Uniform 5-polytope

<h2 id="history-of-discovery">History of discovery</h2>
<ul><li><a href="/facts/Regular_polytope/RYTnEF6Q">Regular polytopes</a>: (convex faces)
<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 3 regular polytopes in 5 or more <a href="/facts/Dimension/0ktmUyac">dimensions</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 facets (<a href="/facts/Convex_regular_4-polytope/lFf8IH1C">convex regular 4-polytopes</a>) in his publication <i>On the Regular and Semi-Regular Figures in Space of n Dimensions</i>.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a></li></ul></li>
<li>Convex uniform polytopes:
<ul><li>1940-1988: 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, II, and III</i>.</li>
<li>1966: <a href="/facts/Norman_W._Johnson/bQtL9w56">Norman W. Johnson</a> completed his  Ph.D. Dissertation under Coxeter, <i>The Theory of Uniform Polytopes and Honeycombs</i>, University of Toronto</li></ul></li>
<li>Non-convex uniform polytopes:
<ul><li>1966: Johnson describes two non-convex uniform antiprisms in 5-space in his dissertation.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a></li>
<li>2000-2024: Jonathan Bowers and other researchers search for other non-convex uniform 5-polytopes,<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> with a current count of 1348 known uniform 5-polytopes outside infinite families (convex and non-convex), excluding the prisms of the uniform 4-polytopes. The list is not proven complete.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a><a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a></li></ul></li></ul>
<h2 id="regular-5-polytopes">Regular 5-polytopes</h2>
<p class="note">Main article: <a href="/facts/List_of_regular_polytopes/Kng25Ymq">List of regular polytopes § Five Dimensions</a></p>
<p>Regular 5-polytopes can be represented by the <a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli symbol</a> {p,q,r,s}, with s {p,q,r} 4-polytope <a href="/facts/Facet_(mathematics)/IreRHGJ7">facets</a> around each <a href="/facts/Face_(geometry)/T2E6BTU4">face</a>. There are exactly three such regular polytopes, all convex:
</p>
<ul><li>{3,3,3,3} - <a href="/facts/5-simplex/KIfBFfhl">5-simplex</a></li>
<li>{4,3,3,3} - <a href="/facts/5-cube/XABdhjof">5-cube</a></li>
<li>{3,3,3,4} - <a href="/facts/5-orthoplex/vU1TvdQ1">5-orthoplex</a></li></ul>
<p>There are no nonconvex regular polytopes in 5 dimensions or above.
</p>
<h2 id="convex-uniform-5-polytopes">Convex uniform 5-polytopes</h2>

Unsolved problem in mathematics
What is the complete set of convex uniform 5-polytopes?<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a>
<a href="/facts/List_of_unsolved_problems_in_mathematics/RroxGPpP">More unsolved problems in mathematics</a>

