List of uniform polychora (Meta, 11)
From Hi.gher. Space
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- | This page tabulates simple data for each of the uniform polychora. | + | <[#ontology [kind meta] [cats 4D Uniform Polytope]]> |
+ | This page tabulates simple data for each of the ([[convex]], [[Euclidean]], non-[[prismatic]]) [[uniform]] [[polychora]]. | ||
== Families == | == Families == | ||
=== Pyromorphs === | === Pyromorphs === | ||
<table class='shapelist dx'><tr> | <table class='shapelist dx'><tr> | ||
- | <th style='width:5%;' class='key' rowspan='2'>Dx</th> | + | <th style='width:5%;' class='key' rowspan='2'>[[Dx number|Dx]]</th> |
- | <th style='width: | + | <th style='width:6%;' class='key' rowspan='2'>[[Coxeter-Dynkin symbol|CD]]</th> |
+ | <th style='width:29%;' class='key' rowspan='2'>Variant</th> | ||
<th colspan='4'>[[Vertex figure|Verf]] cell counts</th> | <th colspan='4'>[[Vertex figure|Verf]] cell counts</th> | ||
<th colspan='4'>Element counts</th> | <th colspan='4'>Element counts</th> | ||
Line 18: | Line 20: | ||
<th style='width:5%;'>|V|</th> | <th style='width:5%;'>|V|</th> | ||
</tr><tr> | </tr><tr> | ||
- | <td class='cat' colspan=' | + | <td class='cat' colspan='11'></td> |
</tr><tr class='row1'> | </tr><tr class='row1'> | ||
- | <td>1</td><td>[[Pyrochoron|parent | + | <td>1</td><td>xooo</td><td>[[Pyrochoron|parent]]</td> |
<td>[[Tetrahedron|(3<sup>3</sup>)]]<sup>4</sup></td><td>--</td><td>--</td><td>--</td> | <td>[[Tetrahedron|(3<sup>3</sup>)]]<sup>4</sup></td><td>--</td><td>--</td><td>--</td> | ||
<td>5</td><td>10</td><td>10</td><td>5</td> | <td>5</td><td>10</td><td>10</td><td>5</td> | ||
</tr><tr class='row2'> | </tr><tr class='row2'> | ||
- | <td>3</td><td>[[ | + | <td>3</td><td>xxoo</td><td>[[Pyrotomochoron|truncate]]</td> |
<td>[[Tetrahedral truncate|(3·6<sup>2</sup>)]]<sup>3</sup></td><td>--</td><td>--</td><td>[[Tetrahedron|(3<sup>3</sup>)]]<sup>1</sup></td> | <td>[[Tetrahedral truncate|(3·6<sup>2</sup>)]]<sup>3</sup></td><td>--</td><td>--</td><td>[[Tetrahedron|(3<sup>3</sup>)]]<sup>1</sup></td> | ||
<td>10</td><td>30</td><td>40</td><td>20</td> | <td>10</td><td>30</td><td>40</td><td>20</td> | ||
</tr><tr class='row1'> | </tr><tr class='row1'> | ||
- | <td>2</td><td>[[ | + | <td>2</td><td>oxoo</td><td>[[Pyrorectichoron|rectate]]</td> |
<td>[[Octahedron|(3<sup>4</sup>)]]<sup>3</sup></td><td>--</td><td>--</td><td>[[Tetrahedron|(3<sup>3</sup>)]]<sup>2</sup></td> | <td>[[Octahedron|(3<sup>4</sup>)]]<sup>3</sup></td><td>--</td><td>--</td><td>[[Tetrahedron|(3<sup>3</sup>)]]<sup>2</sup></td> | ||
<td>10</td><td>30</td><td>30</td><td>10</td> | <td>10</td><td>30</td><td>30</td><td>10</td> | ||
</tr><tr class='row2'> | </tr><tr class='row2'> | ||
- | <td>6</td><td>[[Pyromesochoron|mesotruncate | + | <td>6</td><td>oxxo</td><td>[[Pyromesochoron|mesotruncate]]</td> |
<td>[[Tetrahedral truncate|(3·6<sup>2</sup>)]]<sup>2</sup></td><td>--</td><td>--</td><td>[[Tetrahedral truncate|(3·6<sup>2</sup>)]]<sup>2</sup></td> | <td>[[Tetrahedral truncate|(3·6<sup>2</sup>)]]<sup>2</sup></td><td>--</td><td>--</td><td>[[Tetrahedral truncate|(3·6<sup>2</sup>)]]<sup>2</sup></td> | ||
- | <td> | + | <td>10</td><td>40</td><td>60</td><td>30</td> |
+ | </tr><tr> | ||
+ | <td class='cat' colspan='11'></td> | ||
</tr><tr class='row1'> | </tr><tr class='row1'> | ||
- | <td>5</td><td>[[ | + | <td>5</td><td>xoxo</td><td>[[Pyrocantichoron|cantellate]]</td> |
<td>[[Cuboctahedron|({3·4}<sup>2</sup>)]]<sup>2</sup></td><td>--</td><td>[[Triangular prism|(3·4<sup>2</sup>)]]<sup>2</sup></td><td>[[Octahedron|(3<sup>4</sup>)]]<sup>1</sup></td> | <td>[[Cuboctahedron|({3·4}<sup>2</sup>)]]<sup>2</sup></td><td>--</td><td>[[Triangular prism|(3·4<sup>2</sup>)]]<sup>2</sup></td><td>[[Octahedron|(3<sup>4</sup>)]]<sup>1</sup></td> | ||
<td>20</td><td>80</td><td>90</td><td>30</td> | <td>20</td><td>80</td><td>90</td><td>30</td> | ||
</tr><tr class='row2'> | </tr><tr class='row2'> | ||
- | <td>7</td><td>[[ | + | <td>7</td><td>xxxo</td><td>[[Pyrocantitomochoron|cantitruncate]]</td> |
- | <td>[[Octahedral truncate|(4·6<sup>2</sup>)<sup>2</sup> | + | <td>[[Octahedral truncate|(4·6<sup>2</sup>)]]<sup>2</sup></td><td>--</td><td>[[Triangular prism|(3·4<sup>2</sup>)]]<sup>1</sup></td><td>[[Tetrahedral truncate|(3·6<sup>2</sup>)]]<sup>1</sup></td> |
<td>20</td><td>80</td><td>120</td><td>60</td> | <td>20</td><td>80</td><td>120</td><td>60</td> | ||
</tr><tr class='row1'> | </tr><tr class='row1'> | ||
- | <td>9</td><td>[[ | + | <td>9</td><td>xoox</td><td>[[Pyroperichoron|runcinate (peritope)]]</td> |
