View source for 1 22 polytope

{{Short description|Uniform 6-polytope}}
{{DISPLAYTITLE:1<sub>&thinsp;22</sub> polytope}}
{| class=wikitable width=560 align=right style="margin-left:0.5em;"
|- align=center valign=top
|[[File:Up 1 22 t0 E6.svg|160px]]<br>1<sub>22</sub><br>{{CDD|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea}}
|[[File:Up 1 22 t1 E6.svg|160px]]<br>Rectified 1<sub>22</sub><br>{{CDD|nodea|3a|nodea|3a|branch_10|3a|nodea|3a|nodea}}
|rowspan=2 valign=center|[[File:Up 1 22 t2 E6.svg|240px]]<br>Birectified 1<sub>22</sub><br>{{CDD|nodea|3a|nodea_1|3a|branch|3a|nodea_1|3a|nodea}}
|- align=center valign=top
|[[File:Up 2 21 t0 E6.svg|160px]]<br>[[2 21 polytope|2<sub>21</sub>]]<br>{{CDD|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea}}
|[[File:Up 2 21 t1 E6.svg|160px]]<br>[[Rectified 2 21 polytope|Rectified 2<sub>21</sub>]]<br>{{CDD|nodea|3a|nodea_1|3a|branch|3a|nodea|3a|nodea}}
|- valign=top
!colspan=3|[[orthogonal projection]]s in E<sub>6</sub> [[Coxeter plane]]
|}
In 6-dimensional [[geometry]], the '''1<sub>22</sub>''' polytope is a [[uniform polytope]], constructed from the [[E6 (mathematics)|E<sub>6</sub>]] group. It was first published in [[E. L. Elte]]'s 1912 listing of semiregular polytopes, named as V<sub>72</sub> (for its 72 vertices).<ref>Elte, 1912</ref>

Its [[Coxeter symbol]] is '''1<sub>22</sub>''', describing its bifurcating [[Coxeter-Dynkin diagram]], with a single ring on the end of the 1-node sequence. There are two rectifications of the 1<sub>22</sub>, constructed by positions points on the elements of 1<sub>22</sub>. The '''rectified 1<sub>22</sub>''' is constructed by points at the mid-edges of the '''1<sub>22</sub>'''. The '''birectified 1<sub>22</sub>''' is constructed by points at the triangle face centers of the '''1<sub>22</sub>'''.

These polytopes are from a family of 39 convex [[uniform 6-polytope|uniform polytopes in 6-dimensions]], made of [[uniform 5-polytope|uniform polytope]] facets and [[vertex figure]]s, defined by all permutations of rings in this [[Coxeter-Dynkin diagram]]: {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea}}.

== 1<sub>22</sub> polytope ==

{| class="wikitable" align="right" style="margin-left:10px" width="280"
!bgcolor=#e7dcc3 colspan=2|1<sub>22</sub> polytope
|-
|bgcolor=#e7dcc3|Type||[[Uniform 6-polytope]]
|-
|bgcolor=#e7dcc3|Family||[[Uniform 1 k2 polytope|1<sub>k2</sub> polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| {3,3<sup>2,2</sup>}
|-
|bgcolor=#e7dcc3|Coxeter symbol|| 1<sub>22</sub>
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]||{{CDD|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea}} or {{CDD|node_1|3|node|split1|nodes|3ab|nodes}}
|-
|bgcolor=#e7dcc3|5-faces||54:<br>27 [[demipenteract|1<sub>21</sub>]] [[File:Demipenteract graph ortho.svg|30px]]<br>27 [[demipenteract|1<sub>21</sub>]] [[File:Demipenteract graph ortho.svg|30px]]
|-
|bgcolor=#e7dcc3|4-faces||702:<br>270 [[16-cell|1<sub>11</sub>]] [[File:Cross graph 4.svg|25px]]<br>432 [[pentachoron|1<sub>20</sub>]] [[File:4-simplex t0.svg|25px]]
|-
|bgcolor=#e7dcc3|Cells||2160:<br>1080 [[tetrahedron|1<sub>10</sub>]] [[File:3-simplex t0.svg|25px]]<br>1080 [[tetrahedron|{3,3}]] [[File:3-simplex t0.svg|25px]]
|-
|bgcolor=#e7dcc3|Faces||2160 [[triangle|{3}]] [[File:2-simplex t0.svg|25px]]
|-
|bgcolor=#e7dcc3|Edges||720
|-
|bgcolor=#e7dcc3|Vertices||72
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[Birectified 5-simplex]]:<br>0<sub>22</sub> [[File:5-simplex t2.svg|50px]]
|-
|bgcolor=#e7dcc3|[[Petrie polygon]]||[[Dodecagon]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||[[E6 (mathematics)|E<sub>6</sub>]], [[3,3<sup>2,2</sup>]], order 103680
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]], [[Isotopic (geometry)|isotopic]]
|}
The 1<sub>22</sub> polytope contains 72 vertices, and 54 [[5-demicube|5-demicubic]] facets. It has a [[birectified 5-simplex]] [[vertex figure]]. Its 72 vertices represent the root vectors of the [[simple Lie group]] [[E6 (mathematics)|E<sub>6</sub>]].

