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| | The writer's title is Christy. I am presently a journey agent. What me and my family adore is bungee jumping but I've been taking on new things recently. Ohio is exactly where his home is and his family enjoys it.<br><br>My web-site ... online psychic readings ([http://cartoonkorea.com/ce002/1093612 cartoonkorea.com]) |
| |+ Graphs of three [[List of regular polytopes#Five-dimensional regular polytopes and higher|regular]] and related [[uniform polytope]]s
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| |- align=center valign=top
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| |colspan=4|[[File:6-simplex t0.svg|100px]]<br/>[[6-simplex]]<BR>{{CDD|node_1|3|node|3|node|3|node|3|node|3|node}}
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| |colspan=4|[[File:6-simplex t01.svg|100px]]<br/>[[Truncated 6-simplex]]<BR>{{CDD|node_1|3|node_1|3|node|3|node|3|node|3|node}}
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| |colspan=4|[[File:6-simplex t1.svg|100px]]<br/>[[Rectified 6-simplex]]<BR>{{CDD|node|3|node_1|3|node|3|node|3|node|3|node}}
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| |- align=center valign=top
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| |colspan=6|[[File:6-simplex t02.svg|150px]]<br/>[[Cantellated 6-simplex]]<BR>{{CDD|node_1|3|node|3|node_1|3|node|3|node|3|node}}
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| |colspan=6|[[File:6-simplex t03.svg|150px]]<br/>[[Runcinated 6-simplex]]<BR>{{CDD|node_1|3|node|3|node|3|node_1|3|node|3|node}}
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| |- align=center valign=top
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| |colspan=6|[[File:6-simplex t04.svg|150px]]<br/>[[Stericated 6-simplex]]<BR>{{CDD|node_1|3|node|3|node|3|node|3|node_1|3|node}}
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| |colspan=6|[[File:6-simplex t05.svg|150px]]<br/>[[Pentellated 6-simplex]]<BR>{{CDD|node_1|3|node|3|node|3|node|3|node|3|node_1}}
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| |- align=center valign=top
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| |colspan=4|[[File:6-cube t5.svg|100px]]<br/>[[6-orthoplex]]<BR>{{CDD|node_1|3|node|3|node|3|node|3|node|4|node}}
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| |colspan=4|[[File:6-cube t45.svg|100px]]<br/>[[Truncated 6-orthoplex]]<BR>{{CDD|node_1|3|node_1|3|node|3|node|3|node|4|node}}
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| |colspan=4|[[File:6-cube t4.svg|100px]]<br/>[[Rectified 6-orthoplex]]<BR>{{CDD|node|3|node_1|3|node|3|node|3|node|4|node}}
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| |- align=center valign=top
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| |colspan=4|[[File:6-cube t35.svg|100px]]<br/>[[Cantellated 6-orthoplex]]<BR>{{CDD|node_1|3|node|3|node_1|3|node|3|node|4|node}}
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| |colspan=4|[[File:6-cube t25.svg|100px]]<br/>[[Runcinated 6-orthoplex]]<BR>{{CDD|node_1|3|node|3|node|3|node_1|3|node|4|node}}
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| |colspan=4|[[File:6-cube t15.svg|100px]]<br/>[[Stericated 6-orthoplex]]<BR>{{CDD|node_1|3|node|3|node|3|node|3|node_1|4|node}}
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| |- align=center valign=top
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| |colspan=6|[[File:6-cube t02.svg|150px]]<br/>[[Cantellated 6-cube]]<BR>{{CDD|node_1|4|node|3|node_1|3|node|3|node|3|node}}
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| |colspan=6|[[File:6-cube t03.svg|150px]]<br/>[[Runcinated 6-cube]]<BR>{{CDD|node_1|4|node|3|node|3|node_1|3|node|3|node}}
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| |- align=center valign=top
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| |colspan=6|[[File:6-cube t04.svg|150px]]<br/>[[Stericated 6-cube]]<BR>{{CDD|node_1|4|node|3|node|3|node|3|node_1|3|node}}
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| |colspan=6|[[File:6-cube t05.svg|150px]]<br/>[[Pentellated 6-cube]]<BR>{{CDD|node_1|4|node|3|node|3|node|3|node|3|node_1}}
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| |- align=center valign=top
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| |colspan=4|[[File:6-cube t0.svg|100px]]<br/>[[6-cube]]<BR>{{CDD|node_1|4|node|3|node|3|node|3|node|3|node}}
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| |colspan=4|[[File:6-cube t01.svg|100px]]<br/>[[Truncated 6-cube]]<BR>{{CDD|node_1|4|node|3|node|3|node|3|node|3|node}}
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| |colspan=4|[[File:6-cube t1.svg|100px]]<br/>[[Rectified 6-cube]]<BR>{{CDD|node_1|4|node_1|3|node|3|node|3|node|3|node}}
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| |- align=center valign=top
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| |colspan=4|[[File:6-demicube t0 D6.svg|100px]]<br/>[[6-demicube]]<BR>{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node}}
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| |colspan=4|[[File:6-demicube t01 D6.svg|100px]]<br/>[[Truncated 6-demicube]]<BR>{{CDD|nodes_10ru|split2|node_1|3|node|3|node|3|node}}
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| |colspan=4|[[File:6-demicube t02 D6.svg|100px]]<br/>[[Cantellated 6-demicube]]<BR>{{CDD|nodes_10ru|split2|node|3|node_1|3|node|3|node}}
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| |- align=center valign=top
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| |colspan=6|[[File:6-demicube t03 D6.svg|150px]]<br/>[[Runcinated 6-demicube]]<BR>{{CDD|nodes_10ru|split2|node|3|node|3|node_1|3|node}}
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| |colspan=6|[[File:6-demicube t04 D6.svg|150px]]<br/>[[Stericated 6-demicube]]<BR>{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node_1}}
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| |- align=center valign=top
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| |colspan=6|[[File:Up 2 21 t0 E6.svg|150px]]<br/>[[2 21 polytope|2<sub>21</sub>]]<BR>{{CDD|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea}}
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| |colspan=6|[[File:Up 1 22 t0 E6.svg|150px]]<br/>[[1 22 polytope|1<sub>22</sub>]]<BR>{{CDD|nodea|3a|nodea|3a|branch_10|3a|nodea|3a|nodea}}
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| |- align=center valign=top
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| |colspan=6|[[File:Up 2 21 t1 E6.svg|150px]]<br/>[[Truncated 2 21 polytope|Truncated 2<sub>21</sub>]]<BR>{{CDD|nodea_1|3a|nodea_1|3a|branch|3a|nodea|3a|nodea}}
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| |colspan=6|[[File:Up 2 21 t2 E6.svg|150px]]<br/>[[Truncated 1 22 polytope|Truncated 1<sub>22</sub>]]<BR>{{CDD|nodea|3a|nodea|3a|branch_11|3a|nodea|3a|nodea}}
| |
| |}
| |
| In [[Six-dimensional space|six-dimensional]] [[geometry]], a '''uniform polypeton'''<ref>A [[5-polytope#A note on generality of terms for n-polytopes and elements|proposed name]] '''polypeton''' (plural: '''polypeta''') has been advocated, from the [[Greek language|Greek]] root ''poly-'' meaning "many", a shortened ''[[Numerical prefix|penta]]-'' meaning "five", and suffix ''-on''. "Five" refers to the dimension of the 5-polytope [[Facet (mathematics)|facets]].</ref><ref>[http://www.steelpillow.com/polyhedra/ditela.html Ditela, polytopes and dyads]</ref> (or '''uniform [[6-polytope]]''') is a six-dimensional [[uniform polytope]]. A uniform polypeton is [[vertex-transitive]], and all [[Facet (geometry)|facets]] are [[uniform polyteron|uniform polytera]].
| |
| | |
| The complete set of '''convex uniform polypeta''' has not been determined, but most can be made as [[Wythoff construction]]s from a small set of [[Coxeter groups|symmetry groups]]. These construction operations are represented by the [[permutation]]s of [[ring (mathematics)|ring]]s of the [[Coxeter-Dynkin diagram]]s. Each combination of at least one ring on every connected group of nodes in the diagram produces a uniform 6-polytope. | |
| | |
| The simplest uniform polypeta are [[regular polytope]]s: the [[6-simplex]] {3,3,3,3,3}, the [[6-cube]] (hexeract) {4,3,3,3,3}, and the [[6-orthoplex]] (hexacross) {3,3,3,3,4}.
| |
| | |
| == Uniform 6-polytopes by fundamental Coxeter groups ==
| |
| | |
| Uniform 6-polytopes with reflective symmetry can be generated by these four Coxeter groups, represented by permutations of rings of the [[Coxeter-Dynkin diagram]]s.
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| | |
| There are four fundamental reflective symmety groups which generate 153 unique uniform 6-polytopes.
| |
| | |
| {| class=wikitable
| |
| |-
| |
| !#
| |
| !colspan=2|[[Coxeter group]]
| |
| ![[Coxeter-Dynkin diagram]]
| |
| |-
| |
| |1||A<sub>6</sub>|| [3,3,3,3,3]||{{CDD|node|3|node|3|node|3|node|3|node|3|node}}
| |
| |-
| |
| |2||B<sub>6</sub>||[3,3,3,3,4]||{{CDD|node|3|node|3|node|3|node|3|node|4|node}}
| |
| |-
| |
| |3||D<sub>6</sub>||[3,3,3,3<sup>1,1</sup>]||{{CDD|node|3|node|3|node|3|node|split1|nodes}}
| |
| |-
| |
| |rowspan=2|4
| |
| |rowspan=2|[[E6 (mathematics)|E<sub>6</sub>]]
| |
| ||[3<sup>2,2,1</sup>]||{{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea}}
| |
| |-
| |
| |[3,3<sup>2,2</sup>]||{{CDD|node|3|node|split1|nodes|3ab|nodes}}
| |
| |}
| |
| | |
| {| class=wikitable width=480
| |
| |[[File:Coxeter diagram finite rank6 correspondence.png|480px]]<BR>Coxeter-Dynkin diagram correspondences between families and higher symmetry within diagrams. Nodes of the same color in each row represent identical mirrors. Black nodes are not active in the correspondence.
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| |}
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| | |
| == Uniform prismatic families ==
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| | |
| '''Uniform prism'''
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| | |
| There are 6 categorical [[Uniform polytope|uniform]] prisms based the [[uniform 5-polytope]]s.
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| | |
| {| class=wikitable
| |
| |-
| |
| !#
| |
| !colspan=2|[[Coxeter group]]
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| ![[Coxeter-Dynkin diagram]]
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| !Notes
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| |-
| |
| |1||A<sub>5</sub>×A<sub>1</sub>|| [3,3,3,3,2]||{{CDD|node|3|node|3|node|3|node|3|node|2|node}}||Prism family based on [[6-simplex]]
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| |-
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| |2||B<sub>5</sub>×A<sub>1</sub>||[4,3,3,3,2]||{{CDD|node|4|node|3|node|3|node|3|node|2|node}}||Prism family based on [[6-cube]]
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| |-
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| |3a||D<sub>5</sub>×A<sub>1</sub>|| [3<sup>2,1,1</sup>,2]||{{CDD|nodes|split2|node|3|node|3|node|2|node}}||Prism family based on [[6-demicube]]
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| |}
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| | |
| {| class=wikitable
| |
| |-
| |
| !#
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| !colspan=2|[[Coxeter group]]
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| ![[Coxeter-Dynkin diagram]]
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| !Notes
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| |-
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| |4||A<sub>3</sub>×I<sub>2</sub>(p)×A<sub>1</sub>|| [3,3,2,p,2]||{{CDD|node|3|node|3|node|2|node|p|node|2|node}}||Prism family based on [[Tetrahedron|tetrahedral]]-p-gonal [[duoprism]]s
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| |-
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| |5||B<sub>3</sub>×I<sub>2</sub>(p)×A<sub>1</sub>|| [4,3,2,p,2]||{{CDD|node|4|node|3|node|2|node|p|node|2|node}}||Prism family based on [[Cube|cubic]]-p-gonal [[duoprism]]s
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| |-
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| |6||H<sub>3</sub>×I<sub>2</sub>(p)×A<sub>1</sub>|| [5,3,2,p,2]||{{CDD|node|5|node|3|node|2|node|p|node|2|node}}||Prism family based on [[Dodecahedron|dodecahedral]]-p-gonal [[duoprism]]s
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| |}
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| | |
| '''Uniform duoprism'''
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| There are 11 categorical [[Uniform polytope|uniform]] [[duoprism]]atic families of polytopes based on [[Cartesian product]]s of lower dimensional uniform polytopes. Five are formed as the product of a [[uniform polychoron]] with a [[regular polygon]], and six are formed by the product of two [[uniform polyhedron|uniform polyhedra]]:
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| {| class=wikitable
| |
| |-
| |
| !#
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| !colspan=2|[[Coxeter group]]
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| ![[Coxeter-Dynkin diagram]]
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| !Notes
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| |-
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| |1||A<sub>4</sub>×I<sub>2</sub>(p)|| [3,3,3,2,p]||{{CDD|node|3|node|3|node|3|node|2|node|p|node}}||Family based on [[5-cell]]-p-gonal duoprisms.