<td>[[Tetrahedron|(3<sup>3</sup>)]]<sup>1</sup></td><td>[[Triangular prism|(3·4<sup>2</sup>)]]<sup>3</sup></td><td>[[Triangular prism|(3·4<sup>2</sup>)]]<sup>3</sup></td><td>[[Tetrahedron|(3<sup>3</sup>)]]<sup>1</sup></td> | <td>[[Tetrahedron|(3<sup>3</sup>)]]<sup>1</sup></td><td>[[Triangular prism|(3·4<sup>2</sup>)]]<sup>3</sup></td><td>[[Triangular prism|(3·4<sup>2</sup>)]]<sup>3</sup></td><td>[[Tetrahedron|(3<sup>3</sup>)]]<sup>1</sup></td> | ||
<td>30</td><td>70</td><td>60</td><td>20</td> | <td>30</td><td>70</td><td>60</td><td>20</td> | ||
</tr><tr class='row2'> | </tr><tr class='row2'> | ||
- | <td>11</td><td>[[ | + | <td>11</td><td>xxox</td><td>[[Pyroruncitomochoron|runcitruncate]]</td> |
<td>[[Tetrahedral truncate|(3·6<sup>2</sup>)]]<sup>1</sup></td><td>[[Hexagonal prism|(6·4<sup>2</sup>)]]<sup>2</sup></td><td>[[Triangular prism|(3·4<sup>2</sup>)]]<sup>1</sup></td><td>[[Cuboctahedron|({3·4}<sup>2</sup>)]]<sup>1</sup></td> | <td>[[Tetrahedral truncate|(3·6<sup>2</sup>)]]<sup>1</sup></td><td>[[Hexagonal prism|(6·4<sup>2</sup>)]]<sup>2</sup></td><td>[[Triangular prism|(3·4<sup>2</sup>)]]<sup>1</sup></td><td>[[Cuboctahedron|({3·4}<sup>2</sup>)]]<sup>1</sup></td> | ||
<td>30</td><td>120</td><td>150</td><td>60</td> | <td>30</td><td>120</td><td>150</td><td>60</td> | ||
</tr><tr class='row1'> | </tr><tr class='row1'> | ||
- | <td>15</td><td>[[ | + | <td>15</td><td>xxxx</td><td>[[Pyropantochoron|omnitruncate (pantome)]]</td> |
<td>[[Octahedral truncate|(4·6<sup>2</sup>)]]<sup>1</sup></td><td>[[Hexagonal prism|(6·4<sup>2</sup>)]]<sup>1</sup></td><td>[[Hexagonal prism|(6·4<sup>2</sup>)]]<sup>1</sup></td><td>[[Octahedral truncate|(4·6<sup>2</sup>)]]<sup>1</sup></td> | <td>[[Octahedral truncate|(4·6<sup>2</sup>)]]<sup>1</sup></td><td>[[Hexagonal prism|(6·4<sup>2</sup>)]]<sup>1</sup></td><td>[[Hexagonal prism|(6·4<sup>2</sup>)]]<sup>1</sup></td><td>[[Octahedral truncate|(4·6<sup>2</sup>)]]<sup>1</sup></td> | ||
<td>30</td><td>150</td><td>240</td><td>120</td> | <td>30</td><td>150</td><td>240</td><td>120</td> | ||
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<table class='shapelist dx'><tr> | <table class='shapelist dx'><tr> | ||
<th style='width:5%;' class='key' rowspan='2'>Dx</th> | <th style='width:5%;' class='key' rowspan='2'>Dx</th> | ||
- | <th style='width: | + | <th style='width:6%;' class='key' rowspan='2'>CD</th> |
+ | <th style='width:29%;' class='key' rowspan='2'>Variant</th> | ||
<th colspan='4'>Verf cell counts</th> | <th colspan='4'>Verf cell counts</th> | ||
<th colspan='4'>Element counts</th> | <th colspan='4'>Element counts</th> | ||
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<th style='width:5%;'>|V|</th> | <th style='width:5%;'>|V|</th> | ||
</tr><tr> | </tr><tr> | ||
- | <td class='cat' colspan='10'></td> | + | <td class='cat' colspan='11'></td> |
+ | </tr><tr class='row1'> | ||
+ | <td>1</td><td>xooo</td><td>[[Geochoron|parent]]</td> | ||
+ | <td>[[Cube|(4<sup>3</sup>)]]<sup>4</sup></td><td>--</td><td>--</td><td>--</td> | ||
+ | <td>8</td><td>24</td><td>32</td><td>16</td> | ||
+ | </tr><tr class='row2'> | ||
+ | <td>3</td><td>xxoo</td><td>[[Geotomochoron|truncate]]</td> | ||
+ | <td>[[Cubic truncate|(3·8<sup>2</sup>)]]<sup>3</sup></td><td>--</td><td>--</td><td>[[Tetrahedron|(3<sup>3</sup>)]]<sup>1</sup></td> | ||
+ | <td>24</td><td>88</td><td>128</td><td>64</td> | ||
+ | </tr><tr class='row1'> | ||
+ | <td>2</td><td>oxoo</td><td>[[Georectichoron|rectate]]</td> | ||
+ | <td>[[Cuboctahedron|({3·4}<sup>2</sup>)]]<sup>3</sup></td><td>--</td><td>--</td><td>[[Tetrahedron|(3<sup>3</sup>)]]<sup>2</sup></td> | ||
+ | <td>24</td><td>88</td><td>96</td><td>32</td> | ||
+ | </tr><tr class='row2'> | ||
+ | <td>6</td><td>oxxo</td><td>[[Stauromesochoron|mesotruncate]]</td> | ||
+ | <td>[[Octahedral truncate|(4·6<sup>2</sup>)]]<sup>2</sup></td><td>--</td><td>--</td><td>[[Tetrahedral truncate|(3·6<sup>2</sup>)]]<sup>2</sup></td> | ||
+ | <td>24</td><td>120</td><td>192</td><td>96</td> | ||
+ | </tr><tr class='row1'> | ||
+ | <td>4</td><td>ooxo</td><td>''[[Xylochoron|dual rectate]]''</td> | ||
+ | <td>[[Octahedron|(3<sup>4</sup>)]]<sup>2</sup></td><td>--</td><td>--</td><td>[[Octahedron|(3<sup>4</sup>)]]<sup>4</sup></td> | ||
+ | <td>24</td><td>96</td><td>96</td><td>24</td> | ||
+ | </tr><tr class='row2'> | ||
+ | <td>12</td><td>ooxx</td><td>[[Aerotomochoron|dual truncate]]</td> | ||
+ | <td>[[Octahedron|(3<sup>4</sup>)]]<sup>1</sup></td><td>--</td><td>--</td><td>[[Tetrahedral truncate|(3·6<sup>2</sup>)]]<sup>4</sup></td> | ||
