/facts/Rectified_5-cell/2UAMuo2P">rectified 5-cell</a> Rectified pentachoron (rap)</td><td></td><td>r{3,3,3}</td><td>(3)<a href="/facts/Octahedron/G38q92xW">(3.3.3.3)</a></td><td></td><td></td><td>(2)<a href="/facts/Tetrahedron/7mkSrl6j">(3.3.3)</a></td><td>10</td><td>30</td><td>30</td><td>10</td></tr><tr><th>3</th><td><a href="/facts/Truncated_5-cell/oqF6jkb7">truncated 5-cell</a> Truncated pentachoron (tip)</td><td></td><td>t{3,3,3}</td><td>(3)<a href="/facts/Truncated_tetrahedron/K9ItvmUK">(3.6.6)</a></td><td></td><td></td><td>(1)<a href="/facts/Tetrahedron/7mkSrl6j">(3.3.3)</a></td><td>10</td><td>30</td><td>40</td><td>20</td></tr><tr><th>4</th><td><a href="/facts/Cantellated_5-cell/pm158bSU">cantellated 5-cell</a> Small rhombated pentachoron (srip)</td><td></td><td>rr{3,3,3}</td><td>(2)<a href="/facts/Cuboctahedron/hD7p0Mem">(3.4.3.4)</a></td><td></td><td>(2)<a href="/facts/Triangular_prism/Yjp4Kw9M">(3.4.4)</a></td><td>(1)<a href="/facts/Octahedron/G38q92xW">(3.3.3.3)</a></td><td>20</td><td>80</td><td>90</td><td>30</td></tr><tr><th>7</th><td><a href="/facts/Cantitruncated_5-cell/pm158bSU">cantitruncated 5-cell</a> Great rhombated pentachoron (grip)</td><td></td><td>tr{3,3,3}</td><td>(2)<a href="/facts/Truncated_octahedron/GA98b4TH">(4.6.6)</a></td><td></td><td>(1)<a href="/facts/Triangular_prism/Yjp4Kw9M">(3.4.4)</a></td><td>(1)<a href="/facts/Truncated_tetrahedron/K9ItvmUK">(3.6.6)</a></td><td>20</td><td>80</td><td>120</td><td>60</td></tr><tr><th>8</th><td><a href="/facts/Runcitruncated_5-cell/7A0II7du">runcitruncated 5-cell</a> Prismatorhombated pentachoron (prip)</td><td></td><td>t0,1,3{3,3,3}</td><td>(1)<a href="/facts/Truncated_tetrahedron/K9ItvmUK">(3.6.6)</a></td><td>(2)<a href="/facts/Hexagonal_prism/oxYc0nF2">(4.4.6)</a></td><td>(1)<a href="/facts/Triangular_prism/Yjp4Kw9M">(3.4.4)</a></td><td>(1)<a href="/facts/Cuboctahedron/hD7p0Mem">(3.4.3.4)</a></td><td>30</td><td>120</td><td>150</td><td>60</td></tr></tbody></table>
[[3,3,3]] uniform polytopes<table><tbody><tr><th rowspan="2">#</th><th rowspan="2">Name Bowers name (and acronym)</th><th rowspan="2"><a href="/facts/Vertex_figure/3u2ySK0C">Vertexfigure</a></th><th rowspan="2"><a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter diagram</a>and <a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli</a>symbols</th><th colspan="3">Cell counts by location</th><th colspan="4">Element counts</th></tr><tr><th>Pos. 3-0(10)</th><th>Pos. 1-2(20)</th><th>Alt</th><th>Cells</th><th>Faces</th><th>Edges</th><th>Vertices</th></tr><tr><th>5</th><td>*<a href="/facts/Runcinated_5-cell/7A0II7du">runcinated 5-cell</a> Small prismatodecachoron (spid)</td><td></td><td>t0,3{3,3,3}</td><td>(2)<a href="/facts/Tetrahedron/7mkSrl6j">(3.3.3)</a></td><td>(6)<a href="/facts/Triangular_prism/Yjp4Kw9M">(3.4.4)</a></td><td></td><td>30</td><td>70</td><td>60</td><td>20</td></tr><tr><th>6</th><td>*<a href="/facts/Bitruncated_5-cell/oqF6jkb7">bitruncated 5-cell</a>Decachoron (deca)</td><td></td><td>2t{3,3,3}</td><td>(4)<a href="/facts/Truncated_tetrahedron/K9ItvmUK">(3.6.6)</a></td><td></td><td></td><td>10</td><td>40</td><td>60</td><td>30</td></tr><tr><th>9</th><td>*<a href="/facts/Omnitruncated_5-cell/7A0II7du">omnitruncated 5-cell</a> Great prismatodecachoron (gippid)</td><td></td><td>t0,1,2,3{3,3,3}</td><td>(2)<a href="/facts/Truncated_octahedron/GA98b4TH">(4.6.6)</a></td><td>(2)<a href="/facts/Hexagonal_prism/oxYc0nF2">(4.4.6)</a></td><td></td><td>30</td><td>150</td><td>240</td><td>120</td></tr><tr><th>Nonuniform</th><td><a href="/facts/Omnisnub_5-cell/7A0II7du">omnisnub 5-cell</a>Snub decachoron (snad)Snub pentachoron (snip)<a class="footnote-ref" id="fnref:20" href="#fn:20"><sup>20</sup></a></td><td></td><td>ht0,1,2,3{3,3,3}</td><td> (2)<a href="/facts/Snub_tetrahedron/1U93vmXV">(3.3.3.3.3)</a></td><td> (2)<a href="/facts/Triangular_antiprism/G38q92xW">(3.3.3.3)</a></td><td> (4)<a href="/facts/Tetrahedron/7mkSrl6j">(3.3.3)</a></td><td>90</td><td>300</td><td>270</td><td>60</td></tr></tbody></table>
<p>The three uniform 4-polytopes forms marked with an <a href="/facts/Asterisk/qwmtPW5V">asterisk</a>, *, have the higher <a href="/facts/Pentachoric_symmetry/FtoaKNqe">extended pentachoric symmetry</a>, of order 240, [[3,3,3]] because the element corresponding to any element of the underlying 5-cell can be exchanged with one of those corresponding to an element of its dual. There is one small index subgroup [3,3,3]+, order 60, or its doubling [[3,3,3]]+, order 120, defining an <a href="/facts/Omnisnub_5-cell/7A0II7du">omnisnub 5-cell</a> which is listed for completeness, but is not uniform.
</p>
<h3>The B4 family</h3>
<p class="note">Further information: <a href="/facts/B4_polytope/XLtUADz6">B4 polytope</a></p>
<p>This family has <a href="/facts/Hexadecachoric_symmetry/FtoaKNqe"><i>diploid hexadecachoric</i> symmetry</a>,<a class="footnote-ref" id="fnref:21" href="#fn:21"><sup>21</sup></a> [4,3,3], of <a href="/facts/Symmetry_order/tmtq9goq">order</a> 24×16=384: 4!=24 permutations of the four axes, 24=16 for reflection in each axis. There are 3 small index subgroups, with the first two generate uniform 4-polytopes which are also repeated in other families, [1+,4,3,3], [4,(3,3)+], and [4,3,3]+, all order 192.
</p>
<h4>Tesseract truncations</h4>
<table><tbody><tr><th rowspan="2">#</th><th rowspan="2">Name(Bowers name and acronym)</th><th rowspan="2"><a href="/facts/Vertex_figure/3u2ySK0C">Vertexfigure</a></th><th rowspan="2"><a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter diagram</a>and <a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli</a>symbols</th><th colspan="5">Cell counts by location</th><th colspan="4">Element counts</th></tr><tr><th>Pos. 3(8)</th><th>Pos. 2(24)</th><th>Pos. 1(32)</th><th>Pos. 0(16)</th><th>Cells</th><th>Faces</th><th>Edges</th><th>Vertices</th></tr><tr><th>10</th><td><a href="/facts/Tesseract/tlqouBoU">tesseract</a> or 8-cellTesseract (tes)</td><td></td><td>{4,3,3}</td><td>(4)<a href="/facts/Cube/65lUaAqN">(4.4.4)</a></td><td></td><td></td><td></td><td>8</td><td>24</td><td>32</td><td>16</td></tr><tr><th>11</th><td><a href="/facts/Rectified_tesseract/xfSdmduC">Rectified tesseract</a> (rit)</td><td></td><td>r{4,3,3}</td><td>(3)<a href="/facts/Cuboctahedron/hD7p0Mem">(3.4.3.4)</a></td><td></td><td></td><td>(2)<a href="/facts/Tetrahedron/7mkSrl6j">(3.3.3)</a></td><td>24</td><td>88</td><td>96</td><td>32</td></tr><tr><th>13</th><td><a href="/facts/Truncated_tesseract/ShQzMKoa">Truncated tesseract</a> (tat)</td><td></td><td>t{4,3,3}</td><td>(3)<a href="/facts/Truncated_cube/XVAUG4fD">(3.8.8)</a></td><td></td><td></td><td>(1)<a href="/facts/Tetrahedron/7mkSrl6j">(3.3.3)</a></td><td>24</td><td>88</td><td>128</td><td>64</td></tr><tr><th>14</th><td><a href="/facts/Cantellated_tesseract/gLYspT03">Cantellated tesseract</a>Small rhombated tesseract (srit)</td><td></td><td>rr{4,3,3}</td><td>(2)<a href="/facts/Small_rhombicuboctahedron/TmxUmEnl">(3.4.4.4)</a></td><td></td><td>(2)<a href="/facts/Triangular_prism/Yjp4Kw9M">(3.4.4)</a></td><td>(1)<a href="/facts/Octahedron/G38q92xW">(3.3.3.3)</a></td><td>56</td><td>248</td><td>288</td><td>96</td></tr><tr><th>15</th><td><i><a href="/facts/Runcinated_tesseract/NqDGHQxp">Runcinated tesseract</a></i>(also <i>runcinated 16-cell</i>)Small disprismatotesseractihexadecachoron (sidpith)</td><td></td><td>t0,3{4,3,3}</td><td>(1)<a href="/facts/Cube/65lUaAqN">(4.4.4)</a></td><td>(3)<a href="/facts/Cube/65lUaAqN">(4.4.4)</a></td><td>(3)<a href="/facts/Triangular_prism/Yjp4Kw9M">(3.4.4)</a></td><td>(1)<a href="/facts/Tetrahedron/7mkSrl6j">(3.3.3)</a></td><td>80</td><td>208</td><td>192</td><td>64</td></tr><tr><th>16</th><td><i><a href="/facts/Bitruncated_tesseract/ShQzMKoa">Bitruncated tesseract</a></i>(also <i>bitruncated 16-cell</i>)Tesseractihexadecachoron (tah)</td><td></td><td>2t{4,3,3}</td><td>(2)<a href="/facts/Truncated_octahedron/GA98b4TH">(4.6.6)</a></td><td></td><td></td><td>(2)<a href="/facts/Truncated_tetrahedron/K9ItvmUK">(3.6.6)</a></td><td>24</td><td>120</td><td>192</td><td>96</td></tr><tr><th>18</th><td><a href="/facts/Cantitruncated_tesseract/gLYspT03">Cantitruncated tesseract</a>Great rhombated tesseract (grit)</td><td></td><td>tr{4,3,3}</td><td>(2)<a href="/facts/Truncated_cuboctahedron/nd2ltxcm">(4.6.8)</a></td><td></td><td>(1)<a href="/facts/Triangular_prism/Yjp4Kw9M">(3.4.4)</a></td><td>(1)<a href="/facts/Truncated_tetrahedron/K9ItvmUK">(3.6.6)</a></td><td>56</td><td>248</td><td>384</td><td>192</td></tr><tr><th>19</th><td><a href="/facts/Runcitruncated_tesseract/NqDGHQxp">Runcitruncated tesseract</a>Prismatorhombated hexadecachoron (proh)</td><td></td><td>t0,1,3{4,3,3}</td><td>(1)<a href="/facts/Truncated_cube/XVAUG4fD">(3.8.8)</a></td><td>(2)<a href="/facts/Octagonal_prism/pMnX2f7F">(4.4.8)</a></td><td>(1)<a href="/facts/Triangular_prism/Yjp4Kw9M">(3.4.4)</a></td><td>(1)<a href="/facts/Cuboctahedron/hD7p0Mem">(3.4.3.4)</a></td><td>80</td><td>368</td><td>480</td><td>192</td></tr><tr><th>21</th><td><i><a href="/facts/Omnitruncated_tesseract/NqDGHQxp">Omnitruncated tesseract</a></i>(also <i>omnitruncated 16-cell</i>)Great disprismatotesseractihexadecachoron (gidpith)</td><td></td><td>t0,1,2,3{3,3,4}</td><td>(1)<a href="/facts/Truncated_cuboctahedron/nd2ltxcm">(4.6.8)</a></td><td>(1)<a href="/facts/Octagonal_prism/pMnX2f7F">(4.4.8)</a></td><td>(1)<a href="/facts/Hexagonal_prism/oxYc0nF2">(4.4.6)</a></td><td>(1)<a href="/facts/Truncated_octahedron/GA98b4TH">(4.6.6)</a></td><td>80</td><td>464</td><td>768</td><td>384</td></tr></tbody></table>
Related half tesseract, [1+,4,3,3] uniform 4-polytopes<table><tbody><tr><th rowspan="2">#</th><th rowspan="2">Name(Bowers style acronym)</th><th rowspan="2"><a href="/facts/Vertex_figure/3u2ySK0C">Vertexfigure</a></th><th rowspan="2"><a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter diagram</a>and <a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli</a>symbols</th><th colspan="5">Cell counts by location</th><th colspan="4">Element counts</th></tr><tr><th>Pos. 3(8)</th><th>Pos. 2(24)</th><th>Pos. 1(32)</th><th>Pos. 0(16)</th><th>Alt</th><th>Cells</th><th>Faces</th><th>Edges</th><th>Vertices</th></tr><tr><th>[12]</th><td>Half tesseractDemitesseract= <a href="/facts/16-cell/X0YAfobw">16-cell</a> (hex)</td><td></td><td> = h{4,3,3}={3,3,4}</td><td>(4)<a href="/facts/Tetrahedron/7mkSrl6j">(3.3.3)</a></td><td></td><td></td><td></td><td>(4)<a href="/facts/Tetrahedron/7mkSrl6j">(3.3.3)</a></td><td>16</td><td>32</td><td>24</td><td>8</td></tr><tr><th>[17]</th><td><a href="/facts/Cantic_tesseract/ShQzMKoa">Cantic tesseract</a>= <a href="/facts/Truncated_16-cell/ShQzMKoa">Truncated 16-cell</a> (thex)</td><td></td><td> = h2{4,3,3}=t{4,3,3}</td><td>(4)<a href="/facts/Truncated_tetrahedron/K9ItvmUK">(6.6.3)</a></td><td></td><td></td><td>(1)<a href="/facts/Octahedron/G38q92xW">(3.3.3.3)</a></td><td></td><td>24</td><td>96</td><td>120</td><td>48</td></tr><tr><th>[11]</th><td><a href="/facts/Runcic_tesseract/xfSdmduC">Runcic tesseract</a>= <a href="/facts/Rectified_tesseract/xfSdmduC">Rectified tesseract</a> (rit)</td><td></td><td> = h3{4,3,3}=r{4,3,3}</td><td>(3)<a href="/facts/Cuboctahedron/hD7p0Mem">(3.4.3.4)</a></td><td></td><td></td><td>(2)<a href="/facts/Tetrahedron/7mkSrl6j">(3.3.3)</a></td><td></td><td>24</td><td>88</td><td>96</td><td>32</td></tr><tr><th>[16]</th><td><a href="/facts/Runcicantic_tesseract/ShQzMKoa">Runcicantic tesseract</a>= <a href="/facts/Bitruncated_tesseract/ShQzMKoa">Bitruncated tesseract</a> (tah)</td><td></td><td> = h2,3{4,3,3}=2t{4,3,3}</td><td>(2)<a href="/facts/Truncated_octahedron/GA98b4TH">(3.4.3.4)</a></td><td></td><td></td><td>(2)<a href="/facts/Truncated_tetrahedron/K9ItvmUK">(3.6.6)</a></td><td></td><td>24</td><td>120</td><td>192</td><td>96</td></tr><tr><th>[11]</th><td>= <a href="/facts/Rectified_tesseract/xfSdmduC">Rectified tesseract</a> (rat)</td><td></td><td> = h1{4,3,3}=r{4,3,3}</td><td></td><td></td><td></td><td></td><td></td><td>24</td><td>88</td><td>96</td><td>32</td></tr><tr><th>[16]</th><td>= <a href="/facts/Bitruncated_tesseract/ShQzMKoa">Bitruncated tesseract</a> (tah)</td><td></td><td> = h1,2{4,3,3}=2t{4,3,3}</td><td></td><td></td><td></td><td></td><td></td><td>24</td><td>120</td><td>192</td><td>96</td></tr><tr><th>[23]</th><td>= <a href="/facts/Rectified_24-cell/YJm98yGE">Rectified 24-cell</a> (rico)</td><td></td><td> = h1,3{4,3,3}=rr{3,3,4}</td><td></td><td></td><td></td><td></td><td></td><td>48</td><td>240</td><td>288</td><td>96</td></tr><tr><th>[24]</th><td>= <a href="/facts/Truncated_24-cell/1dnQysxX">Truncated 24-cell</a> (tico)</td><td></td><td> = h1,2,3{4,3,3}=tr{3,3,4}</td><td></td><td></td><td></td><td></td><td></td><td>48</td><td>240</td><td>384</td><td>192</td></tr></tbody></table>