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| |-
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| |2||B<sub>4</sub>×I<sub>2</sub>(p)|| [4,3,3,2,p]||{{CDD|node|4|node|3|node|3|node|2|node|p|node}}||Family based on [[tesseract]]-p-gonal duoprisms.
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| |-
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| |3||F<sub>4</sub>×I<sub>2</sub>(p)|| [3,4,3,2,p]||{{CDD|node|3|node|4|node|3|node|2|node|p|node}}||Family based on [[24-cell]]-p-gonal duoprisms.
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| |-
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| |4||H<sub>4</sub>×I<sub>2</sub>(p)|| [5,3,3,2,p]||{{CDD|node|5|node|3|node|3|node|2|node|p|node}}||Family based on [[120-cell]]-p-gonal duoprisms.
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| |-
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| |5|| D<sub>4</sub>×I<sub>2</sub>(p)|| [3<sup>1,1,1</sup>,2,p]||{{CDD|nodes|split2|node|3|node|2|node|p|node}}||Family based on [[demitesseract]]-p-gonal duoprisms.
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| |}
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| | |
| {| class=wikitable
| |
| |-
| |
| !#
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| !colspan=2|[[Coxeter group]]
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| ![[Coxeter-Dynkin diagram]]
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| !Notes
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| |-
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| |6||A<sub>3</sub><sup>2</sup>|| [3,3,2,3,3]||{{CDD|node|3|node|3|node|2|node|3|node|3|node}}||Family based on [[tetrahedron|tetrahedral]] duoprisms.
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| |-
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| |7||A<sub>3</sub>×B<sub>3</sub>|| [3,3,2,4,3]||{{CDD|node|3|node|3|node|2|node|4|node|3|node}}||Family based on [[tetrahedron|tetrahedral]]-[[Cube|cubic]] duoprisms.
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| |-
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| |8||A<sub>3</sub>×H<sub>3</sub>|| [3,3,2,5,3]||{{CDD|node|3|node|3|node|2|node|5|node|3|node}}||Family based on [[tetrahedron|tetrahedral]]-[[Dodecahedron|dodecahedral]] duoprisms.
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| |-
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| |9||B<sub>3</sub><sup>2</sup>|| [4,3,2,4,3]||{{CDD|node|4|node|3|node|2|node|4|node|3|node}}||Family based on [[Cube|cubic]] duoprisms.
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| |-
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| |10||B<sub>3</sub>×H<sub>3</sub>|| [4,3,2,5,3]||{{CDD|node|4|node|3|node|2|node|5|node|3|node}}||Family based on [[Cube|cubic]]-[[Dodecahedron|dodecahedral]] duoprisms.
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| |-
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| |11||H<sub>3</sub><sup>2</sup>|| [5,3,2,5,3]||{{CDD|node|5|node|3|node|2|node|5|node|3|node}}||Family based on [[Dodecahedron|dodecahedral]] duoprisms.
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| |}
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| '''Uniform triaprism'''
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| There is one infinite family of [[Uniform polytope|uniform]] [[triaprism]]atic families of polytopes constructed as a [[Cartesian product]]s of three regular polygons. Each combination of at least one ring on every connected group produces a uniform prismatic 6-polytope.
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| {| class=wikitable
| |
| |-
| |
| !#
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| !colspan=2|[[Coxeter group]]
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| ![[Coxeter-Dynkin diagram]]
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| !Notes
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| |-
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| |1||I<sub>2</sub>(p)×I<sub>2</sub>(q)×I<sub>2</sub>(r)|| [p,2,q,2,r]||{{CDD|node|p|node|2|node|q|node|2|node|r|node}}||Family based on p,q,r-gonal triprisms
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| |}
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| == Enumerating the convex uniform 6-polytopes ==
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| * [[Simplex]] family: A<sub>6</sub> [3<sup>4</sup>] - {{CDD|node|3|node|3|node|3|node|3|node|3|node}}
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| ** 35 uniform 6-polytopes as permutations of rings in the group diagram, including one regular:
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| **# {3<sup>4</sup>} - [[6-simplex]] - {{CDD|node_1|3|node|3|node|3|node|3|node|3|node}}
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| * [[Hypercube]]/[[orthoplex]] family: B<sub>6</sub> [4,3<sup>4</sup>] - {{CDD|node|4|node|3|node|3|node|3|node|3|node}}
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| ** 63 uniform 6-polytopes as permutations of rings in the group diagram, including two regular forms:
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| **# {4,3<sup>3</sup>} — [[6-cube]] (hexeract) - {{CDD|node_1|4|node|3|node|3|node|3|node|3|node}}
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| **# {3<sup>3</sup>,4} — [[6-orthoplex]], (hexacross) - {{CDD|node|4|node|3|node|3|node|3|node|3|node_1}}
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| * [[Demihypercube]] D<sub>6</sub> family: [3<sup>3,1,1</sup>] - {{CDD|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea}}
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| ** 47 uniform 6-polytopes (16 unique) as permutations of rings in the group diagram, including:
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| **# {3,3<sup>2,1</sup>}, '''1<sub>21</sub>''' '''[[6-demicube]]''' (demihexeract) - {{CDD|nodea_1|3a|branch|3a|nodea|3a|nodea|3a|nodea}}; also as h{4,3<sup>3</sup>}, {{CDD|node_h|4|node|3|node|3|node|3|node|3|node}}
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| **# {3,3,3<sup>1,1</sup>}, '''2<sub>11</sub>''' '''[[6-orthoplex]]''' - {{CDD|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea_1}}
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| * [[Semiregular k 21 polytope|E<sub>6</sub>]] family: [3<sup>3,1,1</sup>] - {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea}}
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| ** 39 uniform 6-polytopes (16 unique) as permutations of rings in the group diagram, including:
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| **# {3,3,3<sup>2,1</sup>}, '''[[Gosset 2 21 polytope|2<sub>21</sub>]]''' - {{CDD|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea}}
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| **# {3,3<sup>2,2</sup>}, '''[[Gosset 1 22 polytope|1<sub>22</sub>]]''' - {{CDD|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea}}
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| These fundamental families generate 153 nonprismatic convex uniform polypeta.
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| In addition, there are 105 uniform 6-polytope constructions based on prisms of the [[uniform polyteron]]s: [3,3,3,3,2], [4,3,3,3,2], [5,3,3,3,2], [3<sup>2,1,1</sup>,2].
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| In addition, there are infinitely many uniform 6-polytope based on:
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| # Duoprism prism families: [3,3,2,p,2], [4,3,2,p,2], [5,3,2,p,2].
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| # Duoprism families: [3,3,3,2,p], [4,3,3,2,p], [5,3,3,2,p].
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| # Triaprism family: [p,2,q,2,r].
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| | |
| === The A<sub>6</sub> family ===
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| There are 32+4−1=35 forms, derived by marking one or more nodes of the [[Coxeter-Dynkin diagram]].
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| All 35 are enumerated below. They are named by [[Norman Johnson (mathematician)|Norman Johnson]] from the Wythoff construction operations upon regular 6-simplex (heptapeton). Bowers-style acronym names are given in parentheses for cross-referencing.
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| The A<sub>6</sub> family has symmetry of order 5040 (7 [[factorial]]).
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| The coordinates of uniform 6-polytopes with 6-simplex symmetry can be generated as permutations of simple integers in 7-space, all in hyperplanes with [[normal vector]] (1,1,1,1,1,1,1).
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| See also [[list of A6 polytopes]] for graphs of these polytopes.
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| {| class="wikitable"
| |
| |-
| |
| !rowspan=2|#
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| !rowspan=2|[[Coxeter-Dynkin diagram|Coxeter-Dynkin]]
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| !rowspan=2|[[Norman Johnson (mathematician)|Johnson]] naming system<BR>Bowers name and (acronym)
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| !rowspan=2|Base point
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| !colspan=6|Element counts
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| |-
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| ! 5|| 4|| 3|| 2|| 1|| 0
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| |- align=center
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| !1
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| |{{CDD|node|3|node|3|node|3|node|3|node|3|node_1}}
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| | [[6-simplex]]<BR>heptapeton (hop)
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| |(0,0,0,0,0,0,1)