+ | <td>24</td><td>96</td><td>120</td><td>48</td> | ||
+ | </tr><tr class='row1'> | ||
+ | <td>8</td><td>ooox</td><td>[[Aerochoron|dual]]</td> | ||
+ | <td>--</td><td>--</td><td>--</td><td>[[Tetrahedron|(3<sup>3</sup>)]]<sup>8</sup></td> | ||
+ | <td>16</td><td>32</td><td>24</td><td>8</td> | ||
+ | </tr><tr> | ||
+ | <td class='cat' colspan='11'></td> | ||
+ | </tr><tr class='row1'> | ||
+ | <td>5</td><td>xoxo</td><td>[[Geocantichoron|cantellate]]</td> | ||
+ | <td>[[Cuboctahedral rectate|(3·4<sup>3</sup>)]]<sup>2</sup></td><td>--</td><td>[[Triangular prism|(3·4<sup>2</sup>)]]<sup>2</sup></td><td>[[Octahedron|(3<sup>4</sup>)]]<sup>1</sup></td> | ||
+ | <td>56</td><td>248</td><td>288</td><td>96</td> | ||
+ | </tr><tr class='row2'> | ||
+ | <td>10</td><td>oxox</td><td>''[[Xylorectichoron|dual cantellate]]''</td> | ||
+ | <td>[[Cuboctahedron|({3·4}<sup>2</sup>)]]<sup>1</sup></td><td>[[Square prism|(4·4<sup>2</sup>)]]<sup>2</sup></td><td>--</td><td>[[Cuboctahedron|({3·4}<sup>2</sup>)]]<sup>2</sup></td> | ||
+ | <td>48</td><td>240</td><td>288</td><td>96</td> | ||
+ | </tr><tr class='row1'> | ||
+ | <td>7</td><td>xxxo</td><td>[[Geocantitomochoron|cantitruncate]]</td> | ||
+ | <td>[[Cuboctahedral truncate|(4·6·8)]]<sup>2</sup></td><td>--</td><td>[[Triangular prism|(3·4<sup>2</sup>)]]<sup>1</sup></td><td>[[Tetrahedral truncate|(3·6<sup>2</sup>)]]<sup>1</sup></td> | ||
+ | <td>56</td><td>248</td><td>384</td><td>192</td> | ||
+ | </tr><tr class='row2'> | ||
+ | <td>14</td><td>oxxx</td><td>''[[Xylotomochoron|dual cantitruncate]]''</td> | ||
+ | <td>[[Octahedral truncate|(4·6<sup>2</sup>)]]<sup>1</sup></td><td>[[Square prism|(4·4<sup>2</sup>)]]<sup>1</sup></td><td>--</td><td>[[Octahedral truncate|(4·6<sup>2</sup>)]]<sup>2</sup></td> | ||
+ | <td>48</td><td>240</td><td>384</td><td>192</td> | ||
+ | </tr><tr class='row1'> | ||
+ | <td>9</td><td>xoox</td><td>[[Stauroperichoron|runcinate (peritope)]]</td> | ||
+ | <td>[[Cube|(4<sup>3</sup>)]]<sup>1</sup></td><td>[[Square prism|(4·4<sup>2</sup>)]]<sup>3</sup></td><td>[[Triangular prism|(3·4<sup>2</sup>)]]<sup>3</sup></td><td>[[Tetrahedron|(3<sup>3</sup>)]]<sup>1</sup></td> | ||
+ | <td>80</td><td>208</td><td>192</td><td>64</td> | ||
+ | </tr><tr class='row2'> | ||
+ | <td>11</td><td>xxox</td><td>[[Georuncitomochoron|runcitruncate]]</td> | ||
+ | <td>[[Cubic truncate|(3·8<sup>2</sup>)]]<sup>1</sup></td><td>[[Octagonal prism|(8·4<sup>2</sup>)]]<sup>2</sup></td><td>[[Triangular prism|(3·4<sup>2</sup>)]]<sup>1</sup></td><td>[[Cuboctahedron|({3·4}<sup>2</sup>)]]<sup>1</sup></td> | ||
+ | <td>80</td><td>368</td><td>480</td><td>192</td> | ||
+ | </tr><tr class='row1'> | ||
+ | <td>13</td><td>xoxx</td><td>[[Aeroruncitomochoron|dual runcitruncate]]</td> | ||
+ | <td>[[Cuboctahedral rectate|(3·4<sup>3</sup>)]]<sup>1</sup></td><td>[[Square prism|(4·4<sup>2</sup>)]]<sup>1</sup></td><td>[[Hexagonal prism|(6·4<sup>2</sup>)]]<sup>2</sup></td><td>[[Tetrahedral truncate|(3·6<sup>2</sup>)]]<sup>1</sup></td> | ||
+ | <td>80</td><td>368</td><td>480</td><td>192</td> | ||
+ | </tr><tr class='row2'> | ||
+ | <td>15</td><td>xxxx</td><td>[[Stauropantochoron|omnitruncate (pantome)]]</td> | ||
+ | <td>[[Cuboctahedral truncate|(4·6·8)]]<sup>1</sup></td><td>[[Octagonal prism|(8·4<sup>2</sup>)]]<sup>1</sup></td><td>[[Hexagonal prism|(6·4<sup>2</sup>)]]<sup>1</sup></td><td>[[Octahedral truncate|(4·6<sup>2</sup>)]]<sup>1</sup></td> | ||
+ | <td>80</td><td>464</td><td>768</td><td>384</td> | ||
</tr></table> | </tr></table> | ||
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<table class='shapelist dx'><tr> | <table class='shapelist dx'><tr> | ||
<th style='width:5%;' class='key' rowspan='2'>Dx</th> | <th style='width:5%;' class='key' rowspan='2'>Dx</th> | ||
- | <th style='width: | + | <th style='width:7%;' class='key' rowspan='2'>CD</th> |
+ | <th style='width:28%;' class='key' rowspan='2'>Variant</th> | ||
<th colspan='4'>Verf cell counts</th> | <th colspan='4'>Verf cell counts</th> | ||
<th colspan='4'>Element counts</th> | <th colspan='4'>Element counts</th> | ||
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<th style='width:5%;'>|V|</th> | <th style='width:5%;'>|V|</th> | ||
</tr><tr> | </tr><tr> | ||
- | <td class='cat' colspan=' | + | <td class='cat' colspan='11'></td> |
+ | </tr><tr class='row1'> | ||
+ | <td>1</td><td>xooo</td><td>[[Xylochoron|parent]]</td> | ||
+ | <td>[[Octahedron|(3<sup>4</sup>)]]<sup>6</sup></td><td>--</td><td>--</td><td>--</td> | ||
+ | <td>24</td><td>96</td><td>96</td><td>24</td> | ||
+ | </tr><tr class='row2'> | ||