<table><tbody><tr><th rowspan="2">#</th><th rowspan="2">Name(Bowers style acronym)</th><th rowspan="2"><a href="/facts/Vertex_figure/3u2ySK0C">Vertexfigure</a></th><th rowspan="2"><a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter diagram</a>and <a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli</a>symbols</th><th colspan="5">Cell counts by location</th><th colspan="4">Element counts</th></tr><tr><th>Pos. 3(8)</th><th>Pos. 2(24)</th><th>Pos. 1(32)</th><th>Pos. 0(16)</th><th>Alt</th><th>Cells</th><th>Faces</th><th>Edges</th><th>Vertices</th></tr><tr><th>Nonuniform</th><td><a href="/facts/Omnisnub_tesseract/NqDGHQxp">omnisnub tesseract</a>Snub tesseract (snet)<a class="footnote-ref" id="fnref:22" href="#fn:22"><sup>22</sup></a>(Or <i>omnisnub 16-cell</i>)</td><td></td><td>ht0,1,2,3{4,3,3}</td><td>(1)<a href="/facts/Snub_cube/pbnr40gk">(3.3.3.3.4)</a></td><td>(1)<a href="/facts/Square_antiprism/4514Q02s">(3.3.3.4)</a></td><td>(1)<a href="/facts/Triangular_antiprism/G38q92xW">(3.3.3.3)</a></td><td>(1)<a href="/facts/Snub_tetrahedron/1U93vmXV">(3.3.3.3.3)</a></td><td>(4)<a href="/facts/Tetrahedron/7mkSrl6j">(3.3.3)</a></td><td>272</td><td>944</td><td>864</td><td>192</td></tr></tbody></table>
<h4>16-cell truncations</h4>
<table><tbody><tr><th rowspan="2">#</th><th rowspan="2">Name (Bowers name and acronym)</th><th rowspan="2"><a href="/facts/Vertex_figure/3u2ySK0C">Vertexfigure</a></th><th rowspan="2"><a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter diagram</a>and <a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli</a>symbols</th><th colspan="5">Cell counts by location</th><th colspan="4">Element counts</th></tr><tr><th>Pos. 3(8)</th><th>Pos. 2(24)</th><th>Pos. 1(32)</th><th>Pos. 0(16)</th><th>Alt</th><th>Cells</th><th>Faces</th><th>Edges</th><th>Vertices</th></tr><tr><th>12</th><td><a href="/facts/16-cell/X0YAfobw">16-cell</a>Hexadecachoron<a class="footnote-ref" id="fnref:23" href="#fn:23"><sup>23</sup></a> (hex)</td><td></td><td>{3,3,4}</td><td></td><td></td><td></td><td>(8)<a href="/facts/Tetrahedron/7mkSrl6j">(3.3.3)</a></td><td></td><td>16</td><td>32</td><td>24</td><td>8</td></tr><tr><th>[22]</th><td>*Rectified 16-cell(Same as <i><a href="/facts/24-cell/tfoGnlAD">24-cell</a></i>) (ico)</td><td></td><td> = r{3,3,4}</td><td>(2)<a href="/facts/Octahedron/G38q92xW">(3.3.3.3)</a></td><td></td><td></td><td>(4)<a href="/facts/Octahedron/G38q92xW">(3.3.3.3)</a></td><td></td><td>24</td><td>96</td><td>96</td><td>24</td></tr><tr><th>17</th><td><a href="/facts/Truncated_16-cell/ShQzMKoa">Truncated 16-cell</a>Truncated hexadecachoron (thex)</td><td></td><td>t{3,3,4}</td><td>(1)<a href="/facts/Octahedron/G38q92xW">(3.3.3.3)</a></td><td></td><td></td><td>(4)<a href="/facts/Truncated_tetrahedron/K9ItvmUK">(3.6.6)</a></td><td></td><td>24</td><td>96</td><td>120</td><td>48</td></tr><tr><th>[23]</th><td>*Cantellated 16-cell(Same as <i><a href="/facts/Rectified_24-cell/YJm98yGE">rectified 24-cell</a></i>) (rico)</td><td></td><td> = rr{3,3,4}</td><td>(1)<a href="/facts/Cuboctahedron/hD7p0Mem">(3.4.3.4)</a></td><td>(2)<a href="/facts/Cube/65lUaAqN">(4.4.4)</a></td><td></td><td>(2)<a href="/facts/Cuboctahedron/hD7p0Mem">(3.4.3.4)</a></td><td></td><td>48</td><td>240</td><td>288</td><td>96</td></tr><tr><th>[15]</th><td><i><a href="/facts/Runcinated_tesseract/NqDGHQxp">Runcinated 16-cell</a></i>(also <i>runcinated tesseract</i>) (sidpith)</td><td></td><td>t0,3{3,3,4}</td><td>(1)<a href="/facts/Cube/65lUaAqN">(4.4.4)</a></td><td>(3)<a href="/facts/Cube/65lUaAqN">(4.4.4)</a></td><td>(3)<a href="/facts/Triangular_prism/Yjp4Kw9M">(3.4.4)</a></td><td>(1)<a href="/facts/Tetrahedron/7mkSrl6j">(3.3.3)</a></td><td></td><td>80</td><td>208</td><td>192</td><td>64</td></tr><tr><th>[16]</th><td><i><a href="/facts/Bitruncated_tesseract/ShQzMKoa">Bitruncated 16-cell</a></i>(also <i>bitruncated tesseract</i>) (tah)</td><td></td><td>2t{3,3,4}</td><td>(2)<a href="/facts/Truncated_octahedron/GA98b4TH">(4.6.6)</a></td><td></td><td></td><td>(2)<a href="/facts/Truncated_tetrahedron/K9ItvmUK">(3.6.6)</a></td><td></td><td>24</td><td>120</td><td>192</td><td>96</td></tr><tr><th>[24]</th><td>*Cantitruncated 16-cell(Same as <i><a href="/facts/Truncated_24-cell/1dnQysxX">truncated 24-cell</a></i>) (tico)</td><td></td><td> = tr{3,3,4}</td><td>(1)<a href="/facts/Truncated_octahedron/GA98b4TH">(4.6.6)</a></td><td>(1)<a href="/facts/Cube/65lUaAqN">(4.4.4)</a></td><td></td><td>(2)<a href="/facts/Truncated_octahedron/GA98b4TH">(4.6.6)</a></td><td></td><td>48</td><td>240</td><td>384</td><td>192</td></tr><tr><th>20</th><td><a href="/facts/Runcitruncated_16-cell/NqDGHQxp">Runcitruncated 16-cell</a>Prismatorhombated tesseract (prit)</td><td></td><td>t0,1,3{3,3,4}</td><td>(1)<a href="/facts/Rhombicuboctahedron/TmxUmEnl">(3.4.4.4)</a></td><td>(1)<a href="/facts/Cube/65lUaAqN">(4.4.4)</a></td><td>(2)<a href="/facts/Hexagonal_prism/oxYc0nF2">(4.4.6)</a></td><td>(1)<a href="/facts/Truncated_tetrahedron/K9ItvmUK">(3.6.6)</a></td><td></td><td>80</td><td>368</td><td>480</td><td>192</td></tr><tr><th>[21]</th><td><i><a href="/facts/Omnitruncated_tesseract/NqDGHQxp">Omnitruncated 16-cell</a></i>(also <i>omnitruncated tesseract</i>) (gidpith)</td><td></td><td>t0,1,2,3{3,3,4}</td><td>(1)<a href="/facts/Truncated_cuboctahedron/nd2ltxcm">(4.6.8)</a></td><td>(1)<a href="/facts/Octagonal_prism/pMnX2f7F">(4.4.8)</a></td><td>(1)<a href="/facts/Hexagonal_prism/oxYc0nF2">(4.4.6)</a></td><td>(1)<a href="/facts/Truncated_octahedron/GA98b4TH">(4.6.6)</a></td><td></td><td>80</td><td>464</td><td>768</td><td>384</td></tr><tr><th>[31]</th><td><i>alternated cantitruncated 16-cell</i>(Same as the <a href="/facts/Snub_24-cell/WbEJ3cPQ">snub 24-cell</a>) (sadi)</td><td></td><td>sr{3,3,4}</td><td>(1)<a href="/facts/Regular_icosahedron/1U93vmXV">(3.3.3.3.3)</a></td><td></td><td>(1)<a href="/facts/Tetrahedron/7mkSrl6j">(3.3.3)</a></td><td>(2)<a href="/facts/Regular_icosahedron/1U93vmXV">(3.3.3.3.3)</a></td><td>(4)<a href="/facts/Tetrahedron/7mkSrl6j">(3.3.3)</a></td><td>144</td><td>480</td><td>432</td><td>96</td></tr><tr><th>Nonuniform</th><td>Runcic snub rectified 16-cellPyritosnub tesseract (pysnet)</td><td></td><td>sr3{3,3,4}</td><td>(1)<a href="/facts/Rhombicuboctahedron/TmxUmEnl">(3.4.4.4)</a></td><td>(2)<a href="/facts/Triangular_prism/Yjp4Kw9M">(3.4.4)</a></td><td>(1)<a href="/facts/Cube/65lUaAqN">(4.4.4)</a></td><td>(1)<a href="/facts/Regular_icosahedron/1U93vmXV">(3.3.3.3.3)</a></td><td>(2)<a href="/facts/Triangular_prism/Yjp4Kw9M">(3.4.4)</a></td><td>176</td><td>656</td><td>672</td><td>192</td></tr></tbody></table>
(*) Just as rectifying the <a href="/facts/Tetrahedron/7mkSrl6j">tetrahedron</a> produces the <a href="/facts/Octahedron/G38q92xW">octahedron</a>, rectifying the 16-cell produces the 24-cell, the regular member of the following family.
<p>The <i>snub 24-cell</i> is repeat to this family for completeness. It is an alternation of the <i>cantitruncated 16-cell</i> or <i>truncated 24-cell</i>, with the half symmetry group [(3,3)+,4]. The truncated octahedral cells become icosahedra. The cubes becomes tetrahedra, and 96 new tetrahedra are created in the gaps from the removed vertices.
</p>
<h3>The F4 family</h3>
<p class="note">Further information: <a href="/facts/F4_polytope/DYg3Rz3d">F4 polytope</a></p>
<p>This family has <a href="/facts/Icositetrachoric_symmetry/FtoaKNqe"><i>diploid icositetrachoric</i> symmetry</a>,<a class="footnote-ref" id="fnref:24" href="#fn:24"><sup>24</sup></a> [3,4,3], of <a href="/facts/Symmetry_order/tmtq9goq">order</a> 24×48=1152: the 48 symmetries of the octahedron for each of the 24 cells. There are 3 small index subgroups, with the first two isomorphic pairs generating uniform 4-polytopes which are also repeated in other families, [3+,4,3], [3,4,3+], and [3,4,3]+, all order 576.
</p>
[3,4,3] uniform 4-polytopes<table><tbody><tr><th rowspan="2">#</th><th rowspan="2">Name</th><th rowspan="2"><a href="/facts/Vertex_figure/3u2ySK0C">Vertexfigure</a></th><th rowspan="2"><a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter diagram</a>and <a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli</a>symbols</th><th colspan="4">Cell counts by location</th><th colspan="4">Element counts</th></tr><tr><th>Pos. 3(24)</th><th>Pos. 2(96)</th><th>Pos. 1(96)</th><th>Pos. 0(24)</th><th>Cells</th><th>Faces</th><th>Edges</th><th>Vertices</th></tr><tr><th>22</th><td><a href="/facts/24-cell/tfoGnlAD">24-cell</a>(Same as <i>rectified 16-cell</i>)Icositetrachoron<a class="footnote-ref" id="fnref:25" href="#fn:25"><sup>25</sup></a> (ico)</td><td></td><td>{3,4,3}</td><td>(6)<a href="/facts/Octahedron/G38q92xW">(3.3.3.3)</a></td><td></td><td></td><td></td><td>24</td><td>96</td><td>96</td><td>24</td></tr><tr><th>23</th><td><a href="/facts/Rectified_24-cell/YJm98yGE">rectified 24-cell</a>(Same as <i>cantellated 16-cell</i>)Rectified icositetrachoron (rico)</td><td></td><td>r{3,4,3}</td><td>(3)<a href="/facts/Cuboctahedron/hD7p0Mem">(3.4.3.4)</a></td><td></td><td></td><td>(2)<a href="/facts/Cube/65lUaAqN">(4.4.4)</a></td><td>48</td><td>240</td><td>288</td><td>96</td></tr><tr><th>24</th><td><a href="/facts/Truncated_24-cell/1dnQysxX">truncated 24-cell</a>(Same as <i>cantitruncated 16-cell</i>)Truncated icositetrachoron (tico)</td><td></td><td>t{3,4,3}</td><td>(3)<a href="/facts/Truncated_octahedron/GA98b4TH">(4.6.6)</a></td><td></td><td></td><td>(1)<a href="/facts/Cube/65lUaAqN">(4.4.4)</a></td><td>48</td><td>240</td><td>384</td><td>192</td></tr><tr><th>25</th><td><a href="/facts/Cantellated_24-cell/vjuEmTjU">cantellated 24-cell</a>Small rhombated icositetrachoron (srico)</td><td></td><td>rr{3,4,3}</td><td>(2)<a href="/facts/Small_rhombicuboctahedron/TmxUmEnl">(3.4.4.4)</a></td><td></td><td>(2)<a href="/facts/Triangular_prism/Yjp4Kw9M">(3.4.4)</a></td><td>(1)<a href="/facts/Cuboctahedron/hD7p0Mem">(3.4.3.4)</a></td><td>144</td><td>720</td><td>864</td><td>288</td></tr><tr><th>28</th><td><a href="/facts/Cantitruncated_24-cell/vjuEmTjU">cantitruncated 24-cell</a>Great rhombated icositetrachoron (grico)</td><td></td><td>tr{3,4,3}</td><td>(2)<a href="/facts/Truncated_cuboctahedron/nd2ltxcm">(4.6.8)</a></td><td></td><td>(1)<a href="/facts/Triangular_prism/Yjp4Kw9M">(3.4.4)</a></td><td>(1)<a href="/facts/Truncated_cube/XVAUG4fD">(3.8.8)</a></td><td>144</td><td>720</td><td>1152</td><td>576</td></tr><tr><th>29</th><td><a href="/facts/Runcitruncated_24-cell/buMulGIL">runcitruncated 24-cell</a>Prismatorhombated icositetrachoron (prico)</td><td></td><td>t0,1,3{3,4,3}</td><td>(1)<a href="/facts/Truncated_octahedron/GA98b4TH">(4.6.6)</a></td><td>(2)<a href="/facts/Hexagonal_prism/oxYc0nF2">(4.4.6)</a></td><td>(1)<a href="/facts/Triangular_prism/Yjp4Kw9M">(3.4.4)</a></td><td>(1)<a href="/facts/Small_rhombicuboctahedron/TmxUmEnl">(3.4.4.4)</a></td><td>240</td><td>1104</td><td>1440</td><td>576</td></tr></tbody></table>
[3+,4,3] uniform 4-polytopes<table><tbody><tr><th rowspan="2">#</th><th rowspan="2">Name</th><th rowspan="2"><a href="/facts/Vertex_figure/3u2ySK0C">Vertexfigure</a></th><th rowspan="2"><a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter diagram</a>and <a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli</a>symbols</th><th colspan="5">Cell counts by location</th><th colspan="4">Element counts</th></tr><tr><th>Pos. 3(24)</th><th>Pos. 2(96)</th><th>Pos. 1(96)</th><th>Pos. 0(24)</th><th>Alt</th><th>Cells</th><th>Faces</th><th>Edges</th><th>Vertices</th></tr><tr><th>31</th><td>†<a href="/facts/Snub_24-cell/WbEJ3cPQ">snub 24-cell</a>Snub disicositetrachoron (sadi)</td><td></td><td>s{3,4,3}</td><td>(3)<a href="/facts/Regular_icosahedron/1U93vmXV">(3.3.3.3.3)</a></td><td></td><td></td><td>(1)<a href="/facts/Tetrahedron/7mkSrl6j">(3.3.3)</a></td><td>(4)<a href="/facts/Tetrahedron/7mkSrl6j">(3.3.3)</a></td><td>144</td><td>480</td><td>432</td><td>96</td></tr><tr><th>Nonuniform</th><td><a href="/facts/Runcic_snub_24-cell/buMulGIL">runcic snub 24-cell</a>Prismatorhombisnub icositetrachoron (prissi)</td><td></td><td>s3{3,4,3}</td><td>(1)<a href="/facts/Regular_icosahedron/1U93vmXV">(3.3.3.3.3)</a></td><td>(2)<a href="/facts/Triangular_prism/Yjp4Kw9M">(3.4.4)</a></td><td></td><td>(1)<a href="/facts/Truncated_tetrahedron/K9ItvmUK">(3.6.6)</a></td><td>(3)<a href="/facts/Triangular_cupola/Pyocot8L">Tricup</a></td><td>240</td><td>960</td><td>1008</td><td>288</td></tr><tr><th>[25]</th><td>cantic snub 24-cell(Same as <a href="/facts/Cantellated_24-cell/vjuEmTjU">cantellated 24-cell</a>) (srico)</td><td></td><td>s2{3,4,3}</td><td>(2)<a href="/facts/Small_rhombicuboctahedron/TmxUmEnl">(3.4.4.4)</a></td><td></td><td></td><td>(1)<a href="/facts/Cuboctahedron/hD7p0Mem">(3.4.3.4)</a></td><td>(2)<a href="/facts/Triangular_prism/Yjp4Kw9M">(3.4.4)</a></td><td>144</td><td>720</td><td>864</td><td>288</td></tr><tr><th>[29]</th><td>runcicantic snub 24-cell(Same as <a href="/facts/Runcitruncated_24-cell/buMulGIL">runcitruncated 24-cell</a>) (prico)</td><td></td><td>s2,3{3,4,3}</td><td>(1)<a href="/facts/Truncated_octahedron/GA98b4TH">(4.6.6)</a></td><td></td><td>(1)<a href="/facts/Triangular_prism/Yjp4Kw9M">(3.4.4)</a></td><td>(1)<a href="/facts/Small_rhombicuboctahedron/TmxUmEnl">(3.4.4.4)</a></td><td>(2)<a href="/facts/Hexagonal_prism/oxYc0nF2">(4.4.6)</a></td><td>240</td><td>1104</td><td>1440</td><td>576</td></tr></tbody></table>
(†) The snub 24-cell here, despite its common name, is not analogous to the <a href="/facts/Snub_cube/pbnr40gk">snub cube</a>; rather, it is derived by an <a href="/facts/Alternation_(geometry)/H2nVs4xj">alternation</a> of the truncated 24-cell. Its <a href="/facts/Symmetry_number/tmtq9goq">symmetry number</a> is only 576, (the <i>ionic diminished icositetrachoric</i> group, [3+,4,3]).