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| |7||21||35||35||21||7
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| |- align=center
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| !2
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| |{{CDD|node|3|node|3|node|3|node|3|node_1|3|node}}
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| | [[Rectified 6-simplex]]<BR>rectified heptapeton (ril)
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| |(0,0,0,0,0,1,1)
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| || 14 || 63 || 140 || 175 || 105 || 21
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| |- align=center
| |
| !3
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| |{{CDD|node|3|node|3|node|3|node|3|node_1|3|node_1}}
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| | [[Truncated 6-simplex]]<BR>truncated heptapeton (til)
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| |(0,0,0,0,0,1,2)
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| || 14 || 63 || 140 || 175 || 126 || 42
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| |- align=center
| |
| !4
| |
| |{{CDD|node|3|node|3|node|3|node_1|3|node|3|node}}
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| | [[Birectified 6-simplex]]<BR>birectified heptapeton (bril)
| |
| |(0,0,0,0,1,1,1)
| |
| || 14 || 84 || 245 || 350 || 210 || 35
| |
| |- align=center
| |
| !5
| |
| |{{CDD|node|3|node|3|node|3|node_1|3|node|3|node_1}}
| |
| | [[Cantellated 6-simplex]]<BR>small rhombated heptapeton (sril)
| |
| |(0,0,0,0,1,1,2)
| |
| || 35 || 210 || 560 || 805 || 525 || 105
| |
| |- align=center
| |
| !6
| |
| |{{CDD|node|3|node|3|node|3|node_1|3|node_1|3|node}}
| |
| | [[Bitruncated 6-simplex]]<BR>bitruncated heptapeton (batal)
| |
| |(0,0,0,0,1,2,2)
| |
| || 14 || 84 || 245 || 385 || 315 || 105
| |
| |- align=center
| |
| !7
| |
| |{{CDD|node|3|node|3|node|3|node_1|3|node_1|3|node_1}}
| |
| | [[Cantitruncated 6-simplex]]<BR>great rhombated heptapeton (gril)
| |
| |(0,0,0,0,1,2,3)
| |
| || 35 || 210 || 560 || 805 || 630 || 210
| |
| |- align=center
| |
| !8
| |
| |{{CDD|node|3|node|3|node_1|3|node|3|node|3|node_1}}
| |
| | [[Runcinated 6-simplex]]<BR>small prismated heptapeton (spil)
| |
| |(0,0,0,1,1,1,2)
| |
| || 70 || 455 || 1330 || 1610 || 840 || 140
| |
| |- align=center
| |
| !9
| |
| |{{CDD|node|3|node|3|node_1|3|node|3|node_1|3|node}}
| |
| | [[Bicantellated 6-simplex]]<BR>small prismated heptapeton (sabril)
| |
| |(0,0,0,1,1,2,2)
| |
| || 70 || 455 || 1295 || 1610 || 840 || 140
| |
| |- align=center
| |
| !10
| |
| |{{CDD|node|3|node|3|node_1|3|node|3|node_1|3|node_1}}
| |
| | [[Runcitruncated 6-simplex]]<BR>prismatotruncated heptapeton (patal)
| |
| |(0,0,0,1,1,2,3)
| |
| || 70 || 560 || 1820 || 2800 || 1890 || 420
| |
| |- style="text-align:center; background:#e0f0e0;"
| |
| !11
| |
| |{{CDD|node|3|node|3|node_1|3|node_1|3|node|3|node}}
| |
| | [[Tritruncated 6-simplex]]<BR>tetradecapeton (fe)
| |
| |(0,0,0,1,2,2,2)
| |
| || 14 || 84 || 280 || 490 || 420 || 140
| |
| |- align=center
| |
| !12
| |
| |{{CDD|node|3|node|3|node_1|3|node_1|3|node|3|node_1}}
| |
| | [[Runcicantellated 6-simplex]]<BR>prismatorhombated heptapeton (pril)
| |
| |(0,0,0,1,2,2,3)
| |
| || 70 || 455 || 1295 || 1960 || 1470 || 420
| |
| |- align=center
| |
| !13
| |
| |{{CDD|node|3|node|3|node_1|3|node_1|3|node_1|3|node}}
| |
| | [[Bicantitruncated 6-simplex]]<BR>great birhombated heptapeton (gabril)
| |
| |(0,0,0,1,2,3,3)
| |
| || 49 || 329 || 980 || 1540 || 1260 || 420
| |
| |- align=center
| |
| !14
| |
| |{{CDD|node|3|node|3|node_1|3|node_1|3|node_1|3|node_1}}
| |
| | [[Runcicantitruncated 6-simplex]]<BR>great prismated heptapeton (gapil)
| |
| |(0,0,0,1,2,3,4)
| |
| || 70 || 560 || 1820 || 3010 || 2520 || 840
| |
| |- align=center
| |
| !15
| |
| |{{CDD|node|3|node_1|3|node|3|node|3|node|3|node_1}}
| |
| | [[Stericated 6-simplex]]<BR>small cellated heptapeton (scal)
| |
| |(0,0,1,1,1,1,2)
| |
| |105||700||1470||1400||630||105
| |
| |- style="text-align:center; background:#e0f0e0;"
| |
| !16
| |
| |{{CDD|node|3|node_1|3|node|3|node|3|node_1|3|node}}
| |
| | [[Biruncinated 6-simplex]]<BR>small biprismato-tetradecapeton (sibpof)
| |
| |(0,0,1,1,1,2,2)
| |
| || 84 || 714 || 2100 || 2520 || 1260 || 210
| |
| |- align=center
| |
| !17
| |
| |{{CDD|node|3|node_1|3|node|3|node|3|node_1|3|node_1}}
| |
| | [[Steritruncated 6-simplex]]<BR>cellitruncated heptapeton (catal)
| |
| |(0,0,1,1,1,2,3)
| |
| || 105 || 945 || 2940 || 3780 || 2100 || 420
| |
| |- align=center
| |
| !18
| |
| |{{CDD|node|3|node_1|3|node|3|node_1|3|node|3|node_1}}
| |
| | [[Stericantellated 6-simplex]]<BR>cellirhombated heptapeton (cral)
| |
| |(0,0,1,1,2,2,3)
| |
| || 105 || 1050 || 3465 || 5040 || 3150 || 630
| |
| |- align=center
| |
| !19
| |
| |{{CDD|node|3|node_1|3|node|3|node_1|3|node_1|3|node}}
| |
| | [[Biruncitruncated 6-simplex]]<BR>biprismatorhombated heptapeton (bapril)
| |
| |(0,0,1,1,2,3,3)
| |
| || 84 || 714 || 2310 || 3570 || 2520 || 630
| |
| |- align=center
| |
| !20
| |
| |{{CDD|node|3|node_1|3|node|3|node_1|3|node_1|3|node_1}}
| |
| | [[Stericantitruncated 6-simplex]]<BR>celligreatorhombated heptapeton (cagral)
| |
| |(0,0,1,1,2,3,4)
| |
| || 105 || 1155 || 4410 || 7140 || 5040 || 1260
| |
| |- align=center
| |
| !21
| |
| |{{CDD|node|3|node_1|3|node_1|3|node|3|node|3|node_1}}
| |
| | [[Steriruncinated 6-simplex]]<BR>celliprismated heptapeton (copal)
| |
| |(0,0,1,2,2,2,3)
| |
| || 105 || 700 || 1995 || 2660 || 1680 || 420
| |
| |- align=center
| |
| !22
| |
| |{{CDD|node|3|node_1|3|node_1|3|node|3|node_1|3|node_1}}
| |
| |[[Steriruncitruncated 6-simplex]]<BR>celliprismatotruncated heptapeton (captal)
| |
| |(0,0,1,2,2,3,4)
| |
| || 105 || 945 || 3360 || 5670 || 4410 || 1260
| |
| |- align=center
| |
| !23
| |
| |{{CDD|node|3|node_1|3|node_1|3|node_1|3|node|3|node_1}}
| |
| | [[Steriruncicantellated 6-simplex]]<BR>celliprismatorhombated heptapeton (copril)
| |
| |(0,0,1,2,3,3,4)
| |
| || 105 || 1050 || 3675 || 5880 || 4410 || 1260
| |
| |- style="text-align:center; background:#e0f0e0;"
| |
| !24
| |
| |{{CDD|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node}}
| |
| | [[Biruncicantitruncated 6-simplex]]<BR>great biprismato-tetradecapeton (gibpof)
| |
| |(0,0,1,2,3,4,4)
| |
| || 84 || 714 || 2520 || 4410 || 3780 || 1260
| |
| |- align=center
| |
| !25
| |
| |{{CDD|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}
| |
| | [[Steriruncicantitruncated 6-simplex]]<BR>great cellated heptapeton (gacal)
| |
| |(0,0,1,2,3,4,5)
| |
| || 105 || 1155 || 4620 || 8610 || 7560 || 2520
| |
| |- style="text-align:center; background:#e0f0e0;"
| |
| !26
| |
| |{{CDD|node_1|3|node|3|node|3|node|3|node|3|node_1}}
| |
| | [[Pentellated 6-simplex]]<BR>small teri-tetradecapeton (staf)
| |
| |(0,1,1,1,1,1,2)
| |
| || 126 || 434 || 630 || 490 || 210 || 42
| |
| |- align=center
| |
| !27
| |
| |{{CDD|node_1|3|node|3|node|3|node|3|node_1|3|node_1}}
| |
| | [[Pentitruncated 6-simplex]]<BR>teracellated heptapeton (tocal)
| |
| |(0,1,1,1,1,2,3)
| |
| || 126 || 826 || 1785 || 1820 || 945 || 210
| |
| |- align=center
| |
| !28
| |
| |{{CDD|node_1|3|node|3|node|3|node_1|3|node|3|node_1}}
| |
| | [[Penticantellated 6-simplex]]<BR>teriprismated heptapeton (topal)
| |
| |(0,1,1,1,2,2,3)
| |
| || 126 || 1246 || 3570 || 4340 || 2310 || 420
| |
| |- align=center
| |
| !29
| |
| |{{CDD|node_1|3|node|3|node|3|node_1|3|node_1|3|node_1}}
| |
| | [[Penticantitruncated 6-simplex]]<BR>terigreatorhombated heptapeton (togral)
| |
| |(0,1,1,1,2,3,4)
| |
| || 126 || 1351 || 4095 || 5390 || 3360 || 840
| |
| |- align=center
| |
| !30
| |
| |{{CDD|node_1|3|node|3|node_1|3|node|3|node_1|3|node_1}}
| |
| | [[Pentiruncitruncated 6-simplex]]<BR>tericellirhombated heptapeton (tocral)
| |
| |(0,1,1,2,2,3,4)
| |
| || 126 || 1491 || 5565 || 8610 || 5670 || 1260
| |
| |- style="text-align:center; background:#e0f0e0;"
| |
| !31
| |
| |{{CDD|node_1|3|node|3|node_1|3|node_1|3|node|3|node_1}}
| |
| | [[Pentiruncicantellated 6-simplex]]<BR>teriprismatorhombi-tetradecapeton (taporf)
| |
| |(0,1,1,2,3,3,4)
| |
| || 126 || 1596 || 5250 || 7560 || 5040 || 1260
| |
| |- align=center
| |
| !32
| |
| |{{CDD|node_1|3|node|3|node_1|3|node_1|3|node_1|3|node_1}}
| |
| | [[Pentiruncicantitruncated 6-simplex]]<BR>terigreatoprismated heptapeton (tagopal)
| |
| |(0,1,1,2,3,4,5)
| |
| || 126 || 1701 || 6825 || 11550 || 8820 || 2520
| |
| |- style="text-align:center; background:#e0f0e0;"
| |
| !33
| |
| |{{CDD|node_1|3|node_1|3|node|3|node|3|node_1|3|node_1}}
| |
| | [[Pentisteritruncated 6-simplex]]<BR>tericellitrunki-tetradecapeton (tactaf)
| |
| |(0,1,2,2,2,3,4)
| |
| || 126 || 1176 || 3780 || 5250 || 3360 || 840
| |
| |- align=center
| |
| !34
| |
| |{{CDD|node_1|3|node_1|3|node|3|node_1|3|node_1|3|node_1}}
| |
| | [[Pentistericantitruncated 6-simplex]]<BR>tericelligreatorhombated heptapeton (tacogral)
| |
| |(0,1,2,2,3,4,5)
| |
| || 126 || 1596 || 6510 || 11340 || 8820 || 2520
| |
| |- style="text-align:center; background:#e0f0e0;"
| |
| !35
| |
| |{{CDD|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}
| |
| | [[Omnitruncated 6-simplex]]<BR>great teri-tetradecapeton (gotaf)
| |
| |(0,1,2,3,4,5,6)
| |
| || 126 || 1806 || 8400 || 16800 || 15120 || 5040
| |
| |}
| |
| | |
| === The B<sub>6</sub> family ===
| |
| There are 63 forms based on all permutations of the [[Coxeter-Dynkin diagram]]s with one or more rings.
| |
| | |
| The B<sub>6</sub> family has symmetry of order 46080 (6 [[factorial]] x 2<sup>6</sup>).
| |
| | |
| They are named by [[Norman Johnson (mathematician)|Norman Johnson]] from the Wythoff construction operations upon the regular 6-cube and 6-orthoplex. Bowers names and acronym names are given for cross-referencing.
| |
| | |
| See also [[list of B6 polytopes]] for graphs of these polytopes.
