List of uniform polychora (Meta, 11)

From Hi.gher. Space

(Difference between revisions)
(Uniform polyhedra usage statistics)
m (ontology)
 
(23 intermediate revisions not shown)
Line 1: Line 1:
-
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:35%;' class='key' rowspan='2'>Variant</th>
+
<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='10'></td>
+
<td class='cat' colspan='11'></td>
</tr><tr class='row1'>
</tr><tr class='row1'>
-
<td>1</td><td>[[Pyrochoron|parent (1-tome)]]</td>
+
<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>[[2-pyrotomochoron|truncate (2-tome)]]</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>[[3-pyrotomochoron|hemicate (3-tome)]]</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 (4-tome)]]</td>
+
<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>30</td><td>70</td><td>60</td><td>20</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>[[cantellated 5-cell|cantellate (xoxo)]]</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>[[cantitruncated 5-cell|cantitruncate (xxxo)]]</td>
+
<td>7</td><td>xxxo</td><td>[[Pyrocantitomochoron|cantitruncate]]</td>
-
<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>[[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>[[runcinated 5-cell|runcinate (xoox)]]</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>[[runcitruncated 5-cell|runcitruncate (xxox)]]</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>[[pyropantomochoron|omnitruncate (pantome)]]</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>
Line 60: Line 64:
<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:35%;' class='key' rowspan='2'>Variant</th>
+
<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>
Line 73: Line 78:
<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>
Line 79: Line 146:
<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:35%;' class='key' rowspan='2'>Variant</th>
+
<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 92: Line 160:
<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>[[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:35%;' class='key' rowspan='2'>Variant</th>
+
<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 ==
Line 125: Line 311:
<td class='key'>1</td>
<td class='key'>1</td>
<td>[[Tetrahedron]]</td>
<td>[[Tetrahedron]]</td>
 +
<td>5</td><td>4</td><td></td><td>4</td><td class='key'>13</td>
</tr><tr class='row2'>
</tr><tr class='row2'>
<td class='key'>6</td>
<td class='key'>6</td>
<td>[[Tetrahedral truncate]]</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'>
</tr><tr class='row1'>
<td class='key'>111</td>
<td class='key'>111</td>
<td>[[Triangular prism]]</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'>
</tr><tr class='row2'>
<td class='key'>112</td>
<td class='key'>112</td>
<td>[[Hexagonal prism]]</td>
<td>[[Hexagonal prism]]</td>
 +
<td>3</td><td>2</td><td>3</td><td>2</td><td class='key'>10</td>
</tr><tr>
</tr><tr>
<td class='cat' colspan='7'>Octahedral</td>
<td class='cat' colspan='7'>Octahedral</td>
Line 139: Line 329:
<td class='key'>2</td>
<td class='key'>2</td>
<td>[[Cube]]</td>
<td>[[Cube]]</td>
 +
<td></td><td>6</td><td>2</td><td></td><td class='key'>8</td>
</tr><tr class='row2'>
</tr><tr class='row2'>
<td class='key'>3</td>
<td class='key'>3</td>
<td>[[Octahedron]]</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'>
</tr><tr class='row1'>
<td class='key'>7</td>
<td class='key'>7</td>
<td>[[Cuboctahedron]]</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'>
</tr><tr class='row2'>
<td class='key'>8</td>
<td class='key'>8</td>
<td>[[Cubic truncate]]</td>
<td>[[Cubic truncate]]</td>
 +
<td></td><td>2</td><td>3</td><td></td><td class='key'>5</td>
</tr><tr class='row1'>
</tr><tr class='row1'>
<td class='key'>9</td>
<td class='key'>9</td>
<td>[[Octahedral truncate]]</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'>
</tr><tr class='row2'>
<td class='key'>10</td>
<td class='key'>10</td>
<td>[[Cuboctahedral rectate]]</td>
<td>[[Cuboctahedral rectate]]</td>
 +
<td></td><td>2</td><td>2</td><td></td><td class='key'>4</td>
</tr><tr class='row1'>
</tr><tr class='row1'>
<td class='key'>11</td>
<td class='key'>11</td>
<td>[[Cuboctahedral truncate]]</td>
<td>[[Cuboctahedral truncate]]</td>
 +
<td></td><td>2</td><td>3</td><td></td><td class='key'>5</td>
</tr><tr class='row2'>
</tr><tr class='row2'>
-
<td class='key'>2</td>
 