<p>Like the 5-cell, the 24-cell is self-dual, and so the following three forms have twice as many symmetries, bringing their total to 2304 (<a href="/facts/Icositetrachoric_symmetry/FtoaKNqe">extended icositetrachoric symmetry</a> [[3,4,3]]).
</p>
[[3,4,3]] uniform 4-polytopes<table><tbody><tr><th rowspan="2">#</th><th rowspan="2">Name</th><th rowspan="2"><a href="/facts/Vertex_figure/3u2ySK0C">Vertexfigure</a></th><th rowspan="2"><a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter diagram</a>and <a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli</a>symbols</th><th colspan="2">Cell counts by location</th><th colspan="4">Element counts</th></tr><tr><th>Pos. 3-0(48)</th><th>Pos. 2-1(192)</th><th>Cells</th><th>Faces</th><th>Edges</th><th>Vertices</th></tr><tr><th>26</th><td><a href="/facts/Runcinated_24-cell/buMulGIL">runcinated 24-cell</a>Small prismatotetracontoctachoron (spic)</td><td></td><td>t0,3{3,4,3}</td><td>(2)<a href="/facts/Octahedron/G38q92xW">(3.3.3.3)</a></td><td>(6)<a href="/facts/Triangular_prism/Yjp4Kw9M">(3.4.4)</a></td><td>240</td><td>672</td><td>576</td><td>144</td></tr><tr><th>27</th><td><a href="/facts/Bitruncated_24-cell/1dnQysxX">bitruncated 24-cell</a>Tetracontoctachoron (cont)</td><td></td><td>2t{3,4,3}</td><td>(4)<a href="/facts/Truncated_cube/XVAUG4fD">(3.8.8)</a></td><td></td><td>48</td><td>336</td><td>576</td><td>288</td></tr><tr><th>30</th><td><a href="/facts/Omnitruncated_24-cell/buMulGIL">omnitruncated 24-cell</a>Great prismatotetracontoctachoron (gippic)</td><td></td><td>t0,1,2,3{3,4,3}</td><td>(2)<a href="/facts/Truncated_cuboctahedron/nd2ltxcm">(4.6.8)</a></td><td>(2)<a href="/facts/Hexagonal_prism/oxYc0nF2">(4.4.6)</a></td><td>240</td><td>1392</td><td>2304</td><td>1152</td></tr></tbody></table>
[[3,4,3]]+ isogonal 4-polytope<table><tbody><tr><th rowspan="2">#</th><th rowspan="2">Name</th><th rowspan="2"><a href="/facts/Vertex_figure/3u2ySK0C">Vertexfigure</a></th><th rowspan="2"><a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter diagram</a>and <a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli</a>symbols</th><th colspan="3">Cell counts by location</th><th colspan="4">Element counts</th></tr><tr><th>Pos. 3-0(48)</th><th>Pos. 2-1(192)</th><th>Alt</th><th>Cells</th><th>Faces</th><th>Edges</th><th>Vertices</th></tr><tr><th>Nonuniform</th><td><a href="/facts/Omnisnub_24-cell/buMulGIL">omnisnub 24-cell</a>Snub tetracontoctachoron (snoc)Snub icositetrachoron (sni)<a class="footnote-ref" id="fnref:26" href="#fn:26"><sup>26</sup></a></td><td></td><td>ht0,1,2,3{3,4,3}</td><td>(2)<a href="/facts/Snub_cube/pbnr40gk">(3.3.3.3.4)</a></td><td>(2)<a href="/facts/Octahedron/G38q92xW">(3.3.3.3)</a></td><td>(4)<a href="/facts/Tetrahedron/7mkSrl6j">(3.3.3)</a></td><td>816</td><td>2832</td><td>2592</td><td>576</td></tr></tbody></table>
<h3>The H4 family</h3>
<p class="note">Further information: <a href="/facts/H4_polytope/kAjXKGk2">H4 polytope</a></p>
<p>This family has <a href="/facts/Hexacosichoric_symmetry/FtoaKNqe"><i>diploid hexacosichoric</i> symmetry</a>,<a class="footnote-ref" id="fnref:27" href="#fn:27"><sup>27</sup></a> [5,3,3], of <a href="/facts/Symmetry_order/tmtq9goq">order</a> 120×120=24×600=14400: 120 for each of the 120 dodecahedra, or 24 for each of the 600 tetrahedra. There is one small index subgroups [5,3,3]+, all order 7200.
</p>
<h4>120-cell truncations</h4>
<table><tbody><tr><th rowspan="2">#</th><th rowspan="2">Name(Bowers name and acronym)</th><th rowspan="2"><a href="/facts/Vertex_figure/3u2ySK0C">Vertexfigure</a></th><th rowspan="2"><a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter diagram</a>and <a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli</a>symbols</th><th colspan="5">Cell counts by location</th><th colspan="4">Element counts</th></tr><tr><th>Pos. 3(120)</th><th>Pos. 2(720)</th><th>Pos. 1(1200)</th><th>Pos. 0(600)</th><th>Alt</th><th>Cells</th><th>Faces</th><th>Edges</th><th>Vertices</th></tr><tr><th>32</th><td><a href="/facts/120-cell/W4nroBjJ">120-cell</a>(hecatonicosachoron or dodecacontachoron)<a class="footnote-ref" id="fnref:28" href="#fn:28"><sup>28</sup></a>Hecatonicosachoron (hi)</td><td></td><td>{5,3,3}</td><td>(4)<a href="/facts/Dodecahedron/7r519eCl">(5.5.5)</a></td><td></td><td></td><td></td><td></td><td>120</td><td>720</td><td>1200</td><td>600</td></tr><tr><th>33</th><td><a href="/facts/Rectified_120-cell/Sre0krvH">rectified 120-cell</a>Rectified hecatonicosachoron (rahi)</td><td></td><td>r{5,3,3}</td><td>(3)<a href="/facts/Icosidodecahedron/P0JluaVA">(3.5.3.5)</a></td><td></td><td></td><td>(2)<a href="/facts/Tetrahedron/7mkSrl6j">(3.3.3)</a></td><td></td><td>720</td><td>3120</td><td>3600</td><td>1200</td></tr><tr><th>36</th><td><a href="/facts/Truncated_120-cell/FMfWH0Gb">truncated 120-cell</a>Truncated hecatonicosachoron (thi)</td><td></td><td>t{5,3,3}</td><td>(3)<a href="/facts/Truncated_dodecahedron/PVTAsUWQ">(3.10.10)</a></td><td></td><td></td><td>(1)<a href="/facts/Tetrahedron/7mkSrl6j">(3.3.3)</a></td><td></td><td>720</td><td>3120</td><td>4800</td><td>2400</td></tr><tr><th>37</th><td><a href="/facts/Cantellated_120-cell/2AQNHrKQ">cantellated 120-cell</a>Small rhombated hecatonicosachoron (srahi)</td><td></td><td>rr{5,3,3}</td><td>(2)<a href="/facts/Small_rhombicosidodecahedron/AaDckLGP">(3.4.5.4)</a></td><td></td><td>(2)<a href="/facts/Triangular_prism/Yjp4Kw9M">(3.4.4)</a></td><td>(1)<a href="/facts/Octahedron/G38q92xW">(3.3.3.3)</a></td><td></td><td>1920</td><td>9120</td><td>10800</td><td>3600</td></tr><tr><th>38</th><td><i><a href="/facts/Runcinated_120-cell/3yzQlUMW">runcinated 120-cell</a></i>(also <i>runcinated 600-cell</i>)Small disprismatohexacosihecatonicosachoron (sidpixhi)</td><td></td><td>t0,3{5,3,3}</td><td>(1)<a href="/facts/Dodecahedron/7r519eCl">(5.5.5)</a></td><td>(3)<a href="/facts/Pentagonal_prism/CtvKLxcy">(4.4.5)</a></td><td>(3)<a href="/facts/Triangular_prism/Yjp4Kw9M">(3.4.4)</a></td><td>(1)<a href="/facts/Tetrahedron/7mkSrl6j">(3.3.3)</a></td><td></td><td>2640</td><td>7440</td><td>7200</td><td>2400</td></tr><tr><th>39</th><td><i><a href="/facts/Bitruncated_120-cell/FMfWH0Gb">bitruncated 120-cell</a></i>(also <i>bitruncated 600-cell</i>)Hexacosihecatonicosachoron (xhi)</td><td></td><td>2t{5,3,3}</td><td>(2)<a href="/facts/Truncated_icosahedron/LR8ugaY9">(5.6.6)</a></td><td></td><td></td><td>(2)<a href="/facts/Truncated_tetrahedron/K9ItvmUK">(3.6.6)</a></td><td></td><td>720</td><td>4320</td><td>7200</td><td>3600</td></tr><tr><th>42</th><td><a href="/facts/Cantitruncated_120-cell/2AQNHrKQ">cantitruncated 120-cell</a>Great rhombated hecatonicosachoron (grahi)</td><td></td><td>tr{5,3,3}</td><td>(2)<a href="/facts/Truncated_icosidodecahedron/d5v9VpFY">(4.6.10)</a></td><td></td><td>(1)<a href="/facts/Triangular_prism/Yjp4Kw9M">(3.4.4)</a></td><td>(1)<a href="/facts/Truncated_tetrahedron/K9ItvmUK">(3.6.6)</a></td><td></td><td>1920</td><td>9120</td><td>14400</td><td>7200</td></tr><tr><th>43</th><td><a href="/facts/Runcitruncated_120-cell/3yzQlUMW">runcitruncated 120-cell</a>Prismatorhombated hexacosichoron (prix)</td><td></td><td>t0,1,3{5,3,3}</td><td>(1)<a href="/facts/Truncated_dodecahedron/PVTAsUWQ">(3.10.10)</a></td><td>(2)<a href="/facts/Decagonal_prism/dIjtR9zW">(4.4.10)</a></td><td>(1)<a href="/facts/Triangular_prism/Yjp4Kw9M">(3.4.4)</a></td><td>(1)<a href="/facts/Cuboctahedron/hD7p0Mem">(3.4.3.4)</a></td><td></td><td>2640</td><td>13440</td><td>18000</td><td>7200</td></tr><tr><th>46</th><td><i><a href="/facts/Omnitruncated_120-cell/3yzQlUMW">omnitruncated 120-cell</a></i>(also <i>omnitruncated 600-cell</i>)Great disprismatohexacosihecatonicosachoron (gidpixhi)</td><td></td><td>t0,1,2,3{5,3,3}</td><td>(1)<a href="/facts/Truncated_icosidodecahedron/d5v9VpFY">(4.6.10)</a></td><td>(1)<a href="/facts/Decagonal_prism/dIjtR9zW">(4.4.10)</a></td><td>(1)<a href="/facts/Hexagonal_prism/oxYc0nF2">(4.4.6)</a></td><td>(1)<a href="/facts/Truncated_octahedron/GA98b4TH">(4.6.6)</a></td><td></td><td>2640</td><td>17040</td><td>28800</td><td>14400</td></tr><tr><th>Nonuniform</th><td><a href="/facts/Omnisnub_120-cell/3yzQlUMW">omnisnub 120-cell</a>Snub hecatonicosachoron (snixhi)<a class="footnote-ref" id="fnref:29" href="#fn:29"><sup>29</sup></a>(Same as the <i>omnisnub 600-cell</i>)</td><td></td><td>ht0,1,2,3{5,3,3}</td><td> (1)<a href="/facts/Snub_dodecahedron/oouEPVxD">(3.3.3.3.5)</a></td><td> (1)<a href="/facts/Pentagonal_antiprism/gR8aMBKN">(3.3.3.5)</a></td><td> (1)<a href="/facts/Triangular_antiprism/G38q92xW">(3.3.3.3)</a></td><td> (1)<a href="/facts/Snub_tetrahedron/1U93vmXV">(3.3.3.3.3)</a></td><td> (4)<a href="/facts/Tetrahedron/7mkSrl6j">(3.3.3)</a></td><td>9840</td><td>35040</td><td>32400</td><td>7200</td></tr></tbody></table>
<h4>600-cell truncations</h4>
<table><tbody><tr><th rowspan="2">#</th><th rowspan="2">Name(Bowers style acronym)</th><th rowspan="2"><a href="/facts/Vertex_figure/3u2ySK0C">Vertexfigure</a></th><th rowspan="2"><a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter diagram</a>and <a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli</a>symbols</th><th rowspan="2">Symmetry</th><th colspan="4">Cell counts by location</th><th colspan="4">Element counts</th></tr><tr><th>Pos. 3(120)</th><th>Pos. 2(720)</th><th>Pos. 1(1200)</th><th>Pos. 0(600)</th><th>Cells</th><th>Faces</th><th>Edges</th><th>Vertices</th></tr><tr><th>35</th><td><a href="/facts/600-cell/tNBtDUe9">600-cell</a>Hexacosichoron<a class="footnote-ref" id="fnref:30" href="#fn:30"><sup>30</sup></a> (ex)</td><td></td><td>{3,3,5}</td><td>[5,3,3]order 14400</td><td></td><td></td><td></td><td>(20)<a href="/facts/Tetrahedron/7mkSrl6j">(3.3.3)</a></td><td>600</td><td>1200</td><td>720</td><td>120</td></tr><tr><th>[47]</th><td><a href="/facts/600-cell/tNBtDUe9">20-diminished 600-cell</a>= <a href="/facts/Grand_antiprism/aohzsR7w">Grand antiprism</a> (gap)</td><td></td><td>Nonwythoffianconstruction</td><td>[[10,2+,10]]order 400Index 36</td><td>(2)<a href="/facts/Pentagonal_antiprism/gR8aMBKN">(3.3.3.5)</a></td><td></td><td></td><td>(12)<a href="/facts/Tetrahedron/7mkSrl6j">(3.3.3)</a></td><td>320</td><td>720</td><td>500</td><td>100</td></tr><tr><th>[31]</th><td><a href="/facts/600-cell/tNBtDUe9">24-diminished 600-cell</a>= <a href="/facts/Snub_24-cell/WbEJ3cPQ">Snub 24-cell</a> (sadi)</td><td></td><td>Nonwythoffianconstruction</td><td>[3+,4,3]order 576index 25</td><td>(3)<a href="/facts/Regular_icosahedron/1U93vmXV">(3.3.3.3.3)</a></td><td></td><td></td><td>(5)<a href="/facts/Tetrahedron/7mkSrl6j">(3.3.3)</a></td><td>144</td><td>480</td><td>432</td><td>96</td></tr><tr><th>Nonuniform</th><td><a href="/facts/600-cell/tNBtDUe9">bi-24-diminished 600-cell</a>Bi-icositetradiminished hexacosichoron (bidex)</td><td></td><td>Nonwythoffianconstruction</td><td>order 144index 100</td><td>(6)<a href="/facts/Tridiminished_icosahedron/gCVHC0zw">tdi</a></td><td></td><td></td><td></td><td>48</td><td>192</td><td>216</td><td>72</td></tr><tr><th>34</th><td><a href="/facts/Rectified_600-cell/S2DeX7LG">rectified 600-cell</a>Rectified hexacosichoron (rox)</td><td></td><td>r{3,3,5}</td><td>[5,3,3]</td><td>(2)<a href="/facts/Regular_icosahedron/1U93vmXV">(3.3.3.3.3)</a></td><td></td><td></td><td>(5)<a href="/facts/Octahedron/G38q92xW">(3.3.3.3)</a></td><td>720</td><td>3600</td><td>3600</td><td>720</td></tr><tr><th>Nonuniform</th><td><a href="/facts/Rectified_600-cell/S2DeX7LG">120-diminished rectified 600-cell</a>Swirlprismatodiminished rectified hexacosichoron (spidrox)</td><td></td><td>Nonwythoffianconstruction</td><td>order 1200index 12</td><td>(2)<a href="/facts/Pentagonal_antiprism/gR8aMBKN">3.3.3.5</a></td><td>(2)<a href="/facts/Pentagonal_prism/CtvKLxcy">4.4.5</a></td><td></td><td>(5)<a href="/facts/Square_pyramid/wDetDohM">P4</a></td><td>840</td><td>2640</td><td>2400</td><td>600</td></tr><tr><th>41</th><td><a href="/facts/Truncated_600-cell/FMfWH0Gb">truncated 600-cell</a>Truncated hexacosichoron (tex)</td><td></td><td>t{3,3,5}</td><td>[5,3,3]</td><td>(1)<a href="/facts/Regular_icosahedron/1U93vmXV">(3.3.3.3.3)</a></td><td></td><td></td><td>(5)<a href="/facts/Truncated_tetrahedron/K9ItvmUK">(3.6.6)</a></td><td>720</td><td>3600</td><td>4320</td><td>1440</td></tr><tr><th>40</th><td><a href="/facts/Cantellated_600-cell/2AQNHrKQ">cantellated 600-cell</a>Small rhombated hexacosichoron (srix)</td><td></td><td>rr{3,3,5}</td><td>[5,3,3]</td><td>(1)<a href="/facts/Icosidodecahedron/P0JluaVA">(3.5.3.5)</a></td><td>(2)<a href="/facts/Pentagonal_prism/CtvKLxcy">(4.4.5)</a></td><td></td><td>(1)<a href="/facts/Cuboctahedron/hD7p0Mem">(3.4.3.4)</a></td><td>1440</td><td>8640</td><td>10800</td><td>3600</td></tr><tr><th>[38]</th><td><i><a href="/facts/Runcinated_600-cell/3yzQlUMW">runcinated 600-cell</a></i>(also <i>runcinated 120-cell</i>) (sidpixhi)</td><td></td><td>t0,3{3,3,5}</td><td>[5,3,3]</td><td>(1)<a href="/facts/Dodecahedron/7r519eCl">(5.5.5)</a></td><td>(3)<a href="/facts/Pentagonal_prism/CtvKLxcy">(4.4.5)</a></td><td>(3)<a href="/facts/Triangular_prism/Yjp4Kw9M">(3.4.4)</a></td><td>(1)<a href="/facts/Tetrahedron/7mkSrl6j">(3.3.3)</a></td><td>2640</td><td>7440</td><td>7200</td><td>2400</td></tr><tr><th>[39]</th><td><i><a href="/facts/Bitruncated_600-cell/FMfWH0Gb">bitruncated 600-cell</a></i>(also <i>bitruncated 120-cell</i>) (xhi)</td><td></td><td>2t{3,3,5}</td><td>[5,3,3]</td><td>(2)<a href="/facts/Truncated_icosahedron/LR8ugaY9">(5.6.6)</a></td><td></td><td></td><td>(2)<a href="/facts/Truncated_tetrahedron/K9ItvmUK">(3.6.6)</a></td><td>720</td><td>4320</td><td>7200</td><td>3600</td></tr><tr><th>45</th><td><a href="/facts/Cantitruncated_600-cell/2AQNHrKQ">cantitruncated 600-cell</a>Great rhombated hexacosichoron (grix)</td><td></td><td>tr{3,3,5}</td><td>[5,3,3]</td><td>(1)<a href="/facts/Truncated_icosahedron/LR8ugaY9">(5.6.6)</a></td><td>(1)<a href="/facts/Pentagonal_prism/CtvKLxcy">(4.4.5)</a></td><td></td><td>(2)<a href="/facts/Truncated_octahedron/GA98b4TH">(4.6.6)</a></td><td>1440</td><td>8640</td><td>14400</td><td>7200</td></tr><tr><th>44</th><td><a href="/facts/Runcitruncated_600-cell/3yzQlUMW">runcitruncated 600-cell</a>Prismatorhombated hecatonicosachoron (prahi)</td><td></td><td>t0,1,3{3,3,5}</td><td>[5,3,3]</td><td>(1)<a href="/facts/Small_rhombicosidodecahedron/AaDckLGP">(3.4.5.4)</a></td><td>(1)<a href="/facts/Pentagonal_prism/CtvKLxcy">(4.4.5)</a></td><td>(2)<a href="/facts/Hexagonal_prism/oxYc0nF2">(4.4.6)</a></td><td>(1)<a href="/facts/Truncated_tetrahedron/K9ItvmUK">(3.6.6)</a></td><td>2640</td><td>13440</td><td>18000</td><td>7200</td></tr><tr><th>[46]</th><td><i><a href="/facts/Omnitruncated_600-cell/3yzQlUMW">omnitruncated 600-cell</a></i>(also <i>omnitruncated 120-cell</i>) (gidpixhi)</td><td></td><td>t0,1,2,3{3,3,5}</td><td>[5,3,3]</td><td>(1)<a href="/facts/Truncated_icosidodecahedron/d5v9VpFY">(4.6.10)</a></td><td>(1)<a href="/facts/Decagonal_prism/dIjtR9zW">(4.4.10)</a></td><td>(1)<a href="/facts/Hexagonal_prism/oxYc0nF2">(4.4.6)</a></td><td>(1)<a href="/facts/Truncated_octahedron/GA98b4TH">(4.6.6)</a></td><td>2640</td><td>17040</td><td>28800</td><td>14400</td></tr></tbody></table>
<h3>The D4 family</h3>
<p class="note">Further information: <a href="/facts/D4_polytope/YwrboFJR">D4 polytope</a></p>
<p>This <a href="/facts/Demitesseractic_symmetry/FtoaKNqe">demitesseract family</a>, [31,1,1], introduces no new uniform 4-polytopes, but it is worthy to repeat these alternative constructions. This family has <a href="/facts/Symmetry_order/tmtq9goq">order</a> 12×16=192: 4!/2=12 permutations of the four axes, half as alternated, 24=16 for reflection in each axis. There is one small index subgroups that generating uniform 4-polytopes, [31,1,1]+, order 96.