+ | <td>3</td><td>xxoo</td><td>[[Xylotomochoron|truncate]]</td> | ||
+ | <td>[[Octahedral truncate|(4·6<sup>2</sup>)]]<sup>3</sup></td><td>--</td><td>--</td><td>[[Cube|(4<sup>3</sup>)]]<sup>1</sup></td> | ||
+ | <td>48</td><td>240</td><td>384</td><td>192</td> | ||
+ | </tr><tr class='row1'> | ||
+ | <td>2</td><td>oxoo</td><td>[[Xylorectichoron|rectate]]</td> | ||
+ | <td>[[Cuboctahedron|({3·4}<sup>2</sup>)]]<sup>3</sup></td><td>--</td><td>--</td><td>[[Cube|(4<sup>3</sup>)]]<sup>2</sup></td> | ||
+ | <td>48</td><td>240</td><td>288</td><td>96</td> | ||
+ | </tr><tr class='row2'> | ||
+ | <td>6</td><td>oxxo</td><td>[[Xylomesochoron|mesotruncate]]</td> | ||
+ | <td>[[Cubic truncate|(3·8<sup>2</sup>)]]<sup>2</sup></td><td>--</td><td>--</td><td>[[Cubic truncate|(3·8<sup>2</sup>)]]<sup>2</sup></td> | ||
+ | <td>48</td><td>366</td><td>576</td><td>288</td> | ||
+ | </tr><tr> | ||
+ | <td class='cat' colspan='11'></td> | ||
+ | </tr><tr class='row1'> | ||
+ | <td>5</td><td>xoxo</td><td>[[Xylocantichoron|cantellate]]</td> | ||
+ | <td>[[Cuboctahedral rectate|(3·4<sup>3</sup>)]]<sup>2</sup></td><td>--</td><td>[[Triangular prism|(3·4<sup>2</sup>)]]<sup>2</sup></td><td>[[Cuboctahedron|({3·4}<sup>2</sup>)]]<sup>1</sup></td> | ||
+ | <td>144</td><td>720</td><td>864</td><td>288</td> | ||
+ | </tr><tr class='row2'> | ||
+ | <td>7</td><td>xxxo</td><td>[[Xylocantitomochoron|cantitruncate]]</td> | ||
+ | <td>[[Cuboctahedral truncate|(4·6·8)]]<sup>2</sup></td><td>--</td><td>[[Triangular prism|(3·4<sup>2</sup>)]]<sup>1</sup></td><td>[[Cubic truncate|(3·8<sup>2</sup>)]]<sup>1</sup></td> | ||
+ | <td>144</td><td>720</td><td>1152</td><td>576</td> | ||
+ | </tr><tr class='row1'> | ||
+ | <td>9</td><td>xoox</td><td>[[Xyloperichoron|runcinate (peritope)]]</td> | ||
+ | <td>[[Octahedron|(3<sup>4</sup>)]]<sup>1</sup></td><td>[[Triangular prism|(3·4<sup>2</sup>)]]<sup>3</sup></td><td>[[Triangular prism|(3·4<sup>2</sup>)]]<sup>3</sup></td><td>[[Octahedron|(3<sup>4</sup>)]]<sup>1</sup></td> | ||
+ | <td>240</td><td>672</td><td>576</td><td>144</td> | ||
+ | </tr><tr class='row2'> | ||
+ | <td>11</td><td>xxox</td><td>[[Xyloruncitomochoron|runcitruncate]]</td> | ||
+ | <td>[[Octahedral truncate|(4·6<sup>2</sup>)]]<sup>1</sup></td><td>[[Hexagonal prism|(6·4<sup>2</sup>)]]<sup>2</sup></td><td>[[Triangular prism|(3·4<sup>2</sup>)]]<sup>1</sup></td><td>[[Cuboctahedral rectate|(3·4<sup>3</sup>)]]<sup>1</sup></td> | ||
+ | <td>240</td><td>1104</td><td>1440</td><td>576</td> | ||
+ | </tr><tr class='row1'> | ||
+ | <td>15</td><td>xxxx</td><td>[[Xylopantochoron|omnitruncate (pantome)]]</td> | ||
+ | <td>[[Cuboctahedral truncate|(4·6·8)]]<sup>1</sup></td><td>[[Hexagonal prism|(6·4<sup>2</sup>)]]<sup>1</sup></td><td>[[Hexagonal prism|(6·4<sup>2</sup>)]]<sup>1</sup></td><td>[[Cuboctahedral truncate|(4·6·8)]]<sup>1</sup></td> | ||
+ | <td>240</td><td>1392</td><td>2304</td><td>1152</td> | ||
</tr></table> | </tr></table> | ||
Line 98: | Line 204: | ||
<table class='shapelist dx'><tr> | <table class='shapelist dx'><tr> | ||
<th style='width:5%;' class='key' rowspan='2'>Dx</th> | <th style='width:5%;' class='key' rowspan='2'>Dx</th> | ||
- | <th style='width: | + | <th style='width:7%;' class='key' rowspan='2'>CD</th> |
+ | <th style='width:28%;' class='key' rowspan='2'>Variant</th> | ||
<th colspan='4'>Verf cell counts</th> | <th colspan='4'>Verf cell counts</th> | ||
<th colspan='4'>Element counts</th> | <th colspan='4'>Element counts</th> | ||
Line 111: | Line 218: | ||
<th style='width:5%;'>|V|</th> | <th style='width:5%;'>|V|</th> | ||
</tr><tr> | </tr><tr> | ||
- | <td class='cat' colspan='10'></td> | + | <td class='cat' colspan='11'></td> |
+ | </tr><tr class='row1'> | ||
+ | <td>1</td><td>xooo</td><td>[[Cosmochoron|parent]]</td> | ||
+ | <td>[[Dodecahedron|(5<sup>3</sup>)]]<sup>4</sup></td><td>--</td><td>--</td><td>--</td> | ||
+ | <td>120</td><td>720</td><td>1200</td><td>600</td> | ||
+ | </tr><tr class='row2'> | ||
+ | <td>3</td><td>xxoo</td><td>[[Cosmotomochoron|truncate]]</td> | ||
+ | <td>[[Dodecahedral truncate|(3·10<sup>2</sup>)]]<sup>3</sup></td><td>--</td><td>--</td><td>[[Tetrahedron|(3<sup>3</sup>)]]<sup>1</sup></td> | ||
+ | <td>720</td><td>3120</td><td>4800</td><td>2400</td> | ||
+ | </tr><tr class='row1'> | ||
+ | <td>2</td><td>oxoo</td><td>[[Cosmorectichoron|rectate]]</td> | ||
+ | <td>[[Icosidodecahedron|({3·5}<sup>2</sup>)]]<sup>3</sup></td><td>--</td><td>--</td><td>[[Tetrahedron|(3<sup>3</sup>)]]<sup>2</sup></td> | ||
+ | <td>720</td><td>3120</td><td>3600</td><td>1200</td> | ||
+ | </tr><tr class='row2'> | ||