<p>There are 104 known convex uniform 5-polytopes, plus a number of infinite families of <a href="/facts/Duoprism/D73YMRth">duoprism</a> prisms, and polygon-polyhedron duoprisms. All except the <i>grand antiprism prism</i> are based on <a href="/facts/Wythoff_construction/8yTYjp5e">Wythoff constructions</a>, reflection symmetry generated with <a href="/facts/Coxeter_group/bgzHFm1Q">Coxeter groups</a>.
</p>
<h3>Symmetry of uniform 5-polytopes in four dimensions</h3>
<p>The <a href="/facts/5-simplex/KIfBFfhl">5-simplex</a> is the regular form in the A5 family. The <a href="/facts/5-cube/XABdhjof">5-cube</a> and <a href="/facts/5-orthoplex/vU1TvdQ1">5-orthoplex</a> are the regular forms in the B5 family. The bifurcating graph of the D5 family contains the <a href="/facts/5-orthoplex/vU1TvdQ1">5-orthoplex</a>, as well as a <a href="/facts/5-demicube/zvCeNbU2">5-demicube</a> which is an <a href="/facts/Alternation_(geometry)/H2nVs4xj">alternated</a> <a href="/facts/5-cube/XABdhjof">5-cube</a>.
</p><p>Each reflective uniform 5-polytope can be constructed in one or more reflective point group in 5 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,b,a], have an extended symmetry, [[a,b,b,a]], like [3,3,3,3], doubling the symmetry order. 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 5-polytope with chiral symmetry, shown as "empty" circled nodes", but the geometry is not generally adjustable to create uniform solutions.</p>
Fundamental families<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a>
<table><tbody><tr><th>Groupsymbol</th><th><a href="/facts/Order_(group_theory)/Cl3tKGFg">Order</a></th><th colspan="2"><a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxetergraph</a></th><th><a href="/facts/Coxeter_notation/ArUw6goe">Bracketnotation</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="2">Reflections<i>m</i>=5/2 <i>h</i><a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a></th></tr><tr><th>A5</th><td>720</td><td></td><td></td><td>[3,3,3,3]</td><td>[3,3,3,3]+</td><td>6</td><td></td><td>15 </td></tr><tr><th>D5</th><td>1920</td><td></td><td></td><td>[3,3,31,1]</td><td rowspan="2">[3,3,31,1]+</td><td>8</td><td></td><td>20 </td></tr><tr><th>B5</th><td>3840</td><td></td><td></td><td>[4,3,3,3]</td><td>10</td><td>5 </td><td>20 </td></tr></tbody></table>
Uniform prisms
<p>There are 5 finite categorical <a href="/facts/Uniform_polytope/nFcNAzLy">uniform</a> <a href="/facts/Prismatic_polytope/baIRRkCy">prismatic</a> families of polytopes based on the nonprismatic <a href="/facts/Uniform_4-polytope/NHxomMxC">uniform 4-polytopes</a>. There is one infinite family of 5-polytopes based on prisms of the uniform <a href="/facts/Duoprism/D73YMRth">duoprisms</a> {p}×{q}×{ }.
</p>
<table><tbody><tr><th><a href="/facts/Coxeter_group/bgzHFm1Q">Coxetergroup</a></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 colspan="5">Reflections</th></tr><tr><th>A4A1</th><td>120</td><td></td><td></td><td>[3,3,3,2] = [3,3,3]×[ ]</td><td>[3,3,3]+</td><td></td><td></td><td>10 </td><td></td><td>1 </td></tr><tr><th>D4A1</th><td>384</td><td></td><td></td><td>[31,1,1,2] =  [31,1,1]×[ ]</td><td rowspan="2">[31,1,1]+</td><td></td><td></td><td>12 </td><td></td><td>1 </td></tr><tr><th>B4A1</th><td>768</td><td></td><td></td><td>[4,3,3,2] = [4,3,3]×[ ]</td><td></td><td>4 </td><td>12 </td><td></td><td>1 </td></tr><tr><th>F4A1</th><td>2304</td><td></td><td></td><td>[3,4,3,2] = [3,4,3]×[ ]</td><td>[3+,4,3+]</td><td></td><td>12 </td><td>12 </td><td></td><td>1 </td></tr><tr><th>H4A1</th><td>28800</td><td></td><td></td><td>[5,3,3,2] = [3,4,3]×[ ]</td><td>[5,3,3]+</td><td></td><td></td><td>60 </td><td></td><td>1 </td></tr><tr><th colspan="12">Duoprismatic prisms (use 2p and 2q for evens)</th></tr><tr><th>I2(<i>p</i>)I2(<i>q</i>)A1</th><td>8<i>pq</i></td><td></td><td></td><td>[p,2,q,2] = [p]×[q]×[ ]</td><td rowspan="3">[p+,2,q+]</td><td></td><td><i>p</i> </td><td><i>q</i> </td><td></td><td>1 </td></tr><tr><th>I2(2<i>p</i>)I2(<i>q</i>)A1</th><td>16<i>pq</i></td><td></td><td></td><td>[2p,2,q,2] = [2p]×[q]×[ ]</td><td>p </td><td><i>p</i> </td><td><i>q</i> </td><td></td><td>1 </td></tr><tr><th>I2(2<i>p</i>)I2(2<i>q</i>)A1</th><td>32<i>pq</i></td><td></td><td></td><td>[2p,2,2q,2] = [2p]×[2q]×[ ]</td><td><i>p</i> </td><td><i>p</i> </td><td><i>q</i> </td><td><i>q</i> </td><td>1 </td></tr></tbody></table>
Uniform duoprisms
<p>There are 3 categorical <a href="/facts/Uniform_polytope/nFcNAzLy">uniform</a> <a href="/facts/Duoprism/D73YMRth">duoprismatic</a> families of polytopes based on <a href="/facts/Cartesian_product/BfqBWzdL">Cartesian products</a> of the <a href="/facts/Uniform_polyhedron/PHnprqRC">uniform polyhedra</a> and <a href="/facts/Regular_polygon/ws89Nhpt">regular polygons</a>: {<i>q</i>,<i>r</i>}×{<i>p</i>}.
</p>
<table><tbody><tr><th><a href="/facts/Coxeter_group/bgzHFm1Q">Coxetergroup</a></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 colspan="4">Reflections</th></tr><tr><th colspan="12">Prismatic groups (use 2p for even)</th></tr><tr><th>A3<i>I</i>2(<i>p</i>)</th><td>48<i>p</i></td><td></td><td></td><td>[3,3,2,<i>p</i>] = [3,3]×[<i>p</i>]</td><td rowspan="4">[(3,3)+,2,<i>p</i>+]</td><td></td><td>6 </td><td><i>p</i> </td><td></td></tr><tr><th>A3<i>I</i>2(<i>2p</i>)</th><td>96<i>p</i></td><td></td><td></td><td>[3,3,2,2<i>p</i>] = [3,3]×[2<i>p</i>]</td><td></td><td>6 </td><td><i>p</i> </td><td><i>p</i> </td></tr><tr><th>B3<i>I</i>2(<i>p</i>)</th><td>96<i>p</i></td><td></td><td></td><td>[4,3,2,<i>p</i>] = [4,3]×[<i>p</i>]</td><td>3 </td><td>6</td><td><i>p</i> </td></tr><tr><th>B3<i>I</i>2(<i>2p</i>)</th><td>192<i>p</i></td><td></td><td></td><td>[4,3,2,2<i>p</i>] = [4,3]×[2<i>p</i>]</td><td>3 </td><td>6 </td><td><i>p</i> </td><td><i>p</i> </td></tr><tr><th>H3<i>I</i>2(<i>p</i>)</th><td>240<i>p</i></td><td></td><td></td><td>[5,3,2,<i>p</i>] = [5,3]×[<i>p</i>]</td><td rowspan="2">[(5,3)+,2,<i>p</i>+]</td><td></td><td>15 </td><td><i>p</i> </td></tr><tr><th>H3<i>I</i>2(<i>2p</i>)</th><td>480<i>p</i></td><td></td><td></td><td>[5,3,2,2<i>p</i>] = [5,3]×[2<i>p</i>]</td><td></td><td>15 </td><td><i>p</i> </td><td><i>p</i> </td></tr></tbody></table>
<h3>Enumerating the convex uniform 5-polytopes</h3>
<ul><li><a href="/facts/Simplex/0FCfNRN8">Simplex</a> family: A5 [34]
<ul><li>19 uniform 5-polytopes</li></ul></li>
<li><a href="/facts/Hypercube/q2mRPbIz">Hypercube</a>/<a href="/facts/Orthoplex/erk0byF6">Orthoplex</a> family: B5 [4,33]
<ul><li>31 uniform 5-polytopes</li></ul></li>
<li><a href="/facts/Demihypercube/ackHKsEa">Demihypercube</a> D5/E5 family: [32,1,1]
<ul><li>23 uniform 5-polytopes (8 unique)</li></ul></li>
<li>Polychoral prisms:
<ul><li>56 uniform 5-polytope (45 unique) constructions based on prismatic families: [3,3,3]×[ ], [4,3,3]×[ ], [5,3,3]×[ ], [31,1,1]×[ ].</li>
<li>One <a href="/facts/Non-Wythoffian/8yTYjp5e">non-Wythoffian</a> - The grand antiprism prism is the only known non-Wythoffian convex uniform 5-polytope, constructed from two <a href="/facts/Grand_antiprism/aohzsR7w">grand antiprisms</a> connected by polyhedral prisms.</li></ul></li></ul>
<p>That brings the tally to: 19+31+8+45+1=104
</p><p>In addition there are:
</p>
<ul><li>Infinitely many uniform 5-polytope constructions based on duoprism prismatic families: [<i>p</i>]×[<i>q</i>]×[ ].</li>
<li>Infinitely many uniform 5-polytope constructions based on duoprismatic families: [3,3]×[<i>p</i>], [4,3]×[<i>p</i>], [5,3]×[<i>p</i>].</li></ul>
<h3>The A5 family</h3>
<p class="note">Further information: <a href="/facts/A5_polytope/dXEYDxCA">A5 polytope</a></p>
<p>There are 19 forms based on all permutations of the <a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter diagrams</a> with one or more rings. (16+4-1 cases)
</p><p>They are named by <a href="/facts/Norman_Johnson_(mathematician)/bQtL9w56">Norman Johnson</a> from the Wythoff construction operations upon regular 5-simplex (hexateron).
</p><p>The <a href="/facts/Coxeter_group/bgzHFm1Q">A5 family</a> has symmetry of order 720 (6 <a href="/facts/Factorial/kkfy1Bwe">factorial</a>). 7 of the 19 figures, with symmetrically ringed Coxeter diagrams have doubled symmetry, order 1440.
</p><p>The coordinates of uniform 5-polytopes with 5-simplex symmetry can be generated as permutations of simple integers in 6-space, all in hyperplanes with normal vector (1,1,1,1,1,1).
</p>
<table><tbody><tr><th rowspan="2">#</th><th rowspan="2">Base point</th><th rowspan="2"><a href="/facts/Norman_Johnson_(mathematician)/bQtL9w56">Johnson</a> naming systemBowers name and (acronym)<a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter diagram</a></th><th colspan="5">k-face element counts</th><th rowspan="2"><a href="/facts/Vertex_figure/3u2ySK0C">Vertexfigure</a></th><th colspan="6">Facet counts by location: [3,3,3,3]</th></tr><tr><th>4</th><th>3</th><th>2</th><th>1</th><th>0</th><th>[3,3,3](6)</th><th>[3,3,2](15)</th><th>[3,2,3](20)</th><th>[2,3,3](15)</th><th>[3,3,3](6)</th><th>Alt</th></tr><tr><th>1</th><td>(0,0,0,0,0,1) or (0,1,1,1,1,1)</td><td><a href="/facts/5-simplex/KIfBFfhl">5-simplex</a>hexateron (hix)</td><td>6</td><td>15</td><td>20</td><td>15</td><td>6</td><td><a href="/facts/5-cell/Rh6lwDcc">{3,3,3}</a></td><td><a href="/facts/5-cell/Rh6lwDcc">{3,3,3}</a></td><td>-</td><td>-</td><td>-</td><td>-</td><td></td></tr><tr><th>2</th><td>(0,0,0,0,1,1) or (0,0,1,1,1,1)</td><td><a href="/facts/Rectified_5-simplex/H5kx0rFW">Rectified 5-simplex</a>rectified hexateron (rix)</td><td>12</td><td>45</td><td>80</td><td>60</td><td>15</td><td><a href="/facts/Truncated_tetrahedral_prism/7Fc4E1FP">t{3,3}×{ }</a></td><td><a href="/facts/Rectified_5-cell/2UAMuo2P">r{3,3,3}</a></td><td>-</td><td>-</td><td>-</td><td><a href="/facts/5-cell/Rh6lwDcc">{3,3,3}</a></td></tr><tr><th>3</th><td>(0,0,0,0,1,2) or (0,1,2,2,2,2)</td><td><a href="/facts/Truncated_5-simplex/GFrpum4c">Truncated 5-simplex</a>truncated hexateron (tix)</td><td>12</td><td>45</td><td>80</td><td>75</td><td>30</td><td><a href="/facts/5-cell/Rh6lwDcc">Tetrah.pyr</a></td><td><a href="/facts/Truncated_5-cell/oqF6jkb7">t{3,3,3}</a></td><td>-</td><td>-</td><td>-</td><td><a href="/facts/5-cell/Rh6lwDcc">{3,3,3}</a></td><td></td></tr><tr><th>4</th><td>(0,0,0,1,1,2) or (0,1,1,2,2,2)</td><td><a href="/facts/Cantellated_5-simplex/httz7gDl">Cantellated 5-simplex</a>small rhombated hexateron (sarx)</td><td>27</td><td>135</td><td>290</td><td>240</td><td>60</td><td>prism-wedge</td><td><a href="/facts/Cantellated_5-cell/pm158bSU">rr{3,3,3}</a></td><td>-</td><td>-</td><td><a href="/facts/Tetrahedral_prism/oQ720tUe">{ }×{3,3}</a></td><td><a href="/facts/Rectified_5-cell/2UAMuo2P">r{3,3,3}</a></td><td></td></tr><tr><th>5</th><td>(0,0,0,1,2,2) or (0,0,1,2,2,2)</td><td><a href="/facts/Bitruncated_5-simplex/GFrpum4c">Bitruncated 5-simplex</a>	bitruncated hexateron (bittix)</td><td>12</td><td>60</td><td>140</td><td>150</td><td>60</td><td></td><td><a href="/facts/Bitruncated_5-cell/oqF6jkb7">2t{3,3,3}</a></td><td>-</td><td>-</td><td>-</td><td><a href="/facts/Truncated_5-cell/oqF6jkb7">t{3,3,3}</a></td><td></td></tr><tr><th>6</th><td>(0,0,0,1,2,3) or (0,1,2,3,3,3)</td><td><a href="/facts/Cantitruncated_5-simplex/httz7gDl">Cantitruncated 5-simplex</a>great rhombated hexateron (garx)</td><td>27</td><td>135</td><td>290</td><td>300</td><td>120</td><td></td><td><a href="/facts/Cantitruncated_5-cell/pm158bSU">tr{3,3,3}</a></td><td>-</td><td>-</td><td><a href="/facts/Tetrahedral_prism/oQ720tUe">{ }×{3,3}</a></td><td><a href="/facts/Truncated_5-cell/oqF6jkb7">t{3,3,3}</a></td><td></td></tr><tr><th>7</th><td>(0,0,1,1,1,2) or (0,1,1,1,2,2)</td><td><a href="/facts/Runcinated_5-simplex/tylUERyx">Runcinated 5-simplex</a>small prismated hexateron (spix)</td><td>47</td><td>255</td><td>420</td><td>270</td><td>60</td><td></td><td><a href="/facts/Runcinated_5-cell/7A0II7du">t0,3{3,3,3}</a></td><td>-</td><td><a href="/facts/3-3_duoprism/B1dhsgrg">{3}×{3}</a></td><td><a href="/facts/Octahedral_prism/8UVHkxvl">{ }×r{3,3}</a></td><td><a href="/facts/Rectified_5-cell/2UAMuo2P">r{3,3,3}</a></td><td></td></tr><tr><th>8</th><td>(0,0,1,1,2,3) or (0,1,2,2,3,3)</td><td><a href="/facts/Runcitruncated_5-simplex/tylUERyx">Runcitruncated 5-simplex</a>prismatotruncated hexateron (pattix)</td><td>47</td><td>315</td><td>720</td><td>630</td><td>180</td><td></td><td><a href="/facts/Runcitruncated_5-cell/7A0II7du">t0,1,3{3,3,3}</a></td><td>-</td><td><a href="/facts/3-6_duoprism/Map8UCwv">{6}×{3}</a></td><td><a href="/facts/Octahedral_prism/8UVHkxvl">{ }×r{3,3}</a></td><td><a href="/facts/Cantellated_5-cell/pm158bSU">rr{3,3,3}</a></td><td></td></tr><tr><th>9</th><td>(0,0,1,2,2,3) or (0,1,1,2,3,3)</td><td><a href="/facts/Runcicantellated_5-simplex/tylUERyx">Runcicantellated 5-simplex</a>prismatorhombated hexateron (pirx)</td><td>47</td><td>255</td><td>570</td><td>540</td><td>180</td><td></td><td><a href="/facts/Runcitruncated_5-cell/7A0II7du">t0,1,3{3,3,3}</a></td><td>-</td><td><a href="/facts/3-3_duoprism/B1dhsgrg">{3}×{3}</a></td><td><a href="/facts/Truncated_tetrahedral_prism/7Fc4E1FP">{ }×t{3,3}</a></td><td><a href="/facts/Bitruncated_5-cell/oqF6jkb7">2t{3,3,3}</a></td><td></td></tr><tr><th>10</th><td>(0,0,1,2,3,4) or (0,1,2,3,4,4)</td><td><a href="/facts/Runcicantitruncated_5-simplex/tylUERyx">Runcicantitruncated 5-simplex</a>great prismated hexateron (gippix)</td><td>47</td><td>315</td><td>810</td><td>900</td><td>360</td><td>Irr.<a href="/facts/5-cell/Rh6lwDcc">5-cell</a></td><td><a href="/facts/Omnitruncated_5-cell/7A0II7du">t0,1,2,3{3,3,3}</a></td><td>-</td><td><a href="/facts/3-6_duoprism/Map8UCwv">{3}×{6}</a></td><td><a href="/facts/Truncated_tetrahedral_prism/7Fc4E1FP">{ }×t{3,3}</a></td><td><a href="/facts/Cantitruncated_5-cell/pm158bSU">tr{3,3,3}</a></td><td></td></tr><tr><th>11</th><td>(0,1,1,1,2,3) or (0,1,2,2,2,3)</td><td><a href="/facts/Steritruncated_5-simplex/eUVYy57D">Steritruncated 5-simplex</a>celliprismated hexateron (cappix)</td><td>62</td><td>330</td><td>570</td><td>420</td><td>120</td><td></td><td><a href="/facts/Truncated_5-cell/oqF6jkb7">t{3,3,3}</a></td><td><a href="/facts/Truncated_tetrahedral_prism/7Fc4E1FP">{ }×t{3,3}</a></td><td><a href="/facts/3-6_duoprism/Map8UCwv">{3}×{6}</a></td><td><a href="/facts/Tetrahedral_prism/oQ720tUe">{ }×{3,3}</a></td><td><a href="/facts/Runcinated_5-cell/7A0II7du">t0,3{3,3,3}</a></td><td></td></tr><tr><th>12</th><td>(0,1,1,2,3,4) or (0,1,2,3,3,4)</td><td><a href="/facts/Stericantitruncated_5-simplex/eUVYy57D">Stericantitruncated 5-simplex</a>celligreatorhombated hexateron (cograx)</td><td>62</td><td>480</td><td>1140</td><td>1080</td><td>360</td><td></td><td><a href="/facts/Cantitruncated_5-cell/pm158bSU">tr{3,3,3}</a></td><td><a href="/facts/Truncated_octahedral_prism/gCIURDjm">{ }×tr{3,3}</a></td><td><a href="/facts/3-6_duoprism/Map8UCwv">{3}×{6}</a></td><td><a href="/facts/Cuboctahedral_prism/fFab3Nyp">{ }×rr{3,3}</a></td><td><a href="/facts/Runcitruncated_5-cell/7A0II7du">t0,1,3{3,3,3}</a></td><td></td></tr><tr><th>13</th><td>(0,0,0,1,1,1)</td><td><a href="/facts/Birectified_5-simplex/H5kx0rFW">Birectified 5-simplex</a>dodecateron (dot)</td><td>12</td><td>60</td><td>120</td><td>90</td><td>20</td><td><a href="/facts/3-3_duoprism/B1dhsgrg">{3}×{3}</a></td><td><a href="/facts/Rectified_5-cell/2UAMuo2P">r{3,3,3}</a></td><td>-</td><td>-</td><td>-</td><td><a href="/facts/Rectified_5-cell/2UAMuo2P">r{3,3,3}</a></td><td></td></tr><tr><th>14</th><td>(0,0,1,1,2,2)</td><td><a href="/facts/Bicantellated_5-simplex/httz7gDl">Bicantellated 5-simplex</a>small birhombated dodecateron (sibrid)</td><td>32</td><td>180</td><td>420</td><td>360</td><td>90</td><td></td><td><a href="/facts/Cantellated_5-cell/pm158bSU">rr{3,3,3}</a></td><td>-</td><td><a href="/facts/3-3_duoprism/B1dhsgrg">{3}×{3}</a></td><td>-</td><td><a href="/facts/Cantellated_5-cell/pm158bSU">rr{3,3,3}</a></td><td></td></tr><tr><th>15</th><td>(0,0,1,2,3,3)</td><td><a href="/facts/Bicantitruncated_5-simplex/httz7gDl">Bicantitruncated 5-simplex</a>great birhombated dodecateron (gibrid)</td><td>32</td><td>180</td><td>420</td><td>450</td><td>180</td><td></td><td><a href="/facts/Cantitruncated_5-cell/pm158bSU">tr{3,3,3}</a></td><td>-</td><td><a href="/facts/3-3_duoprism/B1dhsgrg">{3}×{3}</a></td><td>-</td><td><a href="/facts/Cantitruncated_5-cell/pm158bSU">tr{3,3,3}</a></td><td></td></tr><tr><th>16</th><td>(0,1,1,1,1,2)</td><td><a href="/facts/Stericated_5-simplex/eUVYy57D">Stericated 5-simplex</a>small cellated dodecateron (scad)</td><td>62</td><td>180</td><td>210</td><td>120</td><td>30</td><td>Irr.