</p>
[31,1,1] uniform 4-polytopes<table><tbody><tr><th rowspan="2">#</th><th rowspan="2">Name (Bowers style acronym)</th><th rowspan="2"><a href="/facts/Vertex_figure/3u2ySK0C">Vertexfigure</a></th><th rowspan="2"><a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter diagram</a> =  = </th><th colspan="5">Cell counts by location</th><th colspan="4">Element counts</th></tr><tr><th>Pos. 0(8)</th><th>Pos. 2(24)</th><th>Pos. 1(8)</th><th>Pos. 3(8)</th><th>Pos. Alt(96)</th><th>3</th><th>2</th><th>1</th><th>0</th></tr><tr><th>[12]</th><td>demitesseracthalf tesseract(Same as <i><a href="/facts/16-cell/X0YAfobw">16-cell</a></i>) (hex)</td><td></td><td> = h{4,3,3}</td><td></td><td></td><td>(4)<a href="/facts/Tetrahedron/7mkSrl6j">(3.3.3)</a></td><td>(4)<a href="/facts/Tetrahedron/7mkSrl6j">(3.3.3)</a></td><td></td><td>16</td><td>32</td><td>24</td><td>8</td></tr><tr><th>[17]</th><td>cantic tesseract(Same as <i><a href="/facts/Truncated_16-cell/ShQzMKoa">truncated 16-cell</a></i>) (thex)</td><td></td><td> = h2{4,3,3}</td><td>(1)<a href="/facts/Octahedron/G38q92xW">(3.3.3.3)</a></td><td></td><td>(2)<a href="/facts/Truncated_tetrahedron/K9ItvmUK">(3.6.6)</a></td><td>(2)<a href="/facts/Truncated_tetrahedron/K9ItvmUK">(3.6.6)</a></td><td></td><td>24</td><td>96</td><td>120</td><td>48</td></tr><tr><th>[11]</th><td>runcic tesseract(Same as <i><a href="/facts/Rectified_tesseract/xfSdmduC">rectified tesseract</a></i>) (rit)</td><td></td><td> = h3{4,3,3}</td><td>(1)<a href="/facts/Tetrahedron/7mkSrl6j">(3.3.3)</a></td><td></td><td>(1)<a href="/facts/Tetrahedron/7mkSrl6j">(3.3.3)</a></td><td>(3)<a href="/facts/Cuboctahedron/hD7p0Mem">(3.4.3.4)</a></td><td></td><td>24</td><td>88</td><td>96</td><td>32</td></tr><tr><th>[16]</th><td>runcicantic tesseract(Same as <i><a href="/facts/Bitruncated_tesseract/ShQzMKoa">bitruncated tesseract</a></i>) (tah)</td><td></td><td> = h2,3{4,3,3}</td><td>(1)<a href="/facts/Truncated_tetrahedron/K9ItvmUK">(3.6.6)</a></td><td></td><td>(1)<a href="/facts/Truncated_tetrahedron/K9ItvmUK">(3.6.6)</a></td><td>(2)<a href="/facts/Truncated_octahedron/GA98b4TH">(4.6.6)</a></td><td></td><td>24</td><td>96</td><td>96</td><td>24</td></tr></tbody></table>
<p>When the 3 bifurcated branch nodes are identically ringed, the symmetry can be increased by 6, as [3[31,1,1]] = [3,4,3], and thus these polytopes are repeated from the <a href="/facts/24-cell/tfoGnlAD">24-cell</a> family.
</p>
[3[31,1,1]] uniform 4-polytopes<table><tbody><tr><th rowspan="2">#</th><th rowspan="2">Name (Bowers style acronym)</th><th rowspan="2"><a href="/facts/Vertex_figure/3u2ySK0C">Vertexfigure</a></th><th rowspan="2"><a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter diagram</a> =  = </th><th colspan="3">Cell counts by location</th><th colspan="4">Element counts</th></tr><tr><th>Pos. 0,1,3(24)</th><th>Pos. 2(24)</th><th>Pos. Alt(96)</th><th>3</th><th>2</th><th>1</th><th>0</th></tr><tr><th>[22]</th><td>rectified 16-cell(Same as <i><a href="/facts/24-cell/tfoGnlAD">24-cell</a></i>) (ico)</td><td></td><td> =  =  = {31,1,1} = r{3,3,4} = {3,4,3}</td><td>(6)<a href="/facts/Octahedron/G38q92xW">(3.3.3.3)</a></td><td></td><td></td><td>48</td><td>240</td><td>288</td><td>96</td></tr><tr><th>[23]</th><td>cantellated 16-cell(Same as <i><a href="/facts/Rectified_24-cell/YJm98yGE">rectified 24-cell</a></i>) (rico)</td><td></td><td> =  =  = r{31,1,1} = rr{3,3,4} = r{3,4,3}</td><td>(3)<a href="/facts/Cuboctahedron/hD7p0Mem">(3.4.3.4)</a></td><td>(2)<a href="/facts/Cube/65lUaAqN">(4.4.4)</a></td><td></td><td>24</td><td>120</td><td>192</td><td>96</td></tr><tr><th>[24]</th><td>cantitruncated 16-cell(Same as <i><a href="/facts/Truncated_24-cell/1dnQysxX">truncated 24-cell</a></i>) (tico)</td><td></td><td> =  =  = t{31,1,1} = tr{3,3,4} = t{3,4,3}</td><td>(3)<a href="/facts/Truncated_octahedron/GA98b4TH">(4.6.6)</a></td><td>(1)<a href="/facts/Cube/65lUaAqN">(4.4.4)</a></td><td></td><td>48</td><td>240</td><td>384</td><td>192</td></tr><tr><th>[31]</th><td><a href="/facts/Snub_24-cell/WbEJ3cPQ">snub 24-cell</a> (sadi)</td><td></td><td> =  =  = s{31,1,1} = sr{3,3,4} = s{3,4,3}</td><td>(3)<a href="/facts/Regular_icosahedron/1U93vmXV">(3.3.3.3.3)</a></td><td>(1)<a href="/facts/Tetrahedron/7mkSrl6j">(3.3.3)</a></td><td>(4)<a href="/facts/Tetrahedron/7mkSrl6j">(3.3.3)</a></td><td>144</td><td>480</td><td>432</td><td>96</td></tr></tbody></table>
<p>Here again the <i>snub 24-cell</i>, with the symmetry group [31,1,1]+ this time, represents an alternated truncation of the truncated 24-cell creating 96 new tetrahedra at the position of the deleted vertices. In contrast to its appearance within former groups as partly snubbed 4-polytope, only within this symmetry group it has the full analogy to the Kepler snubs, i.e. the <a href="/facts/Snub_cube/pbnr40gk">snub cube</a> and the <a href="/facts/Snub_dodecahedron/oouEPVxD">snub dodecahedron</a>.
</p>
<h3>The grand antiprism</h3>
<p>There is one non-Wythoffian uniform convex 4-polytope, known as the <a href="/facts/Grand_antiprism/aohzsR7w">grand antiprism</a>, consisting of 20 <a href="/facts/Pentagonal_antiprism/gR8aMBKN">pentagonal antiprisms</a> forming two perpendicular rings joined by 300 <a href="/facts/Tetrahedron/7mkSrl6j">tetrahedra</a>.  It is loosely analogous to the three-dimensional <a href="/facts/Antiprism/FgP5jIVd">antiprisms</a>, which consist of two parallel <a href="/facts/Polygon/yAq7RjPz">polygons</a> joined by a band of <a href="/facts/Triangle/cRXUwsMn">triangles</a>. Unlike them, however, the grand antiprism is not a member of an infinite family of uniform polytopes.
</p><p>Its symmetry is the <i>ionic diminished Coxeter group</i>, [[10,2+,10]], order 400.
</p>
<table><tbody><tr><th rowspan="2">#</th><th rowspan="2">Name (Bowers style acronym)</th><th rowspan="2"><a href="/facts/Schlegel_diagram/qwsDv1H8">Picture</a></th><th rowspan="2"><a href="/facts/Vertex_figure/3u2ySK0C">Vertexfigure</a></th><th rowspan="2"><a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter diagram</a>and <a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli</a>symbols</th><th rowspan="2" colspan="2">Cells by type</th><th colspan="4">Element counts</th><th rowspan="2">Net</th></tr><tr><th>Cells</th><th>Faces</th><th>Edges</th><th>Vertices</th></tr><tr><th>47</th><th><a href="/facts/Grand_antiprism/aohzsR7w">grand antiprism</a> (gap)</th><td></td><td></td><td>No symbol</td><td>300 (<i><a href="/facts/Tetrahedron/7mkSrl6j">3.3.3</a></i>)</td><td>20 (<i><a href="/facts/Pentagonal_antiprism/gR8aMBKN">3.3.3.5</a></i>)</td><td>320</td><td>20 <a href="/facts/Pentagon/jeHEARHH">{5}</a>700 <a href="/facts/Triangle/cRXUwsMn">{3}</a></td><td>500</td><td>100</td><td></td></tr></tbody></table>
<h3>Prismatic uniform 4-polytopes</h3>
<p>A prismatic polytope is a <a href="/facts/Cartesian_product/BfqBWzdL">Cartesian product</a> of two polytopes of lower dimension; familiar examples are the 3-dimensional <a href="/facts/Prism_(geometry)/baIRRkCy">prisms</a>, which are products of a <a href="/facts/Polygon/yAq7RjPz">polygon</a> and a <a href="/facts/Line_segment/V8E0M4qk">line segment</a>.  The prismatic uniform 4-polytopes consist of two infinite families:
</p>
<ul><li><i>Polyhedral prisms</i>: products of a line segment and a uniform polyhedron.  This family is infinite because it includes prisms built on 3-dimensional prisms and <a href="/facts/Antiprism/FgP5jIVd">antiprisms</a>.</li>
<li><i>Duoprisms</i>: products of two polygons.</li></ul>
<h4>Convex polyhedral prisms</h4>
<p>The most obvious family of prismatic 4-polytopes is the <i>polyhedral prisms,</i> i.e. products of a polyhedron with a <a href="/facts/Line_segment/V8E0M4qk">line segment</a>.  The cells of such a 4-polytopes are two identical uniform polyhedra lying in parallel <a href="/facts/Hyperplane/2s2ycYYd">hyperplanes</a> (the <i>base</i> cells) and a layer of prisms joining them (the <i>lateral</i> cells).  This family includes prisms for the 75 nonprismatic <a href="/facts/Uniform_polyhedron/PHnprqRC">uniform polyhedra</a> (of which 18 are convex; one of these, the cube-prism, is listed above as the <i>tesseract</i>).
</p><p>There are <i>18 convex polyhedral prisms</i> created from 5 <a href="/facts/Platonic_solid/jraREXFD">Platonic solids</a> and 13 <a href="/facts/Archimedean_solid/wYWeHDYT">Archimedean solids</a> as well as for the infinite families of three-dimensional <a href="/facts/Prism_(geometry)/baIRRkCy">prisms</a> and <a href="/facts/Antiprism/FgP5jIVd">antiprisms</a>.  The symmetry number of a polyhedral prism is twice that of the base polyhedron.
</p>
<h4>Tetrahedral prisms: A3 × A1</h4>
<p>This <a href="/facts/Hexadecachoric_symmetry/FtoaKNqe">prismatic tetrahedral symmetry</a> is [3,3,2], order 48. There are two index 2 subgroups, [(3,3)+,2] and [3,3,2]+, but the second doesn't generate a uniform 4-polytope.