+ | <td>6</td><td>oxxo</td><td>[[Rhodomesochoron|mesotruncate]]</td> | ||
+ | <td>[[Icosahedral truncate|(5·6<sup>2</sup>)]]<sup>2</sup></td><td>--</td><td>--</td><td>[[Tetrahedral truncate|(3·6<sup>2</sup>)]]<sup>2</sup></td> | ||
+ | <td>720</td><td>4320</td><td>7200</td><td>3600</td> | ||
+ | </tr><tr class='row1'> | ||
+ | <td>4</td><td>ooxo</td><td>[[Hydrorectichoron|dual rectate]]</td> | ||
+ | <td>[[Icosahedron|(3<sup>5</sup>)]]<sup>2</sup></td><td>--</td><td>--</td><td>[[Octahedron|(3<sup>4</sup>)]]<sup>5</sup></td> | ||
+ | <td>720</td><td>3600</td><td>3600</td><td>720</td> | ||
+ | </tr><tr class='row2'> | ||
+ | <td>12</td><td>ooxx</td><td>[[Hydrotomochoron|dual truncate]]</td> | ||
+ | <td>[[Icosahedron|(3<sup>5</sup>)]]<sup>1</sup></td><td>--</td><td>--</td><td>[[Tetrahedral truncate|(3·6<sup>2</sup>)]]<sup>5</sup></td> | ||
+ | <td>720</td><td>3600</td><td>4320</td><td>1440</td> | ||
+ | </tr><tr class='row1'> | ||
+ | <td>8</td><td>ooox</td><td>[[Hydrochoron|dual]]</td> | ||
+ | <td>--</td><td>--</td><td>--</td><td>[[Tetrahedron|(3<sup>3</sup>)]]<sup>20</sup></td> | ||
+ | <td>600</td><td>1200</td><td>720</td><td>120</td> | ||
+ | </tr><tr> | ||
+ | <td class='cat' colspan='11'></td> | ||
+ | </tr><tr class='row1'> | ||
+ | <td>5</td><td>xoxo</td><td>[[Cosmocantichoron|cantellate]]</td> | ||
+ | <td>[[Icosidodecahedral rectate|(3·4·5·4)]]<sup>2</sup></td><td>--</td><td>[[Triangular prism|(3·4<sup>2</sup>)]]<sup>2</sup></td><td>[[Octahedron|(3<sup>4</sup>)]]<sup>1</sup></td> | ||
+ | <td>1920</td><td>9120</td><td>10800</td><td>3600</td> | ||
+ | </tr><tr class='row2'> | ||
+ | <td>10</td><td>oxox</td><td>[[Hydrocantichoron|dual cantellate]]</td> | ||
+ | <td>[[Icosidodecahedron|({3·5}<sup>2</sup>)]]<sup>1</sup></td><td>[[Pentagonal prism|(5·4<sup>2</sup>)]]<sup>2</sup></td><td>--</td><td>[[Cuboctahedron|({3·4}<sup>2</sup>)]]<sup>2</sup></td> | ||
+ | <td>1440</td><td>8640</td><td>10800</td><td>3600</td> | ||
+ | </tr><tr class='row1'> | ||
+ | <td>7</td><td>xxxo</td><td>[[Cosmocantitomochoron|cantitruncate]]</td> | ||
+ | <td>[[Icosidodecahedral truncate|(4·6·10)]]<sup>2</sup></td><td>--</td><td>[[Triangular prism|(3·4<sup>2</sup>)]]<sup>1</sup></td><td>[[Tetrahedral truncate|(3·6<sup>2</sup>)]]<sup>1</sup></td> | ||
+ | <td>1920</td><td>9120</td><td>14400</td><td>7200</td> | ||
+ | </tr><tr class='row2'> | ||
+ | <td>14</td><td>oxxx</td><td>[[Hydrocantitomochoron|dual cantitruncate]]</td> | ||
+ | <td>[[Icosahedral truncate|(5·6<sup>2</sup>)]]<sup>1</sup></td><td>[[Pentagonal prism|(5·4<sup>2</sup>)]]<sup>1</sup></td><td>--</td><td>[[Octahedral truncate|(4·6<sup>2</sup>)]]<sup>2</sup></td> | ||
+ | <td>1440</td><td>8640</td><td>14400</td><td>7200</td> | ||
+ | </tr><tr class='row1'> | ||
+ | <td>9</td><td>xoox</td><td>[[Rhodoperichoron|runcinate (peritope)]]</td> | ||
+ | <td>[[Dodecahedron|(5<sup>3</sup>)]]<sup>1</sup></td><td>[[Pentagonal prism|(5·4<sup>2</sup>)]]<sup>3</sup></td><td>[[Triangular prism|(3·4<sup>2</sup>)]]<sup>3</sup></td><td>[[Tetrahedron|(3<sup>3</sup>)]]<sup>1</sup></td> | ||
+ | <td>2640</td><td>7440</td><td>7200</td><td>2400</td> | ||
+ | </tr><tr class='row2'> | ||
+ | <td>11</td><td>xxox</td><td>[[Cosmoruncitomochoron|runcitruncate]]</td> | ||
+ | <td>[[Dodecahedral truncate|(3·10<sup>2</sup>)]]<sup>1</sup></td><td>[[Decagonal prism|(10·4<sup>2</sup>)]]<sup>2</sup></td><td>[[Triangular prism|(3·4<sup>2</sup>)]]<sup>1</sup></td><td>[[Cuboctahedron|({3·4}<sup>2</sup>)]]<sup>1</sup></td> | ||
+ | <td>2640</td><td>13440</td><td>18000</td><td>7200</td> | ||
+ | </tr><tr class='row1'> | ||
+ | <td>13</td><td>xoxx</td><td>[[Hydroruncitomochoron|dual runcitruncate]]</td> | ||
+ | <td>[[Icosidodecahedral rectate|(3·4·5·4)]]<sup>1</sup></td><td>[[Pentagonal prism|(5·4<sup>2</sup>)]]<sup>1</sup></td><td>[[Hexagonal prism|(6·4<sup>2</sup>)]]<sup>2</sup></td><td>[[Tetrahedral truncate|(3·6<sup>2</sup>)]]<sup>1</sup></td> | ||
+ | <td>2640</td><td>13440</td><td>18000</td><td>7200</td> | ||
+ | </tr><tr class='row2'> | ||
+ | <td>15</td><td>xxxx</td><td>[[Rhodopantochoron|omnitruncate (pantome)]]</td> | ||
+ | <td>[[Icosidodecahedral truncate|(4·6·10)]]<sup>1</sup></td><td>[[Decagonal prism|(10·4<sup>2</sup>)]]<sup>1</sup></td><td>[[Hexagonal prism|(6·4<sup>2</sup>)]]<sup>1</sup></td><td>[[Octahedral truncate|(4·6<sup>2</sup>)]]<sup>1</sup></td> | ||
+ | <td>2640</td><td>17040</td><td>28800</td><td>14400</td> | ||
</tr></table> | </tr></table> | ||
+ | |||
+ | == Singularities == | ||
+ | === The snub demitesseract === | ||