<a href="/facts/16-cell/X0YAfobw">16-cell</a></td><td><a href="/facts/5-cell/Rh6lwDcc">{3,3,3}</a></td><td><a href="/facts/Tetrahedral_prism/oQ720tUe">{ }×{3,3}</a></td><td><a href="/facts/3-3_duoprism/B1dhsgrg">{3}×{3}</a></td><td><a href="/facts/Tetrahedral_prism/oQ720tUe">{ }×{3,3}</a></td><td><a href="/facts/5-cell/Rh6lwDcc">{3,3,3}</a></td><td></td></tr><tr><th>17</th><td>(0,1,1,2,2,3)</td><td><a href="/facts/Stericantellated_5-simplex/eUVYy57D">Stericantellated 5-simplex</a>small cellirhombated dodecateron (card)</td><td>62</td><td>420</td><td>900</td><td>720</td><td>180</td><td></td><td><a href="/facts/Cantellated_5-cell/pm158bSU">rr{3,3,3}</a></td><td><a href="/facts/Cuboctahedral_prism/fFab3Nyp">{ }×rr{3,3}</a></td><td><a href="/facts/3-3_duoprism/B1dhsgrg">{3}×{3}</a></td><td><a href="/facts/Cuboctahedral_prism/fFab3Nyp">{ }×rr{3,3}</a></td><td><a href="/facts/Cantellated_5-cell/pm158bSU">rr{3,3,3}</a></td><td></td></tr><tr><th>18</th><td>(0,1,2,2,3,4)</td><td><a href="/facts/Steriruncitruncated_5-simplex/eUVYy57D">Steriruncitruncated 5-simplex</a>celliprismatotruncated dodecateron (captid)</td><td>62</td><td>450</td><td>1110</td><td>1080</td><td>360</td><td></td><td><a href="/facts/Runcitruncated_5-cell/7A0II7du">t0,1,3{3,3,3}</a></td><td><a href="/facts/Truncated_tetrahedral_prism/7Fc4E1FP">{ }×t{3,3}</a></td><td><a href="/facts/6-6_duoprism/Rgcg9TrJ">{6}×{6}</a></td><td><a href="/facts/Truncated_tetrahedral_prism/7Fc4E1FP">{ }×t{3,3}</a></td><td><a href="/facts/Runcitruncated_5-cell/7A0II7du">t0,1,3{3,3,3}</a></td><td></td></tr><tr><th>19</th><td>(0,1,2,3,4,5)</td><td><a href="/facts/Omnitruncated_5-simplex/eUVYy57D">Omnitruncated 5-simplex</a>great cellated dodecateron (gocad)</td><td>62</td><td>540</td><td>1560</td><td>1800</td><td>720</td><td><a href="/facts/5-cell/Rh6lwDcc">Irr. {3,3,3}</a></td><td><a href="/facts/Omnitruncated_5-cell/7A0II7du">t0,1,2,3{3,3,3}</a></td><td><a href="/facts/Truncated_octahedral_prism/gCIURDjm">{ }×tr{3,3}</a></td><td><a href="/facts/6-6_duoprism/Rgcg9TrJ">{6}×{6}</a></td><td><a href="/facts/Truncated_octahedral_prism/gCIURDjm">{ }×tr{3,3}</a></td><td><a href="/facts/Omnitruncated_5-cell/7A0II7du">t0,1,2,3{3,3,3}</a></td><td></td></tr><tr><th>Nonuniform</th><td></td><td><a href="/facts/Stericated_5-simplexes/eUVYy57D">Omnisnub 5-simplex</a>snub dodecateron (snod)snub hexateron (snix)</td><td>422</td><td>2340</td><td>4080</td><td>2520</td><td>360</td><td></td><td><a href="/facts/Runcinated_5-cell/7A0II7du">ht0,1,2,3{3,3,3}</a></td><td><a href="/facts/Truncated_octahedral_prism/gCIURDjm">ht0,1,2,3{3,3,2}</a></td><td><a href="/facts/Duoprism/D73YMRth">ht0,1,2,3{3,2,3}</a></td><td><a href="/facts/Truncated_octahedral_prism/gCIURDjm">ht0,1,2,3{3,3,2}</a></td><td><a href="/facts/Runcinated_5-cell/7A0II7du">ht0,1,2,3{3,3,3}</a></td><td>(360)<a href="/facts/5-cell/Rh6lwDcc">Irr. {3,3,3}</a></td></tr></tbody></table>
<h3>The B5 family</h3>
<p class="note">Further information: <a href="/facts/B5_polytope/D9C1sh7N">B5 polytope</a></p>
<p>The <a href="/facts/Coxeter_group/bgzHFm1Q">B5 family</a> has symmetry of order 3840 (5!×25).
</p><p>This family has 25−1=31 Wythoffian uniform polytopes generated by marking one or more nodes of the <a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter diagram</a>. Also added are 8 uniform polytopes generated as alternations with half the symmetry, which form a complete duplicate of the D5 family as ... = ..... (There are more alternations that are not listed because they produce only repetitions, as ... = .... and ... = .... These would give a complete duplication of the uniform 5-polytopes numbered 20 through 34 with symmetry broken in half.)
</p><p>For simplicity it is divided into two subgroups, each with 12 forms, and 7 "middle" forms which equally belong in both.
</p><p>The 5-cube family of 5-polytopes are given by the convex hulls of the base points listed in the following table, with all permutations of coordinates and sign taken. Each base point generates a distinct uniform 5-polytope. All coordinates correspond with uniform 5-polytopes of edge length 2.
</p>
<table><tbody><tr><th rowspan="2">#</th><th rowspan="2">Base point</th><th rowspan="2">Name<a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter diagram</a></th><th colspan="5">Element counts</th><th rowspan="2"><a href="/facts/Vertex_figure/3u2ySK0C">Vertexfigure</a></th><th colspan="6">Facet counts by location: [4,3,3,3]</th></tr><tr><th>4</th><th>3</th><th>2</th><th>1</th><th>0</th><th>[4,3,3](10)</th><th>[4,3,2](40)</th><th>[4,2,3](80)</th><th>[2,3,3](80)</th><th>[3,3,3](32)</th><th>Alt</th></tr><tr><th>20</th><td>(0,0,0,0,1)√2</td><td><a href="/facts/5-orthoplex/vU1TvdQ1">5-orthoplex</a>triacontaditeron (tac)</td><td>32</td><td>80</td><td>80</td><td>40</td><td>10</td><td><a href="/facts/16-cell/X0YAfobw">{3,3,4}</a></td><td>-</td><td>-</td><td>-</td><td>-</td><td><a href="/facts/5-cell/Rh6lwDcc">{3,3,3}</a></td><td></td></tr><tr><th>21</th><td>(0,0,0,1,1)√2</td><td><a href="/facts/Rectified_5-orthoplex/BVp334zE">Rectified 5-orthoplex</a>rectified triacontaditeron (rat)</td><td>42</td><td>240</td><td>400</td><td>240</td><td>40</td><td><a href="/facts/Octahedral_prism/8UVHkxvl">{ }×{3,4}</a></td><td><a href="/facts/16-cell/X0YAfobw">{3,3,4}</a></td><td>-</td><td>-</td><td>-</td><td><a href="/facts/Rectified_5-cell/2UAMuo2P">r{3,3,3}</a></td><td></td></tr><tr><th>22</th><td>(0,0,0,1,2)√2</td><td><a href="/facts/Truncated_5-orthoplex/EMvxIKtY">Truncated 5-orthoplex</a>truncated triacontaditeron (tot)</td><td>42</td><td>240</td><td>400</td><td>280</td><td>80</td><td>(Octah.pyr)</td><td><a href="/facts/16-cell/X0YAfobw">{3,3,4}</a></td><td>-</td><td>-</td><td>-</td><td><a href="/facts/Truncated_5-cell/oqF6jkb7">t{3,3,3}</a></td><td></td></tr><tr><th>23</th><td>(0,0,1,1,1)√2</td><td><a href="/facts/Birectified_5-cube/81cgLLCG">Birectified 5-cube</a>penteractitriacontaditeron (nit)(Birectified 5-orthoplex)</td><td>42</td><td>280</td><td>640</td><td>480</td><td>80</td><td><a href="/facts/3-4_duoprism/jlvkbaXz">{4}×{3}</a></td><td><a href="/facts/24-cell/tfoGnlAD">r{3,3,4}</a></td><td>-</td><td>-</td><td>-</td><td><a href="/facts/Rectified_5-cell/2UAMuo2P">r{3,3,3}</a></td><td></td></tr><tr><th>24</th><td>(0,0,1,1,2)√2</td><td><a href="/facts/Cantellated_5-orthoplex/7bWh33Qz">Cantellated 5-orthoplex</a>small rhombated triacontaditeron (sart)</td><td>82</td><td>640</td><td>1520</td><td>1200</td><td>240</td><td>Prism-wedge</td><td><a href="/facts/24-cell/tfoGnlAD">r{3,3,4}</a></td><td><a href="/facts/Octahedral_prism/8UVHkxvl">{ }×{3,4}</a></td><td>-</td><td>-</td><td><a href="/facts/Cantellated_5-cell/pm158bSU">rr{3,3,3}</a></td><td></td></tr><tr><th>25</th><td>(0,0,1,2,2)√2</td><td><a href="/facts/Bitruncated_5-orthoplex/EMvxIKtY">Bitruncated 5-orthoplex</a>bitruncated triacontaditeron (bittit)</td><td>42</td><td>280</td><td>720</td><td>720</td><td>240</td><td></td><td><a href="/facts/Truncated_16-cell/ShQzMKoa">t{3,3,4}</a></td><td>-</td><td>-</td><td>-</td><td><a href="/facts/Bitruncated_5-cell/oqF6jkb7">2t{3,3,3}</a></td><td></td></tr><tr><th>26</th><td>(0,0,1,2,3)√2</td><td><a href="/facts/Cantitruncated_5-orthoplex/7bWh33Qz">Cantitruncated 5-orthoplex</a>great rhombated triacontaditeron (gart)</td><td>82</td><td>640</td><td>1520</td><td>1440</td><td>480</td><td></td><td><a href="/facts/Truncated_16-cell/ShQzMKoa">t{3,3,4}</a></td><td><a href="/facts/Octahedral_prism/8UVHkxvl">{ }×{3,4}</a></td><td>-</td><td>-</td><td><a href="/facts/Runcitruncated_5-cell/7A0II7du">t0,1,3{3,3,3}</a></td><td></td></tr><tr><th>27</th><td>(0,1,1,1,1)√2</td><td><a href="/facts/Rectified_5-cube/81cgLLCG">Rectified 5-cube</a>rectified penteract (rin)</td><td>42</td><td>200</td><td>400</td><td>320</td><td>80</td><td><a href="/facts/Tetrahedral_prism/oQ720tUe">{3,3}×{ }</a></td><td><a href="/facts/Rectified_tesseract/xfSdmduC">r{4,3,3}</a></td><td>-</td><td>-</td><td>-</td><td><a href="/facts/5-cell/Rh6lwDcc">{3,3,3}</a></td><td></td></tr><tr><th>28</th><td>(0,1,1,1,2)√2</td><td><a href="/facts/Runcinated_5-orthoplex/IcsgoFAF">Runcinated 5-orthoplex</a>small prismated triacontaditeron (spat)</td><td>162</td><td>1200</td><td>2160</td><td>1440</td><td>320</td><td></td><td><a href="/facts/Rectified_tesseract/xfSdmduC">r{4,3,3}</a></td><td><a href="/facts/Cuboctahedral_prism/fFab3Nyp">{ }×r{3,4}</a></td><td><a href="/facts/3-4_duoprism/jlvkbaXz">{3}×{4}</a></td><td></td><td><a href="/facts/Runcinated_5-cell/7A0II7du">t0,3{3,3,3}</a></td><td></td></tr><tr><th>29</th><td>(0,1,1,2,2)√2</td><td><a href="/facts/Bicantellated_5-cube/SNt8QftF">Bicantellated 5-cube</a>small birhombated penteractitriacontaditeron (sibrant)(Bicantellated 5-orthoplex)</td><td>122</td><td>840</td><td>2160</td><td>1920</td><td>480</td><td></td><td><a href="/facts/Rectified_24-cell/YJm98yGE">rr{3,3,4}</a></td><td>-</td><td><a href="/facts/3-4_duoprism/jlvkbaXz">{4}×{3}</a></td><td>-</td><td><a href="/facts/Cantellated_5-cell/pm158bSU">rr{3,3,3}</a></td><td></td></tr><tr><th>30</th><td>(0,1,1,2,3)√2</td><td><a href="/facts/Runcitruncated_5-orthoplex/IcsgoFAF">Runcitruncated 5-orthoplex</a>prismatotruncated triacontaditeron (pattit)</td><td>162</td><td>1440</td><td>3680</td><td>3360</td><td>960</td><td></td><td><a href="/facts/Rectified_24-cell/YJm98yGE">rr{3,3,4}</a></td><td><a href="/facts/Cuboctahedral_prism/fFab3Nyp">{ }×r{3,4}</a></td><td><a href="/facts/4-6_duoprism/cWuVUKco">{6}×{4}</a></td><td>-</td><td><a href="/facts/Runcitruncated_5-cell/7A0II7du">t0,1,3{3,3,3}</a></td><td></td></tr><tr><th>31</th><td>(0,1,2,2,2)√2</td><td><a href="/facts/Bitruncated_5-cube/kLk4Dwvl">Bitruncated 5-cube</a>bitruncated penteract (bittin)</td><td>42</td><td>280</td><td>720</td><td>800</td><td>320</td><td></td><td><a href="/facts/Bitruncated_tesseract/ShQzMKoa">2t{4,3,3}</a></td><td>-</td><td>-</td><td>-</td><td><a href="/facts/Truncated_5-cell/oqF6jkb7">t{3,3,3}</a></td><td></td></tr><tr><th>32</th><td>(0,1,2,2,3)√2</td><td><a href="/facts/Runcicantellated_5-orthoplex/IcsgoFAF">Runcicantellated 5-orthoplex</a>prismatorhombated triacontaditeron (pirt)</td><td>162</td><td>1200</td><td>2960</td><td>2880</td><td>960</td><td></td><td><a href="/facts/Bitruncated_tesseract/ShQzMKoa">2t{4,3,3}</a></td><td><a href="/facts/Truncated_octahedral_prism/gCIURDjm">{ }×t{3,4}</a></td><td><a href="/facts/3-4_duoprism/jlvkbaXz">{3}×{4}</a></td><td>-</td><td><a href="/facts/Runcitruncated_5-cell/7A0II7du">t0,1,3{3,3,3}</a></td><td></td></tr><tr><th>33</th><td>(0,1,2,3,3)√2</td><td><a href="/facts/Bicantitruncated_5-cube/SNt8QftF">Bicantitruncated 5-cube</a>great birhombated triacontaditeron (gibrant)(Bicantitruncated 5-orthoplex)</td><td>122</td><td>840</td><td>2160</td><td>2400</td><td>960</td><td></td><td><a href="/facts/Truncated_24-cell/1dnQysxX">tr{3,3,4}</a></td><td>-</td><td><a href="/facts/3-4_duoprism/jlvkbaXz">{4}×{3}</a></td><td>-</td><td><a href="/facts/Cantellated_5-cell/pm158bSU">rr{3,3,3}</a></td><td></td></tr><tr><th>34</th><td>(0,1,2,3,4)√2</td><td><a href="/facts/Runcicantitruncated_5-orthoplex/IcsgoFAF">Runcicantitruncated 5-orthoplex</a>great prismated triacontaditeron (gippit)</td><td>162</td><td>1440</td><td>4160</td><td>4800</td><td>1920</td><td></td><td><a href="/facts/Truncated_24-cell/1dnQysxX">tr{3,3,4}</a></td><td><a href="/facts/Truncated_octahedral_prism/gCIURDjm">{ }×t{3,4}</a></td><td><a href="/facts/4-6_duoprism/cWuVUKco">{6}×{4}</a></td><td>-</td><td><a href="/facts/Omnitruncated_5-cell/7A0II7du">t0,1,2,3{3,3,3}</a></td><td></td></tr><tr><th>35</th><td>(1,1,1,1,1)</td><td><a href="/facts/5-cube/XABdhjof">5-cube</a>penteract (pent)</td><td>10</td><td>40</td><td>80</td><td>80</td><td>32</td><td><a href="/facts/5-cell/Rh6lwDcc">{3,3,3}</a></td><td><a href="/facts/Tesseract/tlqouBoU">{4,3,3}</a></td><td>-</td><td>-</td><td>-</td><td>-</td><td></td></tr><tr><th>36</th><td>(1,1,1,1,1)+ (0,0,0,0,1)√2</td><td><a href="/facts/Stericated_5-cube/DCXXtmGr">Stericated 5-cube</a>small cellated penteractitriacontaditeron (scant)(Stericated 5-orthoplex)</td><td>242</td><td>800</td><td>1040</td><td>640</td><td>160</td><td>Tetr.antiprm</td><td><a href="/facts/Tesseract/tlqouBoU">{4,3,3}</a></td><td><a href="/facts/Tesseract/tlqouBoU">{4,3}×{ }</a></td><td><a href="/facts/3-4_duoprism/jlvkbaXz">{4}×{3}</a></td><td><a href="/facts/Tetrahedral_prism/oQ720tUe">{ }×{3,3}</a></td><td><a href="/facts/5-cell/Rh6lwDcc">{3,3,3}</a></td><td></td></tr><tr><th>37</th><td>(1,1,1,1,1)+ (0,0,0,1,1)√2</td><td><a href="/facts/Runcinated_5-cube/bYexRjL1">Runcinated 5-cube</a>small prismated penteract (span)</td><td>202</td><td>1240</td><td>2160</td><td>1440</td><td>320</td><td></td><td><a href="/facts/Runcinated_tesseract/NqDGHQxp">t0,3{4,3,3}</a></td><td>-</td><td><a href="/facts/3-4_duoprism/jlvkbaXz">{4}×{3}</a></td><td><a href="/facts/Rectified_tetrahedral_prism/8UVHkxvl">{ }×r{3,3}</a></td><td><a href="/facts/Rectified_5-cell/2UAMuo2P">r{3,3,3}</a></td><td></td></tr><tr><th>38</th><td>(1,1,1,1,1)+ (0,0,0,1,2)√2</td><td><a href="/facts/Steritruncated_5-orthoplex/DCXXtmGr">Steritruncated 5-orthoplex</a>celliprismated triacontaditeron (cappin)</td><td>242</td><td>1520</td><td>2880</td><td>2240</td><td>640</td><td></td><td><a href="/facts/Runcinated_tesseract/NqDGHQxp">t0,3{4,3,3}</a></td><td><a href="/facts/Tesseract/tlqouBoU">{4,3}×{ }</a></td><td><a href="/facts/4-6_duoprism/cWuVUKco">{6}×{4}</a></td><td><a href="/facts/Truncated_tetrahedral_prism/7Fc4E1FP">{ }×t{3,3}</a></td><td><a href="/facts/Truncated_5-cell/oqF6jkb7">t{3,3,3}</a></td><td></td></tr><tr><th>39</th><td>(1,1,1,1,1)+ (0,0,1,1,1)√2</td><td><a href="/facts/Cantellated_5-cube/SNt8QftF">Cantellated 5-cube</a>small rhombated penteract (sirn)</td><td>122</td><td>680</td><td>1520</td><td>1280</td><td>320</td><td>Prism-wedge</td><td><a href="/facts/Cantellated_tesseract/gLYspT03">rr{4,3,3}</a></td><td>-</td><td>-</td><td><a href="/facts/Tetrahedral_prism/oQ720tUe">{ }×{3,3}</a></td><td><a href="/facts/Rectified_5-cell/2UAMuo2P">r{3,3,3}</a></td><td></td></tr><tr><th>40</th><td>(1,1,1,1,1)+ (0,0,1,1,2)√2</td><td><a href="/facts/Stericantellated_5-cube/DCXXtmGr">Stericantellated 5-cube</a>cellirhombated penteractitriacontaditeron (carnit)(Stericantellated 5-orthoplex)</td><td>242</td><td>2080</td><td>4720</td><td>3840</td><td>960</td><td></td><td><a href="/facts/Cantellated_tesseract/gLYspT03">rr{4,3,3}</a></td><td><a href="/facts/Rhombicuboctahedral_prism/4MJfAP4R">rr{4,3}×{ }</a></td><td><a href="/facts/3-4_duoprism/jlvkbaXz">{4}×{3}</a></td><td><a href="/facts/Cantellated_tetrahedral_prism/fFab3Nyp">{ }×rr{3,3}</a></td><td><a href="/facts/Cantellated_5-cell/pm158bSU">rr{3,3,3}</a></td><td></td></tr><tr><th>41</th><td>(1,1,1,1,1)+ (0,0,1,2,2)√2</td><td><a href="/facts/Runcicantellated_5-cube/bYexRjL1">Runcicantellated 5-cube</a>prismatorhombated penteract (prin)</td><td>202</td><td>1240</td><td>2960</td><td>2880</td><td>960</td><td></td><td><a href="/facts/Runcitruncated_16-cell/NqDGHQxp">t0,2,3{4,3,3}</a></td><td>-</td><td><a href="/facts/3-4_duoprism/jlvkbaXz">{4}×{3}</a></td><td><a href="/facts/Truncated_tetrahedral_prism/7Fc4E1FP">{ }×t{3,3}</a></td><td><a href="/facts/Bitruncated_5-cell/oqF6jkb7">2t{3,3,3}</a></td><td></td></tr><tr><th>42</th><td>(1,1,1,1,1)+ (0,0,1,2,3)√2</td><td><a href="/facts/Stericantitruncated_5-orthoplex/DCXXtmGr">Stericantitruncated 5-orthoplex</a>celligreatorhombated triacontaditeron (cogart)</td><td>242</td><td>2320</td><td>5920</td><td>5760</td><td>1920</td><td></td><td><a href="/facts/Runcitruncated_16-cell/NqDGHQxp">t0,2,3{4,3,3}</a></td><td><a href="/facts/Rhombicuboctahedral_prism/4MJfAP4R">rr{4,3}×{ }</a></td><td><a href="/facts/4-6_duoprism/cWuVUKco">{6}×{4}</a></td><td><a href="/facts/Truncated_octahedral_prism/gCIURDjm">{ }×tr{3,3}</a></td><td><a href="/facts/Cantitruncated_5-cell/pm158bSU">tr{3,3,3}</a></td><td></td></tr><tr><th>43</th><td>(1,1,1,1,1)+ (0,1,1,1,1)√2</td><td><a href="/facts/Truncated_5-cube/kLk4Dwvl">Truncated 5-cube</a>truncated penteract (tan)</td><td>42</td><td>200</td><td>400</td><td>400</td><td>160</td><td><a href="/facts/5-cell/Rh6lwDcc">Tetrah.pyr</a></td><td><a href="/facts/Truncated_tesseract/ShQzMKoa">t{4,3,3}</a></td><td>-</td><td>-</td><td>-</td><td><a href="/facts/5-cell/Rh6lwDcc">{3,3,3}</a></td><td></td></tr><tr><th>44</th><td>(1,1,1,1,1)+ (0,1,1,1,2)√2</td><td><a href="/facts/Steritruncated_5-cube/DCXXtmGr">Steritruncated 5-cube</a>celliprismated triacontaditeron (capt)</td><td>242</td><td>1600</td><td>2960</td><td>2240</td><td>640</td><td></td><td><a href="/facts/Truncated_tesseract/ShQzMKoa">t{4,3,3}</a></td><td><a href="/facts/Truncated_cubic_prism/u7WBPerC">t{4,3}×{ }</a></td><td><a href="/facts/3-8_duoprism/lr0l84cc">{8}×{3}</a></td><td><a href="/facts/Tetrahedral_prism/oQ720tUe">{ }×{3,3}</a></td><td><a href="/facts/Runcinated_5-cell/7A0II7du">t0,3{3,3,3}</a></td><td></td></tr><tr><th>45</th><td>(1,1,1,1,1)+ (0,1,1,2,2)√2</td><td><a href="/facts/Runcitruncated_5-cube/bYexRjL1">Runcitruncated 5-cube</a>prismatotruncated