=== Alternate names ===
* '''Pentacontatetra-peton''' (Acronym Mo) - 54-facetted polypeton (Jonathan Bowers)<ref>Klitzing, (o3o3o3o3o *c3x - [http://bendwavy.org/klitzing/incmats/mo.htm mo])</ref>

=== Images ===

{| class=wikitable width=480
|+ [[Coxeter plane]] [[orthographic projection]]s
|-
<!--!2×E6<br>[18]-->
!E6<br>[12]
!D5<br>[8]
!D4 / A2<br>[6]
|- valign=top align=center
<!--|[[File:Polytope 1 22.png|120px]]<br>(1)-->
|[[File:up 1 22 t0 E6.svg|120px]]<br>(1,2)
|[[File:up 1 22 t0 D5.svg|120px]]<br>(1,3)
|[[File:up 1 22 t0 D4.svg|120px]]<br>(1,9,12)
|-
!B6<br>[12/2]
!A5<br>[6]
!A4<br>{{brackets|5}} = [10]
!A3 / D3<br>[4]
|- valign=top align=center
|[[File:up 1 22 t0 B6.svg|120px]]<br>(1,2)
|[[File:up 1 22 t0 A5.svg|120px]]<br>(2,3,6)
|[[File:up 1 22 t0 A4.svg|120px]]<br>(1,2)
|[[File:up 1 22 t0 D3.svg|120px]]<br>(1,6,8,12)
|}

=== Construction ===

It is created by a [[Wythoff construction]] upon a set of 6 [[hyperplane]] mirrors in 6-dimensional space.

The facet information can be extracted from its [[Coxeter-Dynkin diagram]], {{CDD|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea}}.

Removing the node on either of 2-length branches leaves the [[5-demicube]], 1<sub>31</sub>, {{CDD|nodea|3a|branch_01lr|3a|nodea|3a|nodea}}.

The [[vertex figure]] is determined by removing the ringed node and ringing the neighboring node. This makes the [[birectified 5-simplex]], 0<sub>22</sub>, {{CDD|node|3|node|3|node_1|3|node|3|node}}.