</p>
[3,3,2] uniform 4-polytopes<table><tbody><tr><th rowspan="2">#</th><th rowspan="2">Name (Bowers style acronym)</th><th rowspan="2"><a href="/facts/Schlegel_diagram/qwsDv1H8">Picture</a></th><th rowspan="2"><a href="/facts/Vertex_figure/3u2ySK0C">Vertexfigure</a></th><th rowspan="2"><a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter diagram</a>and <a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli</a>symbols</th><th rowspan="2" colspan="3">Cells by type</th><th colspan="4">Element counts</th><th rowspan="2">Net</th></tr><tr><th>Cells</th><th>Faces</th><th>Edges</th><th>Vertices</th></tr><tr><th>48</th><th><a href="/facts/Tetrahedral_prism/oQ720tUe">Tetrahedral prism</a> (tepe)</th><td></td><td></td><td>{3,3}×{ }t0,3{3,3,2}</td><td>2 <a href="/facts/Tetrahedron/7mkSrl6j">3.3.3</a></td><td>4 <a href="/facts/Triangular_prism/Yjp4Kw9M">3.4.4</a></td><td></td><td>6</td><td>8 {3}6 {4}</td><td>16</td><td>8</td><td></td></tr><tr><th>49</th><th><a href="/facts/Truncated_tetrahedral_prism/7Fc4E1FP">Truncated tetrahedral prism</a> (tuttip)</th><td></td><td></td><td>t{3,3}×{ }t0,1,3{3,3,2}</td><td>2 <a href="/facts/Truncated_tetrahedron/K9ItvmUK">3.6.6</a></td><td>4 <a href="/facts/Triangular_prism/Yjp4Kw9M">3.4.4</a></td><td>4 <a href="/facts/Hexagonal_prism/oxYc0nF2">4.4.6</a></td><td>10</td><td>8 {3}18 {4}8 {6}</td><td>48</td><td>24</td><td></td></tr></tbody></table>
[[3,3],2] uniform 4-polytopes<table><tbody><tr><th rowspan="2">#</th><th rowspan="2">Name (Bowers style acronym)</th><th rowspan="2"><a href="/facts/Schlegel_diagram/qwsDv1H8">Picture</a></th><th rowspan="2"><a href="/facts/Vertex_figure/3u2ySK0C">Vertexfigure</a></th><th rowspan="2"><a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter diagram</a>and <a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli</a>symbols</th><th rowspan="2" colspan="3">Cells by type</th><th colspan="4">Element counts</th><th rowspan="2">Net</th></tr><tr><th>Cells</th><th>Faces</th><th>Edges</th><th>Vertices</th></tr><tr><th>[51]</th><td><i>Rectified tetrahedral prism</i>(Same as <a href="/facts/Octahedral_prism/8UVHkxvl">octahedral prism</a>) (ope)</td><td></td><td></td><td>r{3,3}×{ }t1,3{3,3,2}</td><td>2 <a href="/facts/Octahedron/G38q92xW">3.3.3.3</a></td><td>4 <a href="/facts/Triangular_prism/Yjp4Kw9M">3.4.4</a></td><td></td><td>6</td><td>16 {3}12 {4}</td><td>30</td><td>12</td><td></td></tr><tr><th>[50]</th><td><i>Cantellated tetrahedral prism</i>(Same as <a href="/facts/Cuboctahedral_prism/fFab3Nyp">cuboctahedral prism</a>) (cope)</td><td></td><td></td><td>rr{3,3}×{ }t0,2,3{3,3,2}</td><td>2 <a href="/facts/Cuboctahedron/hD7p0Mem">3.4.3.4</a></td><td>8 <a href="/facts/Triangular_prism/Yjp4Kw9M">3.4.4</a></td><td>6 <a href="/facts/Cube/65lUaAqN">4.4.4</a></td><td>16</td><td>16 {3}36 {4}</td><td>60</td><td>24</td><td></td></tr><tr><th>[54]</th><td><i>Cantitruncated tetrahedral prism</i>(Same as <a href="/facts/Truncated_octahedral_prism/gCIURDjm">truncated octahedral prism</a>) (tope)</td><td></td><td></td><td>tr{3,3}×{ }t0,1,2,3{3,3,2}</td><td>2 <a href="/facts/Truncated_octahedron/GA98b4TH">4.6.6</a></td><td>8 <a href="/facts/Hexagonal_prism/oxYc0nF2">6.4.4</a></td><td>6 <a href="/facts/Cube/65lUaAqN">4.4.4</a></td><td>16</td><td>48 {4}16 {6}</td><td>96</td><td>48</td><td></td></tr><tr><th>[59]</th><td><i>Snub tetrahedral prism</i>(Same as <a href="/facts/Icosahedral_prism/mpSGUN0y">icosahedral prism</a>) (ipe)</td><td></td><td></td><td>sr{3,3}×{ }</td><td>2 <a href="/facts/Regular_icosahedron/1U93vmXV">3.3.3.3.3</a></td><td>20 <a href="/facts/Triangular_prism/Yjp4Kw9M">3.4.4</a></td><td></td><td>22</td><td>40 {3}30 {4}</td><td>72</td><td>24</td><td></td></tr><tr><th>Nonuniform</th><td><a href="/facts/Omnisnub_tetrahedral_antiprism/gCIURDjm">omnisnub tetrahedral antiprism</a>Pyritohedral icosahedral antiprism (pikap)</td><td></td><td></td><td>                    s                  {                                                                      3                                                                              3                                                                              2                                                              }                      {\displaystyle s\left\{{\begin{array}{l}3\\3\\2\end{array}}\right\}}  </td><td>2 <a href="/facts/Snub_tetrahedron/1U93vmXV">3.3.3.3.3</a></td><td>8 <a href="/facts/Triangular_antiprism/G38q92xW">3.3.3.3</a></td><td>6+24 <a href="/facts/Tetrahedron/7mkSrl6j">3.3.3</a></td><td>40</td><td>16+96 {3}</td><td>96</td><td>24</td><td></td></tr></tbody></table>
<h4>Octahedral prisms: B3 × A1</h4>
<p>This <a href="/facts/Hexadecachoric_symmetry/FtoaKNqe">prismatic octahedral family symmetry</a> is [4,3,2], order 96. There are 6 subgroups of index 2, order 48 that are expressed in alternated 4-polytopes below. <a href="/facts/Coxeter_notation/ArUw6goe">Symmetries</a> are [(4,3)+,2], [1+,4,3,2], [4,3,2+], [4,3+,2], [4,(3,2)+], and [4,3,2]+.
</p>
<table><tbody><tr><th rowspan="2">#</th><th rowspan="2">Name (Bowers style acronym)</th><th rowspan="2"><a href="/facts/Schlegel_diagram/qwsDv1H8">Picture</a></th><th rowspan="2"><a href="/facts/Vertex_figure/3u2ySK0C">Vertexfigure</a></th><th rowspan="2"><a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter diagram</a>and <a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli</a>symbols</th><th rowspan="2" colspan="4">Cells by type</th><th colspan="4">Element counts</th><th rowspan="2">Net</th></tr><tr><th>Cells</th><th>Faces</th><th>Edges</th><th>Vertices</th></tr><tr><th>[10]</th><td><i>Cubic prism</i>(Same as <i><a href="/facts/Tesseract/tlqouBoU">tesseract</a></i>)(Same as <i>4-4 duoprism</i>) (tes)</td><td></td><td></td><td>{4,3}×{ }t0,3{4,3,2}</td><td>2 <a href="/facts/Cube/65lUaAqN">4.4.4</a></td><td>6 <a href="/facts/Cube/65lUaAqN">4.4.4</a></td><td></td><td></td><td>8</td><td>24 {4}</td><td>32</td><td>16</td><td></td></tr><tr><th>50</th><td><a href="/facts/Cuboctahedral_prism/fFab3Nyp">Cuboctahedral prism</a>(Same as <i>cantellated tetrahedral prism</i>) (cope)</td><td></td><td></td><td>r{4,3}×{ }t1,3{4,3,2}</td><td>2 <a href="/facts/Cuboctahedron/hD7p0Mem">3.4.3.4</a></td><td>8 <a href="/facts/Triangular_prism/Yjp4Kw9M">3.4.4</a></td><td>6 <a href="/facts/Cube/65lUaAqN">4.4.4</a></td><td></td><td>16</td><td>16 {3}36 {4}</td><td>60</td><td>24</td><td></td></tr><tr><th>51</th><td><i><a href="/facts/Octahedral_prism/8UVHkxvl">Octahedral prism</a></i>(Same as <i>rectified tetrahedral prism</i>)(Same as <i>triangular antiprismatic prism</i>) (ope)</td><td></td><td></td><td>{3,4}×{ }t2,3{4,3,2}</td><td>2 <a href="/facts/Octahedron/G38q92xW">3.3.3.3</a></td><td>8 <a href="/facts/Triangular_prism/Yjp4Kw9M">3.4.4</a></td><td></td><td></td><td>10</td><td>16 {3}12 {4}</td><td>30</td><td>12</td><td></td></tr><tr><th>52</th><td><a href="/facts/Rhombicuboctahedral_prism/4MJfAP4R">Rhombicuboctahedral prism</a> (sircope)</td><td></td><td></td><td>rr{4,3}×{ }t0,2,3{4,3,2}</td><td>2 <a href="/facts/Rhombicuboctahedron/TmxUmEnl">3.4.4.4</a></td><td>8 <a href="/facts/Triangular_prism/Yjp4Kw9M">3.4.4</a></td><td>18 <a href="/facts/Cube/65lUaAqN">4.4.4</a></td><td></td><td>28</td><td>16 {3}84 {4}</td><td>120</td><td>48</td><td></td></tr><tr><th>53</th><td><a href="/facts/Truncated_cubic_prism/u7WBPerC">Truncated cubic prism</a> (ticcup)</td><td></td><td></td><td>t{4,3}×{ }t0,1,3{4,3,2}</td><td>2 <a href="/facts/Truncated_cube/XVAUG4fD">3.8.8</a></td><td>8 <a href="/facts/Triangular_prism/Yjp4Kw9M">3.4.4</a></td><td>6 <a href="/facts/Octagonal_prism/pMnX2f7F">4.4.8</a></td><td></td><td>16</td><td>16 {3}36 {4}12 {8}</td><td>96</td><td>48</td><td></td></tr><tr><th>54</th><td><a href="/facts/Truncated_octahedral_prism/gCIURDjm">Truncated octahedral prism</a>(Same as <i>cantitruncated tetrahedral prism</i>) (tope)</td><td></td><td></td><td>t{3,4}×{ }t1,2,3{4,3,2}</td><td>2 <a href="/facts/Truncated_octahedron/GA98b4TH">4.6.6</a></td><td>6 <a href="/facts/Cube/65lUaAqN">4.4.4</a></td><td>8 <a href="/facts/Hexagonal_prism/oxYc0nF2">4.4.6</a></td><td></td><td>16</td><td>48 {4}16 {6}</td><td>96</td><td>48</td><td></td></tr><tr><th>55</th><td><a href="/facts/Truncated_cuboctahedral_prism/y28TNUGc">Truncated cuboctahedral prism</a> (gircope)</td><td></td><td></td><td>tr{4,3}×{ }t0,1,2,3{4,3,2}</td><td>2 <a href="/facts/Truncated_cuboctahedron/nd2ltxcm">4.6.8</a></td><td>12 <a href="/facts/Cube/65lUaAqN">4.4.4</a></td><td>8 <a href="/facts/Hexagonal_prism/oxYc0nF2">4.4.6</a></td><td>6 <a href="/facts/Octagonal_prism/pMnX2f7F">4.4.8</a></td><td>28</td><td>96 {4}16 {6}12 {8}</td><td>192</td><td>96</td><td></td></tr><tr><th>56</th><td><a href="/facts/Snub_cubic_prism/sDkaMLCv">Snub cubic prism</a> (sniccup)</td><td></td><td></td><td>sr{4,3}×{ }</td><td>2 <a href="/facts/Snub_cube/pbnr40gk">3.3.3.3.4</a></td><td>32 <a href="/facts/Triangular_prism/Yjp4Kw9M">3.4.4</a></td><td>6 <a href="/facts/Cube/65lUaAqN">4.4.4</a></td><td></td><td>40</td><td>64 {3}72 {4}</td><td>144</td><td>48</td><td></td></tr><tr><th>[48]</th><td><a href="/facts/Tetrahedral_prism/oQ720tUe">Tetrahedral prism</a> (tepe)</td><td></td><td></td><td>h{4,3}×{ }</td><td>2 <a href="/facts/Tetrahedron/7mkSrl6j">3.3.3</a></td><td>4 <a href="/facts/Triangular_prism/Yjp4Kw9M">3.4.4</a></td><td></td><td></td><td>6</td><td>8 {3}6 {4}</td><td>16</td><td>8</td><td></td></tr><tr><th>[49]</th><td><a href="/facts/Truncated_tetrahedral_prism/7Fc4E1FP">Truncated tetrahedral prism</a> (tuttip)</td><td></td><td></td><td>h2{4,3}×{ }</td><td>2 <a href="/facts/Truncated_tetrahedron/K9ItvmUK">3.3.6</a></td><td>4 <a href="/facts/Triangular_prism/Yjp4Kw9M">3.4.4</a></td><td>4 <a href="/facts/Hexagonal_prism/oxYc0nF2">4.4.6</a></td><td></td><td>6</td><td>8 {3}6 {4}</td><td>16</td><td>8</td><td></td></tr><tr><th>[50]</th><td><a href="/facts/Cuboctahedral_prism/fFab3Nyp">Cuboctahedral prism</a> (cope)</td><td></td><td></td><td>rr{3,3}×{ }</td><td>2 <a href="/facts/Cuboctahedron/hD7p0Mem">3.4.3.4</a></td><td>8 <a href="/facts/Triangular_prism/Yjp4Kw9M">3.4.4</a></td><td>6 <a href="/facts/Cube/65lUaAqN">4.4.4</a></td><td></td><td>16</td><td>16 {3}36 {4}</td><td>60</td><td>24</td><td></td></tr><tr><th>[52]</th><td><a href="/facts/Rhombicuboctahedral_prism/4MJfAP4R">Rhombicuboctahedral prism</a> (sircope)</td><td></td><td></td><td>s2{3,4}×{ }</td><td>2 <a href="/facts/Rhombicuboctahedron/TmxUmEnl">3.4.4.4</a></td><td>8 <a href="/facts/Triangular_prism/Yjp4Kw9M">3.4.4</a></td><td>18 <a href="/facts/Cube/65lUaAqN">4.4.4</a></td><td></td><td>28</td><td>16 {3}84 {4}</td><td>120</td><td>48</td><td></td></tr><tr><th>[54]</th><td><a href="/facts/Truncated_octahedral_prism/gCIURDjm">Truncated octahedral prism</a> (tope)</td><td></td><td></td><td>tr{3,3}×{ }</td><td>2 <a href="/facts/Truncated_octahedron/GA98b4TH">4.6.6</a></td><td>6 <a href="/facts/Cube/65lUaAqN">4.4.4</a></td><td>8 <a href="/facts/Hexagonal_prism/oxYc0nF2">4.4.6</a></td><td></td><td>16</td><td>48 {4}16 {6}</td><td>96</td><td>48</td><td></td></tr><tr><th>[59]</th><td><a href="/facts/Icosahedral_prism/mpSGUN0y">Icosahedral prism</a> (ipe)</td><td></td><td></td><td>s{3,4}×{ }</td><td>2 <a href="/facts/Regular_icosahedron/1U93vmXV">3.3.3.3.3</a></td><td>20 <a href="/facts/Triangular_prism/Yjp4Kw9M">3.4.4</a></td><td></td><td></td><td>22</td><td>40 {3}30 {4}</td><td>72</td><td>24</td><td></td></tr><tr><th>[12]</th><td><a href="/facts/16-cell/X0YAfobw">16-cell</a> (hex)</td><td></td><td></td><td>s{2,4,3}</td><td>2+6+8 <a href="/facts/Tetrahedron/7mkSrl6j">3.3.3.3</a></td><td></td><td></td><td></td><td>16</td><td>32 {3}</td><td>24</td><td>8</td><td></td></tr><tr><th>Nonuniform</th><td><a href="/facts/Omnisnub_tetrahedral_antiprism/gCIURDjm">Omnisnub tetrahedral antiprism</a>= Pyritohedral icosahedral antiprism (pikap)</td><td></td><td></td><td>sr{2,3,4}</td><td>2 <a href="/facts/Regular_icosahedron/1U93vmXV">3.3.3.3.3</a></td><td>8 <a href="/facts/Octahedron/G38q92xW">3.3.3.3</a></td><td>6+24 <a href="/facts/Tetrahedron/7mkSrl6j">3.3.3</a></td><td></td><td>40</td><td>16+96 {3}</td><td>96</td><td>24</td><td></td></tr><tr><th>Nonuniform</th><td><a href="/facts/Truncated_cuboctahedral_prism/y28TNUGc">Edge-snub octahedral hosochoron</a>Pyritosnub alterprism (pysna)</td><td></td><td></td><td>sr3{2,3,4}</td><td>2 <a href="/facts/Rhombicuboctahedron/TmxUmEnl">3.4.4.4</a></td><td>6 <a href="/facts/Cube/65lUaAqN">4.4.4</a></td><td>8 <a href="/facts/Octahedron/G38q92xW">3.3.3.3</a></td><td>24 <a href="/facts/Triangular_prism/Yjp4Kw9M">3.4.4</a></td><td>40</td><td>16+48 {3}12+12+24+24 {4}</td><td>144</td><td>48</td><td></td></tr><tr><th>Nonuniform</th><td><a href="/facts/Omnisnub_cubic_antiprism/y28TNUGc">Omnisnub cubic antiprism</a>Snub cubic antiprism (sniccap)</td><td></td><td></td><td>                    s                  {                                                                      4                                                                              3                                                                              2                                                              }                      {\displaystyle s\left\{{\begin{array}{l}4\\3\\2\end{array}}\right\}}  </td><td>2 <a href="/facts/Snub_cube/pbnr40gk">3.3.3.3.4</a></td><td>12+48 <a href="/facts/Tetrahedron/7mkSrl6j">3.3.3</a></td><td>8 <a href="/facts/Octahedron/G38q92xW">3.3.3.3</a></td><td>6 <a href="/facts/Square_antiprism/4514Q02s">3.3.3.4</a></td><td>76</td><td>16+192 {3}12 {4}</td><td>192</td><td>48</td><td></td></tr><tr><th>Nonuniform</th><td><a href="/facts/Runcic_snub_cubic_hosochoron/4MJfAP4R">Runcic snub cubic hosochoron</a>Truncated tetrahedral alterprism (tuta)</td><td></td><td></td><td>s3{2,4,3}</td><td>2 <a href="/facts/Truncated_tetrahedron/K9ItvmUK">3.6.6</a></td><td>6 <a href="/facts/Tetrahedron/7mkSrl6j">3.3.3</a></td><td>8 <a href="/facts/Triangular_cupola/Pyocot8L">triangular cupola</a></td><td></td><td>16</td><td>52</td><td>60</td><td>24</td><td></td></tr></tbody></table>
<h4>Icosahedral prisms: H3 × A1</h4>
<p>This <a href="/facts/Hexacosichoric_symmetry/FtoaKNqe">prismatic icosahedral symmetry</a> is [5,3,2], order 240. There are two index 2 subgroups, [(5,3)+,2] and [5,3,2]+, but the second doesn't generate a uniform polychoron.