penteract (pattin)</td><td>202</td><td>1560</td><td>3760</td><td>3360</td><td>960</td><td></td><td><a href="/facts/Runcitruncated_tesseract/NqDGHQxp">t0,1,3{4,3,3}</a></td><td>-</td><td><a href="/facts/3-8_duoprism/lr0l84cc">{8}×{3}</a></td><td><a href="/facts/Octahedral_prism/8UVHkxvl">{ }×r{3,3}</a></td><td><a href="/facts/Cantellated_5-cell/pm158bSU">rr{3,3,3}</a></td><td></td></tr><tr><th>46</th><td>(1,1,1,1,1)+ (0,1,1,2,3)√2</td><td><a href="/facts/Steriruncitruncated_5-cube/DCXXtmGr">Steriruncitruncated 5-cube</a>celliprismatotruncated penteractitriacontaditeron (captint)(Steriruncitruncated 5-orthoplex)</td><td>242</td><td>2160</td><td>5760</td><td>5760</td><td>1920</td><td></td><td><a href="/facts/Runcitruncated_tesseract/NqDGHQxp">t0,1,3{4,3,3}</a></td><td><a href="/facts/Truncated_cubic_prism/u7WBPerC">t{4,3}×{ }</a></td><td><a href="/facts/6-8_duoprism/qIIl5PBx">{8}×{6}</a></td><td><a href="/facts/Truncated_tetrahedral_prism/7Fc4E1FP">{ }×t{3,3}</a></td><td><a href="/facts/Runcitruncated_5-cell/7A0II7du">t0,1,3{3,3,3}</a></td><td></td></tr><tr><th>47</th><td>(1,1,1,1,1)+ (0,1,2,2,2)√2</td><td><a href="/facts/Cantitruncated_5-cube/SNt8QftF">Cantitruncated 5-cube</a>great rhombated penteract (girn)</td><td>122</td><td>680</td><td>1520</td><td>1600</td><td>640</td><td></td><td><a href="/facts/Cantitruncated_tesseract/gLYspT03">tr{4,3,3}</a></td><td>-</td><td>-</td><td><a href="/facts/Tetrahedral_prism/oQ720tUe">{ }×{3,3}</a></td><td><a href="/facts/Truncated_5-cell/oqF6jkb7">t{3,3,3}</a></td><td></td></tr><tr><th>48</th><td>(1,1,1,1,1)+ (0,1,2,2,3)√2</td><td><a href="/facts/Stericantitruncated_5-cube/DCXXtmGr">Stericantitruncated 5-cube</a>celligreatorhombated penteract (cogrin)</td><td>242</td><td>2400</td><td>6000</td><td>5760</td><td>1920</td><td></td><td> <a href="/facts/Cantitruncated_tesseract/gLYspT03">tr{4,3,3}</a></td><td><a href="/facts/Truncated_cuboctahedral_prism/y28TNUGc">tr{4,3}×{ }</a></td><td><a href="/facts/3-8_duoprism/lr0l84cc">{8}×{3}</a></td><td><a href="/facts/Cantellated_tetrahedral_prism/fFab3Nyp">{ }×rr{3,3}</a></td><td><a href="/facts/Runcitruncated_5-cell/7A0II7du">t0,1,3{3,3,3}</a></td><td></td></tr><tr><th>49</th><td>(1,1,1,1,1)+ (0,1,2,3,3)√2</td><td><a href="/facts/Runcicantitruncated_5-cube/bYexRjL1">Runcicantitruncated 5-cube</a>great prismated penteract (gippin)</td><td>202</td><td>1560</td><td>4240</td><td>4800</td><td>1920</td><td></td><td><a href="/facts/Omnitruncated_tesseract/NqDGHQxp">t0,1,2,3{4,3,3}</a></td><td>-</td><td><a href="/facts/3-8_duoprism/lr0l84cc">{8}×{3}</a></td><td><a href="/facts/Truncated_tetrahedral_prism/7Fc4E1FP">{ }×t{3,3}</a></td><td><a href="/facts/Cantitruncated_5-cell/pm158bSU">tr{3,3,3}</a></td><td></td></tr><tr><th>50</th><td>(1,1,1,1,1)+ (0,1,2,3,4)√2</td><td><a href="/facts/Omnitruncated_5-cube/DCXXtmGr">Omnitruncated 5-cube</a>great cellated penteractitriacontaditeron (gacnet)(omnitruncated 5-orthoplex)</td><td>242</td><td>2640</td><td>8160</td><td>9600</td><td>3840</td><td><a href="/facts/5-cell/Rh6lwDcc">Irr. {3,3,3}</a></td><td><a href="/facts/Truncated_cuboctahedral_prism/y28TNUGc">tr{4,3}×{ }</a></td><td><a href="/facts/Truncated_cuboctahedral_prism/y28TNUGc">tr{4,3}×{ }</a></td><td><a href="/facts/6-8_duoprism/qIIl5PBx">{8}×{6}</a></td><td><a href="/facts/Omnitruncated_tetrahedral_prism/gCIURDjm">{ }×tr{3,3}</a></td><td><a href="/facts/Omnitruncated_5-cell/7A0II7du">t0,1,2,3{3,3,3}</a></td><td></td></tr><tr><th>51</th><td></td><td><a href="/facts/5-demicube/zvCeNbU2">5-demicube</a>hemipenteract (hin) = </td><td>26</td><td>120</td><td>160</td><td>80</td><td>16</td><td><a href="/facts/Rectified_5-cell/2UAMuo2P">r{3,3,3}</a></td><td><a href="/facts/16-cell/X0YAfobw">h{4,3,3}</a></td><td>-</td><td>-</td><td>-</td><td>-</td><td>(16)<a href="/facts/5-cell/Rh6lwDcc">{3,3,3}</a></td></tr><tr><th>52</th><td></td><td><a href="/facts/Cantic_5-cube/gkfd7877">Cantic 5-cube</a>Truncated hemipenteract (thin) = </td><td>42</td><td>280</td><td>640</td><td>560</td><td>160</td><td></td><td><a href="/facts/Truncated_16-cell/ShQzMKoa">h2{4,3,3}</a></td><td>-</td><td>-</td><td>-</td><td>(16)<a href="/facts/Rectified_5-cell/2UAMuo2P">r{3,3,3}</a></td><td>(16)<a href="/facts/Truncated_5-cell/oqF6jkb7">t{3,3,3}</a></td></tr><tr><th>53</th><td></td><td><a href="/facts/Runcic_5-cube/530DDo9X">Runcic 5-cube</a>Small rhombated hemipenteract (sirhin) = </td><td>42</td><td>360</td><td>880</td><td>720</td><td>160</td><td></td><td><a href="/facts/Rectified_tesseract/xfSdmduC">h3{4,3,3}</a></td><td>-</td><td>-</td><td>-</td><td>(16)<a href="/facts/Rectified_5-cell/2UAMuo2P">r{3,3,3}</a></td><td>(16)<a href="/facts/Cantellated_5-cell/pm158bSU">rr{3,3,3}</a></td></tr><tr><th>54</th><td></td><td><a href="/facts/Steric_5-cube/LlxgkUSu">Steric 5-cube</a>Small prismated hemipenteract (siphin) = </td><td>82</td><td>480</td><td>720</td><td>400</td><td>80</td><td></td><td><a href="/facts/16-cell/X0YAfobw">h{4,3,3}</a></td><td><a href="/facts/Tetrahedral_prism/oQ720tUe">h{4,3}×{}</a></td><td>-</td><td>-</td><td>(16)<a href="/facts/5-cell/Rh6lwDcc">{3,3,3}</a></td><td>(16)<a href="/facts/Runcinated_5-cell/7A0II7du">t0,3{3,3,3}</a></td></tr><tr><th>55</th><td></td><td><a href="/facts/Runcicantic_5-cube/530DDo9X">Runcicantic 5-cube</a>Great rhombated hemipenteract (girhin) = </td><td>42</td><td>360</td><td>1040</td><td>1200</td><td>480</td><td></td><td><a href="/facts/Bitruncated_tesseract/ShQzMKoa">h2,3{4,3,3}</a></td><td>-</td><td>-</td><td>-</td><td>(16)<a href="/facts/Bitruncated_5-cell/oqF6jkb7">2t{3,3,3}</a></td><td>(16)<a href="/facts/Cantitruncated_5-cell/pm158bSU">tr{3,3,3}</a></td></tr><tr><th>56</th><td></td><td><a href="/facts/Stericantic_5-cube/LlxgkUSu">Stericantic 5-cube</a>Prismatotruncated hemipenteract (pithin) = </td><td>82</td><td>720</td><td>1840</td><td>1680</td><td>480</td><td></td><td><a href="/facts/Truncated_16-cell/ShQzMKoa">h2{4,3,3}</a></td><td><a href="/facts/Truncated_tetrahedral_prism/7Fc4E1FP">h2{4,3}×{}</a></td><td>-</td><td>-</td><td>(16)<a href="/facts/Cantitruncated_5-cell/pm158bSU">rr{3,3,3}</a></td><td>(16)<a href="/facts/Runcitruncated_5-cell/7A0II7du">t0,1,3{3,3,3}</a></td></tr><tr><th>57</th><td></td><td><a href="/facts/Steriruncic_5-cube/LlxgkUSu">Steriruncic 5-cube</a>Prismatorhombated hemipenteract (pirhin) = </td><td>82</td><td>560</td><td>1280</td><td>1120</td><td>320</td><td></td><td><a href="/facts/Rectified_tesseract/xfSdmduC">h3{4,3,3}</a></td><td><a href="/facts/Tetrahedral_prism/oQ720tUe">h{4,3}×{}</a></td><td>-</td><td>-</td><td>(16)<a href="/facts/Truncated_5-cell/oqF6jkb7">t{3,3,3}</a></td><td>(16)<a href="/facts/Runcitruncated_5-cell/7A0II7du">t0,1,3{3,3,3}</a></td></tr><tr><th>58</th><td></td><td><a href="/facts/Steriruncicantic_5-cube/LlxgkUSu">Steriruncicantic 5-cube</a>Great prismated hemipenteract (giphin) = </td><td>82</td><td>720</td><td>2080</td><td>2400</td><td>960</td><td></td><td><a href="/facts/Bitruncated_tesseract/ShQzMKoa">h2,3{4,3,3}</a></td><td><a href="/facts/Truncated_tetrahedral_prism/7Fc4E1FP">h2{4,3}×{}</a></td><td>-</td><td>-</td><td>(16)<a href="/facts/Cantitruncated_5-cell/pm158bSU">tr{3,3,3}</a></td><td>(16)<a href="/facts/Omnitruncated_5-cell/7A0II7du">t0,1,2,3{3,3,3}</a></td></tr><tr><th>Nonuniform</th><td></td><td>Alternated runcicantitruncated 5-orthoplexSnub prismatotriacontaditeron (snippit)Snub hemipenteract (snahin) = </td><td>1122</td><td>6240</td><td>10880</td><td>6720</td><td>960</td><td></td><td><a href="/facts/Snub_24-cell/WbEJ3cPQ">sr{3,3,4}</a></td><td><a href="/facts/Truncated_octahedral_prism/gCIURDjm">sr{2,3,4}</a></td><td><a href="/facts/Duoprism/D73YMRth">sr{3,2,4}</a></td><td>-</td><td><a href="/facts/Runcinated_5-cell/7A0II7du">ht0,1,2,3{3,3,3}</a></td><td>(960)<a href="/facts/5-cell/Rh6lwDcc">Irr. {3,3,3}</a></td></tr><tr><th>Nonuniform</th><td></td><td>Edge-snub 5-orthoplexPyritosnub penteract (pysnan)</td><td>1202</td><td>7920</td><td>15360</td><td>10560</td><td>1920</td><td></td><td>sr3{3,3,4}</td><td><a href="/facts/Truncated_cuboctahedral_prism/y28TNUGc">sr3{2,3,4}</a></td><td>sr3{3,2,4}</td><td><a href="/facts/Icosahedral_prism/mpSGUN0y">s{3,3}×{ }</a></td><td><a href="/facts/Runcinated_5-cell/7A0II7du">ht0,1,2,3{3,3,3}</a></td><td>(960)<a href="/facts/Tetrahedral_prism/oQ720tUe">Irr. {3,3}×{ }</a></td></tr><tr><th>Nonuniform</th><td></td><td>Snub 5-cubeSnub penteract (snan)</td><td>2162</td><td>12240</td><td>21600</td><td>13440</td><td>960</td><td></td><td><a href="/facts/Runcinated_tesseracts/NqDGHQxp">ht0,1,2,3{3,3,4}</a></td><td><a href="/facts/Truncated_cuboctahedral_prism/y28TNUGc">ht0,1,2,3{2,3,4}</a></td><td><a href="/facts/Duoprism/D73YMRth">ht0,1,2,3{3,2,4}</a></td><td><a href="/facts/Truncated_octahedral_prism/gCIURDjm">ht0,1,2,3{3,3,2}</a></td><td><a href="/facts/Runcinated_5-cell/7A0II7du">ht0,1,2,3{3,3,3}</a></td><td>(1920)<a href="/facts/5-cell/Rh6lwDcc">Irr. {3,3,3}</a></td></tr></tbody></table>
<h3>The D5 family</h3>
<p class="note">Further information: <a href="/facts/D5_polytope/0FoPPFld">D5 polytope</a></p>
<p>The <a href="/facts/Coxeter_group/bgzHFm1Q">D5 family</a> has symmetry of order 1920 (5! x 24).
</p><p>This family has 23 Wythoffian uniform polytopes, from <i>3×8-1</i> permutations of the D5 <a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter diagram</a> with one or more rings. 15 (2×8-1) are repeated from the B5 family and 8 are unique to this family, though even those 8 duplicate the alternations from the B5 family.
</p><p>In the 15 repeats, both of the nodes terminating the length-1 branches are ringed, so the two kinds of  element are identical and the symmetry doubles: the relations are ... = .... and ... = ..., creating a complete duplication of the uniform 5-polytopes 20 through 34 above. The 8 new forms have one such node ringed and one not, with the relation ... = ... duplicating uniform 5-polytopes 51 through 58 above.
</p>
<table><tbody><tr><th rowspan="2">#</th><th rowspan="2"><a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter diagram</a><a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli symbol</a> symbolsJohnson and Bowers names</th><th colspan="5">Element counts</th><th rowspan="2"><a href="/facts/Vertex_figure/3u2ySK0C">Vertexfigure</a></th><th colspan="6">Facets by location:  [31,2,1]</th></tr><tr><th>4</th><th>3</th><th>2</th><th>1</th><th>0</th><th>[3,3,3](16)</th><th>[31,1,1](10)</th><th>[3,3]×[ ](40)</th><th>[ ]×[3]×[ ](80)</th><th>[3,3,3](16)</th><th>Alt</th></tr><tr><th>[51]</th><td> = h{4,3,3,3}, <a href="/facts/5-demicube/zvCeNbU2">5-demicube</a>Hemipenteract (hin)</td><td>26</td><td>120</td><td>160</td><td>80</td><td>16</td><td><a href="/facts/Rectified_5-cell/2UAMuo2P">r{3,3,3}</a></td><td><a href="/facts/5-cell/Rh6lwDcc">{3,3,3}</a></td><td><a href="/facts/16-cell/X0YAfobw">h{4,3,3}</a></td><td>-</td><td>-</td><td>-</td><td></td></tr><tr><th>[52]</th><td> = h2{4,3,3,3}, <a href="/facts/Cantic_5-cube/gkfd7877">cantic 5-cube</a>Truncated hemipenteract (thin)</td><td>42</td><td>280</td><td>640</td><td>560</td><td>160</td><td></td><td><a href="/facts/Truncated_5-cell/oqF6jkb7">t{3,3,3}</a></td><td><a href="/facts/Truncated_16-cell/ShQzMKoa">h2{4,3,3}</a></td><td>-</td><td>-</td><td><a href="/facts/Rectified_5-cell/2UAMuo2P">r{3,3,3}</a></td><td></td></tr><tr><th>[53]</th><td> = h3{4,3,3,3}, <a href="/facts/Runcic_5-cube/530DDo9X">runcic 5-cube</a>Small rhombated hemipenteract (sirhin)</td><td>42</td><td>360</td><td>880</td><td>720</td><td>160</td><td></td><td><a href="/facts/Cantellated_5-cell/pm158bSU">rr{3,3,3}</a></td><td><a href="/facts/Rectified_tesseract/xfSdmduC">h3{4,3,3}</a></td><td>-</td><td>-</td><td><a href="/facts/Rectified_5-cell/2UAMuo2P">r{3,3,3}</a></td><td></td></tr><tr><th>[54]</th><td> = h4{4,3,3,3}, <a href="/facts/Steric_5-cube/LlxgkUSu">steric 5-cube</a>Small prismated hemipenteract (siphin)</td><td>82</td><td>480</td><td>720</td><td>400</td><td>80</td><td></td><td><a href="/facts/Runcinated_5-cell/7A0II7du">t0,3{3,3,3}</a></td><td><a href="/facts/16-cell/X0YAfobw">h{4,3,3}</a></td><td><a href="/facts/Tetrahedral_prism/oQ720tUe">h{4,3}×{}</a></td><td>-</td><td><a href="/facts/5-cell/Rh6lwDcc">{3,3,3}</a></td><td></td></tr><tr><th>[55]</th><td> = h2,3{4,3,3,3}, <a href="/facts/Runcicantic_5-cube/530DDo9X">runcicantic 5-cube</a>Great rhombated hemipenteract (girhin)</td><td>42</td><td>360</td><td>1040</td><td>1200</td><td>480</td><td></td><td><a href="/facts/Bitruncated_5-cell/oqF6jkb7">2t{3,3,3}</a></td><td><a href="/facts/Bitruncated_tesseract/ShQzMKoa">h2,3{4,3,3}</a></td><td>-</td><td>-</td><td><a href="/facts/Cantitruncated_5-cell/pm158bSU">tr{3,3,3}</a></td><td></td></tr><tr><th>[56]</th><td> = h2,4{4,3,3,3}, <a href="/facts/Stericantic_5-cube/LlxgkUSu">stericantic 5-cube</a>Prismatotruncated hemipenteract (pithin)</td><td>82</td><td>720</td><td>1840</td><td>1680</td><td>480</td><td></td><td><a href="/facts/Runcitruncated_5-cell/7A0II7du">t0,1,3{3,3,3}</a></td><td><a href="/facts/Truncated_16-cell/ShQzMKoa">h2{4,3,3}</a></td><td><a href="/facts/Truncated_tetrahedral_prism/7Fc4E1FP">h2{4,3}×{}</a></td><td>-</td><td><a href="/facts/Cantitruncated_5-cell/pm158bSU">rr{3,3,3}</a></td><td></td></tr><tr><th>[57]</th><td> = h3,4{4,3,3,3}, <a href="/facts/Steriruncic_5-cube/LlxgkUSu">steriruncic 5-cube</a>Prismatorhombated hemipenteract (pirhin)</td><td>82</td><td>560</td><td>1280</td><td>1120</td><td>320</td><td></td><td><a href="/facts/Runcitruncated_5-cell/7A0II7du">t0,1,3{3,3,3}</a></td><td><a href="/facts/Rectified_tesseract/xfSdmduC">h3{4,3,3}</a></td><td><a href="/facts/Tetrahedral_prism/oQ720tUe">h{4,3}×{}</a></td><td>-</td><td><a href="/facts/Truncated_5-cell/oqF6jkb7">t{3,3,3}</a></td><td></td></tr><tr><th>[58]</th><td> = h2,3,4{4,3,3,3}, <a href="/facts/Steriruncicantic_5-cube/LlxgkUSu">steriruncicantic 5-cube</a>Great prismated hemipenteract (giphin)</td><td>82</td><td>720</td><td>2080</td><td>2400</td><td>960</td><td></td><td><a href="/facts/Omnitruncated_5-cell/7A0II7du">t0,1,2,3{3,3,3}</a></td><td><a href="/facts/Bitruncated_tesseract/ShQzMKoa">h2,3{4,3,3}</a></td><td><a href="/facts/Truncated_tetrahedral_prism/7Fc4E1FP">h2{4,3}×{}</a></td><td>-</td><td><a href="/facts/Cantitruncated_5-cell/pm158bSU">tr{3,3,3}</a></td><td></td></tr><tr><th>Nonuniform</th><td> = ht0,1,2,3{3,3,3,4}, alternated runcicantitruncated 5-orthoplexSnub hemipenteract (snahin)</td><td>1122</td><td>6240</td><td>10880</td><td>6720</td><td>960</td><td></td><td><a href="/facts/Runcinated_5-cell/7A0II7du">ht0,1,2,3{3,3,3}</a></td><td><a href="/facts/Snub_24-cell/WbEJ3cPQ">sr{3,3,4}</a></td><td><a href="/facts/Truncated_octahedral_prism/gCIURDjm">sr{2,3,4}</a></td><td><a href="/facts/Duoprism/D73YMRth">sr{3,2,4}</a></td><td><a href="/facts/Runcinated_5-cell/7A0II7du">ht0,1,2,3{3,3,3}</a></td><td>(960)<a href="/facts/5-cell/Rh6lwDcc">Irr. {3,3,3}</a></td></tr></tbody></table>
<h3>Uniform prismatic forms</h3>
<p>There are 5 finite categorical <a href="/facts/Uniform_polytope/nFcNAzLy">uniform</a> <a href="/facts/Prismatic_polytope/baIRRkCy">prismatic</a> families of polytopes based on the nonprismatic uniform <a href="/facts/4-polytope/mY8vBpec">4-polytopes</a>. For simplicity, most alternations are not shown.
</p>
<h4>A4 × A1</h4>
<p>This prismatic family has <a href="/facts/Uniform_4-polytope/NHxomMxC">9 forms</a>:
</p><p>The <a href="/facts/Coxeter_group/bgzHFm1Q">A1 x A4 family</a> has symmetry of order 240 (2*5!).
</p>
<table><tbody><tr><th rowspan="2">#</th><th rowspan="2"><a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter diagram</a> and <a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli</a>symbolsName</th><th colspan="5" rowspan="1">Element counts</th></tr><tr><th>Facets</th><th>Cells</th><th>Faces</th><th>Edges</th><th>Vertices</th></tr><tr><td>59</td><td> = {3,3,3}×{ }<a href="/facts/5-cell_prism/mqxsoqwY">5-cell prism</a> (penp)</td><td>7</td><td>20</td><td>30</td><td>25</td><td>10</td></tr><tr><td>60</td><td> = r{3,3,3}×{ }Rectified 5-cell prism (rappip)</td><td>12</td><td>50</td><td>90</td><td>70</td><td>20</td></tr><tr><td>61</td><td> = t{3,3,3}×{ }Truncated 5-cell prism (tippip)</td><td>12</td><td>50</td><td>100</td><td>100</td><td>40</td></tr><tr><td>62</td><td> = rr{3,3,3}×{ }Cantellated 5-cell prism (srippip)</td><td>22</td><td>120</td><td>250</td><td>210</td><td>60</td></tr><tr><td>63</td><td> = t0,3{3,3,3}×{ }Runcinated 5-cell prism (spiddip)</td><td>32</td><td>130</td><td>200</td><td>140</td><td>40</td></tr><tr><td>64</td><td> = 2t{3,3,3}×{ }Bitruncated 5-cell prism (decap)</td><td>12</td><td>60</td><td>140</td><td>150</td><td>60</td></tr><tr><td>65</td><td> = tr{3,3,3}×{ }Cantitruncated 5-cell prism (grippip)</td><td>22</td><td>120</td><td>280</td><td>300</td><td>120</td></tr><tr><td>66</td><td> = t0,1,3{3,3,3}×{ }Runcitruncated 5-cell prism (prippip)</td><td>32</td><td>180</td><td>390</td><td>360</td><td>120</td></tr><tr><td>67</td><td> = t0,1,2,3{3,3,3}×{ }Omnitruncated 5-cell prism (gippiddip)</td><td>32</td><td>210</td><td>540</td><td>600</td><td>240</td></tr></tbody></table>