| |
| | |
| {| class="wikitable"
| |
| !rowspan=2|#
| |
| !rowspan=2|[[Coxeter-Dynkin diagram]]
| |
| !rowspan=2|[[Schläfli symbol]]
| |
| !rowspan=2|Names
| |
| !colspan=6|Element counts
| |
| |-
| |
| ! 5|| 4|| 3|| 2|| 1|| 0
| |
| |- align=center BGCOLOR="#f0e0e0"
| |
| !36
| |
| |<!-- [x3o3o3o3o4o] -->{{CDD|node|4|node|3|node|3|node|3|node|3|node_1}}||t<sub>0</sub>{3,3,3,3,4}||[[6-orthoplex]]<BR>Hexacontatetrapeton (gee)||64||192||240||160||60||12
| |
| |- align=center BGCOLOR="#f0e0e0"
| |
| !37
| |
| |<!-- [o3x3o3o3o4o] -->{{CDD|node|4|node|3|node|3|node|3|node_1|3|node}}||t<sub>1</sub>{3,3,3,3,4}||[[Rectified 6-orthoplex]]<BR>Rectified hexacontatetrapeton (rag)||76||576||1200||1120||480||60
| |
| |- align=center BGCOLOR="#f0e0e0"
| |
| !38
| |
| |<!-- [o3o3x3o3o4o] -->{{CDD|node|4|node|3|node|3|node_1|3|node|3|node}}||t<sub>2</sub>{3,3,3,3,4}||[[Birectified 6-orthoplex]]<BR>Birectified hexacontatetrapeton (brag)||76||636||2160||2880||1440||160
| |
| |- align=center BGCOLOR="#e0e0f0"
| |
| !39
| |
| |<!-- [o3o3o3x3o4o] -->{{CDD|node|4|node|3|node_1|3|node|3|node|3|node}}||t<sub>2</sub>{4,3,3,3,3}||[[Birectified 6-cube]]<BR>Birectified hexeract (brox)||76||636||2080||3200||1920||240
| |
| |- align=center BGCOLOR="#e0e0f0"
| |
| !40
| |
| |<!-- [o3o3o3o3x4o] -->{{CDD|node|4|node_1|3|node|3|node|3|node|3|node}}||t<sub>1</sub>{4,3,3,3,3}||[[Rectified 6-cube]]<BR>Rectified hexeract (rax)||76||576||1200||1120||480||60
| |
| |- align=center BGCOLOR="#e0e0f0"
| |
| !41
| |
| |<!-- [o3o3o3o3o4x] -->{{CDD|node_1|4|node|3|node|3|node|3|node|3|node}}||t<sub>0</sub>{4,3,3,3,3}||[[6-cube]]<BR>Hexeract (ax)||12||60||160||240||192||64
| |
| |- align=center BGCOLOR="#f0e0e0"
| |
| !42
| |
| |<!-- [x3x3o3o3o4o] -->{{CDD|node|4|node|3|node|3|node|3|node_1|3|node_1}}||t<sub>0,1</sub>{3,3,3,3,4}||[[Truncated 6-orthoplex]]<BR>Truncated hexacontatetrapeton (tag)||76||576||1200||1120||540||120
| |
| |- align=center BGCOLOR="#f0e0e0"
| |
| !43
| |
| |<!-- [x3o3x3o3o4o] -->{{CDD|node|4|node|3|node|3|node_1|3|node|3|node_1}}||t<sub>0,2</sub>{3,3,3,3,4}||[[Cantellated 6-orthoplex]]<BR>Small rhombated hexacontatetrapeton (srog)||136||1656||5040||6400||3360||480
| |
| |- align=center BGCOLOR="#f0e0e0"
| |
| !44
| |
| |<!-- [o3x3x3o3o4o] -->{{CDD|node|4|node|3|node|3|node_1|3|node_1|3|node}}||t<sub>1,2</sub>{3,3,3,3,4}||[[Bitruncated 6-orthoplex]]<BR>Bitruncated hexacontatetrapeton (botag)|| || || || ||1920||480
| |
| |- align=center BGCOLOR="#f0e0e0"
| |
| !45
| |
| |<!-- [x3o3o3x3o4o] -->{{CDD|node|4|node|3|node_1|3|node|3|node|3|node_1}}||t<sub>0,3</sub>{3,3,3,3,4}||[[Runcinated 6-orthoplex]]<BR>Small prismated hexacontatetrapeton (spog)|| || || || ||7200||960
| |
| |- align=center BGCOLOR="#f0e0e0"
| |
| !46
| |
| |<!-- [o3x3o3x3o4o] -->{{CDD|node|4|node|3|node_1|3|node|3|node_1|3|node}}||t<sub>1,3</sub>{3,3,3,3,4}||[[Bicantellated 6-orthoplex]]<BR>Small birhombated hexacontatetrapeton (siborg)|| || || || ||8640||1440
| |
| |- align=center BGCOLOR="#e0f0e0"
| |
| !47
| |
| |<!-- [o3o3x3x3o4o] -->{{CDD|node|4|node|3|node_1|3|node_1|3|node|3|node}}||t<sub>2,3</sub>{4,3,3,3,3}||[[Tritruncated 6-cube]]<BR>Hexeractihexacontitetrapeton (xog)|| || || || ||3360||960
| |
| |- align=center BGCOLOR="#f0e0e0"
| |
| !48
| |
| |<!-- [x3o3o3o3x4o] -->{{CDD|node|4|node_1|3|node|3|node|3|node|3|node_1}}||t<sub>0,4</sub>{3,3,3,3,4}||[[Stericated 6-orthoplex]]<BR>Small cellated hexacontatetrapeton (scag)|| || || || ||5760||960
| |
| |- align=center BGCOLOR="#e0f0e0"
| |
| !49
| |
| |<!-- [o3x3o3o3x4o] -->{{CDD|node|4|node_1|3|node|3|node|3|node_1|3|node}}||t<sub>1,4</sub>{4,3,3,3,3}||[[Biruncinated 6-cube]]<BR>Small biprismato-hexeractihexacontitetrapeton (sobpoxog)|| || || || ||11520||1920
| |
| |- align=center BGCOLOR="#e0e0f0"
| |
| !50
| |
| |<!-- [o3o3x3o3x4o] -->{{CDD|node|4|node_1|3|node|3|node_1|3|node|3|node}}||t<sub>1,3</sub>{4,3,3,3,3}||[[Bicantellated 6-cube]]<BR>Small birhombated hexeract (saborx)|| || || || ||9600||1920
| |
| |- align=center BGCOLOR="#e0e0f0"
| |
| !51
| |
| |<!-- [o3o3o3x3x4o] -->{{CDD|node|4|node_1|3|node_1|3|node|3|node|3|node}}||t<sub>1,2</sub>{4,3,3,3,3}||[[Bitruncated 6-cube]]<BR>Bitruncated hexeract (botox)|| || || || ||2880||960
| |
| |- align=center BGCOLOR="#e0f0e0"
| |
| !52
| |
| |<!-- [x3o3o3o3o4x] -->{{CDD|node_1|4|node|3|node|3|node|3|node|3|node_1}}||t<sub>0,5</sub>{4,3,3,3,3}||[[Pentellated 6-cube]]<BR>Small teri-hexeractihexacontitetrapeton (stoxog)|| || || || ||1920||384
| |
| |- align=center BGCOLOR="#e0e0f0"
| |
| !53
| |
| |<!-- [o3x3o3o3o4x] -->{{CDD|node_1|4|node|3|node|3|node|3|node_1|3|node}}||t<sub>0,4</sub>{4,3,3,3,3}||[[Stericated 6-cube]]<BR>Small cellated hexeract (scox)|| || || || ||5760||960
| |
| |- align=center BGCOLOR="#e0e0f0"
| |
| !54
| |
| |<!-- [o3o3x3o3o4x] -->{{CDD|node_1|4|node|3|node|3|node_1|3|node|3|node}}||t<sub>0,3</sub>{4,3,3,3,3}||[[Runcinated 6-cube]]<BR>Small prismated hexeract (spox)|| || || || ||7680||1280
| |
| |- align=center BGCOLOR="#e0e0f0"
| |
| !55
| |
| |<!-- [o3o3o3x3o4x] -->{{CDD|node_1|4|node|3|node_1|3|node|3|node|3|node}}||t<sub>0,2</sub>{4,3,3,3,3}||[[Cantellated 6-cube]]<BR>Small rhombated hexeract (srox)|| || || || ||4800||960
| |
| |- align=center BGCOLOR="#e0e0f0"
| |
| !56
| |
| |<!-- [o3o3o3o3x4x] -->{{CDD|node_1|4|node_1|3|node|3|node|3|node|3|node}}||t<sub>0,1</sub>{4,3,3,3,3}||[[Truncated 6-cube]]<BR>Truncated hexeract (tox)||76||444||1120||1520||1152||384
| |
| |- align=center BGCOLOR="#f0e0e0"
| |
| !57
| |
| |<!-- [x3x3x3o3o4o] -->{{CDD|node|4|node|3|node|3|node_1|3|node_1|3|node_1}}||t<sub>0,1,2</sub>{3,3,3,3,4}||[[Cantitruncated 6-orthoplex]]<BR>Great rhombated hexacontatetrapeton (grog)|| || || || ||3840||960
| |
| |- align=center BGCOLOR="#f0e0e0"
| |
| !58
| |
| |<!-- [x3x3o3x3o4o] -->{{CDD|node|4|node|3|node_1|3|node|3|node_1|3|node_1}}||t<sub>0,1,3</sub>{3,3,3,3,4}||[[Runcitruncated 6-orthoplex]]<BR>Prismatotruncated hexacontatetrapeton (potag)|| || || || ||15840||2880
| |
| |- align=center BGCOLOR="#f0e0e0"
| |
| !59
| |
| |<!-- [x3o3x3x3o4o] -->{{CDD|node|4|node|3|node_1|3|node_1|3|node|3|node_1}}||t<sub>0,2,3</sub>{3,3,3,3,4}||[[Runcicantellated 6-orthoplex]]<BR>Prismatorhombated hexacontatetrapeton (prog)|| || || || ||11520||2880
| |
| |- align=center BGCOLOR="#f0e0e0"
| |
| !60
| |
| |<!-- [o3x3x3x3o4o] -->{{CDD|node|4|node|3|node_1|3|node_1|3|node_1|3|node}}||t<sub>1,2,3</sub>{3,3,3,3,4}||[[Bicantitruncated 6-orthoplex]]<BR>Great birhombated hexacontatetrapeton (gaborg)|| || || || ||10080||2880
| |
| |- align=center BGCOLOR="#f0e0e0"
| |
| !61
| |
| |<!-- [x3x3o3o3x4o] -->{{CDD|node|4|node_1|3|node|3|node|3|node_1|3|node_1}}||t<sub>0,1,4</sub>{3,3,3,3,4}||[[Steritruncated 6-orthoplex]]<BR>Cellitruncated hexacontatetrapeton (catog)|| || || || ||19200||3840
| |
| |- align=center BGCOLOR="#f0e0e0"
| |
| !62
| |
| |<!-- [x3o3x3o3x4o] -->{{CDD|node|4|node_1|3|node|3|node_1|3|node|3|node_1}}||t<sub>0,2,4</sub>{3,3,3,3,4}||[[Stericantellated 6-orthoplex]]<BR>Cellirhombated hexacontatetrapeton (crag)|| || || || ||28800||5760
| |
| |- align=center BGCOLOR="#f0e0e0"
| |
| !63
| |
| |<!-- [o3x3x3o3x4o] -->{{CDD|node|4|node_1|3|node|3|node_1|3|node_1|3|node}}||t<sub>1,2,4</sub>{3,3,3,3,4}||[[Biruncitruncated 6-orthoplex]]<BR>Biprismatotruncated hexacontatetrapeton (boprax)|| || || || ||23040||5760
| |
| |- align=center BGCOLOR="#f0e0e0"
| |
| !64
| |
| |<!-- [x3o3o3x3x4o] -->{{CDD|node|4|node_1|3|node_1|3|node|3|node|3|node_1}}||t<sub>0,3,4</sub>{3,3,3,3,4}||[[Steriruncinated 6-orthoplex]]<BR>Celliprismated hexacontatetrapeton (copog)|| || || || ||15360||3840
| |
| |- align=center BGCOLOR="#e0e0f0"
| |
| !65
| |
| |<!-- [o3x3o3x3x4o] -->{{CDD|node|4|node_1|3|node_1|3|node|3|node_1|3|node}}||t<sub>1,2,4</sub>{4,3,3,3,3}||[[Biruncitruncated 6-cube]]<BR>Biprismatotruncated hexeract (boprag)|| || || || ||23040||5760
| |
| |- align=center BGCOLOR="#e0e0f0"
| |
| !66
| |
| |<!-- [o3o3x3x3x4o] -->{{CDD|node|4|node_1|3|node_1|3|node_1|3|node|3|node}}||t<sub>1,2,3</sub>{4,3,3,3,3}||[[Bicantitruncated 6-cube]]<BR>Great birhombated hexeract (gaborx)|| || || || ||11520||3840
| |
| |- align=center BGCOLOR="#f0e0e0"
| |
| !67
| |
| |<!-- [x3x3o3o3o4x] -->{{CDD|node_1|4|node|3|node|3|node|3|node_1|3|node_1}}||t<sub>0,1,5</sub>{3,3,3,3,4}||[[Pentitruncated 6-orthoplex]]<BR>Teritruncated hexacontatetrapeton (tacox)|| || || || ||8640||1920
| |
| |- align=center BGCOLOR="#f0e0e0"
| |
| !68
| |
| |<!-- [x3o3x3o3o4x] -->{{CDD|node_1|4|node|3|node|3|node_1|3|node|3|node_1}}||t<sub>0,2,5</sub>{3,3,3,3,4}||[[Penticantellated 6-orthoplex]]<BR>Terirhombated hexacontatetrapeton (tapox)|| || || || ||21120||3840
| |
| |- align=center BGCOLOR="#e0e0f0"
| |
| !69
| |