-
<td>[[Square prism]]</td>
 
-
</tr><tr class='row1'>
 
<td class='key'>113</td>
<td class='key'>113</td>
<td>[[Octagonal prism]]</td>
<td>[[Octagonal prism]]</td>
 +
<td></td><td>2</td><td></td><td></td><td class='key'>2</td>
</tr><tr>
</tr><tr>
<td class='cat' colspan='7'>Icosahedral</td>
<td class='cat' colspan='7'>Icosahedral</td>
Line 168: Line 363:
<td class='key'>4</td>
<td class='key'>4</td>
<td>[[Dodecahedron]]</td>
<td>[[Dodecahedron]]</td>
 +
<td></td><td></td><td></td><td>2</td><td class='key'>2</td>
</tr><tr class='row2'>
</tr><tr class='row2'>
<td class='key'>5</td>
<td class='key'>5</td>
<td>[[Icosahedron]]</td>
<td>[[Icosahedron]]</td>
 +
<td></td><td></td><td></td><td>2</td><td class='key'>2</td>
</tr><tr class='row1'>
</tr><tr class='row1'>
<td class='key'>13</td>
<td class='key'>13</td>
<td>[[Icosidodecahedron]]</td>
<td>[[Icosidodecahedron]]</td>
 +
<td></td><td></td><td></td><td>2</td><td class='key'>2</td>
</tr><tr class='row2'>
</tr><tr class='row2'>
<td class='key'>14</td>
<td class='key'>14</td>
<td>[[Dodecahedral truncate]]</td>
<td>[[Dodecahedral truncate]]</td>
 +
<td></td><td></td><td></td><td>2</td><td class='key'>2</td>
</tr><tr class='row1'>
</tr><tr class='row1'>
<td class='key'>15</td>
<td class='key'>15</td>
<td>[[Icosahedral truncate]]</td>
<td>[[Icosahedral truncate]]</td>
 +
<td></td><td></td><td></td><td>2</td><td class='key'>2</td>
</tr><tr class='row2'>
</tr><tr class='row2'>
<td class='key'>16</td>
<td class='key'>16</td>
<td>[[Icosidodecahedral rectate]]</td>
<td>[[Icosidodecahedral rectate]]</td>
 +
<td></td><td></td><td></td><td>2</td><td class='key'>2</td>
</tr><tr class='row1'>
</tr><tr class='row1'>
<td class='key'>17</td>
<td class='key'>17</td>
<td>[[Icosidodecahedral truncate]]</td>
<td>[[Icosidodecahedral truncate]]</td>
 +
<td></td><td></td><td></td><td>2</td><td class='key'>2</td>
</tr><tr class='row2'>
</tr><tr class='row2'>
<td class='key'>115</td>
<td class='key'>115</td>
<td>[[Pentagonal prism]]</td>
<td>[[Pentagonal prism]]</td>
 +
<td></td><td></td><td></td><td>4</td><td class='key'>4</td>
</tr><tr class='row1'>
</tr><tr class='row1'>
<td class='key'>116</td>
<td class='key'>116</td>
<td>[[Decagonal prism]]</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>
</tr></table>
-
 
-
[[Category:Uniform polychora|*]]
 

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|
1xoooparent (33)4------ 510105
3xxootruncate (3·62)3----(33)1 10304020
2oxoorectate (34)3----(33)2 10303010
6oxxomesotruncate (3·62)2----(3·62)2 10406030
5xoxocantellate ({3·4}2)2--(3·42)2(34)1 20809030
7xxxocantitruncate (4·62)2--(3·42)1(3·62)1 208012060
9xooxruncinate (peritope) (33)1(3·42)3(3·42)3(33)1 30706020
11xxoxruncitruncate (3·62)1(6·42)2(3·42)1({3·4}2)1 3012015060
15xxxxomnitruncate (pantome) (4·62)1(6·42)1(6·42)1(4·62)1 30150240120