<h4>B4 × A1</h4>
<p>This prismatic family has <a href="/facts/Uniform_4-polytope/NHxomMxC">16 forms</a>. (Three are shared with [3,4,3]×[ ] family)
</p><p>The <a href="/facts/Coxeter_group/bgzHFm1Q">A1×B4 family</a> has symmetry of order 768 (254!).
</p><p>The last three snubs can be realised with equal-length edges, but turn out nonuniform anyway because some of their 4-faces are not uniform 4-polytopes.
</p>
<table><tbody><tr><th rowspan="2">#</th><th rowspan="2"><a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter diagram</a> and <a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli</a>symbolsName</th><th colspan="5" rowspan="1">Element counts</th></tr><tr><th>Facets</th><th>Cells</th><th>Faces</th><th>Edges</th><th>Vertices</th></tr><tr><td>[16]</td><td> = {4,3,3}×{ }Tesseractic prism (pent)(Same as <a href="/facts/5-cube/XABdhjof">5-cube</a>)</td><td>10</td><td>40</td><td>80</td><td>80</td><td>32</td></tr><tr><td>68</td><td> = r{4,3,3}×{ }Rectified tesseractic prism (rittip)</td><td>26</td><td>136</td><td>272</td><td>224</td><td>64</td></tr><tr><td>69</td><td> = t{4,3,3}×{ }Truncated tesseractic prism (tattip)</td><td>26</td><td>136</td><td>304</td><td>320</td><td>128</td></tr><tr><td>70</td><td> = rr{4,3,3}×{ }Cantellated tesseractic prism (srittip)</td><td>58</td><td>360</td><td>784</td><td>672</td><td>192</td></tr><tr><td>71</td><td> = t0,3{4,3,3}×{ }Runcinated tesseractic prism (sidpithip)</td><td>82</td><td>368</td><td>608</td><td>448</td><td>128</td></tr><tr><td>72</td><td> = 2t{4,3,3}×{ }Bitruncated tesseractic prism (tahp)</td><td>26</td><td>168</td><td>432</td><td>480</td><td>192</td></tr><tr><td>73</td><td> = tr{4,3,3}×{ }Cantitruncated tesseractic prism (grittip)</td><td>58</td><td>360</td><td>880</td><td>960</td><td>384</td></tr><tr><td>74</td><td> = t0,1,3{4,3,3}×{ }Runcitruncated tesseractic prism (prohp)</td><td>82</td><td>528</td><td>1216</td><td>1152</td><td>384</td></tr><tr><td>75</td><td> = t0,1,2,3{4,3,3}×{ }Omnitruncated tesseractic prism (gidpithip)</td><td>82</td><td>624</td><td>1696</td><td>1920</td><td>768</td></tr><tr><td>76</td><td> = {3,3,4}×{ }<a href="/facts/16-cell_prism/mqxsoqwY">16-cell prism</a> (hexip)</td><td>18</td><td>64</td><td>88</td><td>56</td><td>16</td></tr><tr><td>77</td><td> = r{3,3,4}×{ }Rectified 16-cell prism (icope)(Same as 24-cell prism)</td><td>26</td><td>144</td><td>288</td><td>216</td><td>48</td></tr><tr><td>78</td><td> = t{3,3,4}×{ }Truncated 16-cell prism (thexip)</td><td>26</td><td>144</td><td>312</td><td>288</td><td>96</td></tr><tr><td>79</td><td> = rr{3,3,4}×{ }Cantellated 16-cell prism (ricope)(Same as rectified 24-cell prism)</td><td>50</td><td>336</td><td>768</td><td>672</td><td>192</td></tr><tr><td>80</td><td> = tr{3,3,4}×{ }Cantitruncated 16-cell prism (ticope)(Same as truncated 24-cell prism)</td><td>50</td><td>336</td><td>864</td><td>960</td><td>384</td></tr><tr><td>81</td><td> = t0,1,3{3,3,4}×{ }Runcitruncated 16-cell prism (prittip)</td><td>82</td><td>528</td><td>1216</td><td>1152</td><td>384</td></tr><tr><td>82</td><td> = sr{3,3,4}×{ }snub 24-cell prism (sadip)</td><td>146</td><td>768</td><td>1392</td><td>960</td><td>192</td></tr><tr><td>Nonuniform</td><td>rectified tesseractic alterprism (rita)</td><td>50</td><td>288</td><td>464</td><td>288</td><td>64</td></tr><tr><td>Nonuniform</td><td>truncated 16-cell alterprism (thexa)</td><td>26</td><td>168</td><td>384</td><td>336</td><td>96</td></tr><tr><td>Nonuniform</td><td>bitruncated tesseractic alterprism (taha)</td><td>50</td><td>288</td><td>624</td><td>576</td><td>192</td></tr></tbody></table>
<h4>F4 × A1</h4>
<p>This prismatic family has <a href="/facts/Uniform_4-polytope/NHxomMxC">10 forms</a>.
</p><p>The <a href="/facts/Coxeter_group/bgzHFm1Q">A1 x F4 family</a> has symmetry of order 2304 (2*1152). Three polytopes 85, 86 and 89 (green background) have double symmetry [[3,4,3],2], order 4608. The last one, snub 24-cell prism, (blue background) has [3+,4,3,2] symmetry, order 1152.
</p>
<table><tbody><tr><th rowspan="2">#</th><th rowspan="2"><a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter diagram</a> and <a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli</a>symbolsName</th><th colspan="5" rowspan="1">Element counts</th></tr><tr><th>Facets</th><th>Cells</th><th>Faces</th><th>Edges</th><th>Vertices</th></tr><tr><td>[77]</td><td> = {3,4,3}×{ }<a href="/facts/24-cell_prism/mqxsoqwY">24-cell prism</a> (icope)</td><td>26</td><td>144</td><td>288</td><td>216</td><td>48</td></tr><tr><td>[79]</td><td> = r{3,4,3}×{ }rectified 24-cell prism (ricope)</td><td>50</td><td>336</td><td>768</td><td>672</td><td>192</td></tr><tr><td>[80]</td><td> = t{3,4,3}×{ }truncated 24-cell prism (ticope)</td><td>50</td><td>336</td><td>864</td><td>960</td><td>384</td></tr><tr><td>83</td><td> = rr{3,4,3}×{ }cantellated 24-cell prism (sricope)</td><td>146</td><td>1008</td><td>2304</td><td>2016</td><td>576</td></tr><tr><td>84</td><td> = t0,3{3,4,3}×{ }runcinated 24-cell prism (spiccup)</td><td>242</td><td>1152</td><td>1920</td><td>1296</td><td>288</td></tr><tr><td>85</td><td> = 2t{3,4,3}×{ } bitruncated 24-cell prism (contip)</td><td>50</td><td>432</td><td>1248</td><td>1440</td><td>576</td></tr><tr><td>86</td><td> = tr{3,4,3}×{ }cantitruncated 24-cell prism (gricope)</td><td>146</td><td>1008</td><td>2592</td><td>2880</td><td>1152</td></tr><tr><td>87</td><td> = t0,1,3{3,4,3}×{ }runcitruncated 24-cell prism (pricope)</td><td>242</td><td>1584</td><td>3648</td><td>3456</td><td>1152</td></tr><tr><td>88</td><td> = t0,1,2,3{3,4,3}×{ } omnitruncated 24-cell prism (gippiccup)</td><td>242</td><td>1872</td><td>5088</td><td>5760</td><td>2304</td></tr><tr><td>[82]</td><td> = s{3,4,3}×{ }snub 24-cell prism (sadip)</td><td>146</td><td>768</td><td>1392</td><td>960</td><td>192</td></tr></tbody></table>
<h4>H4 × A1</h4>
<p>This prismatic family has <a href="/facts/Uniform_4-polytope/NHxomMxC">15 forms</a>:
</p><p>The <a href="/facts/Coxeter_group/bgzHFm1Q">A1 x H4 family</a> has symmetry of order 28800 (2*14400).
</p>
<table><tbody><tr><th rowspan="2">#</th><th rowspan="2"><a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter diagram</a> and <a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli</a>symbolsName</th><th colspan="5" rowspan="1">Element counts</th></tr><tr><th>Facets</th><th>Cells</th><th>Faces</th><th>Edges</th><th>Vertices</th></tr><tr><td>89</td><td> = {5,3,3}×{ }<a href="/facts/120-cell_prism/mqxsoqwY">120-cell prism</a> (hipe)</td><td>122</td><td>960</td><td>2640</td><td>3000</td><td>1200</td></tr><tr><td>90</td><td> = r{5,3,3}×{ }Rectified 120-cell prism (rahipe)</td><td>722</td><td>4560</td><td>9840</td><td>8400</td><td>2400</td></tr><tr><td>91</td><td> = t{5,3,3}×{ }Truncated 120-cell prism (thipe)</td><td>722</td><td>4560</td><td>11040</td><td>12000</td><td>4800</td></tr><tr><td>92</td><td> = rr{5,3,3}×{ }Cantellated 120-cell prism (srahip)</td><td>1922</td><td>12960</td><td>29040</td><td>25200</td><td>7200</td></tr><tr><td>93</td><td> = t0,3{5,3,3}×{ }Runcinated 120-cell prism (sidpixhip)</td><td>2642</td><td>12720</td><td>22080</td><td>16800</td><td>4800</td></tr><tr><td>94</td><td> = 2t{5,3,3}×{ }Bitruncated 120-cell prism (xhip)</td><td>722</td><td>5760</td><td>15840</td><td>18000</td><td>7200</td></tr><tr><td>95</td><td> = tr{5,3,3}×{ }Cantitruncated 120-cell prism (grahip)</td><td>1922</td><td>12960</td><td>32640</td><td>36000</td><td>14400</td></tr><tr><td>96</td><td> = t0,1,3{5,3,3}×{ }Runcitruncated 120-cell prism (prixip)</td><td>2642</td><td>18720</td><td>44880</td><td>43200</td><td>14400</td></tr><tr><td>97</td><td> = t0,1,2,3{5,3,3}×{ }Omnitruncated 120-cell prism (gidpixhip)</td><td>2642</td><td>22320</td><td>62880</td><td>72000</td><td>28800</td></tr><tr><td>98</td><td> = {3,3,5}×{ }<a href="/facts/600-cell_prism/mqxsoqwY">600-cell prism</a> (exip)</td><td>602</td><td>2400</td><td>3120</td><td>1560</td><td>240</td></tr><tr><td>99</td><td> = r{3,3,5}×{ }Rectified 600-cell prism (roxip)</td><td>722</td><td>5040</td><td>10800</td><td>7920</td><td>1440</td></tr><tr><td>100</td><td> = t{3,3,5}×{ }Truncated 600-cell prism (texip)</td><td>722</td><td>5040</td><td>11520</td><td>10080</td><td>2880</td></tr><tr><td>101</td><td> = rr{3,3,5}×{ }Cantellated 600-cell prism (srixip)</td><td>1442</td><td>11520</td><td>28080</td><td>25200</td><td>7200</td></tr><tr><td>102</td><td> = tr{3,3,5}×{ }Cantitruncated 600-cell prism (grixip)</td><td>1442</td><td>11520</td><td>31680</td><td>36000</td><td>14400</td></tr><tr><td>103</td><td> = t0,1,3{3,3,5}×{ }Runcitruncated 600-cell prism (prahip)</td><td>2642</td><td>18720</td><td>44880</td><td>43200</td><td>14400</td></tr></tbody></table>
<h4>Duoprism prisms</h4>
<p>Uniform duoprism prisms, {<i>p</i>}×{<i>q</i>}×{ }, form an infinite class for all integers <i>p</i>,<i>q</i>>2. {4}×{4}×{ } makes a lower symmetry form of the <a href="/facts/5-cube/XABdhjof">5-cube</a>.
</p><p>The extended <a href="/facts/F-vector/oVFESPjJ">f-vector</a> of {<i>p</i>}×{<i>q</i>}×{ } is computed as (<i>p</i>,<i>p</i>,1)*(<i>q</i>,<i>q</i>,1)*(2,1) = (2<i>pq</i>,5<i>pq</i>,4<i>pq</i>+2<i>p</i>+2<i>q</i>,3<i>pq</i>+3<i>p</i>+3<i>q</i>,<i>p</i>+<i>q</i>+2,1).
</p>
<table><tbody><tr><th rowspan="2"><a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter diagram</a></th><th rowspan="2">Names</th><th colspan="6">Element counts</th></tr><tr><th>4-faces</th><th>Cells</th><th>Faces</th><th>Edges</th><th>Vertices</th></tr><tr><td></td><td>{<i>p</i>}×{<i>q</i>}×{ }<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a></td><td><i>p</i>+<i>q</i>+2</td><td>3<i>pq</i>+3<i>p</i>+3<i>q</i></td><td>4<i>pq</i>+2<i>p</i>+2<i>q</i></td><td>5<i>pq</i></td><td>2<i>pq</i></td></tr><tr><td></td><td>{<i>p</i>}2×{ }</td><td>2(<i>p</i>+1)</td><td>3<i>p</i>(<i>p</i>+1)</td><td>4<i>p</i>(<i>p</i>+1)</td><td>5<i>p</i>2</td><td>2<i>p</i>2</td></tr><tr><td></td><td>{3}2×{ }</td><td>8</td><td>36</td><td>48</td><td>45</td><td>18</td></tr><tr><td></td><td>{4}2×{ } = <a href="/facts/5-cube/XABdhjof">5-cube</a></td><td>10</td><td>40</td><td>80</td><td>80</td><td>32</td></tr></tbody></table>
<h4>Grand antiprism prism</h4>
<p>The grand antiprism prism is the only known convex non-Wythoffian uniform 5-polytope. It has 200 vertices, 1100 edges, 1940 faces (40 pentagons, 500 squares, 1400 triangles), 1360 cells (600 <a href="/facts/Tetrahedron/7mkSrl6j">tetrahedra</a>, 40 <a href="/facts/Pentagonal_antiprism/gR8aMBKN">pentagonal antiprisms</a>, 700 <a href="/facts/Triangular_prism/Yjp4Kw9M">triangular prisms</a>, 20 <a href="/facts/Pentagonal_prism/CtvKLxcy">pentagonal prisms</a>), and 322 hypercells (2 <a href="/facts/Grand_antiprism/aohzsR7w">grand antiprisms</a> , 20 <a href="/facts/Pentagonal_antiprism/gR8aMBKN">pentagonal antiprism</a> prisms , and 300 <a href="/facts/Tetrahedral_prism/oQ720tUe">tetrahedral prisms</a> ).
</p>
<table><tbody><tr><th rowspan="2">#</th><th rowspan="2">Name</th><th colspan="5">Element counts</th></tr><tr><th>Facets</th><th>Cells</th><th>Faces</th><th>Edges</th><th>Vertices</th></tr><tr><td>104</td><td>grand antiprism prism (gappip)<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a></td><td>322</td><td>1360</td><td>1940</td><td>1100</td><td>200</td></tr></tbody></table>
<h2 id="notes-on-the-wythoff-construction-for-the-uniform-5-polytopes">Notes on the Wythoff construction for the uniform 5-polytopes</h2>
<p>Construction of the reflective 5-dimensional <a href="/facts/Uniform_polytope/nFcNAzLy">uniform polytopes</a> are done through a <a href="/facts/Wythoff_construction/8yTYjp5e">Wythoff construction</a> process, and represented through a <a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter diagram</a>, where each node represents a mirror. Nodes are ringed to imply which mirrors are active. The full set of uniform polytopes generated are based on the unique permutations of ringed nodes. Uniform 5-polytopes are named in relation to the <a href="/facts/Regular_polytope/RYTnEF6Q">regular polytopes</a> in each family. Some families have two regular constructors and thus may have two ways of naming them.
</p><p>Here are the primary operators available for constructing and naming the uniform 5-polytopes.
</p><p>The last operation, the snub, and more generally the alternation, are the operations that can create nonreflective forms. These are drawn with "hollow rings" at the nodes.
</p><p>The prismatic forms and bifurcating graphs can use the same truncation indexing notation, but require an explicit numbering system on the nodes for clarity.
</p>
<table><tbody><tr><th>Operation</th><th colspan="2">Extended<a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli symbol</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,s}</td><td>{p,q,r,s}</td><td></td><td>Any regular 5-polytope</td></tr><tr><th><a href="/facts/Rectification_(geometry)/PkoUdq53">Rectified</a></th><td>t1{p,q,r,s}</td><td>r{p,q,r,s}</td><td></td><td>The edges are fully truncated into single points. The 5-polytope now has the combined faces of the parent and dual.</td></tr><tr><th><a href="/facts/Rectification_(geometry)/PkoUdq53">Birectified</a></th><td>t2{p,q,r,s}</td><td>2r{p,q,r,s}</td><td></td><td>Birectification reduces faces to points, <a href="/facts/Cell_(geometry)/T2E6BTU4">cells</a> to their <a href="/facts/Dual_polytope/rkIHbmAf">duals</a>.</td></tr><tr><th><a href="/facts/Rectification_(geometry)/PkoUdq53">Trirectified</a></th><td>t3{p,q,r,s}</td><td>3r{p,q,r,s}</td><td></td><td>Trirectification reduces cells to points. (Dual rectification)</td></tr><tr><th><a href="/facts/Rectification_(geometry)/PkoUdq53">Quadrirectified</a></th><td>t4{p,q,r,s}</td><td>4r{p,q,r,s}</td><td></td><td>Quadrirectification reduces 4-faces to points. (Dual)</td></tr><tr><th><a href="/facts/Truncation_(geometry)/DYb589Je">Truncated</a></th><td>t0,1{p,q,r,s}</td><td>t{p,q,r,s}</td><td></td><td>Each original vertex is cut off, with a new face filling the gap. Truncation has a degree of freedom, which has one solution that creates a uniform truncated 5-polytope. The 5-polytope has its original faces doubled in sides, and contains the faces of the dual.</td></tr><tr><th><a href="/facts/Cantellation_(geometry)/kS24Tde3">Cantellated</a></th><td>t0,2{p,q,r,s}</td><td>rr{p,q,r,s}</td><td></td><td>In addition to vertex truncation, each original edge is <i>beveled</i> with new rectangular faces appearing in their place. </td></tr><tr><th><a href="/facts/Runcination_(geometry)/nwUxfUIX">Runcinated</a></th><td colspan="2">t0,3{p,q,r,s}</td><td></td><td>Runcination reduces cells and creates new cells at the vertices and edges.</td></tr><tr><th><a href="/facts/Sterication_(geometry)/nFcNAzLy">Stericated</a></th><td>t0,4{p,q,r,s}</td><td>2r2r{p,q,r,s}</td><td></td><td>Sterication reduces facets and creates new facets (hypercells) at the vertices and edges in the gaps. (Same as <a href="/facts/Expansion_(geometry)/eMpoRlnm">expansion</a> operation for 5-polytopes.)</td></tr><tr><th><a href="/facts/Omnitruncation_(geometry)/hmbMrRm6">Omnitruncated</a></th><td colspan="2">t0,1,2,3,4{p,q,r,s}</td><td></td><td>All four operators, truncation, cantellation, runcination, and sterication are applied.</td></tr><tr><th>Half</th><td colspan="2">h{2p,3,q,r}</td><td></td><td><a href="/facts/Alternation_(geometry)/H2nVs4xj">Alternation</a>, same as </td></tr><tr><th>Cantic</th><td colspan="2">h2{2p,3,q,r}</td><td></td><td>Same as </td></tr><tr><th>Runcic</th><td colspan="2">h3{2p,3,q,r}</td><td></td><td>Same as </td></tr><tr><th>Runcicantic</th><td colspan="2">h2,3{2p,3,q,r}</td><td></td><td>Same as </td></tr><tr><th>Steric</th><td colspan="2">h4{2p,3,q,r}</td><td></td><td>Same as </td></tr><tr><th>Steriruncic</th><td colspan="2">h3,4{2p,3,q,r}</td><td></td><td>Same as </td></tr><tr><th>Stericantic</th><td colspan="2">h2,4{2p,3,q,r}</td><td></td><td>Same as </td></tr><tr><th>Steriruncicantic</th><td colspan="2">h2,3,4{2p,3,q,r}</td><td></td><td>Same as </td></tr><tr><th>Snub</th><td colspan="2">s{p,2q,r,s}</td><td></td><td>Alternated truncation</td></tr><tr><th>Snub rectified</th><td colspan="2">sr{p,q,2r,s}</td><td></td><td>Alternated truncated rectification</td></tr><tr><th></th><td colspan="2">ht0,1,2,3{p,q,r,s}</td><td></td><td>Alternated runcicantitruncation</td></tr><tr><th>Full snub</th><td colspan="2">ht0,1,2,3,4{p,q,r,s}</td><td></td><td>Alternated omnitruncation</td></tr></tbody></table>
<h2 id="regular-and-uniform-honeycombs">Regular and uniform honeycombs</h2>