Seen in a [[configuration (polytope)|configuration matrix]], the element counts can be derived by mirror removal and ratios of [[Coxeter group#Properties|Coxeter group]] orders.<ref name=:0>Coxeter, Regular Polytopes, 11.8 Gossett figures in  six, seven, and eight dimensions, p. 202-203</ref>
{| class=wikitable
!E<sub>6</sub>||width=60|{{CDD|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea}}||k-face|| f<sub>k</sub> || f<sub>0</sub> || f<sub>1</sub>||f<sub>2</sub>||colspan=2|f<sub>3</sub>||colspan=3|f<sub>4</sub>||colspan=2|f<sub>5</sub>|| ''k''-figure|| notes
|- align=right
|A<sub>5</sub> ||{{CDD|nodea|3a|nodea|3a|nodes_0x|3a|nodea|3a|nodea}}|| ( )
! f<sub>0</sub>
|BGCOLOR="#ffe0ff"|'''72'''||20||90||60||60||15||15||30||6||6||[[Birectified 5-cell|r{3,3,3}]]  || E<sub>6</sub>/A<sub>5</sub> = 72*6!/6! = 72
|- align=right
|A<sub>2</sub>A<sub>2</sub>A<sub>1</sub> ||{{CDD|nodea|3a|nodea|2|nodes_x1|2|nodea|3a|nodea}}|| { }
! f<sub>1</sub>
||2||BGCOLOR="#ffe0e0"|'''720'''||9||9||9||3||3||9||3||3||[[3-3 duoprism|{3}×{3}]] || E<sub>6</sub>/A<sub>2</sub>A<sub>2</sub>A<sub>1</sub> = 72*6!/3!/3!/2 = 720
|- align=right
|A<sub>2</sub>A<sub>1</sub>A<sub>1</sub> ||{{CDD|nodea|2|nodea_x|2|branch_01|2|nodea_x|2|nodea}}|| [[triangle|{3}]]
! f<sub>2</sub>
||3||3||BGCOLOR="#ffffe0"|'''2160'''||2||2||1||1||4||2||2||[[Tetragonal disphenoid|s{2,4}]] || E<sub>6</sub>/A<sub>2</sub>A<sub>1</sub>A<sub>1</sub> = 72*6!/3!/2/2 = 2160
|- align=right
|rowspan=2|A<sub>3</sub>A<sub>1</sub> ||{{CDD|nodea_x|2|nodea|3a|branch_01l|2|nodea_x|2|nodea}}|| rowspan=2|[[tetrahedron|{3,3}]]
! rowspan=2|f<sub>3</sub>
||4||6||4||BGCOLOR="#e0ffe0"|'''1080'''||BGCOLOR="#e0ffe0"|*||1||0||2||2||1||rowspan=2|[[Isosceles triangle|{ }∨( )]] ||rowspan=2| E<sub>6</sub>/A<sub>3</sub>A<sub>1</sub> = 72*6!/4!/2 = 1080
|- align=right
|{{CDD|nodea|2|nodea_x|2|branch_01r|3a|nodea|2|nodea_x}}
||4||6||4||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|'''1080'''||0||1||2||1||2
|- align=right
|rowspan=2|A<sub>4</sub>A<sub>1</sub> ||{{CDD|nodea|2|nodea_x|2|branch_01r|3a|nodea|3a|nodea}}||rowspan=2| [[5-cell|{3,3,3}]]
!rowspan=3|f<sub>4</sub>
||5||10||10||5||0||BGCOLOR="#e0ffff"|'''216'''||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|*||2||0||rowspan=3|{ } ||rowspan=2| E<sub>6</sub>/A<sub>4</sub>A<sub>1</sub> = 72*6!/5!/2 = 216
|- align=right
|{{CDD|nodea|3a|nodea|3a|branch_01l|2|nodea_x|2|nodea}}
||5||10||10||0||5||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|'''216'''||BGCOLOR="#e0ffff"|*||0||2
|- align=right
|D<sub>4</sub> ||{{CDD|nodea_x|2|nodea|3a|branch_01lr|3a|nodea|2|nodea_x}}|| [[16-cell|h{4,3,3}]]
||8||24||32||8||8||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|'''270'''||1||1 || E<sub>6</sub>/D<sub>4</sub> = 72*6!/8/4! = 270
|- align=right
|rowspan=2|D<sub>5</sub> ||{{CDD|nodea|3a|nodea|3a|branch_01lr|3a|nodea|2|nodea_x}}|| rowspan=2|[[5-demicube|h{4,3,3,3}]]
!rowspan=2|f<sub>5</sub>
||16||80||160||80||40||16||0||10||BGCOLOR="#e0e0ff"|'''27'''||BGCOLOR="#e0e0ff"|*||rowspan=2|( ) ||rowspan=2| E<sub>6</sub>/D<sub>5</sub> = 72*6!/16/5! = 27
|- align=right
|{{CDD|nodea_x|2|nodea|3a|branch_01lr|3a|nodea|3a|nodea}}|| 16||80||160||40||80||0||16||10||BGCOLOR="#e0e0ff"|*||BGCOLOR="#e0e0ff"|'''27'''
|}