</p>
<table><tbody><tr><th rowspan="2">#</th><th rowspan="2">Name (Bowers name and acronym)</th><th rowspan="2"><a href="/facts/Schlegel_diagram/qwsDv1H8">Picture</a></th><th rowspan="2"><a href="/facts/Vertex_figure/3u2ySK0C">Vertexfigure</a></th><th rowspan="2"><a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter diagram</a>and <a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli</a>symbols</th><th rowspan="2" colspan="4">Cells by type</th><th colspan="4">Element counts</th><th rowspan="2">Net</th></tr><tr><th>Cells</th><th>Faces</th><th>Edges</th><th>Vertices</th></tr><tr><th>57</th><td><a href="/facts/Dodecahedral_prism/dYnjHXzr">Dodecahedral prism</a> (dope)</td><td></td><td></td><td>{5,3}×{ }t0,3{5,3,2}</td><td>2 <a href="/facts/Dodecahedron/7r519eCl">5.5.5</a></td><td>12 <a href="/facts/Pentagonal_prism/CtvKLxcy">4.4.5</a></td><td></td><td></td><td>14</td><td>30 {4}24 {5}</td><td>80</td><td>40</td><td></td></tr><tr><th>58</th><td><a href="/facts/Icosidodecahedral_prism/VscFj34Y">Icosidodecahedral prism</a> (iddip)</td><td></td><td></td><td>r{5,3}×{ }t1,3{5,3,2}</td><td>2 <a href="/facts/Icosidodecahedron/P0JluaVA">3.5.3.5</a></td><td>20 <a href="/facts/Triangular_prism/Yjp4Kw9M">3.4.4</a></td><td>12 <a href="/facts/Pentagonal_prism/CtvKLxcy">4.4.5</a></td><td></td><td>34</td><td>40 {3}60 {4}24 {5}</td><td>150</td><td>60</td><td></td></tr><tr><th>59</th><td><a href="/facts/Icosahedral_prism/mpSGUN0y">Icosahedral prism</a>(same as <i>snub tetrahedral prism</i>) (ipe)</td><td></td><td></td><td>{3,5}×{ }t2,3{5,3,2}</td><td>2 <a href="/facts/Regular_icosahedron/1U93vmXV">3.3.3.3.3</a></td><td>20 <a href="/facts/Triangular_prism/Yjp4Kw9M">3.4.4</a></td><td></td><td></td><td>22</td><td>40 {3}30 {4}</td><td>72</td><td>24</td><td></td></tr><tr><th>60</th><td><a href="/facts/Truncated_dodecahedral_prism/kfIzrUKL">Truncated dodecahedral prism</a> (tiddip)</td><td></td><td></td><td>t{5,3}×{ }t0,1,3{5,3,2}</td><td>2 <a href="/facts/Truncated_dodecahedron/PVTAsUWQ">3.10.10</a></td><td>20 <a href="/facts/Triangular_prism/Yjp4Kw9M">3.4.4</a></td><td>12 <a href="/facts/Decagonal_prism/dIjtR9zW">4.4.10</a></td><td></td><td>34</td><td>40 {3}90 {4}24 {10}</td><td>240</td><td>120</td><td></td></tr><tr><th>61</th><td><a href="/facts/Rhombicosidodecahedral_prism/c0QIs8L3">Rhombicosidodecahedral prism</a> (sriddip)</td><td></td><td></td><td>rr{5,3}×{ }t0,2,3{5,3,2}</td><td>2 <a href="/facts/Rhombicosidodecahedron/AaDckLGP">3.4.5.4</a></td><td>20 <a href="/facts/Triangular_prism/Yjp4Kw9M">3.4.4</a></td><td>30 <a href="/facts/Cube/65lUaAqN">4.4.4</a></td><td>12 <a href="/facts/Pentagonal_prism/CtvKLxcy">4.4.5</a></td><td>64</td><td>40 {3}180 {4}24 {5}</td><td>300</td><td>120</td><td></td></tr><tr><th>62</th><td><a href="/facts/Truncated_icosahedral_prism/gRMq75Sm">Truncated icosahedral prism</a> (tipe)</td><td></td><td></td><td>t{3,5}×{ }t1,2,3{5,3,2}</td><td>2 <a href="/facts/Truncated_icosahedron/LR8ugaY9">5.6.6</a></td><td>12 <a href="/facts/Pentagonal_prism/CtvKLxcy">4.4.5</a></td><td>20 <a href="/facts/Hexagonal_prism/oxYc0nF2">4.4.6</a></td><td></td><td>34</td><td>90 {4}24 {5}40 {6}</td><td>240</td><td>120</td><td></td></tr><tr><th>63</th><td><a href="/facts/Truncated_icosidodecahedral_prism/PbvDF6gP">Truncated icosidodecahedral prism</a> (griddip)</td><td></td><td></td><td>tr{5,3}×{ }t0,1,2,3{5,3,2}</td><td>2 <a href="/facts/Truncated_icosidodecahedron/d5v9VpFY">4.6.10</a></td><td>30 <a href="/facts/Cube/65lUaAqN">4.4.4</a></td><td>20 <a href="/facts/Hexagonal_prism/oxYc0nF2">4.4.6</a></td><td>12 <a href="/facts/Decagonal_prism/dIjtR9zW">4.4.10</a></td><td>64</td><td>240 {4}40 {6}24 {10}</td><td>480</td><td>240</td><td></td></tr><tr><th>64</th><td><a href="/facts/Snub_dodecahedral_prism/TlglgI8u">Snub dodecahedral prism</a> (sniddip)</td><td></td><td></td><td>sr{5,3}×{ }</td><td>2 <a href="/facts/Snub_dodecahedron/oouEPVxD">3.3.3.3.5</a></td><td>80 <a href="/facts/Triangular_prism/Yjp4Kw9M">3.4.4</a></td><td>12 <a href="/facts/Pentagonal_prism/CtvKLxcy">4.4.5</a></td><td></td><td>94</td><td>160 {3}150 {4}24 {5}</td><td>360</td><td>120</td><td></td></tr><tr><th>Nonuniform</th><td><a href="/facts/Omnisnub_dodecahedral_antiprism/PbvDF6gP">Omnisnub dodecahedral antiprism</a>Snub dodecahedral antiprism (sniddap)</td><td></td><td></td><td>                    s                  {                                                                      5                                                                              3                                                                              2                                                              }                      {\displaystyle s\left\{{\begin{array}{l}5\\3\\2\end{array}}\right\}}  </td><td>2 <a href="/facts/Snub_dodecahedron/oouEPVxD">3.3.3.3.5</a></td><td>30+120 <a href="/facts/Tetrahedron/7mkSrl6j">3.3.3</a></td><td>20 <a href="/facts/Octahedron/G38q92xW">3.3.3.3</a></td><td>12 <a href="/facts/Pentagonal_antiprism/gR8aMBKN">3.3.3.5</a></td><td>184</td><td>20+240 {3}24 {5}</td><td>220</td><td>120</td><td></td></tr></tbody></table>
<h4>Duoprisms: [p] × [q]</h4>

<p>The second is the infinite family of <a href="/facts/Duoprism/D73YMRth">uniform duoprisms</a>, products of two <a href="/facts/Regular_polygon/ws89Nhpt">regular polygons</a>. A duoprism's <a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter-Dynkin diagram</a> is . Its <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</a> is a <a href="/facts/Disphenoid_tetrahedron/7rU7jc7U">disphenoid tetrahedron</a>, .
</p><p>This family overlaps with the first: when one of the two "factor" polygons is a square, the product is equivalent to a hyperprism whose base is a three-dimensional prism.  The symmetry number of a duoprism whose factors are a <i>p</i>-gon and a <i>q</i>-gon (a "<i>p,q</i>-duoprism") is 4<i>pq</i> if <i>p</i>≠<i>q</i>; if the factors are both <i>p</i>-gons, the symmetry number is 8<i>p</i>2. The tesseract can also be considered a 4,4-duoprism.
</p><p>The extended <a href="/facts/F-vector/oVFESPjJ">f-vector</a> of {<i>p</i>}×{<i>q</i>} is (<i>p</i>,<i>p</i>,1)*(<i>q</i>,<i>q</i>,1) = (<i>pq</i>,2<i>pq</i>,<i>pq</i>+<i>p</i>+<i>q</i>,<i>p</i>+<i>q</i>).
</p>
<ul><li>Cells: <i>p</i> <i>q</i>-gonal prisms, <i>q</i> <i>p</i>-gonal prisms</li>
<li>Faces: <i>pq</i> squares, <i>p</i> <i>q</i>-gons, <i>q</i> <i>p</i>-gons</li>
<li>Edges: <i>2pq</i></li>
<li>Vertices: <i>pq</i></li></ul>
<p>There is no uniform analogue in four dimensions to the infinite family of three-dimensional <a href="/facts/Antiprism/FgP5jIVd">antiprisms</a>.
</p><p>Infinite set of <i>p-q duoprism</i> -  - p <i>q</i>-gonal prisms, q <i>p</i>-gonal prisms:
</p>
<table><tbody><tr><th>Name</th><th>Coxeter graph</th><th>Cells</th><th>Images</th><th>Net</th></tr><tr><th><a href="/facts/3-3_duoprism/B1dhsgrg">3-3 duoprism</a> (triddip)</th><td></td><td>3+3 triangular prisms</td><td></td><td></td></tr><tr><th><a href="/facts/3-4_duoprism/jlvkbaXz">3-4 duoprism</a> (tisdip)</th><td></td><td>3 cubes4 triangular prisms</td><td> </td><td></td></tr><tr><th><a href="/facts/4-4_duoprism/tlqouBoU">4-4 duoprism</a> (tes)(same as tesseract)</th><td></td><td>4+4 cubes</td><td></td><td></td></tr><tr><th><a href="/facts/3-5_duoprism/D73YMRth">3-5 duoprism</a> (trapedip)</th><td></td><td>3 pentagonal prisms5 triangular prisms</td><td> </td><td></td></tr><tr><th><a href="/facts/4-5_duoprism/D73YMRth">4-5 duoprism</a> (squipdip)</th><td></td><td>4 pentagonal prisms5 cubes</td><td> </td><td></td></tr><tr><th><a href="/facts/5-5_duoprism/J7TrIYEa">5-5 duoprism</a> (pedip)</th><td></td><td>5+5 pentagonal prisms</td><td></td><td></td></tr><tr><th><a href="/facts/3-6_duoprism/Map8UCwv">3-6 duoprism</a> (thiddip)</th><td></td><td>3 hexagonal prisms6 triangular prisms</td><td> </td><td></td></tr><tr><th><a href="/facts/4-6_duoprism/cWuVUKco">4-6 duoprism</a> (shiddip)</th><td></td><td>4 hexagonal prisms6 cubes</td><td> </td><td></td></tr><tr><th><a href="/facts/5-6_duoprism/D73YMRth">5-6 duoprism</a> (phiddip)</th><td></td><td>5 hexagonal prisms6 pentagonal prisms</td><td> </td><td></td></tr><tr><th><a href="/facts/6-6_duoprism/Rgcg9TrJ">6-6 duoprism</a> (hiddip)</th><td></td><td>6+6 hexagonal prisms</td><td></td><td></td></tr></tbody></table>
<table><tbody><tr><td><a href="/facts/3-3_duoprism/B1dhsgrg">3-3</a></td><td><a href="/facts/3-4_duoprism/jlvkbaXz">3-4</a></td><td>3-5</td><td>3-6</td><td>3-7</td><td>3-8</td></tr><tr><td><a href="/facts/3-4_duoprism/jlvkbaXz">4-3</a></td><td>4-4</td><td>4-5</td><td>4-6</td><td>4-7</td><td>4-8</td></tr><tr><td>5-3</td><td>5-4</td><td>5-5</td><td>5-6</td><td>5-7</td><td>5-8</td></tr><tr><td>6-3</td><td>6-4</td><td>6-5</td><td>6-6</td><td>6-7</td><td>6-8</td></tr><tr><td>7-3</td><td>7-4</td><td>7-5</td><td>7-6</td><td>7-7</td><td>7-8</td></tr><tr><td>8-3</td><td>8-4</td><td>8-5</td><td>8-6</td><td>8-7</td><td>8-8</td></tr></tbody></table>
<p>Alternations are possible.  =  gives the family of <i>duoantiprisms</i>, but they generally cannot be made uniform. p=q=2 is the only <i>convex</i> case that can be made uniform, giving the regular 16-cell. p=5, q=5/3 is the only nonconvex case that can be made uniform, giving the so-called <a href="/facts/Great_duoantiprism/sAPm9o6b">great duoantiprism</a>.  gives the p-2q-gonal <i>prismantiprismoid</i> (an edge-alternation of the 2p-4q duoprism), but this cannot be made uniform in any cases.<a class="footnote-ref" id="fnref:31" href="#fn:31"><sup>31</sup></a>
</p>
<h4>Polygonal prismatic prisms: [p] × [ ] × [ ]</h4>
<p>The infinite set of uniform prismatic prisms overlaps with the 4-p duoprisms: (p≥3) -  - <i>p</i> cubes and 4 <i>p</i>-gonal prisms - (All are the same as 4-p duoprism) The second polytope in the series is a lower symmetry of the regular <a href="/facts/Tesseract/tlqouBoU">tesseract</a>, {4}×{4}.
</p>
Convex <i>p</i>-gonal prismatic prisms<table><tbody><tr><th>Name</th><th><a href="/facts/3-4_duoprism/jlvkbaXz">{3}×{4}</a></th><th><a href="/facts/Tesseract/tlqouBoU">{4}×{4}</a></th><th><a href="/facts/Pentagonal_prismatic_prism/D73YMRth">{5}×{4}</a></th><th><a href="/facts/4-6_duoprism/cWuVUKco">{6}×{4}</a></th><th><a href="/facts/Heptagonal_prismatic_prism/D73YMRth">{7}×{4}</a></th><th><a href="/facts/4-8_duoprism/ENR9TBVX">{8}×{4}</a></th><th><a href="/facts/Uniform_prismatic_prism/NHxomMxC">{p}×{4}</a></th></tr><tr><th><a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeterdiagrams</a></th><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>Image</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th><a href="/facts/Cell_(geometry)/T2E6BTU4">Cells</a></th><td>3 <a href="/facts/Cube/65lUaAqN">{4}×{}</a> 4 <a href="/facts/Triangular_prism/Yjp4Kw9M">{3}×{}</a> </td><td>4 <a href="/facts/Cube/65lUaAqN">{4}×{}</a> 4 <a href="/facts/Cube/65lUaAqN">{4}×{}</a> </td><td>5 <a href="/facts/Cube/65lUaAqN">{4}×{}</a> 4 <a href="/facts/Pentagonal_prism/CtvKLxcy">{5}×{}</a> </td><td>6 <a href="/facts/Cube/65lUaAqN">{4}×{}</a> 4 <a href="/facts/Hexagonal_prism/oxYc0nF2">{6}×{}</a> </td><td>7 <a href="/facts/Cube/65lUaAqN">{4}×{}</a> 4 <a href="/facts/Heptagonal_prism/vdIlUSVA">{7}×{}</a> </td><td>8 <a href="/facts/Cube/65lUaAqN">{4}×{}</a> 4 <a href="/facts/Octagonal_prism/pMnX2f7F">{8}×{}</a> </td><td>p <a href="/facts/Cube/65lUaAqN">{4}×{}</a> 4 <a href="/facts/Prism_(geometry)/baIRRkCy">{p}×{}</a></td></tr><tr><th><a href="/facts/Polyhedral_net/lU8fmGJs">Net</a></th><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table>

<h4>Polygonal antiprismatic prisms: [p] × [ ] × [ ]</h4>
<p>The infinite sets of <i><a href="/facts/Uniform_antiprismatic_prism/thA0prtm">uniform antiprismatic prisms</a></i> are constructed from two parallel uniform <a href="/facts/Antiprism/FgP5jIVd">antiprisms</a>): (p≥2) -  - 2 <i>p</i>-gonal antiprisms, connected by 2 <i>p</i>-gonal prisms and <i>2p</i> triangular prisms. 
</p>
Convex <i>p</i>-gonal antiprismatic prisms<table><tbody><tr><th>Name</th><th><a href="/facts/Tetrahedral_prism/oQ720tUe">s{2,2}×{}</a></th><th><a href="/facts/Octahedral_prism/8UVHkxvl">s{2,3}×{}</a></th><th><a href="/facts/Square_antiprismatic_prism/thA0prtm">s{2,4}×{}</a></th><th><a href="/facts/Pentagonal_antiprismatic_prism/thA0prtm">s{2,5}×{}</a></th><th><a href="/facts/Hexagonal_antiprismatic_prism/thA0prtm">s{2,6}×{}</a></th><th><a href="/facts/Heptagonal_antiprismatic_prism/thA0prtm">s{2,7}×{}</a></th><th><a href="/facts/Octagonal_antiprismatic_prism/thA0prtm">s{2,8}×{}</a></th><th><a href="/facts/Uniform_antiprismatic_prism/thA0prtm">s{2,p}×{}</a></th></tr><tr><th><a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeterdiagram</a></th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>Image</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th><a href="/facts/Vertex_figure/3u2ySK0C">Vertexfigure</a></th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>Cells</th><td>2 <a href="/facts/Tetrahedron/7mkSrl6j">s{2,2}</a>(2) {2}×{}=<a href="/facts/Square/JHJeMvmL">{4}</a>4 <a href="/facts/Triangular_prism/Yjp4Kw9M">{3}×{}</a></td><td>2 <a href="/facts/Octahedron/G38q92xW">s{2,3}</a>2 <a href="/facts/Triangular_prism/Yjp4Kw9M">{3}×{}</a>6 <a href="/facts/Triangular_prism/Yjp4Kw9M">{3}×{}</a></td><td>2 <a href="/facts/Square_antiprism/4514Q02s">s{2,4}</a>2 <a href="/facts/Cube/65lUaAqN">{4}×{}</a>8 <a href="/facts/Triangular_prism/Yjp4Kw9M">{3}×{}</a></td><td>2 <a href="/facts/Pentagonal_antiprism/gR8aMBKN">s{2,5}</a>2 <a href="/facts/Pentagonal_prism/CtvKLxcy">{5}×{}</a>10 <a href="/facts/Triangular_prism/Yjp4Kw9M">{3}×{}</a></td><td>2 <a href="/facts/Hexagonal_antiprism/bt1W4cs6">s{2,6}</a>2 <a href="/facts/Hexagonal_prism/oxYc0nF2">{6}×{}</a>12 <a href="/facts/Triangular_prism/Yjp4Kw9M">{3}×{}</a></td><td>2 <a href="/facts/Heptagonal_antiprism/gh9eMN8a">s{2,7}</a>2 <a href="/facts/Heptagonal_prism/vdIlUSVA">{7}×{}</a>14 <a href="/facts/Triangular_prism/Yjp4Kw9M">{3}×{}</a></td><td>2 <a href="/facts/Octagonal_antiprism/4XCvwXUF">s{2,8}</a>2 <a href="/facts/Octagonal_prism/pMnX2f7F">{8}×{}</a>16 <a href="/facts/Triangular_prism/Yjp4Kw9M">{3}×{}</a></td><td>2 s{2,p}2 {p}×{}2<i>p</i> {3}×{}</td></tr><tr><th>Net</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table>
<p>A <i>p-gonal antiprismatic prism</i> has <i>4p</i> triangle, <i>4p</i> square and <i>4</i> p-gon faces. It has <i>10p</i> edges, and <i>4p</i> vertices.
</p>
<h3>Nonuniform alternations</h3>

<p>Coxeter showed only two uniform solutions for rank 4 Coxeter groups with all rings <a href="/facts/Alternation_(geometry)/H2nVs4xj">alternated</a> (shown with empty circle nodes). The first is , s{21,1,1} which represented an index 24 subgroup (<a href="/facts/Coxeter_notation/ArUw6goe">symmetry</a> [2,2,2]+, order 8) form of the <a href="/facts/Demitesseract/X0YAfobw">demitesseract</a>, , h{4,3,3} (symmetry [1+,4,3,3] = [31,1,1], order 192). The second is , s{31,1,1}, which is an index 6 subgroup (symmetry [31,1,1]+, order 96) form of the <a href="/facts/Snub_24-cell/WbEJ3cPQ">snub 24-cell</a>, , s{3,4,3}, (symmetry [3+,4,3], order 576).