| |<!-- [o3x3x3o3o4x] -->{{CDD|node_1|4|node|3|node|3|node_1|3|node_1|3|node}}||t<sub>0,3,4</sub>{4,3,3,3,3}||[[Steriruncinated 6-cube]]<BR>Celliprismated hexeract (copox)|| || || || ||15360||3840
| |
| |- align=center BGCOLOR="#e0e0f0"
| |
| !70
| |
| |<!-- [x3o3o3x3o4x] -->{{CDD|node_1|4|node|3|node_1|3|node|3|node|3|node_1}}||t<sub>0,2,5</sub>{4,3,3,3,3}||[[Penticantellated 6-cube]]<BR>Terirhombated hexeract (topag)|| || || || ||21120||3840
| |
| |- align=center BGCOLOR="#e0e0f0"
| |
| !71
| |
| |<!-- [o3x3o3x3o4x] -->{{CDD|node_1|4|node|3|node_1|3|node|3|node_1|3|node}}||t<sub>0,2,4</sub>{4,3,3,3,3}||[[Stericantellated 6-cube]]<BR>Cellirhombated hexeract (crax)|| || || || ||28800||5760
| |
| |- align=center BGCOLOR="#e0e0f0"
| |
| !72
| |
| |<!-- [o3o3x3x3o4x] -->{{CDD|node_1|4|node|3|node_1|3|node_1|3|node|3|node}}||t<sub>0,2,3</sub>{4,3,3,3,3}||[[Runcicantellated 6-cube]]<BR>Prismatorhombated hexeract (prox)|| || || || ||13440||3840
| |
| |- align=center BGCOLOR="#e0e0f0"
| |
| !73
| |
| |<!-- [x3o3o3o3x4x] -->{{CDD|node_1|4|node_1|3|node|3|node|3|node|3|node_1}}||t<sub>0,1,5</sub>{4,3,3,3,3}||[[Pentitruncated 6-cube]]<BR>Teritruncated hexeract (tacog)|| || || || ||8640||1920
| |
| |- align=center BGCOLOR="#e0e0f0"
| |
| !74
| |
| |<!-- [o3x3o3o3x4x] -->{{CDD|node_1|4|node_1|3|node|3|node|3|node_1|3|node}}||t<sub>0,1,4</sub>{4,3,3,3,3}||[[Steritruncated 6-cube]]<BR>Cellitruncated hexeract (catax)|| || || || ||19200||3840
| |
| |- align=center BGCOLOR="#e0e0f0"
| |
| !75
| |
| |<!-- [o3o3x3o3x4x] -->{{CDD|node_1|4|node_1|3|node|3|node_1|3|node|3|node}}||t<sub>0,1,3</sub>{4,3,3,3,3}||[[Runcitruncated 6-cube]]<BR>Prismatotruncated hexeract (potax)|| || || || ||17280||3840
| |
| |- align=center BGCOLOR="#e0e0f0"
| |
| !76
| |
| |<!-- [o3o3o3x3x4x] -->{{CDD|node_1|4|node_1|3|node_1|3|node|3|node|3|node}}||t<sub>0,1,2</sub>{4,3,3,3,3}||[[Cantitruncated 6-cube]]<BR>Great rhombated hexeract (grox)|| || || || ||5760||1920
| |
| |- align=center BGCOLOR="#f0e0e0"
| |
| !77
| |
| |<!-- [x3x3x3x3o4o] -->{{CDD|node|4|node|3|node_1|3|node_1|3|node_1|3|node_1}}||t<sub>0,1,2,3</sub>{3,3,3,3,4}||[[Runcicantitruncated 6-orthoplex]]<BR>Great prismated hexacontatetrapeton (gopog)|| || || || ||20160||5760
| |
| |- align=center BGCOLOR="#f0e0e0"
| |
| !78
| |
| |<!-- [x3x3x3o3x4o] -->{{CDD|node|4|node_1|3|node|3|node_1|3|node_1|3|node_1}}||t<sub>0,1,2,4</sub>{3,3,3,3,4}||[[Stericantitruncated 6-orthoplex]]<BR>Celligreatorhombated hexacontatetrapeton (cagorg)|| || || || ||46080||11520
| |
| |- align=center BGCOLOR="#f0e0e0"
| |
| !79
| |
| |<!-- [x3x3o3x3x4o] -->{{CDD|node|4|node_1|3|node_1|3|node|3|node_1|3|node_1}}||t<sub>0,1,3,4</sub>{3,3,3,3,4}||[[Steriruncitruncated 6-orthoplex]]<BR>Celliprismatotruncated hexacontatetrapeton (captog)|| || || || ||40320||11520
| |
| |- align=center BGCOLOR="#f0e0e0"
| |
| !80
| |
| |<!-- [x3o3x3x3x4o] -->{{CDD|node|4|node_1|3|node_1|3|node_1|3|node|3|node_1}}||t<sub>0,2,3,4</sub>{3,3,3,3,4}||[[Steriruncicantellated 6-orthoplex]]<BR>Celliprismatorhombated hexacontatetrapeton (coprag)|| || || || ||40320||11520
| |
| |- align=center BGCOLOR="#e0f0e0"
| |
| !81
| |
| |<!-- [o3x3x3x3x4o] -->{{CDD|node|4|node_1|3|node_1|3|node_1|3|node_1|3|node}}||t<sub>1,2,3,4</sub>{4,3,3,3,3}||[[Biruncicantitruncated 6-cube]]<BR>Great biprismato-hexeractihexacontitetrapeton (gobpoxog)|| || || || ||34560||11520
| |
| |- align=center BGCOLOR="#f0e0e0"
| |
| !82
| |
| |<!-- [x3x3x3o3o4x] -->{{CDD|node_1|4|node|3|node|3|node_1|3|node_1|3|node_1}}||t<sub>0,1,2,5</sub>{3,3,3,3,4}||[[Penticantitruncated 6-orthoplex]]<BR>Terigreatorhombated hexacontatetrapeton (togrig)|| || || || ||30720||7680
| |
| |- align=center BGCOLOR="#f0e0e0"
| |
| !83
| |
| |<!-- [x3x3o3x3o4x] -->{{CDD|node_1|4|node|3|node_1|3|node|3|node_1|3|node_1}}||t<sub>0,1,3,5</sub>{3,3,3,3,4}||[[Pentiruncitruncated 6-orthoplex]]<BR>Teriprismatotruncated hexacontatetrapeton (tocrax)|| || || || ||51840||11520
| |
| |- align=center BGCOLOR="#e0f0e0"
| |
| !84
| |
| |<!-- [x3o3x3x3o4x] -->{{CDD|node_1|4|node|3|node_1|3|node_1|3|node|3|node_1}}||t<sub>0,2,3,5</sub>{4,3,3,3,3}||[[Pentiruncicantellated 6-cube]]<BR>Teriprismatorhombi-hexeractihexacontitetrapeton (tiprixog)|| || || || ||46080||11520
| |
| |- align=center BGCOLOR="#e0e0f0"
| |
| !85
| |
| |<!-- [o3x3x3x3o4x] -->{{CDD|node_1|4|node|3|node_1|3|node_1|3|node_1|3|node}}||t<sub>0,2,3,4</sub>{4,3,3,3,3}||[[Steriruncicantellated 6-cube]]<BR>Celliprismatorhombated hexeract (coprix)|| || || || ||40320||11520
| |
| |- align=center BGCOLOR="#e0f0e0"
| |
| !86
| |
| |<!-- [x3x3o3o3x4x] -->{{CDD|node_1|4|node_1|3|node|3|node|3|node_1|3|node_1}}||t<sub>0,1,4,5</sub>{4,3,3,3,3}||[[Pentisteritruncated 6-cube]]<BR>Tericelli-hexeractihexacontitetrapeton (tactaxog)|| || || || ||30720||7680
| |
| |- align=center BGCOLOR="#e0e0f0"
| |
| !87
| |
| |<!-- [x3o3x3o3x4x] -->{{CDD|node_1|4|node_1|3|node|3|node_1|3|node|3|node_1}}||t<sub>0,1,3,5</sub>{4,3,3,3,3}||[[Pentiruncitruncated 6-cube]]<BR>Teriprismatotruncated hexeract (tocrag)|| || || || ||51840||11520
| |
| |- align=center BGCOLOR="#e0e0f0"
| |
| !88
| |
| |<!-- [o3x3x3o3x4x] -->{{CDD|node_1|4|node_1|3|node|3|node_1|3|node_1|3|node}}||t<sub>0,1,3,4</sub>{4,3,3,3,3}||[[Steriruncitruncated 6-cube]]<BR>Celliprismatotruncated hexeract (captix)|| || || || ||40320||11520
| |
| |- align=center BGCOLOR="#e0e0f0"
| |
| !89
| |
| |<!-- [x3o3o3x3x4x] -->{{CDD|node_1|4|node_1|3|node_1|3|node|3|node|3|node_1}}||t<sub>0,1,2,5</sub>{4,3,3,3,3}||[[Penticantitruncated 6-cube]]<BR>Terigreatorhombated hexeract (togrix)|| || || || ||30720||7680
| |
| |- align=center BGCOLOR="#e0e0f0"
| |
| !90
| |
| |<!-- [o3x3o3x3x4x] -->{{CDD|node_1|4|node_1|3|node_1|3|node|3|node_1|3|node}}||t<sub>0,1,2,4</sub>{4,3,3,3,3}||[[Stericantitruncated 6-cube]]<BR>Celligreatorhombated hexeract (cagorx)|| || || || ||46080||11520
| |
| |- align=center BGCOLOR="#e0e0f0"
| |
| !91
| |
| |<!-- [o3o3x3x3x4x] -->{{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node|3|node}}||t<sub>0,1,2,3</sub>{4,3,3,3,3}||[[Runcicantitruncated 6-cube]]<BR>Great prismated hexeract (gippox)|| || || || ||23040||7680
| |
| |- align=center BGCOLOR="#f0e0e0"
| |
| !92
| |
| |<!-- [x3x3x3x3x4o] -->{{CDD|node|4|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}||t<sub>0,1,2,3,4</sub>{3,3,3,3,4}||[[Steriruncicantitruncated 6-orthoplex]]<BR>Great cellated hexacontatetrapeton (gocog)|| || || || ||69120||23040
| |
| |- align=center BGCOLOR="#f0e0e0"
| |
| !93
| |
| |<!-- [x3x3x3x3o4x] -->{{CDD|node_1|4|node|3|node_1|3|node_1|3|node_1|3|node_1}}||t<sub>0,1,2,3,5</sub>{3,3,3,3,4}||[[Pentiruncicantitruncated 6-orthoplex]]<BR>Terigreatoprismated hexacontatetrapeton (tagpog)|| || || || ||80640||23040
| |
| |- align=center BGCOLOR="#f0e0e0"
| |
| !94
| |
| |<!-- [x3x3x3o3x4x] -->{{CDD|node_1|4|node_1|3|node|3|node_1|3|node_1|3|node_1}}||t<sub>0,1,2,4,5</sub>{3,3,3,3,4}||[[Pentistericantitruncated 6-orthoplex]]<BR>Tericelligreatorhombated hexacontatetrapeton (tecagorg)|| || || || ||80640||23040
| |
| |- align=center BGCOLOR="#e0e0f0"
| |
| !95
| |
| |<!-- [x3x3o3x3x4x] -->{{CDD|node_1|4|node_1|3|node_1|3|node|3|node_1|3|node_1}}||t<sub>0,1,2,4,5</sub>{4,3,3,3,3}||[[Pentistericantitruncated 6-cube]]<BR>Tericelligreatorhombated hexeract (tocagrax)|| || || || ||80640||23040
| |
| |- align=center BGCOLOR="#e0e0f0"
| |
| !96
| |
| |<!-- [x3o3x3x3x4x] -->{{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node|3|node_1}}||t<sub>0,1,2,3,5</sub>{4,3,3,3,3}||[[Pentiruncicantitruncated 6-cube]]<BR>Terigreatoprismated hexeract (tagpox)|| || || || ||80640||23040
| |
| |- align=center BGCOLOR="#e0e0f0"
| |
| !97
| |
| |<!-- [o3x3x3x3x4x] -->{{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node_1|3|node}}||t<sub>0,1,2,3,4</sub>{4,3,3,3,3}||[[Steriruncicantitruncated 6-cube]]<BR>Great cellated hexeract (gocax)|| || || || ||69120||23040
| |
| |- align=center BGCOLOR="#e0f0e0"
| |
| !98
| |
| |<!-- [x3x3x3x3x4x] -->{{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}||t<sub>0,1,2,3,4,5</sub>{4,3,3,3,3}||[[Omnitruncated 6-cube]]<BR>Great teri-hexeractihexacontitetrapeton (gotaxog)|| || || || ||138240||46080
| |
| | |
| |}
| |
| | |
| === The D<sub>6</sub> family ===
| |
| | |
| The D<sub>6</sub> family has symmetry of order 23040 (6 [[factorial]] x 2<sup>5</sup>).
| |
| | |
| This family has 3×16−1=47 Wythoffian uniform polytopes, generated by marking one or more nodes of the D<sub>6</sub> [[Coxeter-Dynkin diagram]]. Of these, 31 (2×16−1) are repeated from the B<sub>6</sub> family and 16 are unique to this family. The 16 unique forms are enumerated below. Bowers-style acronym names are given for cross-referencing.
| |
| | |
| See [[list of D6 polytopes]] for Coxeter plane graphs of these polytopes.