<p>There are five fundamental affine <a href="/facts/Coxeter_groups/bgzHFm1Q">Coxeter groups</a>, and 13 prismatic groups that generate regular and uniform tessellations in Euclidean 4-space.<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a><a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a>
</p>
Fundamental groups<table><tbody><tr><th>#</th><th colspan="3"><a href="/facts/Coxeter_group/bgzHFm1Q">Coxeter group</a></th><th><a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter diagram</a></th><th>Forms</th></tr><tr><td>1</td><td>                                                                        A                ~                                                          4                                {\displaystyle {\tilde {A}}_{4}}  </td><td>[3[5]]</td><td>[(3,3,3,3,3)]</td><td></td><td>7</td></tr><tr><td>2</td><td>                                                                        C                ~                                                          4                                {\displaystyle {\tilde {C}}_{4}}  </td><td>[4,3,3,4]</td><td></td><td></td><td>19</td></tr><tr><td>3</td><td>                                                                        B                ~                                                          4                                {\displaystyle {\tilde {B}}_{4}}  </td><td>[4,3,31,1]</td><td>[4,3,3,4,1+]</td><td> = </td><td>23 (8 new)</td></tr><tr><td>4</td><td>                                                                        D                ~                                                          4                                {\displaystyle {\tilde {D}}_{4}}  </td><td>[31,1,1,1]</td><td>[1+,4,3,3,4,1+]</td><td> = </td><td>9 (0 new)</td></tr><tr><td>5</td><td>                                                                        F                ~                                                          4                                {\displaystyle {\tilde {F}}_{4}}  </td><td>[3,4,3,3]</td><td></td><td></td><td>31 (21 new)</td></tr></tbody></table>
<p>There are three <a href="/facts/List_of_regular_polytopes/Kng25Ymq">regular honeycombs</a> of Euclidean 4-space:
</p>
<ul><li><a href="/facts/Tesseractic_honeycomb/fpucRJJn">tesseractic honeycomb</a>, with symbols {4,3,3,4},  = . There are 19 uniform honeycombs in this family.</li>
<li><a href="/facts/24-cell_honeycomb/foFoK5YX">24-cell honeycomb</a>, with symbols {3,4,3,3}, . There are 31 reflective uniform honeycombs in this family, and one alternated form.
<ul><li><a href="/facts/Truncated_24-cell_honeycomb/uQN2kXg1">Truncated 24-cell honeycomb</a> with symbols t{3,4,3,3}, </li>
<li><a href="/facts/Snub_24-cell_honeycomb/bzFwdh7X">Snub 24-cell honeycomb</a>, with symbols s{3,4,3,3},  and  constructed by four <a href="/facts/Snub_24-cell/WbEJ3cPQ">snub 24-cell</a>, one <a href="/facts/16-cell/X0YAfobw">16-cell</a>, and five <a href="/facts/5-cell/Rh6lwDcc">5-cells</a> at each vertex.</li></ul></li>
<li><a href="/facts/16-cell_honeycomb/lbktV8Nr">16-cell honeycomb</a>, with symbols {3,3,4,3}, </li></ul>
<p>Other families that generate uniform honeycombs:
</p>
<ul><li>There are 23 uniquely ringed forms, 8 new ones in the <a href="/facts/16-cell_honeycomb/lbktV8Nr">16-cell honeycomb</a> family. With symbols h{4,32,4} it is geometrically identical to the <a href="/facts/16-cell_honeycomb/lbktV8Nr">16-cell honeycomb</a>,  = </li>
<li>There are 7 uniquely ringed forms from the 
  