=== Related complex polyhedron ===
[[File:Complex polyhedron 3-3-3-4-2.png|thumb|Orthographic projection in Aut(E6) Coxeter plane with 18-gonal symmetry for complex polyhedron, <sub>3</sub>{3}<sub>3</sub>{4}<sub>2</sub>. It has 72 vertices, 216 3-edges, and 54 3{3}3 faces.]]
The [[regular complex polytope|regular complex polyhedron]] <sub>3</sub>{3}<sub>3</sub>{4}<sub>2</sub>, {{CDD|3node_1|3|3node|4|node}}, in <math>\mathbb{C}^2</math> has a real representation as the ''1<sub>22</sub>'' polytope in 4-dimensional space. It has 72 vertices, 216 3-edges, and 54 3{3}3 faces. Its [[complex reflection group]] is <sub>3</sub>[3]<sub>3</sub>[4]<sub>2</sub>, order 1296. It has a half-symmetry quasiregular construction as {{CDD|3node|3|3node_1|3|3node}}, as a [[rectification (geometry)|rectification]] of the [[Hessian polyhedron]], {{CDD|3node_1|3|3node|3|3node}}.<ref> Coxeter, H. S. M., ''Regular Complex Polytopes'', second edition, Cambridge University Press, (1991). p.30 and p.47</ref>
=== Related polytopes and honeycomb ===
Along with the semiregular polytope, '''[[Gosset 2 21 polytope|2<sub>21</sub>]]''', it is also one of a family of 39 convex [[uniform polytope]]s in 6-dimensions, made of [[uniform polytope]] facets and [[vertex figure]]s, defined by all permutations of rings in this [[Coxeter-Dynkin diagram]]: {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea}}.

{{1 k2 polytopes}}

==== Geometric folding ====

The '''1<sub>22</sub>''' is related to the [[24-cell]] by a geometric [[Folding (Dynkin diagram)|folding]] E6 → F4 of [[Coxeter-Dynkin diagram]]s, E6 corresponding to '''1<sub>22</sub>''' in 6 dimensions, F4 to the 24-cell in 4 dimensions. This can be seen in the [[Coxeter plane]] projections. The 24 vertices of the 24-cell are projected in the same two rings as seen in the 1<sub>22</sub>.
{| class=wikitable
|-
!colspan=2|E6/F4 Coxeter planes
|- align=center valign=top
|[[File:Up 1 22 t0 E6.svg|160px]]<br>'''1<sub>22</sub>'''
|[[File:24-cell t3 F4.svg|160px]]<br>24-cell
|-
!colspan=2|D4/B4 Coxeter planes
|- align=center valign=top
|[[File:up 1 22 t0 D4.svg|160px]]<br>'''1<sub>22</sub>'''
|[[File:24-cell t3 B3.svg|160px]]<br>24-cell
|}

==== Tessellations ====
This polytope is the [[vertex figure]] for a [[Uniform honeycomb|uniform tessellation]] of 6-dimensional space, '''[[2 22 honeycomb|2<sub>22</sub>]]''', {{CDD|node_1|3|node|3|node|split1|nodes|3ab|nodes}}.

== Rectified 1<sub>22</sub> polytope ==
{| class="wikitable" align="right" style="margin-left:10px" width="280"
!bgcolor=#e7dcc3 colspan=2|Rectified 1<sub>22</sub>
|-
|bgcolor=#e7dcc3|Type||[[Uniform 6-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| 2r{3,3,3<sup>2,1</sup>}<br>r{3,3<sup>2,2</sup>}
|-
|bgcolor=#e7dcc3|[[Coxeter symbol]]|| 0<sub>221</sub>
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]||{{CDD|nodea|3a|nodea|3a|branch_10|3a|nodea|3a|nodea}}<br>or {{CDD|node|3|node_1|split1|nodes|3ab|nodes}}
|-
|bgcolor=#e7dcc3|5-faces||126
|-
|bgcolor=#e7dcc3|4-faces||1566
|-
|bgcolor=#e7dcc3|Cells||6480
|-
|bgcolor=#e7dcc3|Faces||6480
|-
|bgcolor=#e7dcc3|Edges||6480
|-
|bgcolor=#e7dcc3|Vertices||720
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||3-3 duoprism prism
|-
|bgcolor=#e7dcc3|[[Petrie polygon]]||[[Dodecagon]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||[[E6 (mathematics)|E<sub>6</sub>]], [[3,3<sup>2,2</sup>]], order 103680
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}
The '''rectified 1<sub>22</sub>''' polytope (also called '''0<sub>221</sub>''') can tessellate 6-dimensional space as the [[Voronoi cell]] of the [[E6 honeycomb|E6* honeycomb]] lattice (dual of E6 lattice).<ref>[http://home.digital.net/~pervin/publications/vermont.html The Voronoi Cells of the E6* and E7* Lattices] {{Webarchive|url=https://web.archive.org/web/20160130193811/http://home.digital.net/~pervin/publications/vermont.html |date=2016-01-30 }}, Edward Pervin</ref>