+ | The [[snub demitesseract]], incorrectly referred to as the ''snub 24-cell'', can be formed by the construction of ''alternated truncated 24-cell'' or ''alternated cantitruncated 16-cell'' - but in order to be a snub of one of these families, the alternation would need to be of the parent's omnitruncate, and it is not. | ||
+ | |||
+ | When correctly represented, the snub demitesseract's Coxeter-Dynkin symbol is that of a three-pronged star, with all four nodes as snub rings. | ||
+ | |||
+ | The snub demitesseract has 144 cells, 480 faces, 432 edges and 96 vertices. | ||
+ | *Its cells are 24 [[icosahedra]] and 120 [[tetrahedra]]. They are joined with three icosahedra and five tetrahedra around each vertex. | ||
+ | *Its faces are all [[triangle]]s. | ||
+ | |||
+ | === The grand antiprism === | ||
+ | There is one (convex, Euclidean, non-prismatic) uniform polychoron not part of any of the above families, known as the [[grand antiprism]]. Because it doesn't exist within any family above, it is non-Wythoffian and has no Dx number, CD or Schlaefli symbols. It is loosely analogous to the three-dimensional antiprisms, which consist of two parallel polygons joined by a band of triangles. Unlike them, however, the grand antiprism is not a member of an infinite family of uniform polytopes. | ||
+ | |||
+ | Its elements are 320 cells, 720 faces, 500 edges and 100 vertices. | ||
+ | *Its cells are 20 [[pentagonal antiprism]]s forming two perpendicular rings joined by 300 tetrahedra. | ||
+ | *Its faces are 20 [[pentagon]]s and 700 triangles. | ||
== Uniform polyhedra usage statistics == | == Uniform polyhedra usage statistics == | ||
- | [[ | + | <table class='shapelist'><tr> |
+ | <th rowspan='2' class='key' style='width:5%'>#</th><th rowspan='2' class='key' style='width:35%'>Polyhedron</th><th colspan='5'>Unique uses</th> | ||
+ | </tr><tr> | ||
+ | <th style='width:12%'>Pyromorphs</th><th style='width:12%'>Stauromorphs</th><th style='width:12%'>Xylomorphs</th><th style='width:12%'>Rhodomorphs</th><th style='width:12%'>Total</th> | ||
+ | </tr><tr> | ||
+ | <td class='cat' colspan='7'>Tetrahedral</td> | ||
+ | </tr><tr class='row1'> | ||
+ | <td class='key'>1</td> | ||
+ | <td>[[Tetrahedron]]</td> | ||
+ | <td>5</td><td>4</td><td></td><td>4</td><td class='key'>13</td> | ||
+ | </tr><tr class='row2'> | ||
+ | <td class='key'>6</td> | ||
+ | <td>[[Tetrahedral truncate]]</td> | ||
+ | <td>5</td><td>4</td><td></td><td>4</td><td class='key'>13</td> | ||
+ | </tr><tr class='row1'> | ||
+ | <td class='key'>111</td> | ||
+ | <td>[[Triangular prism]]</td> | ||
+ | <td>5</td><td>4</td><td>5</td><td>4</td><td class='key'>18</td> | ||
+ | </tr><tr class='row2'> | ||
+ | <td class='key'>112</td> | ||
+ | <td>[[Hexagonal prism]]</td> | ||
+ | <td>3</td><td>2</td><td>3</td><td>2</td><td class='key'>10</td> | ||
+ | </tr><tr> | ||
+ | <td class='cat' colspan='7'>Octahedral</td> | ||
+ | </tr><tr class='row1'> | ||
+ | <td class='key'>2</td> | ||
+ | <td>[[Cube]]</td> | ||
+ | <td></td><td>6</td><td>2</td><td></td><td class='key'>8</td> | ||
+ | </tr><tr class='row2'> | ||
+ | <td class='key'>3</td> | ||
+ | <td>[[Octahedron]]</td> | ||
+ | <td>2</td><td>4</td><td>3</td><td>2</td><td class='key'>11</td> | ||
+ | </tr><tr class='row1'> | ||
+ | <td class='key'>7</td> | ||
+ | <td>[[Cuboctahedron]]</td> | ||
+ | <td>2</td><td>4</td><td>2</td><td>2</td><td class='key'>10</td> | ||
+ | </tr><tr class='row2'> | ||
+ | <td class='key'>8</td> | ||
+ | <td>[[Cubic truncate]]</td> | ||
+ | <td></td><td>2</td><td>3</td><td></td><td class='key'>5</td> | ||
+ | </tr><tr class='row1'> | ||
+ | <td class='key'>9</td> | ||
+ | <td>[[Octahedral truncate]]</td> | ||
+ | <td>3</td><td>4</td><td>2</td><td>2</td><td class='key'>11</td> | ||
+ | </tr><tr class='row2'> | ||
+ | <td class='key'>10</td> | ||
+ | <td>[[Cuboctahedral rectate]]</td> | ||
+ | <td></td><td>2</td><td>2</td><td></td><td class='key'>4</td> | ||
+ | </tr><tr class='row1'> | ||
+ | <td class='key'>11</td> | ||
+ | <td>[[Cuboctahedral truncate]]</td> | ||
+ | <td></td><td>2</td><td>3</td><td></td><td class='key'>5</td> | ||
+ | </tr><tr class='row2'> | ||
+ | <td class='key'>113</td> | ||
+ | <td>[[Octagonal prism]]</td> | ||
+ | <td></td><td>2</td><td></td><td></td><td class='key'>2</td> | ||
+ | </tr><tr> | ||
+ | <td class='cat' colspan='7'>Icosahedral</td> | ||
+ | </tr><tr class='row1'> | ||
+ | <td class='key'>4</td> | ||
+ | <td>[[Dodecahedron]]</td> | ||
+ | <td></td><td></td><td></td><td>2</td><td class='key'>2</td> | ||