| |
| | |
| {| class="wikitable"
| |
| !rowspan=2|#
| |
| !rowspan=2|[[Coxeter diagram]]
| |
| !rowspan=2|Names
| |
| !rowspan=2|Base point<BR>(Alternately signed)
| |
| !colspan=6|Element counts
| |
| !rowspan=2|Circumrad
| |
| |-
| |
| !5||4||3||2||1||0
| |
| |- align=center
| |
| !99
| |
| ||{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node}} = {{CDD|node_h1|4|node|3|node|3|node|3|node|3|node}}||[[6-demicube]]<BR>Hemihexeract (hax)||(1,1,1,1,1,1)||44||252||640||640||240||32||0.8660254
| |
| |- align=center
| |
| !100
| |
| ||{{CDD|nodes_10ru|split2|node_1|3|node|3|node|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node|3|node}}||[[Cantic 6-cube]]<BR>Truncated hemihexeract (thax)||(1,1,3,3,3,3)||76||636||2080||3200||2160||480||2.1794493
| |
| |- align=center
| |
| !101
| |
| ||{{CDD|nodes_10ru|split2|node|3|node_1|3|node|3|node}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node|3|node}}||[[Runcic 6-cube]]<BR>Small rhombated hemihexeract (sirhax)||(1,1,1,3,3,3)|| || || || ||3840||640||1.9364916
| |
| |- align=center
| |
| !102
| |
| ||{{CDD|nodes_10ru|split2|node|3|node|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node|3|node|3|node_1|3|node}}||[[Steric 6-cube]]<BR>Small prismated hemihexeract (sophax)||(1,1,1,1,3,3)|| || || || ||3360||480||1.6583123
| |
| |- align=center
| |
| !103
| |
| ||{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node|3|node|3|node_1}}||[[Pentic 6-cube]]<BR>Small cellated demihexeract (sochax)||(1,1,1,1,1,3)|| || || || ||1440||192||1.3228756
| |
| |- align=center
| |
| !104
| |
| ||{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node|3|node}}||[[Runcicantic 6-cube]]<BR>Great rhombated hemihexeract (girhax)||(1,1,3,5,5,5)|| || || || ||5760||1920||3.2787192
| |
| |- align=center
| |
| !105
| |
| ||{{CDD|nodes_10ru|split2|node_1|3|node|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node_1|3|node}}||[[Stericantic 6-cube]]<BR>Prismatotruncated hemihexeract (pithax)||(1,1,3,3,5,5)|| || || || ||12960||2880||2.95804
| |
| |- align=center
| |
| !106
| |
| ||{{CDD|nodes_10ru|split2|node|3|node_1|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node_1|3|node}}||[[Steriruncic 6-cube]]<BR>Prismatorhombated hemihexeract (prohax)||(1,1,1,3,5,5)|| || || || ||7680||1920||2.7838821
| |
| |- align=center
| |
| !107
| |
| ||{{CDD|nodes_10ru|split2|node_1|3|node|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node|3|node_1}}||[[Penticantic 6-cube]]<BR>Cellitruncated hemihexeract (cathix)||(1,1,3,3,3,5)|| || || || ||9600||1920||2.5980761
| |
| |- align=center
| |
| !108
| |
| ||{{CDD|nodes_10ru|split2|node|3|node_1|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node|3|node_1}}||[[Pentiruncic 6-cube]]<BR>Cellirhombated hemihexeract (crohax)||(1,1,1,3,3,5)|| || || || ||10560||1920||2.3979158
| |
| |- align=center
| |
| !109
| |
| ||{{CDD|nodes_10ru|split2|node|3|node|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node|3|node_1|3|node_1}}||[[Pentisteric 6-cube]]<BR>Celliprismated hemihexeract (cophix)||(1,1,1,1,3,5)|| || || || ||5280||960||2.1794496
| |
| |- align=center
| |
| !110
| |
| ||{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node_1|3|node_1}}||[[Steriruncicantic 6-cube]]<BR>Great prismated hemihexeract (gophax)||(1,1,3,5,7,7)|| || || || ||17280||5760||4.0926762
| |
| |- align=center
| |
| !111
| |
| ||{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node|3|node_1}}||[[Pentiruncicantic 6-cube]]<BR>Celligreatorhombated hemihexeract (cagrohax)||(1,1,3,5,5,7)|| || || || ||20160||5760||3.7080991
| |
| |- align=center
| |
| !112
| |
| ||{{CDD|nodes_10ru|split2|node_1|3|node|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node_1|3|node_1}}||[[Pentistericantic 6-cube]]<BR>Celliprismatotruncated hemihexeract (capthix)||(1,1,3,3,5,7)|| || || || ||23040||5760||3.4278274
| |
| |- align=center
| |
| !113
| |
| ||{{CDD|nodes_10ru|split2|node|3|node_1|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node_1|3|node_1}}||[[Pentisteriruncic 6-cube]]<BR>Celliprismatorhombated hemihexeract (caprohax)||(1,1,1,3,5,7)|| || || || ||15360||3840||3.2787192
| |
| |- align=center
| |
| !114
| |
| ||{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node_1|3|node_1}}||[[Pentisteriruncicantic 6-cube]]<BR>Great cellated hemihexeract (gochax)||(1,1,3,5,7,9)|| || || || ||34560||11520||4.5552168
| |
| |}
| |
| | |
| === The E<sub>6</sub> family ===
| |
| There are 39 forms based on all permutations of the [[Coxeter-Dynkin diagram]]s with one or more rings. Bowers-style acronym names are given for cross-referencing. The [[E6 (mathematics)|E<sub>6</sub>]] family has symmetry of order 51,840.
| |
| | |
| See also [[list of E6 polytopes]] for graphs of these polytopes.
| |
| | |
| {| class="wikitable"
| |
| |-
| |
| !rowspan=2|#
| |
| !rowspan=2|[[Coxeter-Dynkin diagram]]<br/>[[Schläfli symbol]]
| |
| !rowspan=2|Names
| |
| !colspan=6|Element counts
| |
| |-
| |
| ! 5-faces
| |
| ! 4-faces
| |
| ! Cells
| |
| ! Faces
| |
| ! Edges
| |
| ! Vertices
| |
| |- align=center
| |
| |115||{{CDD|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea}}||[[2 21 polytope|2<sub>21</sub>]]<BR>Icosiheptaheptacontidipeton (jak)||99||648||1080||720||216||27
| |
| |- align=center
| |
| |116||{{CDD|nodea|3a|nodea_1|3a|branch|3a|nodea|3a|nodea}}||[[Rectified 2 21 polytope|Rectified 2<sub>21</sub>]]<BR>Rectified icosiheptaheptacontidipeton (rojak)||126||1350||4320||5040||2160||216
| |
| |- style="text-align:center; background:#e0f0e0;"
| |
| |117||{{CDD|nodea|3a|nodea|3a|branch_10|3a|nodea|3a|nodea}}||[[Rectified 1 22 polytope|Rectified 1<sub>22</sub>]]<BR>Rectified pentacontatetrapeton (ram)||126||1566||6480||10800||6480||720
| |
| |- style="text-align:center; background:#e0f0e0;"
| |
| |118||{{CDD|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea}}||[[1 22 polytope|1<sub>22</sub>]]<BR>Pentacontatetrapeton (mo)||54||702||2160||2160||720||72
| |
| |- align=center
| |
| |119||{{CDD|nodea_1|3a|nodea_1|3a|branch|3a|nodea|3a|nodea}}||[[Truncated 2 21 polytope|Truncated 2<sub>21</sub>]]<BR>Truncated icosiheptaheptacontidipeton (tojak)||126||1350||4320||5040||2376||432
| |
| |- align=center
| |
| |120||{{CDD|nodea_1|3a|nodea|3a|branch_10|3a|nodea|3a|nodea}}||[[Cantellasted 2 21 polytope|Cantellated 2<sub>21</sub>]]<BR>Small rhombated icosiheptaheptacontidipeton (sirjak)||342||3942||15120||24480||15120||2160
| |
| |- align=center
| |
| |121||{{CDD|nodea_1|3a|nodea|3a|branch|3a|nodea_1|3a|nodea}}||[[Runcinated 2 21 polytope|Runcinated 2<sub>21</sub>]]<BR>Small demiprismated icosiheptaheptacontidipeton (shopjak)||342||4662||16200||19440||8640||1080
| |
| |- style="text-align:center; background:#e0f0e0;"
| |
| |122||{{CDD|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea_1}}||[[Stericated 2 21 polytope|Stericated 2<sub>21</sub>]]<BR>Trirectified pentacontatetrapeton (trim)||558||4608||8640||6480||2160||270
| |
| |- align=center
| |
| |123||{{CDD|nodea_1|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea}}||Demified icosiheptaheptacontidipeton (hejak)||342||2430||7200||7920||3240||432
| |
| |- align=center
| |
| |124||{{CDD|nodea|3a|nodea_1|3a|branch_10|3a|nodea|3a|nodea}}||[[Bitruncated 2 21 polytope|Bitruncated 2<sub>21</sub>]]<BR>Bitruncated icosiheptaheptacontidipeton (botajik)||||||||||||2160
| |
| |- style="text-align:center; background:#e0f0e0;"
| |
| |125||{{CDD|nodea|3a|nodea_1|3a|branch|3a|nodea_1|3a|nodea}}||[[1_22_polytope#Birectified_1_22_polytope|Bicantellated 2<sub>21</sub>]]<BR>Birectified pentacontatetrapeton (barm)|| || || || ||12960||2160
| |
| |- align=center
| |
| |126||{{CDD|nodea|3a|nodea_1|3a|branch_01lr|3a|nodea|3a|nodea}}||Demirectified icosiheptaheptacontidipeton (harjak)||||||||||||1080
| |
| |- style="text-align:center; background:#e0f0e0;"
| |
| |127||{{CDD|nodea|3a|nodea|3a|branch_11|3a|nodea|3a|nodea}}||Truncated pentacontatetrapeton (tim)|| || || || ||13680||1440
| |
| |- align=center
| |
| |128||{{CDD|nodea_1|3a|nodea_1|3a|branch_10|3a|nodea|3a|nodea}}||[[Cantitruncated 2 21 polytope|Cantitruncated 2<sub>21</sub>]]<BR>Great rhombated icosiheptaheptacontidipeton (girjak)||||||||||||4320
| |
| |- align=center
| |
| |129||{{CDD|nodea_1|3a|nodea_1|3a|branch|3a|nodea_1|3a|nodea}}||[[Runcitruncated 2 21 polytope|Runcitruncated 2<sub>21</sub>]]<BR>Demiprismatotruncated icosiheptaheptacontidipeton (hopitjak)||||||||||||4320
| |
| |- align=center
| |
| |130||{{CDD|nodea_1|3a|nodea_1|3a|branch|3a|nodea|3a|nodea_1}}||[[Steritruncated 2 21 polytope|Steritruncated 2<sub>21</sub>]]<BR>Cellitruncated icosiheptaheptacontidipeton (catjak)||||||||||||2160
| |
| |- align=center
| |
| |131||{{CDD|nodea_1|3a|nodea_1|3a|branch_01lr|3a|nodea|3a|nodea}}||Demitruncated icosiheptaheptacontidipeton (hotjak)||||||||||||2160
| |
| |- align=center
| |
| |132||{{CDD|nodea_1|3a|nodea|3a|branch_10|3a|nodea_1|3a|nodea}}||[[Runcicantellated 2 21 polytope|Runcicantellated 2<sub>21</sub>]]<BR>Demiprismatorhombated icosiheptaheptacontidipeton (haprojak)||||||||||||6480
| |
| |- style="text-align:center; background:#e0f0e0;"
| |
| |133||{{CDD|nodea_1|3a|nodea|3a|branch_10|3a|nodea|3a|nodea_1}}||[[Stericantellated 2 21 polytope|Stericantellated 2<sub>21</sub>]]<BR>Small birhombated pentacontatetrapeton (sabrim)||||||||||||6480
| |
| |- align=center
| |
| |134||{{CDD|nodea_1|3a|nodea|3a|branch_11|3a|nodea|3a|nodea}}||Small demirhombated icosiheptaheptacontidipeton (shorjak)||||||||||||4320
| |
| |- align=center
| |
| |135||{{CDD|nodea_1|3a|nodea|3a|branch_01lr|3a|nodea_1|3a|nodea}}||Small prismated icosiheptaheptacontidipeton (spojak)||||||||||||4320
| |
| |- style="text-align:center; background:#e0f0e0;"
| |
| |136||{{CDD|nodea_1|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea_1}}||Small prismated pentacontatetrapeton (spam)||||||||||||2160
| |
| |- style="text-align:center; background:#e0f0e0;"
| |
| |137||{{CDD|nodea|3a|nodea_1|3a|branch_10|3a|nodea_1|3a|nodea}}||Bitruncated pentacontatetrapeton (bitem)||||||||||||6480
| |
| |- align=center
| |
| |138||{{CDD|nodea|3a|nodea_1|3a|branch_11|3a|nodea|3a|nodea}}||Tritruncated icosiheptaheptacontidipeton (titajak)||||||||||||4320
| |
| |- style="text-align:center; background:#e0f0e0;"
| |
| |139||{{CDD|nodea|3a|nodea_1|3a|branch_01lr|3a|nodea_1|3a|nodea}}||Small rhombated pentacontatetrapeton (sram)||||||||||||6480
| |
| |- align=center
| |
| |140||{{CDD|nodea_1|3a|nodea_1|3a|branch_10|3a|nodea_1|3a|nodea}}||[[Runcicantitruncated 2 21 polytope|Runcicantitruncated 2<sub>21</sub>]]<BR>Great demiprismated icosiheptaheptacontidipeton (ghopjak)||||||||||||12960
| |
| |- align=center
| |
| |141||{{CDD|nodea_1|3a|nodea_1|3a|branch_10|3a|nodea|3a|nodea_1}}||[[Stericantitruncated 2 21 polytope|Stericantitruncated 2<sub>21</sub>]]<BR>Celligreatorhombated icosiheptaheptacontidipeton (cograjik)||||||||||||12960