</p><p>Other alternations, such as , as an alternation from the <a href="/facts/Omnitruncated_tesseract/NqDGHQxp">omnitruncated tesseract</a> , can not be made uniform as solving for equal edge lengths are in general <a href="/facts/Overdetermined_system/DMWk4pQR">overdetermined</a> (there are six equations but only four variables). Such nonuniform alternated figures can be constructed as <a href="/facts/Vertex-transitive/5nC8PdBm">vertex-transitive</a> 4-polytopes by the removal of one of two half sets of the vertices of the full ringed figure, but will have unequal edge lengths. Just like uniform alternations, they will have half of the symmetry of uniform figure, like [4,3,3]+, order 192, is the symmetry of the <i>alternated omnitruncated tesseract</i>.<a class="footnote-ref" id="fnref:32" href="#fn:32"><sup>32</sup></a>
</p><p>Wythoff constructions with alternations produce <a href="/facts/Vertex-transitive/5nC8PdBm">vertex-transitive</a> figures that can be made equilateral, but not uniform because the alternated gaps (around the removed vertices) create cells that are not regular or semiregular. A proposed name for such figures is scaliform polytopes.<a class="footnote-ref" id="fnref:33" href="#fn:33"><sup>33</sup></a> This category allows a subset of <a href="/facts/Johnson_solid/8SR7CyNz">Johnson solids</a> as cells, for example <a href="/facts/Triangular_cupola/Pyocot8L">triangular cupola</a>.
</p><p>Each <a href="/facts/Vertex_configuration/wWaBS6t0">vertex configuration</a> within a Johnson solid must exist within the vertex figure. For example, a square pyramid has two vertex configurations: 3.3.4 around the base, and 3.3.3.3 at the apex.
</p><p>The nets and vertex figures of the four convex equilateral cases are given below, along with a list of cells around each vertex.
</p>
Four convex vertex-transitive equilateral 4-polytopes with nonuniform cells<table><tbody><tr><th>Coxeterdiagram</th><th>s3{2,4,3}, </th><th>s3{3,4,3}, </th><th colspan="2">Others</th></tr><tr><th>Relation</th><th>24 of 48 vertices of <a href="/facts/Rhombicuboctahedral_prism/4MJfAP4R">rhombicuboctahedral prism</a></th><th>288 of 576 vertices of<a href="/facts/Runcitruncated_24-cell/buMulGIL">runcitruncated 24-cell</a></th><th>72 of 120 verticesof <a href="/facts/600-cell/tNBtDUe9">600-cell</a></th><th>600 of 720 verticesof <a href="/facts/Rectified_600-cell/S2DeX7LG">rectified 600-cell</a></th></tr><tr><th>Projection</th><td></td><td></td><td></td><td>Two rings of pyramids</td></tr><tr><th>Net</th><td><a href="/facts/Runcic_snub_cubic_hosochoron/4MJfAP4R">runcic snub cubic hosochoron</a><a class="footnote-ref" id="fnref:34" href="#fn:34"><sup>34</sup></a><a class="footnote-ref" id="fnref:35" href="#fn:35"><sup>35</sup></a></td><td><a href="/facts/Runcic_snub_24-cell/buMulGIL">runcic snub 24-cell</a><a class="footnote-ref" id="fnref:36" href="#fn:36"><sup>36</sup></a><a class="footnote-ref" id="fnref:37" href="#fn:37"><sup>37</sup></a></td><td><a class="footnote-ref" id="fnref:38" href="#fn:38"><sup>38</sup></a><a class="footnote-ref" id="fnref:39" href="#fn:39"><sup>39</sup></a><a class="footnote-ref" id="fnref:40" href="#fn:40"><sup>40</sup></a></td><td><a class="footnote-ref" id="fnref:41" href="#fn:41"><sup>41</sup></a><a class="footnote-ref" id="fnref:42" href="#fn:42"><sup>42</sup></a></td></tr><tr><th>Cells</th><th>  </th><th>   </th><th></th><th>  </th></tr><tr><th>Vertexfigure</th><td>(1) 3.4.3.4: <a href="/facts/Triangular_cupola/Pyocot8L">triangular cupola</a>(2) 3.4.6: triangular cupola(1) 3.3.3: <a href="/facts/Tetrahedron/7mkSrl6j">tetrahedron</a>(1) 3.6.6: <a href="/facts/Truncated_tetrahedron/K9ItvmUK">truncated tetrahedron</a></td><td>(1) 3.4.3.4: triangular cupola(2) 3.4.6: triangular cupola(2) 3.4.4: <a href="/facts/Triangular_prism/Yjp4Kw9M">triangular prism</a>(1) 3.6.6: <a href="/facts/Truncated_tetrahedron/K9ItvmUK">truncated tetrahedron</a>(1) 3.3.3.3.3: <a href="/facts/Icosahedron/wWbpnDYp">icosahedron</a></td><td>(2) 3.3.3.5: <a href="/facts/Tridiminished_icosahedron/gCVHC0zw">tridiminished icosahedron</a> (4) 3.5.5: tridiminished icosahedron</td><td>(1) 3.3.3.3: <a href="/facts/Square_pyramid/wDetDohM">square pyramid</a>(4) 3.3.4: square pyramid(2) 4.4.5: <a href="/facts/Pentagonal_prism/CtvKLxcy">pentagonal prism</a>(2) 3.3.3.5 <a href="/facts/Pentagonal_antiprism/gR8aMBKN">pentagonal antiprism</a></td></tr></tbody></table>
<h3>Geometric derivations for 46 nonprismatic Wythoffian uniform polychora</h3>
<p>The 46 Wythoffian 4-polytopes include the six <a href="/facts/Convex_regular_4-polytope/lFf8IH1C">convex regular 4-polytopes</a>. The other forty can be derived from the regular polychora by geometric operations which preserve most or all of their <a href="/facts/Symmetry/hpu7DK8p">symmetries</a>, and therefore may be classified by the <a href="/facts/Symmetry_group/CNqeseMp">symmetry groups</a> that they have in common.
</p>
<table><tbody><tr><td>Summary chart of truncation operations</td><td>Example locations of kaleidoscopic generator point on fundamental domain.</td></tr></tbody></table>
<p>The geometric operations that derive the 40 uniform 4-polytopes from the regular 4-polytopes are <i>truncating</i> operations. A 4-polytope may be truncated at the vertices, edges or faces, leading to addition of cells corresponding to those elements, as shown in the columns of the tables below.
</p><p>The <i><a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter-Dynkin diagram</a></i> shows the four mirrors of the Wythoffian kaleidoscope as nodes, and the edges between the nodes are labeled by an integer showing the angle between the mirrors (<a href="/facts/Pi/vC0UBj1q">π</a>/<i>n</i> <a href="/facts/Radian/Srd38Sxa">radians</a> or 180/<i>n</i> degrees). Circled nodes show which mirrors are active for each form; a mirror is active with respect to a vertex that does not lie on it.
</p>
<table><tbody><tr><th>Operation</th><th><a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli symbol</a></th><th><a href="/facts/Coxeter_notation/ArUw6goe">Symmetry</a></th><th><a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter diagram</a></th><th>Description</th></tr><tr><th>Parent</th><td>t0{p,q,r}</td><td rowspan="15">[p,q,r]</td><td></td><td>Original regular form {p,q,r}</td></tr><tr><th><a href="/facts/Rectification_(geometry)/PkoUdq53">Rectification</a></th><td>t1{p,q,r}</td><td></td><td>Truncation operation applied until the original edges are degenerated into points.</td></tr><tr><th>Birectification(Rectified dual)</th><td>t2{p,q,r}</td><td></td><td>Face are fully truncated to points. Same as rectified dual.</td></tr><tr><th>Trirectification(<a href="/facts/Dual_polytope/rkIHbmAf">dual</a>)</th><td>t3{p,q,r}</td><td></td><td>Cells are truncated to points. Regular dual {r,q,p}</td></tr><tr><th><a href="/facts/Truncation_(geometry)/DYb589Je">Truncation</a></th><td>t0,1{p,q,r}</td><td></td><td>Each vertex is cut off so that the middle of each original edge remains.  Where the vertex was, there appears a new cell, the parent's <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</a>.  Each original cell is likewise truncated.</td></tr><tr><th><a href="/facts/Bitruncation_(geometry)/t0xarbyJ">Bitruncation</a></th><td>t1,2{p,q,r}</td><td></td><td>A truncation between a rectified form and the dual rectified form.</td></tr><tr><th>Tritruncation</th><td>t2,3{p,q,r}</td><td></td><td>Truncated dual {r,q,p}.</td></tr><tr><th><a href="/facts/Cantellation_(geometry)/kS24Tde3">Cantellation</a></th><td>t0,2{p,q,r}</td><td></td><td>A truncation applied to edges and vertices and defines a progression between the regular and dual rectified form.</td></tr><tr><th>Bicantellation</th><td>t1,3{p,q,r}</td><td></td><td>Cantellated dual {r,q,p}.</td></tr><tr><th><a href="/facts/Runcination_(geometry)/nwUxfUIX">Runcination</a>(or <a href="/facts/Expansion_(geometry)/eMpoRlnm">expansion</a>)</th><td>t0,3{p,q,r}</td><td></td><td>A truncation applied to the cells, faces and edges; defines a progression between a regular form and the dual.</td></tr><tr><th>Cantitruncation</th><td>t0,1,2{p,q,r}</td><td></td><td>Both the <i>cantellation</i> and <i>truncation</i> operations applied together.</td></tr><tr><th>Bicantitruncation</th><td>t1,2,3{p,q,r}</td><td></td><td>Cantitruncated dual {r,q,p}.</td></tr><tr><th>Runcitruncation</th><td>t0,1,3{p,q,r}</td><td></td><td>Both the <i>runcination</i> and <i>truncation</i> operations applied together.</td></tr><tr><th>Runcicantellation</th><td>t0,2,3{p,q,r}</td><td></td><td>Runcitruncated dual {r,q,p}.</td></tr><tr><th><a href="/facts/Omnitruncation_(geometry)/hmbMrRm6">Omnitruncation</a>(runcicantitruncation)</th><td>t0,1,2,3{p,q,r}</td><td></td><td>Application of all three operators.</td></tr><tr><th>Half</th><td>h{2p,3,q}</td><td rowspan="4">[1+,2p,3,q]=[(3,p,3),q]</td><td></td><td><a href="/facts/Alternation_(geometry)/H2nVs4xj">Alternation</a> of , same as </td></tr><tr><th>Cantic</th><td>h2{2p,3,q}</td><td></td><td>Same as </td></tr><tr><th>Runcic</th><td>h3{2p,3,q}</td><td></td><td>Same as </td></tr><tr><th>Runcicantic</th><td>h2,3{2p,3,q}</td><td></td><td>Same as </td></tr><tr><th>Quarter</th><td>q{2p,3,2q}</td><td>[1+,2p,3,2q,1+]</td><td></td><td>Same as </td></tr><tr><th>Snub</th><td>s{p,2q,r}</td><td rowspan="4">[p+,2q,r]</td><td></td><td>Alternated truncation</td></tr><tr><th>Cantic snub</th><td>s2{p,2q,r}</td><td></td><td>Cantellated alternated truncation</td></tr><tr><th>Runcic snub</th><td>s3{p,2q,r}</td><td></td><td>Runcinated alternated truncation</td></tr><tr><th>Runcicantic snub</th><td>s2,3{p,2q,r}</td><td></td><td>Runcicantellated alternated truncation</td></tr><tr><th>Snub rectified</th><td>sr{p,q,2r}</td><td>[(p,q)+,2r]</td><td></td><td>Alternated truncated rectification</td></tr><tr><th></th><td>ht0,3{2p,q,2r}</td><td>[(2p,q,2r,2+)]</td><td></td><td>Alternated runcination</td></tr><tr><th>Bisnub</th><td>2s{2p,q,2r}</td><td>[2p,q+,2r]</td><td></td><td>Alternated bitruncation</td></tr><tr><th>Omnisnub</th><td>ht0,1,2,3{p,q,r}</td><td>[p,q,r]+</td><td></td><td>Alternated omnitruncation</td></tr></tbody></table>
<p>See also <a href="/facts/Convex_uniform_honeycomb/78IkTl7A">convex uniform honeycombs</a>, some of which illustrate these operations as applied to the regular <a href="/facts/Cubic_honeycomb/eMxySFur">cubic honeycomb</a>.
</p><p>If two polytopes are <a href="/facts/Dual_polytope/rkIHbmAf">duals</a> of each other (such as the tesseract and 16-cell, or the 120-cell and 600-cell), then <i>bitruncating</i>, <i>runcinating</i> or <i>omnitruncating</i> either produces the same figure as the same operation to the other.  Thus where only the participle appears in the table it should be understood to apply to either parent.
</p>
<h4>Summary of constructions by extended symmetry</h4>
<p>The 46 uniform polychora constructed from the A4, B4, F4, H4 symmetry are given in this table by their full extended symmetry and Coxeter diagrams. The D4 symmetry is also included, though it only creates duplicates. Alternations are grouped by their chiral symmetry. All alternations are given, although the <a href="/facts/Snub_24-cell/WbEJ3cPQ">snub 24-cell</a>, with its 3 constructions from different families is the only one that is uniform. Counts in parentheses are either repeats or nonuniform. The Coxeter diagrams are given with subscript indices 1 through 46. The 3-3 and 4-4 duoprismatic family is included, the second for its relation to the B4 family.