+ | </tr><tr class='row2'> | ||
+ | <td class='key'>5</td> | ||
+ | <td>[[Icosahedron]]</td> | ||
+ | <td></td><td></td><td></td><td>2</td><td class='key'>2</td> | ||
+ | </tr><tr class='row1'> | ||
+ | <td class='key'>13</td> | ||
+ | <td>[[Icosidodecahedron]]</td> | ||
+ | <td></td><td></td><td></td><td>2</td><td class='key'>2</td> | ||
+ | </tr><tr class='row2'> | ||
+ | <td class='key'>14</td> | ||
+ | <td>[[Dodecahedral truncate]]</td> | ||
+ | <td></td><td></td><td></td><td>2</td><td class='key'>2</td> | ||
+ | </tr><tr class='row1'> | ||
+ | <td class='key'>15</td> | ||
+ | <td>[[Icosahedral truncate]]</td> | ||
+ | <td></td><td></td><td></td><td>2</td><td class='key'>2</td> | ||
+ | </tr><tr class='row2'> | ||
+ | <td class='key'>16</td> | ||
+ | <td>[[Icosidodecahedral rectate]]</td> | ||
+ | <td></td><td></td><td></td><td>2</td><td class='key'>2</td> | ||
+ | </tr><tr class='row1'> | ||
+ | <td class='key'>17</td> | ||
+ | <td>[[Icosidodecahedral truncate]]</td> | ||
+ | <td></td><td></td><td></td><td>2</td><td class='key'>2</td> | ||
+ | </tr><tr class='row2'> | ||
+ | <td class='key'>115</td> | ||
+ | <td>[[Pentagonal prism]]</td> | ||
+ | <td></td><td></td><td></td><td>4</td><td class='key'>4</td> | ||
+ | </tr><tr class='row1'> | ||
+ | <td class='key'>116</td> | ||
+ | <td>[[Decagonal prism]]</td> | ||
+ | <td></td><td></td><td></td><td>2</td><td class='key'>2</td> | ||
+ | </tr><tr> | ||
+ | <td class='cat' colspan='2'>Totals</td> | ||
+ | <td class='cat'>25</td><td class='cat'>40</td><td class='cat'>25</td><td class='cat'>40</td><td class='cat'>130</td> | ||
+ | </tr></table> |
Latest revision as of 22:20, 11 February 2014
This page tabulates simple data for each of the (convex, Euclidean, non-prismatic) uniform polychora.
Families
Pyromorphs
Dx | CD | Variant | Verf cell counts | Element counts | ||||||
---|---|---|---|---|---|---|---|---|---|---|
CC (5×) | CF (10×) | CE (10×) | CV (5×) | |C| | |F| | |E| | |V| | |||
1 | xooo | parent | (33)4 | -- | -- | -- | 5 | 10 | 10 | 5 |
3 | xxoo | truncate | (3·62)3 | -- | -- | (33)1 | 10 | 30 | 40 | 20 |
2 | oxoo | rectate | (34)3 | -- | -- | (33)2 | 10 | 30 | 30 | 10 |
6 | oxxo | mesotruncate | (3·62)2 | -- | -- | (3·62)2 | 10 | 40 | 60 | 30 |
5 | xoxo | cantellate | ({3·4}2)2 | -- | (3·42)2 | (34)1 | 20 | 80 | 90 | 30 |
7 | xxxo | cantitruncate | (4·62)2 | -- | (3·42)1 | (3·62)1 | 20 | 80 | 120 | 60 |
9 | xoox | runcinate (peritope) | (33)1 | (3·42)3 | (3·42)3 | (33)1 | 30 | 70 | 60 | 20 |
11 | xxox | runcitruncate | (3·62)1 | (6·42)2 | (3·42)1 | ({3·4}2)1 | 30 | 120 | 150 | 60 |
15 | xxxx | omnitruncate (pantome) | (4·62)1 | (6·42)1 | (6·42)1 | (4·62)1 | 30 | 150 | 240 | 120 |
Stauromorphs
Dx | CD | Variant | Verf cell counts | Element counts | ||||||
---|---|---|---|---|---|---|---|---|---|---|
CC (8×) | CF (24×) | CE (32×) | CV (16×) | |C| | |F| | |E| | |V| | |||
1 | xooo | parent | (43)4 | -- | -- | -- | 8 | 24 | 32 | 16 |
3 | xxoo | truncate | (3·82)3 | -- | -- | (33)1 | 24 | 88 | 128 | 64 |
2 | oxoo | rectate | ({3·4}2)3 | -- | -- | (33)2 | 24 | 88 | 96 | 32 |
6 | oxxo | mesotruncate | (4·62)2 | -- | -- | (3·62)2 | 24 | 120 | 192 | 96 |
4 | ooxo | dual rectate | (34)2 | -- | -- | (34)4 | 24 | 96 | 96 | 24 |
12 | ooxx | dual truncate | (34)1 | -- | -- | (3·62)4 | 24 | 96 | 120 | 48 |
8 | ooox | dual | -- | -- | -- | (33)8 | 16 | 32 | 24 | 8 |
5 | xoxo | cantellate | (3·43)2 | -- | (3·42)2 | (34)1 | 56 | 248 | 288 | 96 |
10 | oxox | dual cantellate | ({3·4}2)1 | (4·42)2 | -- | ({3·4}2)2 | 48 | 240 | 288 | 96 |
7 | xxxo | cantitruncate | (4·6·8)2 | -- | (3·42)1 | (3·62)1 | 56 | 248 | 384 | 192 |
14 | oxxx | dual cantitruncate | (4·62)1 | (4·42)1 | -- | (4·62)2 | 48 | 240 | 384 | 192 |
9 | xoox | runcinate (peritope) | (43)1 | (4·42)3 | (3·42)3 | (33)1 | 80 | 208 | 192 | 64 |
11 | xxox | runcitruncate | (3·82)1 | (8·42)2 | (3·42)1 | ({3·4}2)1 | 80 | 368 | 480 | 192 |
13 | xoxx | dual runcitruncate | (3·43)1 | (4·42)1 | (6·42)2 | (3·62)1 | 80 | 368 | 480 | 192 |
15 | xxxx | omnitruncate (pantome) | (4·6·8)1 | (8·42)1 | (6·42)1 | (4·62)1 | 80 | 464 | 768 | 384 |
Xylomorphs
Dx | CD | Variant | Verf cell counts | Element counts | ||||||
---|---|---|---|---|---|---|---|---|---|---|
CC (24×) | CF (96×) | CE (96×) | CV (24×) | |C| | |F| | |E| | |V| | |||
1 | xooo | parent | (34)6 | -- | -- | -- | 24 | 96 | 96 | 24 |
3 | xxoo | truncate | (4·62)3 | -- | -- | (43)1 | 48 | 240 | 384 | 192 |
2 | oxoo | rectate | ({3·4}2)3 | -- | -- | (43)2 | 48 | 240 | 288 | 96 |
6 | oxxo | mesotruncate | (3·82)2 | -- | -- | (3·82)2 | 48 | 366 | 576 | 288 |