    
      
        
          
            
              
                A
                ~
              
            
          
          
            4
          
        
      
    
    {\displaystyle {\tilde {A}}_{4}}
  
,  family, all new, including:
<ul><li><a href="/facts/4-simplex_honeycomb/YgY6eSTr">4-simplex honeycomb</a> </li>
<li><a href="/facts/Truncated_4-simplex_honeycomb/YgY6eSTr">Truncated 4-simplex honeycomb</a> </li>
<li><a href="/facts/Omnitruncated_4-simplex_honeycomb/YgY6eSTr">Omnitruncated 4-simplex honeycomb</a> </li></ul></li>
<li>There are 9 uniquely ringed forms in the 
  
    
      
        
          
            
              
                D
                ~
              
            
          
          
            4
          
        
      
    
    {\displaystyle {\tilde {D}}_{4}}
  
: [31,1,1,1]  family, two new ones, including the <a href="/facts/Quarter_tesseractic_honeycomb/P266Jc7y">quarter tesseractic honeycomb</a>,  = , and the <a href="/facts/Bitruncated_tesseractic_honeycomb/CVRrn5Fy">bitruncated tesseractic honeycomb</a>,  = .</li></ul>
<p><a href="/facts/Non-Wythoffian/8yTYjp5e">Non-Wythoffian</a> uniform tessellations in 4-space also exist by elongation (inserting layers), and gyration (rotating layers) from these reflective forms.
</p>
Prismatic groups<table><tbody><tr><th>#</th><th colspan="2"><a href="/facts/Coxeter_group/bgzHFm1Q">Coxeter group</a></th><th><a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter diagram</a></th></tr><tr><td>1</td><td>                                                                        C                ~                                                          3                                {\displaystyle {\tilde {C}}_{3}}  ×                                                                        I                ~                                                          1                                {\displaystyle {\tilde {I}}_{1}}  </td><td>[4,3,4,2,∞]</td><td></td></tr><tr><td>2</td><td>                                                                        B                ~                                                          3                                {\displaystyle {\tilde {B}}_{3}}  ×                                                                        I                ~                                                          1                                {\displaystyle {\tilde {I}}_{1}}  </td><td>[4,31,1,2,∞]</td><td></td></tr><tr><td>3</td><td>                                                                        A                ~                                                          3                                {\displaystyle {\tilde {A}}_{3}}  ×                                                                        I                ~                                                          1                                {\displaystyle {\tilde {I}}_{1}}  </td><td>[3[4],2,∞]</td><td></td></tr><tr><td>4</td><td>                                                                        C                ~                                                          2                                {\displaystyle {\tilde {C}}_{2}}  ×                                                                        I                ~                                                          1                                {\displaystyle {\tilde {I}}_{1}}  x                                                                        I                ~                                                          1                                {\displaystyle {\tilde {I}}_{1}}  </td><td>[4,4,2,∞,2,∞]</td><td></td></tr><tr><td>5</td><td>                                                                        H                ~                                                          2                                {\displaystyle {\tilde {H}}_{2}}  ×                                                                        I                ~                                                          1                                {\displaystyle {\tilde {I}}_{1}}  x                                                                        I                ~                                                          1                                {\displaystyle {\tilde {I}}_{1}}  </td><td>[6,3,2,∞,2,∞]</td><td></td></tr><tr><td>6</td><td>                                                                        A                ~                                                          2                                {\displaystyle {\tilde {A}}_{2}}  ×                                                                        I                ~                                                          1                                {\displaystyle {\tilde {I}}_{1}}  x                                                                        I                ~                                                          1                                {\displaystyle {\tilde {I}}_{1}}  </td><td>[3[3],2,∞,2,∞]</td><td></td></tr><tr><td>7</td><td>                                                                        I                ~                                                          1                                {\displaystyle {\tilde {I}}_{1}}  ×                                                                        I                ~                                                          1                                {\displaystyle {\tilde {I}}_{1}}  x                                                                        I                ~                                                          1                                {\displaystyle {\tilde {I}}_{1}}  x                                                                        I                ~                                                          1                                {\displaystyle {\tilde {I}}_{1}}  </td><td>[∞,2,∞,2,∞,2,∞]</td><td></td></tr><tr><td>8</td><td>                                                                        A                ~                                                          2                                {\displaystyle {\tilde {A}}_{2}}  x                                                                        A                ~                                                          2                                {\displaystyle {\tilde {A}}_{2}}  </td><td>[3[3],2,3[3]]</td><td></td></tr><tr><td>9</td><td>                                                                        A                ~                                                          2                                {\displaystyle {\tilde {A}}_{2}}  ×                                                                        B                ~                                                          2                                {\displaystyle {\tilde {B}}_{2}}  </td><td>[3[3],2,4,4]</td><td></td></tr><tr><td>10</td><td>                                                                        A                ~                                                          2                                {\displaystyle {\tilde {A}}_{2}}  ×                                                                        G                ~                                                          2                                {\displaystyle {\tilde {G}}_{2}}  </td><td>[3[3],2,6,3]</td><td></td></tr><tr><td>11</td><td>                                                                        B                ~                                                          2                                {\displaystyle {\tilde {B}}_{2}}  ×                                                                        B                ~                                                          2                                {\displaystyle {\tilde {B}}_{2}}  </td><td>[4,4,2,4,4]</td><td></td></tr><tr><td>12</td><td>                                                                        B                ~                                                          2                                {\displaystyle {\tilde {B}}_{2}}  ×                                                                        G                ~                                                          2                                {\displaystyle {\tilde {G}}_{2}}  </td><td>[4,4,2,6,3]</td><td></td></tr><tr><td>13</td><td>                                                                        G                ~                                                          2                                {\displaystyle {\tilde {G}}_{2}}  ×                                                                        G                ~                                                          2                                {\displaystyle {\tilde {G}}_{2}}  </td><td>[6,3,2,6,3]</td><td></td></tr></tbody></table>
<h3>Regular and uniform hyperbolic honeycombs</h3>
Hyperbolic compact groups
<p>There are 5 <a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">compact hyperbolic Coxeter groups</a> of rank 5, each generating uniform honeycombs in hyperbolic 4-space as permutations of rings of the Coxeter diagrams.
</p>
<table><tbody><tr><td><p>                                                                                          A                  F                                ^                                                          4                                {\displaystyle {\widehat {AF}}_{4}}   = [(3,3,3,3,4)]: </p></td><td><p>                                                                                          D                  H                                ¯                                                          4                                {\displaystyle {\bar {DH}}_{4}}   = [5,3,31,1]: </p></td><td>                                                                        H                ¯                                                          4                                {\displaystyle {\bar {H}}_{4}}   = [3,3,3,5]: <p>                                                                                          B                  H                                ¯                                                          4                                {\displaystyle {\bar {BH}}_{4}}   = [4,3,3,5]:                                                                         K                ¯                                                          4                                {\displaystyle {\bar {K}}_{4}}   = [5,3,3,5]: </p></td></tr></tbody></table>
<p>There are 5 regular compact convex hyperbolic honeycombs in H4 space:<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a>
</p>
Compact regular convex hyperbolic honeycombs<table><tbody><tr><th>Honeycomb name</th><th><a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli</a>Symbol{p,q,r,s}</th><th><a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter diagram</a></th><th>Facettype{p,q,r}</th><th>Celltype{p,q}</th><th>Facetype{p}</th><th>Facefigure{s}</th><th>Edgefigure{r,s}</th><th><a href="/facts/Vertex_figure/3u2ySK0C">Vertexfigure</a>{q,r,s}</th><th><a href="/facts/Dual_polyhedron/rkIHbmAf">Dual</a></th></tr><tr><td><a href="/facts/Order-5_5-cell_honeycomb/eHYIWHfm">Order-5 5-cell</a> (pente)</td><td>{3,3,3,5}</td><td></td><td>{3,3,3}</td><td>{3,3}</td><td>{3}</td><td>{5}</td><td>{3,5}</td><td>{3,3,5}</td><td>{5,3,3,3}</td></tr><tr><td><a href="/facts/Order-3_120-cell_honeycomb/Kng25Ymq">Order-3 120-cell</a> (hitte)</td><td>{5,3,3,3}</td><td></td><td>{5,3,3}</td><td>{5,3}</td><td>{5}</td><td>{3}</td><td>{3,3}</td><td>{3,3,3}</td><td>{3,3,3,5}</td></tr><tr><td><a href="/facts/Order-5_tesseractic_honeycomb/Rq94q2gt">Order-5 tesseractic</a> (pitest)</td><td>{4,3,3,5}</td><td></td><td>{4,3,3}</td><td>{4,3}</td><td>{4}</td><td>{5}</td><td>{3,5}</td><td>{3,3,5}</td><td>{5,3,3,4}</td></tr><tr><td><a href="/facts/Order-4_120-cell_honeycomb/v2xFNTzJ">Order-4 120-cell</a> (shitte)</td><td>{5,3,3,4}</td><td></td><td>{5,3,3}</td><td>{5,3}</td><td>{5}</td><td>{4}</td><td>{3,4}</td><td>{3,3,4}</td><td>{4,3,3,5}</td></tr><tr><td><a href="/facts/Order-5_120-cell_honeycomb/KC6LgT5T">Order-5 120-cell</a> (phitte)</td><td>{5,3,3,5}</td><td></td><td>{5,3,3}</td><td>{5,3}</td><td>{5}</td><td>{5}</td><td>{3,5}</td><td>{3,3,5}</td><td>Self-dual</td></tr></tbody></table>
<p>There are also 4 regular compact hyperbolic star-honeycombs in H4 space:
</p>
Compact regular hyperbolic star-honeycombs<table><tbody><tr><th>Honeycomb name</th><th><a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli</a>Symbol{p,q,r,s}</th><th><a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter diagram</a></th><th>Facettype{p,q,r}</th><th>Celltype{p,q}</th><th>Facetype{p}</th><th>Facefigure{s}</th><th>Edgefigure{r,s}</th><th><a href="/facts/Vertex_figure/3u2ySK0C">Vertexfigure</a>{q,r,s}</th><th><a href="/facts/Dual_polyhedron/rkIHbmAf">Dual</a></th></tr><tr><td><a href="/facts/Small_stellated_120-cell_honeycomb/HEr1I1kq">Order-3 small stellated 120-cell</a></td><td>{5/2,5,3,3}</td><td></td><td>{5/2,5,3}</td><td>{5/2,5}</td><td>{5}</td><td>{5}</td><td>{3,3}</td><td>{5,3,3}</td><td>{3,3,5,5/2}</td></tr><tr><td><a href="/facts/Pentagrammic-order_600-cell_honeycomb/pkWtCx31">Order-5/2 600-cell</a></td><td>{3,3,5,5/2}</td><td></td><td>{3,3,5}</td><td>{3,3}</td><td>{3}</td><td>{5/2}</td><td>{5,5/2}</td><td>{3,5,5/2}</td><td>{5/2,5,3,3}</td></tr><tr><td><a href="/facts/Order-5_icosahedral_120-cell_honeycomb/3vokF37W">Order-5 icosahedral 120-cell</a></td><td>{3,5,5/2,5}</td><td></td><td>{3,5,5/2}</td><td>{3,5}</td><td>{3}</td><td>{5}</td><td>{5/2,5}</td><td>{5,5/2,5}</td><td>{5,5/2,5,3}</td></tr><tr><td><a href="/facts/Great_120-cell_honeycomb/sb9LPxjE">Order-3 great 120-cell</a></td><td>{5,5/2,5,3}</td><td></td><td>{5,5/2,5}</td><td>{5,5/2}</td><td>{5}</td><td>{3}</td><td>{5,3}</td><td>{5/2,5,3}</td><td>{3,5,5/2,5}</td></tr></tbody></table>
Hyperbolic paracompact groups
<p>There are 9 <a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">paracompact hyperbolic Coxeter groups of rank 5</a>, each generating uniform honeycombs in 4-space as permutations of rings of the Coxeter diagrams. Paracompact groups generate honeycombs with infinite <a href="/facts/Facet_(geometry)/IreRHGJ7">facets</a> or <a href="/facts/Vertex_figure/3u2ySK0C">vertex figures</a>.
</p>
<table><tbody><tr><td><p>                                                                        P                ¯                                                          4                                {\displaystyle {\bar {P}}_{4}}   = [3,3[4]]: </p><p>                                                                                          B                  P                                ¯                                                          4                                {\displaystyle {\bar {BP}}_{4}}   = [4,3[4]]:                                                                                           F                  R                                ¯                                                          4                                {\displaystyle {\bar {FR}}_{4}}   = [(3,3,4,3,4)]:                                                                                           D                  P                                ¯                                                          4                                {\displaystyle {\bar {DP}}_{4}}   = [3[3]×[]]: </p></td><td><p>                                                                        N                ¯                                                          4                                {\displaystyle {\bar {N}}_{4}}   = [4,/3\,3,4]:                                                                         O                ¯                                                          4                                {\displaystyle {\bar {O}}_{4}}   = [3,4,31,1]:                                                                         S                ¯                                                          4                                {\displaystyle {\bar {S}}_{4}}   = [4,32,1]:                                                                         M                ¯                                                          4                                {\displaystyle {\bar {M}}_{4}}   = [4,31,1,1]: </p></td><td><p>                                                                        R                ¯                                                          4                                {\displaystyle {\bar {R}}_{4}}   = [3,4,3,4]: </p></td></tr></tbody></table>
<h2 id="notes">Notes</h2>