</p>
<table><tbody><tr><th><a href="/facts/Coxeter_group/bgzHFm1Q">Coxeter group</a></th><th><a href="/facts/Goursat_tetrahedron/kDgl1J5z">Extendedsymmetry</a></th><th colspan="2">Polychora</th><th>Chiralextendedsymmetry</th><th colspan="2">Alternation honeycombs</th></tr><tr><td rowspan="2">[3,3,3]</td><td>[3,3,3](order 120)</td><td>6</td><td>(1) | (2) | (3)(4) | (7) | (8)</td><td colspan="3"></td></tr><tr><td>[2+[3,3,3]](order 240)</td><td>3</td><td>(5)| (6) | (9)</td><td>[2+[3,3,3]]+(order 120)</td><td>(1)</td><td>(−)</td></tr><tr><td rowspan="3">[3,31,1]</td><td>[3,31,1](order 192)</td><td>0</td><td>(none)</td><td colspan="3"></td></tr><tr><td>[1[3,31,1]]=[4,3,3] = (order 384)</td><td>(4)</td><td>(12) | (17) | (11) | (16)</td><td colspan="3"></td></tr><tr><td>[3[31,1,1]]=[3,4,3] = (order 1152)</td><td>(3)</td><td>(22) | (23) | (24)</td><td>[3[3,31,1]]+=[3,4,3]+(order 576)</td><td>(1)</td><td>(31) (= )(−)</td></tr><tr><td rowspan="3">[4,3,3]</td><td>[3[1+,4,3,3]]=[3,4,3] = (order 1152)</td><td>(3)</td><td>(22) | (23) | (24)</td><td colspan="3"></td></tr><tr><td rowspan="2">[4,3,3](order 384)</td><td rowspan="2">12</td><td rowspan="2">(10) | (11) | (12) | (13) | (14)(15) | (16) | (17) | (18) | (19)(20) | (21)</td><td>[1+,4,3,3]+(order 96)</td><td>(2)</td><td>(12) (= )(31)(−)</td></tr><tr><td>[4,3,3]+(order 192)</td><td>(1)</td><td>(−)</td></tr><tr><td rowspan="2">[3,4,3]</td><td>[3,4,3](order 1152)</td><td>6</td><td>(22) | (23) | (24)(25) | (28) | (29)</td><td>[2+[3+,4,3+]](order 576)</td><td>1</td><td>(31)</td></tr><tr><td>[2+[3,4,3]](order 2304)</td><td>3</td><td>(26) | (27) | (30)</td><td>[2+[3,4,3]]+(order 1152)</td><td>(1)</td><td>(−)</td></tr><tr><td>[5,3,3]</td><td>[5,3,3](order 14400)</td><td>15</td><td>(32) | (33) | (34) | (35) | (36)(37) | (38) | (39) | (40)  | (41)(42) | (43) | (44) | (45) | (46)</td><td>[5,3,3]+(order 7200)</td><td>(1)</td><td>(−)</td></tr><tr><td rowspan="4">[3,2,3]</td><td>[3,2,3](order 36)</td><td>0</td><td>(none)</td><td>[3,2,3]+(order 18)</td><td>0</td><td>(none)</td></tr><tr><td>[2+[3,2,3]](order 72)</td><td>0</td><td></td><td>[2+[3,2,3]]+(order 36)</td><td>0</td><td>(none)</td></tr><tr><td>[[3],2,3]=[6,2,3] = (order 72)</td><td>1</td><td></td><td>[1[3,2,3]]=[[3],2,3]+=[6,2,3]+(order 36)</td><td>(1)</td><td></td></tr><tr><td>[(2+,4)[3,2,3]]=[2+[6,2,6]] = (order 288)</td><td>1</td><td></td><td>[(2+,4)[3,2,3]]+=[2+[6,2,6]]+(order 144)</td><td>(1)</td><td></td></tr><tr><td rowspan="5">[4,2,4]</td><td>[4,2,4](order 64)</td><td>0</td><td>(none)</td><td>[4,2,4]+(order 32)</td><td>0</td><td>(none)</td></tr><tr><td>[2+[4,2,4]](order 128)</td><td>0</td><td>(none)</td><td>[2+[(4,2+,4,2+)]](order 64)</td><td>0</td><td>(none)</td></tr><tr><td>[(3,3)[4,2*,4]]=[4,3,3] = (order 384)</td><td>(1)</td><td>(10)</td><td>[(3,3)[4,2*,4]]+=[4,3,3]+(order 192)</td><td>(1)</td><td>(12)</td></tr><tr><td>[[4],2,4]=[8,2,4] = (order 128)</td><td>(1)</td><td></td><td>[1[4,2,4]]=[[4],2,4]+=[8,2,4]+(order 64)</td><td>(1)</td><td></td></tr><tr><td>[(2+,4)[4,2,4]]=[2+[8,2,8]] = (order 512)</td><td>(1)</td><td></td><td>[(2+,4)[4,2,4]]+=[2+[8,2,8]]+(order 256)</td><td>(1)</td><td></td></tr></tbody></table>
<h2 id="uniform-star-polychora">Uniform star polychora</h2>
<p>Other than the aforementioned infinite duoprism and antiprism prism families, which have infinitely many nonconvex members, many uniform star polychora have been discovered. In 1852, Ludwig Schläfli discovered four <i>regular</i> star polychora: {5,3,5/2}, {5/2,3,5}, {3,3,5/2}, and {5/2,3,3}. In 1883, Edmund Hess found the other six: {3,5,5/2}, {5/2,5,3}, {5,5/2,5}, {5/2,5,5/2}, {5,5/2,3}, and {3,5/2,5}. Norman Johnson described three uniform antiprism-like star polychora in his doctoral dissertation of 1966: they are based on the three <a href="/facts/Ditrigonal_polyhedron/b8FVuYbm">ditrigonal polyhedra</a> sharing the edges and vertices of the regular dodecahedron. Many more have been found since then by other researchers, including Jonathan Bowers and George Olshevsky, creating a total count of 2127 known uniform star polychora at present (not counting the infinite set of duoprisms based on star polygons). There is currently no proof of the set's completeness.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Regular_skew_polyhedron/jIyDvyIA">Finite regular skew polyhedra of 4-space</a></li>
<li><a href="/facts/Convex_uniform_honeycomb/78IkTl7A">Convex uniform honeycomb</a> - related infinite 4-polytopes in Euclidean 3-space.</li>
<li><a href="/facts/Convex_uniform_honeycombs_in_hyperbolic_space/n1c2vP9b">Convex uniform honeycombs in hyperbolic space</a> - related infinite 4-polytopes in Hyperbolic 3-space.</li>
<li><a href="/facts/Paracompact_uniform_honeycombs/hnmKAVu0">Paracompact uniform honeycombs</a></li></ul>

<ul><li><a href="/facts/Alicia_Boole_Stott/FoqXBA5T">A. Boole Stott</a>: <i>Geometrical deduction of semiregular from regular polytopes and space fillings</i>, Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910</li>
<li><a href="/facts/Branko_Gr%25C3%25BCnbaum/taVzX1MG">B. Grünbaum</a> <i><a href="/facts/Convex_Polytopes/nQ4YTyVv">Convex Polytopes</a></i>,  New York; London : Springer, c2003. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-00424-6. Second edition prepared by Volker Kaibel, <a href="/facts/Victor_Klee/NMRFBGm6">Victor Klee</a>, and Günter M. Ziegler.</li>
<li>Elte, E. L. (1912), <i>The Semiregular Polytopes of the Hyperspaces</i>, Groningen: University of Groningen, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 1-4181-7968-X {{citation}}: ISBN / Date incompatibility (help) <a href="https://www.amazon.com/Semiregular-Polytopes-Hyperspaces-Emanuel-Lodewijk/dp/141817968X">The semiregular polytopes of the hyperspaces.</a> <a href="http://hdl.handle.net/2027/miun.abr2632.0001.001">The semiregular polytopes of the hyperspaces.</a></li>
<li><a href="/facts/Harold_Scott_MacDonald_Coxeter/D2l5jWGG">H.S.M. Coxeter</a>:
<ul><li>H.S.M. Coxeter, M.S. Longuet-Higgins und J.C.P. Miller: <i>Uniform Polyhedra</i>, Philosophical Transactions of the Royal Society of London, Londen, 1954</li>
<li>H.S.M. Coxeter, <i>Regular Polytopes</i>, 3rd Edition, Dover New York, 1973</li></ul></li>
<li><i><a href="http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html">Kaleidoscopes: Selected Writings of H.S.M. Coxeter</a></i>, editied 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
<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>
<li>H.S.M. Coxeter and W. O. J. Moser. <i>Generators and Relations for Discrete Groups</i> 4th ed, Springer-Verlag. New York. 1980 p. 92, p. 122.</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>
<li><a href="/facts/John_Horton_Conway/g3PA1zoK">John 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 und 39, 1965</li>
<li><a href="/facts/Norman_Johnson_(mathematician)/bQtL9w56">N.W. Johnson</a>: <i>The Theory of Uniform Polytopes and Honeycombs</i>, Ph.D. Dissertation, University of Toronto, 1966</li>
<li>N.W. Johnson: <i>Geometries and Transformations</i>, (2015) Chapter 11: Finite symmetry groups</li>
<li>Richard Klitzing, <i>Snubs, alternated facetings, and Stott-Coxeter-Dynkin diagrams</i>, Symmetry: Culture and Science, Vol. 21, No.4, 329-344, (2010) <a href="http://bendwavy.org/klitzing/pdf/Stott_v8.pdf">[1]</a></li>
<li><a href="/facts/Pieter_Hendrik_Schoute/aWQr1v2I">Schoute, Pieter Hendrik</a> (1911), "Analytic treatment of the polytopes regularly derived from the regular polytopes", <i>Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam</i>, 11 (3): 87 pp <a href="https://books.google.com/books?id=qC5LAAAAYAAJ&dq=Analytic+treatment+of+the+polytopes+regularly+derived+from+the+regular+polytopes&pg=PA357">Googlebook, 370-381</a></li></ul>
<h2 id="external-links">External links</h2>
<ul><li>Convex uniform 4-polytopes
<ul><li><a href="http://www.polytope.de">Uniform, convex polytopes in four dimensions</a>, Marco Möller (in German). Includes alternative names for these figures, including those from Jonathan Bowers, George Olshevsky, and Norman Johnson.</li>
<li><a href="https://web.archive.org/web/20030818121038/http://presh.com/hovinga/regularandsemiregularconvexpolytopesashorthistoricaloverview.html">Regular and semi-regular convex polytopes a short historical overview</a></li>
<li><a href="https://web.archive.org/web/20110718202453/http://public.beuth-hochschule.de/~meiko/pentatope.html">Java3D Applets with sources</a></li></ul></li>
<li>Nonconvex uniform 4-polytopes
<ul><li><a href="http://www.polytope.net/hedrondude/polychora.htm">Uniform polychora</a> by Jonathan Bowers</li>
<li><a href="http://www.software3d.com/Stella.php#stella4D">Stella4D</a> <a href="/facts/Stella_(software)/MDH2P21V">Stella (software)</a> produces interactive views of known uniform polychora including the 64 convex forms and the infinite prismatic families.</li></ul></li>
<li>Klitzing, Richard. <a href="https://bendwavy.org/klitzing/dimensions/polychora.htm">"4D uniform polytopes"</a>.</li>
<li><a href="http://arxiv-web3.library.cornell.edu/abs/1102.1132">4D-Polytopes and Their Dual Polytopes of the Coxeter Group W(A4) Represented by Quaternions</a> International Journal of Geometric Methods in Modern Physics, Vol. 9, No. 4 (2012) Mehmet Koca, Nazife Ozdes Koca, Mudhahir Al-Ajmi (2012) <a href="https://arxiv.org/ftp/arxiv/papers/1102/1102.1132.pdf">[2]</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>N.W. Johnson: Geometries and Transformations, (2018) ISBN 978-1-107-10340-5 Chapter 11: Finite Symmetry Groups, 11.1 Polytopes and Honeycombs, p.224 <a href="/wiki/Norman_Johnson_(mathematician)" target="_blank">/wiki/Norman_Johnson_(mathematician)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900 <a href="/wiki/Thorold_Gosset" target="_blank">/wiki/Thorold_Gosset</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>"Archived copy" (PDF). Archived from the original (PDF) on 2009-12-29. Retrieved 2010-08-13.{{cite web}}:  CS1 maint: archived copy as title (link) <a href="https://web.archive.org/web/20091229071352/http://dissertations.ub.rug.nl/FILES/faculties/science/2007/i.polo.blanco/c5.pdf" target="_blank">https://web.archive.org/web/20091229071352/http://dissertations.ub.rug.nl/FILES/faculties/science/2007/i.polo.blanco/c5.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Elte (1912) <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Uniform Polytopes in Four Dimensions December 6, 1998 oldest archive <a href="https://web.archive.org/web/19981206035238/http://members.aol.com/Polycell/uniform.html" target="_blank">https://web.archive.org/web/19981206035238/http://members.aol.com/Polycell/uniform.html</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes By David Darling, (2004) ASIN: B00SB4TU58 <a href="https://books.google.com/books?id=HrOxRdtYYaMC&q=polychoron&pg=PP1" target="_blank">https://books.google.com/books?id=HrOxRdtYYaMC&q=polychoron&pg=PP1</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Johnson (2015), Chapter 11, section 11.5 Spherical Coxeter groups, 11.5.5 full polychoric groups <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>Uniform Polytopes in Four Dimensions, George Olshevsky. <a href="https://web.archive.org/web/20070204075028/members.aol.com/Polycell/uniform.html" target="_blank">https://web.archive.org/web/20070204075028/members.aol.com/Polycell/uniform.html</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
<li id="fn:9"><p>Möller, Marco (2004). Vierdimensionale Archimedische Polytope (PDF) (Doctoral thesis) (in German). University of Hamburg. <a href="https://ediss.sub.uni-hamburg.de/volltexte/2004/2196/pdf/Dissertation.pdf" target="_blank">https://ediss.sub.uni-hamburg.de/volltexte/2004/2196/pdf/Dissertation.pdf</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li>
<li id="fn:10"><p>Conway (2008) <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li>
<li id="fn:11"><p>Multidimensional Glossary, George Olshevsky <a href="https://web.archive.org/web/20070207021813/http://members.aol.com/Polycell/glossary.html" target="_blank">https://web.archive.org/web/20070207021813/http://members.aol.com/Polycell/glossary.html</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li>
<li id="fn:12"><p>https://www.mit.edu/~hlb/Associahedron/program.pdf Convex and Abstract Polytopes workshop (2005), N.Johnson — "Uniform Polychora" abstract <a href="https://www.mit.edu/~hlb/Associahedron/program.pdf" target="_blank">https://www.mit.edu/~hlb/Associahedron/program.pdf</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li>
<li id="fn:13"><p>"Uniform Polychora". www.polytope.net. Retrieved February 20, 2020. <a href="http://www.polytope.net/hedrondude/polychora.htm" target="_blank">http://www.polytope.net/hedrondude/polychora.htm</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li>
<li id="fn:14"><p>"Uniform Polychora". www.polytope.net. Retrieved February 20, 2020. <a href="http://www.polytope.net/hedrondude/polychora.htm" target="_blank">http://www.polytope.net/hedrondude/polychora.htm</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li>
<li id="fn:15"><p>"Uniform polytope". Polytope Wiki. 6 November 2023. Retrieved 11 November 2023. <a href="https://polytope.miraheze.org/wiki/Uniform_polytope" target="_blank">https://polytope.miraheze.org/wiki/Uniform_polytope</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li>
<li id="fn:16"><p>Coxeter, Regular polytopes, 7.7 Schlaefli's criterion eq 7.78, p.135 <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li>
<li id="fn:17"><p>Johnson (2015), Chapter 11, section 11.5 Spherical Coxeter groups, 11.5.5 full polychoric groups <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li>
<li id="fn:18"><p>Johnson (2015), Chapter 11, section 11.5 Spherical Coxeter groups, 11.5.5 full polychoric groups <a href="#fnref:18" class="footnote-back-ref">↩</a></p></li>
<li id="fn:19"><p>Johnson (2015), Chapter 11, section 11.5 Spherical Coxeter groups, 11.5.5 full polychoric groups <a href="#fnref:19" class="footnote-back-ref">↩</a></p></li>
<li id="fn:20"><p>"S3s3s3s". <a href="http://www.bendwavy.org/klitzing/incmats/s3s3s3s.htm" target="_blank">http://www.bendwavy.org/klitzing/incmats/s3s3s3s.htm</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></p></li>
<li id="fn:21"><p>Johnson (2015), Chapter 11, section 11.5 Spherical Coxeter groups, 11.5.5 full polychoric groups <a href="#fnref:21" class="footnote-back-ref">↩</a></p></li>
<li id="fn:22"><p>"S3s3s4s". <a href="http://www.bendwavy.org/klitzing/incmats/s3s3s4s.htm" target="_blank">http://www.bendwavy.org/klitzing/incmats/s3s3s4s.htm</a> <a href="#fnref:22" class="footnote-back-ref">↩</a></p></li>
<li id="fn:23"><p>Johnson (2015), Chapter 11, section 11.5 Spherical Coxeter groups, 11.5.5 full polychoric groups <a href="#fnref:23" class="footnote-back-ref">↩</a></p></li>
<li id="fn:24"><p>Johnson (2015), Chapter 11, section 11.5 Spherical Coxeter groups, 11.5.5 full polychoric groups <a href="#fnref:24" class="footnote-back-ref">↩</a></p></li>
<li id="fn:25"><p>Johnson (2015), Chapter 11, section 11.5 Spherical Coxeter groups, 11.5.5 full polychoric groups <a href="#fnref:25" class="footnote-back-ref">↩</a></p></li>
<li id="fn:26"><p>"S3s4s3s". <a href="http://www.bendwavy.org/klitzing/incmats/s3s4s3s.htm" target="_blank">http://www.bendwavy.org/klitzing/incmats/s3s4s3s.htm</a> <a href="#fnref:26" class="footnote-back-ref">↩</a></p></li>
<li id="fn:27"><p>Johnson (2015), Chapter 11, section 11.5 Spherical Coxeter groups, 11.5.5 full polychoric groups <a href="#fnref:27" class="footnote-back-ref">↩</a></p></li>
<li id="fn:28"><p>Johnson (2015), Chapter 11, section 11.5 Spherical Coxeter groups, 11.5.5 full polychoric groups <a href="#fnref:28" class="footnote-back-ref">↩</a></p></li>
<li id="fn:29"><p>"S3s3s5s". <a href="http://www.bendwavy.org/klitzing/incmats/s3s3s5s.htm" target="_blank">http://www.bendwavy.org/klitzing/incmats/s3s3s5s.htm</a> <a href="#fnref:29" class="footnote-back-ref">↩</a></p></li>
<li id="fn:30"><p>Johnson (2015), Chapter 11, section 11.5 Spherical Coxeter groups, 11.5.5 full polychoric groups <a href="#fnref:30" class="footnote-back-ref">↩</a></p></li>
<li id="fn:31"><p>sns2s2mx, Richard Klitzing <a href="https://bendwavy.org/klitzing/incmats/sns2s2mx.htm" target="_blank">https://bendwavy.org/klitzing/incmats/sns2s2mx.htm</a> <a href="#fnref:31" class="footnote-back-ref">↩</a></p></li>
<li id="fn:32"><p>H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) p. 582-588 2.7 The four-dimensional analogues of the snub cube <a href="#fnref:32" class="footnote-back-ref">↩</a></p></li>
<li id="fn:33"><p>"Polytope-tree". <a href="http://bendwavy.org/klitzing/explain/polytope-tree.htm#scaliform" target="_blank">http://bendwavy.org/klitzing/explain/polytope-tree.htm#scaliform</a> <a href="#fnref:33" class="footnote-back-ref">↩</a></p></li>
<li id="fn:34"><p>"tuta". <a href="http://bendwavy.org/klitzing/incmats/tut=invtut.htm" target="_blank">http://bendwavy.org/klitzing/incmats/tut=invtut.htm</a> <a href="#fnref:34" class="footnote-back-ref">↩</a></p></li>
<li id="fn:35"><p>Category S1: Simple Scaliforms tutcup <a href="http://www.polytope.net/hedrondude/simplesc.htm" target="_blank">http://www.polytope.net/hedrondude/simplesc.htm</a> <a href="#fnref:35" class="footnote-back-ref">↩</a></p></li>
<li id="fn:36"><p>"Prissi". <a href="http://bendwavy.org/klitzing/incmats/prissi.htm" target="_blank">http://bendwavy.org/klitzing/incmats/prissi.htm</a> <a href="#fnref:36" class="footnote-back-ref">↩</a></p></li>
<li id="fn:37"><p>Category S3: Special Scaliforms prissi <a href="http://www.polytope.net/hedrondude/special.htm" target="_blank">http://www.polytope.net/hedrondude/special.htm</a> <a href="#fnref:37" class="footnote-back-ref">↩</a></p></li>
<li id="fn:38"><p>"bidex". bendwavy.org. Retrieved 11 November 2023. <a href="https://bendwavy.org/klitzing/incmats/bidex.htm" target="_blank">https://bendwavy.org/klitzing/incmats/bidex.htm</a> <a href="#fnref:38" class="footnote-back-ref">↩</a></p></li>
<li id="fn:39"><p>Category S3: Special Scaliforms bidex <a href="http://www.polytope.net/hedrondude/special.htm" target="_blank">http://www.polytope.net/hedrondude/special.htm</a> <a href="#fnref:39" class="footnote-back-ref">↩</a></p></li>
<li id="fn:40"><p>The Bi-icositetradiminished 600-cell <a href="http://eusebeia.dyndns.org/4d/bi24dim600cell" target="_blank">http://eusebeia.dyndns.org/4d/bi24dim600cell</a> <a href="#fnref:40" class="footnote-back-ref">↩</a></p></li>
<li id="fn:41"><p>"spidrox". bendwavy.org. Retrieved 11 November 2023. <a href="https://bendwavy.org/klitzing/incmats/spidrox.htm" target="_blank">https://bendwavy.org/klitzing/incmats/spidrox.htm</a> <a href="#fnref:41" class="footnote-back-ref">↩</a></p></li>
<li id="fn:42"><p>Category S4: Scaliform Swirlprisms spidrox <a href="http://www.polytope.net/hedrondude/scaleswirl.htm" target="_blank">http://www.polytope.net/hedrondude/scaleswirl.htm</a> <a href="#fnref:42" class="footnote-back-ref">↩</a></p></li>
</ol>

Uniform 4-polytope open-in-new

Uniform 4-polytope