5 | xoxo | cantellate | (3·43)2 | -- | (3·42)2 | ({3·4}2)1 | 144 | 720 | 864 | 288 |
7 | xxxo | cantitruncate | (4·6·8)2 | -- | (3·42)1 | (3·82)1 | 144 | 720 | 1152 | 576 |
9 | xoox | runcinate (peritope) | (34)1 | (3·42)3 | (3·42)3 | (34)1 | 240 | 672 | 576 | 144 |
11 | xxox | runcitruncate | (4·62)1 | (6·42)2 | (3·42)1 | (3·43)1 | 240 | 1104 | 1440 | 576 |
15 | xxxx | omnitruncate (pantome) | (4·6·8)1 | (6·42)1 | (6·42)1 | (4·6·8)1 | 240 | 1392 | 2304 | 1152 |
Rhodomorphs
Dx | CD | Variant | Verf cell counts | Element counts | ||||||
---|---|---|---|---|---|---|---|---|---|---|
CC (120×) | CF (720×) | CE (1200×) | CV (600×) | |C| | |F| | |E| | |V| | |||
1 | xooo | parent | (53)4 | -- | -- | -- | 120 | 720 | 1200 | 600 |
3 | xxoo | truncate | (3·102)3 | -- | -- | (33)1 | 720 | 3120 | 4800 | 2400 |
2 | oxoo | rectate | ({3·5}2)3 | -- | -- | (33)2 | 720 | 3120 | 3600 | 1200 |
6 | oxxo | mesotruncate | (5·62)2 | -- | -- | (3·62)2 | 720 | 4320 | 7200 | 3600 |
4 | ooxo | dual rectate | (35)2 | -- | -- | (34)5 | 720 | 3600 | 3600 | 720 |
12 | ooxx | dual truncate | (35)1 | -- | -- | (3·62)5 | 720 | 3600 | 4320 | 1440 |
8 | ooox | dual | -- | -- | -- | (33)20 | 600 | 1200 | 720 | 120 |
5 | xoxo | cantellate | (3·4·5·4)2 | -- | (3·42)2 | (34)1 | 1920 | 9120 | 10800 | 3600 |
10 | oxox | dual cantellate | ({3·5}2)1 | (5·42)2 | -- | ({3·4}2)2 | 1440 | 8640 | 10800 | 3600 |
7 | xxxo | cantitruncate | (4·6·10)2 | -- | (3·42)1 | (3·62)1 | 1920 | 9120 | 14400 | 7200 |
14 | oxxx | dual cantitruncate | (5·62)1 | (5·42)1 | -- | (4·62)2 | 1440 | 8640 | 14400 | 7200 |
9 | xoox | runcinate (peritope) | (53)1 | (5·42)3 | (3·42)3 | (33)1 | 2640 | 7440 | 7200 | 2400 |
11 | xxox | runcitruncate | (3·102)1 | (10·42)2 | (3·42)1 | ({3·4}2)1 | 2640 | 13440 | 18000 | 7200 |
13 | xoxx | dual runcitruncate | (3·4·5·4)1 | (5·42)1 | (6·42)2 | (3·62)1 | 2640 | 13440 | 18000 | 7200 |
15 | xxxx | omnitruncate (pantome) | (4·6·10)1 | (10·42)1 | (6·42)1 | (4·62)1 | 2640 | 17040 | 28800 | 14400 |
Singularities
The snub demitesseract
The snub demitesseract, incorrectly referred to as the snub 24-cell, can be formed by the construction of alternated truncated 24-cell or alternated cantitruncated 16-cell - but in order to be a snub of one of these families, the alternation would need to be of the parent's omnitruncate, and it is not.
When correctly represented, the snub demitesseract's Coxeter-Dynkin symbol is that of a three-pronged star, with all four nodes as snub rings.
The snub demitesseract has 144 cells, 480 faces, 432 edges and 96 vertices.
- Its cells are 24 icosahedra and 120 tetrahedra. They are joined with three icosahedra and five tetrahedra around each vertex.
- Its faces are all triangles.
The grand antiprism
There is one (convex, Euclidean, non-prismatic) uniform polychoron not part of any of the above families, known as the grand antiprism. Because it doesn't exist within any family above, it is non-Wythoffian and has no Dx number, CD or Schlaefli symbols. It is loosely analogous to the three-dimensional antiprisms, which consist of two parallel polygons joined by a band of triangles. Unlike them, however, the grand antiprism is not a member of an infinite family of uniform polytopes.
Its elements are 320 cells, 720 faces, 500 edges and 100 vertices.
- Its cells are 20 pentagonal antiprisms forming two perpendicular rings joined by 300 tetrahedra.
- Its faces are 20 pentagons and 700 triangles.
Uniform polyhedra usage statistics
# | Polyhedron | Unique uses | ||||
---|---|---|---|---|---|---|
Pyromorphs | Stauromorphs | Xylomorphs | Rhodomorphs | Total | ||
Tetrahedral | ||||||
1 | Tetrahedron | 5 | 4 | 4 | 13 | |
6 | Tetrahedral truncate | 5 | 4 | 4 | 13 | |
111 | Triangular prism | 5 | 4 | 5 | 4 | 18 |
112 | Hexagonal prism | 3 | 2 | 3 | 2 | 10 |
Octahedral | ||||||
2 | Cube | 6 | 2 | 8 | ||
3 | Octahedron | 2 | 4 | 3 | 2 | 11 |
7 | Cuboctahedron | 2 | 4 | 2 | 2 | 10 |
8 | Cubic truncate | 2 | 3 | 5 | ||
9 | Octahedral truncate | 3 | 4 | 2 | 2 | 11 |
10 | Cuboctahedral rectate | 2 | 2 | 4 | ||
11 | Cuboctahedral truncate | 2 | 3 | 5 | ||
113 | Octagonal prism | 2 | 2 | |||
Icosahedral | ||||||
4 | Dodecahedron | 2 | 2 | |||
5 | Icosahedron | 2 | 2 | |||
13 | Icosidodecahedron | 2 | 2 | |||
14 | Dodecahedral truncate | 2 | 2 | |||
15 | Icosahedral truncate | 2 | 2 | |||
16 | Icosidodecahedral rectate | 2 | 2 | |||
17 | Icosidodecahedral truncate | 2 | 2 | |||
115 | Pentagonal prism | 4 | 4 | |||
116 | Decagonal prism | 2 | 2 | |||
Totals | 25 | 40 | 25 | 40 | 130 |