| |
| |- align=center
| |
| |142||{{CDD|nodea_1|3a|nodea_1|3a|branch_11|3a|nodea|3a|nodea}}||Great demirhombated icosiheptaheptacontidipeton (ghorjak)||||||||||||8640
| |
| |- style="text-align:center; background:#e0f0e0;"
| |
| |143||{{CDD|nodea_1|3a|nodea_1|3a|branch|3a|nodea_1|3a|nodea_1}}||[[Steriruncitruncated 2 21 polytope|Steriruncitruncated 2<sub>21</sub>]]<BR>Tritruncated pentacontatetrapeton (titam)||||||||||||8640
| |
| |- align=center
| |
| |144||{{CDD|nodea_1|3a|nodea_1|3a|branch_01lr|3a|nodea_1|3a|nodea}}||Prismatotruncated icosiheptaheptacontidipeton (potjak)||||||||||||12960
| |
| |- align=center
| |
| |145||{{CDD|nodea_1|3a|nodea_1|3a|branch_01lr|3a|nodea|3a|nodea_1}}||Demicellitruncated icosiheptaheptacontidipeton (hictijik)||||||||||||8640
| |
| |- align=center
| |
| |146||{{CDD|nodea_1|3a|nodea|3a|branch_11|3a|nodea_1|3a|nodea}}||Prismatorhombated icosiheptaheptacontidipeton (projak)||||||||||||12960
| |
| |- style="text-align:center; background:#e0f0e0;"
| |
| |147||{{CDD|nodea_1|3a|nodea|3a|branch_11|3a|nodea|3a|nodea_1}}||Prismatotruncated pentacontatetrapeton (patom)||||||||||||12960
| |
| |- style="text-align:center; background:#e0f0e0;"
| |
| |148||{{CDD|nodea|3a|nodea_1|3a|branch_11|3a|nodea_1|3a|nodea}}||Great rhombated pentacontatetrapeton (gram)||||||||||||12960
| |
| |- style="text-align:center; background:#e0f0e0;"
| |
| |149||{{CDD|nodea_1|3a|nodea_1|3a|branch_10|3a|nodea_1|3a|nodea_1}}||[[Steriruncicantitruncated 2 21 polytope|Steriruncicantitruncated 2<sub>21</sub>]]<BR>Great birhombated pentacontatetrapeton (gabrim)||||||||||||25920
| |
| |- align=center
| |
| |150||{{CDD|nodea_1|3a|nodea_1|3a|branch_11|3a|nodea_1|3a|nodea}}||Great prismated icosiheptaheptacontidipeton (gapjak)||||||||||||25920
| |
| |- align=center
| |
| |151||{{CDD|nodea_1|3a|nodea_1|3a|branch_11|3a|nodea|3a|nodea_1}}||Demicelligreatorhombated icosiheptaheptacontidipeton (hocgarjik)||||||||||||25920
| |
| |- style="text-align:center; background:#e0f0e0;"
| |
| |152||{{CDD|nodea_1|3a|nodea_1|3a|branch_01lr|3a|nodea_1|3a|nodea_1}}||Prismatorhombated pentacontatetrapeton (prom)||||||||||||25920
| |
| |- style="text-align:center; background:#e0f0e0;"
| |
| |153||{{CDD|nodea_1|3a|nodea_1|3a|branch_11|3a|nodea_1|3a|nodea_1}}||Great prismated pentacontatetrapeton (gopam)|| || || || || ||51840
| |
| |}
| |
| | |
| == Regular and uniform honeycombs ==
| |
| [[File:Coxeter diagram affine rank6 correspondence.png|518px|thumb|Coxeter-Dynkin diagram correspondences between families and higher symmetry within diagrams. Nodes of the same color in each row represent identical mirrors. Black nodes are not active in the correspondence.]]
| |
| There are four fundamental affine [[Coxeter groups]] and 27 prismatic groups that generate regular and uniform tessellations in 5-space:
| |
| {| class=wikitable
| |
| |-
| |
| !#
| |
| !colspan=2|[[Coxeter group]]
| |
| ![[Coxeter diagram]]
| |
| !Forms
| |
| |- align=center
| |
| |1||<math>{\tilde{A}}_5</math>||[3<sup>[6]</sup>]||{{CDD|node|split1|nodes|3ab|nodes|split2|node}}||12
| |
| |- align=center
| |
| |2||<math>{\tilde{C}}_5</math>||[4,3<sup>3</sup>,4]||{{CDD|node|4|node|3|node|3|node|3|node|4|node}}||35
| |
| |- align=center
| |
| |3||<math>{\tilde{B}}_5</math>||[4,3,3<sup>1,1</sup>]<BR>[4,3<sup>3</sup>,4,1<sup>+</sup>]||{{CDD|node|4|node|3|node|3|node|split1|nodes}}<BR>{{CDD|node|4|node|3|node|3|node|3|node|4|node_h0}}||47 (16 new)
| |
| |- align=center
| |
| |4||<math>{\tilde{D}}_5</math>||[3<sup>1,1</sup>,3,3<sup>1,1</sup>]<BR>[1<sup>+</sup>,4,3<sup>3</sup>,4,1<sup>+</sup>]||{{CDD|nodes|split2|node|3|node|split1|nodes}}<BR>{{CDD|node_h0|4|node|3|node|3|node|3|node|4|node_h0}}||20 (3 new)
| |
| |}
| |
| | |
| Regular and uniform honeycombs include:
| |
| * <math>{\tilde{A}}_5</math> There are 12 unique uniform honeycombs, including:
| |
| ** [[5-simplex honeycomb]] {{CDD|node_1|split1|nodes|3ab|nodes|split2|node}}
| |
| ** [[Truncated 5-simplex honeycomb]] {{CDD|branch_11|3ab|nodes|3ab|branch}}
| |
| ** [[Omnitruncated 5-simplex honeycomb]] {{CDD|node_1|split1|nodes_11|3ab|nodes_11|split2|node_1}}
| |
| * <math>{\tilde{C}}_5</math> There are 35 uniform honeycombs, including:
| |
| ** Regular [[hypercube honeycomb]] of Euclidean 5-space, the [[5-cube honeycomb]], with symbols {4,3<sup>3</sup>,4}, {{CDD|node_1|4|node|3|node|3|node|3|node|4|node}} = {{CDD|node_1|4|node|3|node|3|node|split1|nodes}}
| |
| * <math>{\tilde{B}}_5</math> There are 47 uniform honeycombs, 16 new, including:
| |
| ** The uniform [[alternated hypercube honeycomb]], [[5-demicubic honeycomb]], with symbols h{4,3<sup>3</sup>,4}, {{CDD|node_h1|4|node|3|node|3|node|3|node|4|node}} = {{CDD|nodes_10ru|split2|node|3|node|3|node|4|node}} = {{CDD|nodes_10ru|split2|node|3|node|split1|nodes}}
| |
| * <math>{\tilde{D}}_5</math>, [3<sup>1,1</sup>,3,3<sup>1,1</sup>]: There are 20 unique ringed permutations, and 3 new ones. Coxeter calls the first one a [[quarter 5-cubic honeycomb]], with symbols q{4,3<sup>3</sup>,4}, {{CDD|nodes_10ru|split2|node|3|node|split1|nodes_10lu}} = {{CDD|node_h1|4|node|3|node|3|node|3|node|4|node_h1}}. The other two new ones are {{CDD|nodes_10ru|split2|node_1|3|node|split1|nodes_10lu}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node|4|node_h1}}, {{CDD|nodes_10ru|split2|node_1|3|node_1|split1|nodes_10lu}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node|4|node_h1}}.
| |
| | |
| {| class=wikitable
| |
| |+ Prismatic groups
| |
| |-
| |
| !#
| |
| !colspan=2|[[Coxeter group]]
| |
| ![[Coxeter-Dynkin diagram]]
| |
| |-
| |
| |1||<math>{\tilde{A}}_4</math>x<math>{\tilde{I}}_1</math>||[3<sup>[5]</sup>,2,∞]||{{CDD|branch|3ab|nodes|split2|node|2|node|infin|node}}
| |
| |-
| |
| |2||<math>{\tilde{B}}_4</math>x<math>{\tilde{I}}_1</math>||[4,3,3<sup>1,1</sup>,2,∞]||{{CDD|nodes|split2|node|3|node|4|node|2|node|infin|node}}
| |
| |-
| |
| |3||<math>{\tilde{C}}_4</math>x<math>{\tilde{I}}_1</math>||[4,3,3,4,2,∞]||{{CDD|node|4|node|3|node|3|node|4|node|2|node|infin|node}}
| |
| |-
| |
| |4||<math>{\tilde{D}}_4</math>x<math>{\tilde{I}}_1</math>||[3<sup>1,1,1,1</sup>,2,∞]||{{CDD|nodes|split2|node|split1|nodes|2|node|infin|node}}
| |
| |-
| |
| |5||<math>{\tilde{F}}_4</math>x<math>{\tilde{I}}_1</math>||[3,4,3,3,2,∞]||{{CDD|node|3|node|4|node|3|node|3|node|2|node|infin|node}}
| |
| |-
| |
| |6||<math>{\tilde{C}}_3</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>||[4,3,4,2,∞,2,∞]||{{CDD|node|4|node|3|node|4|node|2|node|infin|node|2|node|infin|node}}
| |
| |-
| |
| |7||<math>{\tilde{B}}_3</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>||[4,3<sup>1,1</sup>,2,∞,2,∞]||{{CDD|nodea|3a|branch|3a|4a|nodea|2|node|infin|node|2|node|infin|node}}
| |
| |-
| |
| |8||<math>{\tilde{A}}_3</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>||[3<sup>[4]</sup>,2,∞,2,∞]||{{CDD|branch|3ab|branch|2|node|infin|node|2|node|infin|node}}
| |
| |-
| |
| |9||<math>{\tilde{C}}_2</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>||[4,4,2,∞,2,∞,2,∞]||{{CDD|node|4|node|4|node|2|node|infin|node|2|node|infin|node|2|node|infin|node}}
| |
| |-
| |
| |10||<math>{\tilde{H}}_2</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>||[6,3,2,∞,2,∞,2,∞]||{{CDD|node|6|node|3|node|2|node|infin|node|2|node|infin|node|2|node|infin|node}}
| |
| |-
| |
| |11||<math>{\tilde{A}}_2</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>||[3<sup>[3]</sup>,2,∞,2,∞,2,∞]||{{CDD|node|split1|branch|2|node|infin|node|2|node|infin|node|2|node|infin|node}}
| |
| |-
| |
| |12||<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>||[∞,2,∞,2,∞,2,∞,2,∞]||{{CDD|node|infin|node|2|node|infin|node|2|node|infin|node|2|node|infin|node|2|node|infin|node}}
| |
| |-
| |
| |13||<math>{\tilde{A}}_2</math>x<math>{\tilde{A}}_2</math>x<math>{\tilde{I}}_1</math>||[3<sup>[3]</sup>,2,3<sup>[3]</sup>,2,∞]||{{CDD|node|split1|branch|2|node|split1|branch|2|node|infin|node}}
| |
| |-
| |
| |14||<math>{\tilde{A}}_2</math>x<math>{\tilde{B}}_2</math>x<math>{\tilde{I}}_1</math>||[3<sup>[3]</sup>,2,4,4,2,∞]||{{CDD|node|split1|branch|2|node|4|node|4|node|2|node|infin|node}}
| |
| |-
| |
| |15||<math>{\tilde{A}}_2</math>x<math>{\tilde{G}}_2</math>x<math>{\tilde{I}}_1</math>||[3<sup>[3]</sup>,2,6,3,2,∞]||{{CDD|node|split1|branch|2|node|6|node|3|node|2|node|infin|node}}
| |
| |-
| |
| |16||<math>{\tilde{B}}_2</math>x<math>{\tilde{B}}_2</math>x<math>{\tilde{I}}_1</math>||[4,4,2,4,4,2,∞]||{{CDD|node|4|node|4|node|2|node|4|node|4|node|2|node|infin|node}}
| |
| |-
| |
| |17||<math>{\tilde{B}}_2</math>x<math>{\tilde{G}}_2</math>x<math>{\tilde{I}}_1</math>||[4,4,2,6,3,2,∞]||{{CDD|node|4|node|4|node|2|node|6|node|3|node|2|node|infin|node}}
| |
| |-
| |
| |18||<math>{\tilde{G}}_2</math>x<math>{\tilde{G}}_2</math>x<math>{\tilde{I}}_1</math>||[6,3,2,6,3,2,∞]||{{CDD|node|6|node|3|node|2|node|6|node|3|node|2|node|infin|node}}
| |
| | |
| |-
| |
| |19||<math>{\tilde{A}}_3</math>x<math>{\tilde{A}}_2</math>||[3<sup>[4]</sup>,2,3<sup>[3]</sup>]||{{CDD|branch|3ab|branch|2|node|split1|branch}}
| |
| |-
| |
| |20||<math>{\tilde{B}}_3</math>x<math>{\tilde{A}}_2</math>||[4,3<sup>1,1</sup>,2,3<sup>[3]</sup>]||{{CDD|nodea|3a|branch|3a|4a|nodea|2|node|split1|branch}}
| |
| |-
| |
| |21||<math>{\tilde{C}}_3</math>x<math>{\tilde{A}}_2</math>||[4,3,4,2,3<sup>[3]</sup>]||{{CDD|node|4|node|3|node|4|node|2|node|split1|branch}}
| |
| | |
| |-
| |
| |22||<math>{\tilde{A}}_3</math>x<math>{\tilde{B}}_2</math>||[3<sup>[4]</sup>,2,4,4]||{{CDD|branch|3ab|branch|2|node|4|node|4|node}}
| |
| |-
| |
| |23||<math>{\tilde{B}}_3</math>x<math>{\tilde{B}}_2</math>||[4,3<sup>1,1</sup>,2,4,4]||{{CDD|nodea|3a|branch|3a|4a|nodea|2|node|4|node|4|node}}
| |
| |-
| |
| |24||<math>{\tilde{C}}_3</math>x<math>{\tilde{B}}_2</math>||[4,3,4,2,4,4]||{{CDD|node|4|node|3|node|4|node|2|node|4|node|4|node}}
| |
| | |
| |-
| |
| |25||<math>{\tilde{A}}_3</math>x<math>{\tilde{G}}_2</math>||[3<sup>[4]</sup>,2,6,3]||{{CDD|branch|3ab|branch|2|node|6|node|3|node}}
| |
| |-
| |
| |26||<math>{\tilde{B}}_3</math>x<math>{\tilde{G}}_2</math>||[4,3<sup>1,1</sup>,2,6,3]||{{CDD|nodea|3a|branch|3a|4a|nodea|2|node|6|node|3|node}}
| |
| |-
| |
| |27||<math>{\tilde{C}}_3</math>x<math>{\tilde{G}}_2</math>||[4,3,4,2,6,3]||{{CDD|node|4|node|3|node|4|node|2|node|6|node|3|node}}
| |
| |}
| |
| | |
| === Regular and uniform hyperbolic honeycombs ===
| |
| | |
| There are no compact hyperbolic Coxeter groups of rank 6, groups that can generate honeycombs with all finite facets, and a finite [[vertex figure]]. However there are [[Coxeter-Dynkin_diagram#Rank_4_to_10|12 noncompact hyperbolic Coxeter groups]] of rank 6, each generating uniform honeycombs in 5-space as permutations of rings of the Coxeter diagrams.