<ul><li><a href="/facts/Thorold_Gosset/3wvHtGPn">T. Gosset</a>: <i>On the Regular and Semi-Regular Figures in Space of n Dimensions</i>, <a href="/facts/Messenger_of_Mathematics/anb7QjuX">Messenger of Mathematics</a>, Macmillan, 1900 (3 regular and one semiregular 4-polytope)</li>
<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/Harold_Scott_MacDonald_Coxeter/D2l5jWGG">H.S.M. Coxeter</a>:
<ul><li><a href="/facts/Coxeter/D2l5jWGG">H.S.M. Coxeter</a>, <i>Regular Polytopes</i>, 3rd Edition, Dover New York, 1973 (p. 297 Fundamental regions for irreducible groups generated by reflections, Spherical and Euclidean)</li>
<li><a href="/facts/Coxeter/D2l5jWGG">H.S.M. Coxeter</a>, <i>The Beauty of Geometry: Twelve Essays</i> (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables IV p213)</li></ul></li>
<li>Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-471-01003-6 <a href="http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html">[1]</a>
<ul><li>(Paper 22) H.S.M. Coxeter, <i>Regular and Semi Regular Polytopes I</i>, [Math. Zeit. 46 (1940) 380-407, MR 2,10]</li>
<li>(Paper 23) H.S.M. Coxeter, <i>Regular and Semi-Regular Polytopes II</i>, [Math. Zeit. 188 (1985) 559-591] (p. 287 5D Euclidean groups, p. 298 Four-dimensionsal honeycombs)</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><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>James E. Humphreys, <i>Reflection Groups and Coxeter Groups</i>, Cambridge studies in advanced mathematics, 29 (1990) (Page 141, 6.9 List of hyperbolic Coxeter groups, figure 2) <a href="https://books.google.com/books?id=ODfjmOeNLMUC&dq=%22Reflection%20groups%20and%20Coxeter%20groups%22&pg=PA141">[2]</a></li></ul>
<h2 id="external-links">External links</h2>
<ul><li>Klitzing, Richard. <a href="https://bendwavy.org/klitzing/dimensions/polytera.htm">"5D uniform polytopes (polytera)"</a>. – includes nonconvex forms as well as the duplicate constructions from the B5 and D5 families</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>Uniform 5-polytope</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>
<table><tbody><tr><th colspan="7"><ul><li>v</li><li>t</li><li>e</li></ul>Fundamental convex <a href="/facts/Regular_honeycomb/Kng25Ymq">regular</a> and <a href="/facts/Uniform_honeycomb/fGXtrgGF">uniform honeycombs</a> in dimensions 2–9</th></tr><tr><th>Space</th><th><a href="/facts/Coxeter_group/bgzHFm1Q">Family</a></th><td>                                                                        A                ~                                                          n            −            1                                {\displaystyle {\tilde {A}}_{n-1}}  </td><td>                                                                        C                ~                                                          n            −            1                                {\displaystyle {\tilde {C}}_{n-1}}  </td><td>                                                                        B                ~                                                          n            −            1                                {\displaystyle {\tilde {B}}_{n-1}}  </td><td>                                                                        D                ~                                                          n            −            1                                {\displaystyle {\tilde {D}}_{n-1}}  </td><td>                                                                        G                ~                                                          2                                {\displaystyle {\tilde {G}}_{2}}   /                                                                         F                ~                                                          4                                {\displaystyle {\tilde {F}}_{4}}   /                                                                         E                ~                                                          n            −            1                                {\displaystyle {\tilde {E}}_{n-1}}  </td></tr><tr><th>E2</th><th><a href="/facts/Uniform_tiling/QVK02ksI">Uniform tiling</a></th><td><a href="/facts/Triangular_tiling/Jx8f8Krh">0[3]</a></td><td><a href="/facts/Square_tiling/3ym86mvv">δ3</a></td><td><a href="/facts/Square_tiling/3ym86mvv">hδ3</a></td><td><a href="/facts/Square_tiling/3ym86mvv">qδ3</a></td><td><a href="/facts/Hexagonal_tiling/HwG5F0YH">Hexagonal</a></td></tr><tr><th>E3</th><th><a href="/facts/Uniform_convex_honeycomb/78IkTl7A">Uniform convex honeycomb</a></th><td><a href="/facts/Tetrahedral-octahedral_honeycomb/mX4CxSvp">0[4]</a></td><td><a href="/facts/Cubic_honeycomb/eMxySFur">δ4</a></td><td><a href="/facts/Tetrahedral-octahedral_honeycomb/mX4CxSvp">hδ4</a></td><td><a href="/facts/Quarter_cubic_honeycomb/22qQazbw">qδ4</a></td><td></td></tr><tr><th>E4</th><th><a href="/facts/Uniform_4-honeycomb/mqxsoqwY">Uniform 4-honeycomb</a></th><td><a href="/facts/5-cell_honeycomb/YgY6eSTr">0[5]</a></td><td><a href="/facts/Tesseractic_honeycomb/fpucRJJn">δ5</a></td><td><a href="/facts/16-cell_honeycomb/lbktV8Nr">hδ5</a></td><td><a href="/facts/Rectified_tesseractic_honeycomb/P266Jc7y">qδ5</a></td><td><a href="/facts/24-cell_honeycomb/foFoK5YX">24-cell honeycomb</a></td></tr><tr><th>E5</th><th><a href="/facts/Uniform_5-honeycomb/pxqFTw2F">Uniform 5-honeycomb</a></th><td><a href="/facts/5-simplex_honeycomb/XSLrRsLz">0[6]</a></td><td><a href="/facts/5-cubic_honeycomb/bqAJvIBI">δ6</a></td><td><a href="/facts/5-demicubic_honeycomb/eYp2lMB8">hδ6</a></td><td><a href="/facts/Quarter_5-cubic_honeycomb/r3AAQHbg">qδ6</a></td><td></td></tr><tr><th>E6</th><th><a href="/facts/Uniform_6-honeycomb/suCJSvTN">Uniform 6-honeycomb</a></th><td><a href="/facts/6-simplex_honeycomb/H7LWND7E">0[7]</a></td><td><a href="/facts/6-cubic_honeycomb/2FdezLvD">δ7</a></td><td><a href="/facts/6-demicubic_honeycomb/oxttB0n0">hδ7</a></td><td><a href="/facts/Quarter_6-cubic_honeycomb/vDRQk9kS">qδ7</a></td><td><a href="/facts/2_22_honeycomb/cvXxbk6L">222</a></td></tr><tr><th>E7</th><th><a href="/facts/Uniform_7-honeycomb/M93v3h3c">Uniform 7-honeycomb</a></th><td><a href="/facts/7-simplex_honeycomb/dq8hy2lI">0[8]</a></td><td><a href="/facts/7-cubic_honeycomb/Im2bVBf7">δ8</a></td><td><a href="/facts/7-demicubic_honeycomb/QYm4RCGK">hδ8</a></td><td><a href="/facts/Quarter_7-cubic_honeycomb/GjVHMBLq">qδ8</a></td><td><a href="/facts/1_33_honeycomb/tYL93yF1">133</a> • <a href="/facts/3_31_honeycomb/VlInrcea">331</a></td></tr><tr><th>E8</th><th><a href="/facts/Uniform_8-honeycomb/UdFL9ns2">Uniform 8-honeycomb</a></th><td><a href="/facts/8-simplex_honeycomb/eoAGHg7J">0[9]</a></td><td><a href="/facts/8-cubic_honeycomb/fx3hM5DD">δ9</a></td><td><a href="/facts/8-demicubic_honeycomb/baWsKzQc">hδ9</a></td><td>qδ9</td><td><a href="/facts/1_52_honeycomb/LAJXFvnV">152</a> • <a href="/facts/2_51_honeycomb/XthKHy0m">251</a> • <a href="/facts/5_21_honeycomb/f971VRma">521</a></td></tr><tr><th>E9</th><th><a href="/facts/Uniform_9-honeycomb/Qc5NglV8">Uniform 9-honeycomb</a></th><td>0[10]</td><td>δ10</td><td>hδ10</td><td>qδ10</td><td></td></tr><tr><th>E10</th><th>Uniform 10-honeycomb</th><td>0[11]</td><td>δ11</td><td>hδ11</td><td>qδ11</td><td></td></tr><tr><th>E<i>n</i>−1</th><th>Uniform (<i>n</i>−1)-<a href="/facts/Honeycomb_(geometry)/seyRBcel">honeycomb</a></th><td><a href="/facts/Simplectic_honeycomb/hCmbARWt">0[<i>n</i>]</a></td><td><a href="/facts/Hypercubic_honeycomb/MQxLbDeB">δ<i>n</i></a></td><td><a href="/facts/Alternated_hypercubic_honeycomb/zrU9sG2e">hδ<i>n</i></a></td><td><a href="/facts/Quarter_hypercubic_honeycomb/7Je1f2st">qδ<i>n</i></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></tr></tbody></table>

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

<ol>
<li id="fn:1"><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:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><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:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Bowers, Jonathan (2000). "Uniform Polychora" (PDF). In Reza Sarhagi (ed.). Bridges 2000. Bridges Conference. pp. 239–246. <a href="https://archive.bridgesmathart.org/2000/bridges2000-239.pdf" target="_blank">https://archive.bridgesmathart.org/2000/bridges2000-239.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Uniform Polytera, Jonathan Bowers <a href="http://www.polytope.net/hedrondude/polytera.htm" target="_blank">http://www.polytope.net/hedrondude/polytera.htm</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Uniform polytope <a href="https://polytope.miraheze.org/wiki/Uniform_polytope" target="_blank">https://polytope.miraheze.org/wiki/Uniform_polytope</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>ACW (May 24, 2012), "Convex uniform 5-polytopes", Open Problem Garden, archived from the original on October 5, 2016, retrieved 2016-10-04 <a href="http://www.openproblemgarden.org/op/convex_uniform_5_polytopes" target="_blank">http://www.openproblemgarden.org/op/convex_uniform_5_polytopes</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Regular and semi-regular polytopes III, p.315 Three finite groups of 5-dimensions <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>Coxeter, Regular polytopes, §12.6 The number of reflections, equation 12.61 <a href="/wiki/Coxeter" target="_blank">/wiki/Coxeter</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
<li id="fn:9"><p>"N,k-dippip". <a href="https://bendwavy.org/klitzing/incmats/n-m-dippip.htm" target="_blank">https://bendwavy.org/klitzing/incmats/n-m-dippip.htm</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li>
<li id="fn:10"><p>"Gappip". <a href="https://bendwavy.org/klitzing/incmats/gappip.htm" target="_blank">https://bendwavy.org/klitzing/incmats/gappip.htm</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li>
<li id="fn:11"><p>Regular polytopes, p.297. Table IV, Fundamental regions for irreducible groups generated by reflections. <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li>
<li id="fn:12"><p>Regular and Semiregular polytopes, II, pp.298-302 Four-dimensional honeycombs <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li>
<li id="fn:13"><p>Coxeter, The Beauty of Geometry: Twelve Essays, Chapter 10: Regular honeycombs in hyperbolic space, Summary tables IV p213 <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li>
</ol>

Uniform 5-polytope open-in-new

Uniform 5-polytope