Stauromorphs

Dx CD Variant Verf cell counts Element counts
CC (8×) CF (24×) CE (32×) CV (16×) |C| |F| |E| |V|
1xoooparent (43)4------ 8243216
3xxootruncate (3·82)3----(33)1 248812864
2oxoorectate ({3·4}2)3----(33)2 24889632
6oxxomesotruncate (4·62)2----(3·62)2 2412019296
4ooxodual rectate (34)2----(34)4 24969624
12ooxxdual truncate (34)1----(3·62)4 249612048
8oooxdual ------(33)8 1632248
5xoxocantellate (3·43)2--(3·42)2(34)1 5624828896
10oxoxdual cantellate ({3·4}2)1(4·42)2--({3·4}2)2 4824028896
7xxxocantitruncate (4·6·8)2--(3·42)1(3·62)1 56248384192
14oxxxdual cantitruncate (4·62)1(4·42)1--(4·62)2 48240384192
9xooxruncinate (peritope) (43)1(4·42)3(3·42)3(33)1 8020819264
11xxoxruncitruncate (3·82)1(8·42)2(3·42)1({3·4}2)1 80368480192
13xoxxdual runcitruncate (3·43)1(4·42)1(6·42)2(3·62)1 80368480192
15xxxxomnitruncate (pantome) (4·6·8)1(8·42)1(6·42)1(4·62)1 80464768384

Xylomorphs

Dx CD Variant Verf cell counts Element counts
CC (24×) CF (96×) CE (96×) CV (24×) |C| |F| |E| |V|
1xoooparent (34)6------ 24969624
3xxootruncate (4·62)3----(43)1 48240384192
2oxoorectate ({3·4}2)3----(43)2 4824028896
6oxxomesotruncate (3·82)2----(3·82)2 48366576288
5xoxocantellate (3·43)2--(3·42)2({3·4}2)1 144720864288
7xxxocantitruncate (4·6·8)2--(3·42)1(3·82)1 1447201152576
9xooxruncinate (peritope) (34)1(3·42)3(3·42)3(34)1 240672576144
11xxoxruncitruncate (4·62)1(6·42)2(3·42)1(3·43)1 24011041440576
15xxxxomnitruncate (pantome) (4·6·8)1(6·42)1(6·42)1(4·6·8)1 240139223041152

Rhodomorphs

Dx CD Variant Verf cell counts Element counts
CC (120×) CF (720×) CE (1200×) CV (600×) |C| |F| |E| |V|
1xoooparent (53)4------ 1207201200600
3xxootruncate (3·102)3----(33)1 720312048002400
2oxoorectate ({3·5}2)3----(33)2 720312036001200
6oxxomesotruncate (5·62)2----(3·62)2 720432072003600
4ooxodual rectate (35)2----(34)5 72036003600720
12ooxxdual truncate (35)1----(3·62)5 720360043201440
8oooxdual ------(33)20 6001200720120
5xoxocantellate (3·4·5·4)2--(3·42)2(34)1 19209120108003600
10oxoxdual cantellate ({3·5}2)1(5·42)2--({3·4}2)2 14408640108003600
7xxxocantitruncate (4·6·10)2--(3·42)1(3·62)1 19209120144007200
14oxxxdual cantitruncate (5·62)1(5·42)1--(4·62)2 14408640144007200
9xooxruncinate (peritope) (53)1(5·42)3(3·42)3(33)1 2640744072002400
11xxoxruncitruncate (3·102)1(10·42)2(3·42)1({3·4}2)1 264013440180007200
13xoxxdual runcitruncate (3·4·5·4)1(5·42)1(6·42)2(3·62)1 264013440180007200
15xxxxomnitruncate (pantome) (4·6·10)1(10·42)1(6·42)1(4·62)1 2640170402880014400

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.

Uniform polyhedra usage statistics

#PolyhedronUnique uses
PyromorphsStauromorphsXylomorphsRhodomorphsTotal
Tetrahedral
1 Tetrahedron 54413
6 Tetrahedral truncate 54413
111 Triangular prism 545418
112 Hexagonal prism 323210
Octahedral
2 Cube 628
3 Octahedron 243211
7 Cuboctahedron 242210
8 Cubic truncate 235
9 Octahedral truncate 342211
10 Cuboctahedral rectate 224
11 Cuboctahedral truncate 235
113 Octagonal prism 22
Icosahedral
4 Dodecahedron 22
5 Icosahedron 22
13 Icosidodecahedron 22
14 Dodecahedral truncate 22
15 Icosahedral truncate 22
16 Icosidodecahedral rectate 22
17 Icosidodecahedral truncate 22
115 Pentagonal prism 44
116 Decagonal prism 22
Totals 25402540130