| |
| | |
| {| class=wikitable
| |
| |+ Hyperbolic noncompact groups
| |
| |align=right|
| |
| <math>{\bar{P}}_5</math> = [3,3<sup>[5]</sup>]: {{CDD|branch|3ab|nodes|split2|node|3|node}}<BR>
| |
| <math>{\widehat{AU}}_5</math> = [(3,3,3,3,3,4)]: {{CDD|label4|branch|3ab|nodes|3ab|branch}}
| |
| | |
| <math>{\widehat{AR}}_5</math> = [(3,3,4,3,3,4)]: {{CDD|label4|branch|3ab|nodes|3ab|branch|label4}}
| |
| |align=right|
| |
| <math>{\bar{S}}_5</math> = [4,3,3<sup>2,1</sup>]: {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|4a|nodea}}<BR>
| |
| <math>{\bar{O}}_5</math> = [3,4,3<sup>1,1</sup>]: {{CDD|nodes|split2|node|3|node|4|node|3|node}}<BR>
| |
| <math>{\bar{N}}_5</math> = [3,(3,4)<sup>1,1</sup>]: {{CDD|nodea|4a|nodea|3a|branch|3a|nodea|4a|nodea}}
| |
| |align=right|
| |
| <math>{\bar{U}}_5</math> = [3,3,3,4,3]: {{CDD|node|3|node|3|node|3|node|4|node|3|node}}<BR>
| |
| <math>{\bar{X}}_5</math> = [3,3,4,3,3]: {{CDD|node|3|node|3|node|4|node|3|node|3|node}}<BR>
| |
| <math>{\bar{R}}_5</math> = [3,4,3,3,4]: {{CDD|node|3|node|4|node|3|node|3|node|4|node}}
| |
| |align=right|<math>{\bar{Q}}_5</math> = [3<sup>2,1,1,1</sup>]: {{CDD|nodea|3a|nodes|split2|node|split1|nodes}}<BR>
| |
| <math>{\bar{M}}_5</math> = [4,3,3<sup>1,1,1</sup>]: {{CDD|nodea|4a|nodes|split2|node|split1|nodes}}<BR>
| |
| <math>{\bar{L}}_5</math> = [3<sup>1,1,1,1,1</sup>]: {{CDD|node|branch3|splitsplit2|node|split1|nodes}}
| |
| |}
| |
| | |
| == Notes on the Wythoff construction for the uniform 6-polytopes ==
| |
| | |
| Construction of the reflective 6-dimensional [[uniform polytope]]s are done through a [[Wythoff construction]] process, and represented through a [[Coxeter-Dynkin diagram]], 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 6-polytopes are named in relation to the [[regular polytope]]s in each family. Some families have two regular constructors and thus may have two ways of naming them.
| |
| | |
| Here's the primary operators available for constructing and naming the uniform 6-polytopes.
| |
| | |
| The prismatic forms and bifurcating graphs can use the same truncation indexing notation, but require an explicit numbering system on the nodes for clarity.
| |
| | |
| {|class="wikitable"
| |
| |-
| |
| !Operation
| |
| !Extended<br/>[[Schläfli symbol]]
| |
| !width=110|[[Coxeter-Dynkin diagram|Coxeter-<br/>Dynkin<br/>diagram]]
| |
| !Description
| |
| |-
| |
| ! Parent
| |
| |width=70| t<sub>0</sub>{p,q,r,s,t}
| |
| |{{CDD|node_1|p|node|q|node|r|node|s|node|t|node}}
| |
| | Any regular 6-polytope
| |
| |-
| |
| ! [[Rectification (geometry)|Rectified]]
| |
| | t<sub>1</sub>{p,q,r,s,t}
| |
| |{{CDD|node|p|node_1|q|node|r|node|s|node|t|node}}
| |
| |The edges are fully truncated into single points. The 6-polytope now has the combined faces of the parent and dual.
| |
| |-
| |
| ! Birectified
| |
| | t<sub>2</sub>{p,q,r,s,t}
| |
| |{{CDD|node|p|node|q|node_1|r|node|s|node|t|node}}
| |
| |Birectification reduces [[Cell (geometry)|cells]] to their [[Dual polytope|duals]].
| |
| |-
| |
| ![[Truncation (geometry)|Truncated]]
| |
| | t<sub>0,1</sub>{p,q,r,s,t}
| |
| |{{CDD|node_1|p|node_1|q|node|r|node|s|node|t|node}}
| |
| |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 6-polytope. The 6-polytope has its original faces doubled in sides, and contains the faces of the dual.<br/>[[File:Cube truncation sequence.svg|400px]]
| |
| |-
| |
| ![[Bitruncated]]
| |
| | t<sub>1,2</sub>{p,q,r,s,t}
| |
| |{{CDD|node|p|node_1|q|node_1|r|node|s|node|t|node}}
| |
| |Bitrunction transforms cells to their dual truncation.
| |
| |-
| |
| !Tritruncated
| |
| | t<sub>2,3</sub>{p,q,r,s,t}
| |
| |{{CDD|node|p|node|q|node_1|r|node_1|s|node|t|node}}
| |
| |Tritruncation transforms 4-faces to their dual truncation.
| |
| |-
| |
| ! [[Cantellation (geometry)|Cantellated]]
| |
| | t<sub>0,2</sub>{p,q,r,s,t}
| |
| |{{CDD|node_1|p|node|q|node_1|r|node|s|node|t|node}}
| |
| |In addition to vertex truncation, each original edge is ''beveled'' with new rectangular faces appearing in their place. A uniform cantellation is half way between both the parent and dual forms.<br/>[[File:Cube cantellation sequence.svg|400px]]
| |
| |-
| |
| ! Bicantellated
| |
| | t<sub>1,3</sub>{p,q,r,s,t}
| |
| |{{CDD|node|p|node_1|q|node|r|node_1|s|node|t|node}}
| |
| |In addition to vertex truncation, each original edge is ''beveled'' with new rectangular faces appearing in their place. A uniform cantellation is half way between both the parent and dual forms.
| |
| |-
| |
| ! [[Runcination (geometry)|Runcinated]]
| |
| | t<sub>0,3</sub>{p,q,r,s,t}
| |
| |{{CDD|node_1|p|node|q|node|r|node_1|s|node|t|node}}
| |
| |Runcination reduces cells and creates new cells at the vertices and edges.
| |
| |-
| |
| ! Biruncinated
| |
| | t<sub>1,4</sub>{p,q,r,s,t}
| |
| |{{CDD|node|p|node_1|q|node|r|node|s|node_1|t|node}}
| |
| |Runcination reduces cells and creates new cells at the vertices and edges.
| |
| |-
| |
| ! [[Sterication (geometry)|Stericated]]
| |
| | t<sub>0,4</sub>{p,q,r,s,t}
| |
| |{{CDD|node_1|p|node|q|node|r|node|s|node_1|t|node}}
| |
| |Sterication reduces 4-faces and creates new 4-faces at the vertices, edges, and faces in the gaps.
| |
| |-
| |
| ! Pentellated
| |
| | t<sub>0,5</sub>{p,q,r,s,t}
| |
| |{{CDD|node_1|p|node|q|node|r|node|s|node|t|node_1}}
| |
| |Pentellation reduces 5-faces and creates new 5-faces at the vertices, edges, faces, and cells in the gaps. ([[Expansion (geometry)|expansion]] operation for polypetons)
| |
| |-
| |
| ![[Omnitruncation (geometry)|Omnitruncated]]
| |
| | t<sub>0,1,2,3,4,5</sub>{p,q,r,s,t}
| |
| |{{CDD|node_1|p|node_1|q|node_1|r|node_1|s|node_1|t|node_1}}
| |
| |All five operators, truncation, cantellation, runcination, sterication, and pentellation are applied.
| |
| |}
| |
| | |
| == See also ==
| |
| * [[List of regular polytopes#Higher dimensions]]
| |
| | |
| == Notes ==
| |
| {{reflist}}
| |
| | |
| == References ==
| |
| * [[Thorold Gosset|T. Gosset]]: ''On the Regular and Semi-Regular Figures in Space of n Dimensions'', [[Messenger of Mathematics]], Macmillan, 1900
| |
| * [[Alicia Boole Stott|A. Boole Stott]]: ''Geometrical deduction of semiregular from regular polytopes and space fillings'', Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910
| |
| * [[Harold Scott MacDonald Coxeter|H.S.M. Coxeter]]:
| |
| ** H.S.M. Coxeter, M.S. Longuet-Higgins und J.C.P. Miller: ''Uniform Polyhedra'', Philosophical Transactions of the Royal Society of London, Londne, 1954
| |
| ** H.S.M. Coxeter, ''Regular Polytopes'', 3rd Edition, Dover New York, 1973
| |
| * '''Kaleidoscopes: Selected Writings of H.S.M. Coxeter''', editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html 978-0-471-01003-6]
| |
| ** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380-407, MR 2,10]
| |
| ** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591]
| |
| ** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45]
| |
| * [[Norman Johnson (mathematician)|N.W. Johnson]]: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. Dissertation, University of Toronto, 1966
| |
| * {{KlitzingPolytopes|polypeta.htm|6D|uniform polytopes (polypeta)}}
| |
| * {{KlitzingPolytopes|../explain/polytope-tree.htm#dynkin|Uniform polytopes|truncation operators}}
| |
| | |
| == External links ==
| |
| * [http://www.steelpillow.com/polyhedra/ditela.html Polytope names]
| |
| * [http://www.polytope.net/hedrondude/topes.htm Polytopes of Various Dimensions], Jonathan Bowers
| |
| * [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]
| |
| * {{PolyCell | urlname = glossary.html| title = Glossary for hyperspace}}
| |
| | |
| {{Polytopes}}
| |
| {{Honeycombs}}
| |
| | |
| [[Category:6-polytopes]]
| |