=== Alternate names ===
* Birectified 2<sub>21</sub> polytope
* Rectified pentacontatetrapeton (acronym ''Ram'') - rectified 54-facetted polypeton (Jonathan Bowers)<ref>Klitzing, (o3o3x3o3o *c3o - [http://bendwavy.org/klitzing/incmats/ram.htm ram])</ref>

=== Images ===

Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow.

{| class=wikitable width=480
|+ [[Coxeter plane]] [[orthographic projection]]s
|-
!E6<br>[12]
!D5<br>[8]
!D4 / A2<br>[6]
!B6<br>[12/2]
|- valign=top align=center
|[[File:up 1 22 t1 E6.svg|120px]]
|[[File:up 1 22 t1 D5.svg|120px]]
|[[File:up 1 22 t1 D4.svg|120px]]
|[[File:up 1 22 t1 B6.svg|120px]]
|-
!A5<br>[6]
!A4<br>[5]
!A3 / D3<br>[4]
|- valign=top align=center
|[[File:up 1 22 t1 A5.svg|120px]]
|[[File:up 1 22 t1 A4.svg|120px]]
|[[File:up 1 22 t1 D3.svg|120px]]
|}

=== Construction ===

Its construction is based on the [[E6 (mathematics)|E<sub>6</sub>]] group and information can be extracted from the ringed [[Coxeter-Dynkin diagram]] representing this polytope: {{CDD|nodea|3a|nodea|3a|branch_10|3a|nodea|3a|nodea}}.

Removing the ring on the short branch leaves the [[birectified 5-simplex]], {{CDD|nodea|3a|nodea|3a|nodea_1|3a|nodea|3a|nodea}}.

Removing the ring on the either 2-length branch leaves the [[birectified 5-orthoplex]] in its alternated form: '''t<sub>2</sub>(2<sub>11</sub>)''', {{CDD|nodea|3a|branch_10|3a|nodea|3a|nodea}}.

The [[vertex figure]] is determined by removing the ringed node and ringing the neighboring ring. This makes [[3-3 duoprism prism]], {3}×{3}×{}, {{CDD|node|3|node_1|2|node_1|2|node_1|3|node}}.

Seen in a [[configuration (polytope)|configuration matrix]], the element counts can be derived by mirror removal and ratios of [[Coxeter group#Properties|Coxeter group]] orders.<ref name=":0" /><ref>{{KlitzingPolytopes|polypeta.htm|6D convex uniform polypeta|o3o3x3o3o *c3o - ram}}</ref>
{| class=wikitable
!E<sub>6</sub>||{{CDD|nodea|3a|nodea|3a|branch_10|3a|nodea|3a|nodea}}||k-face|| f<sub>k</sub> || f<sub>0</sub> || f<sub>1</sub>||colspan=3|f<sub>2</sub>||colspan=5|f<sub>3</sub>||colspan=5|f<sub>4</sub>||colspan=3|f<sub>5</sub>|| ''k''-figure|| notes
|- align=right
|A<sub>2</sub>A<sub>2</sub>A<sub>1</sub> ||{{CDD|nodea|3a|nodea|2|nodes_x0|2|nodea|3a|nodea}}|| ( )
!f<sub>0</sub>
|BGCOLOR="#ffe0ff"|'''720'''||18||18||18||9||6||18||9||6||9||6||3||6||9||3||2||3||3||[[3-3 duoprism prism|{3}×{3}×{ }]] || E<sub>6</sub>/A<sub>2</sub>A<sub>2</sub>A<sub>1</sub> = 72*6!/3!/3!/2 = 720
|- align=right
|A<sub>1</sub>A<sub>1</sub>A<sub>1</sub> ||{{CDD|nodea|2|nodea_x|2|nodes_1x|2|nodea_x|2|nodea}}|| { }
!f<sub>1</sub>
||2||BGCOLOR="#ffe0e0"|'''6480'''||2||2||1||1||4||2||1||2||2||1||2||4||1||1||2||2||[[Digonal disphenoid pyramid|{ }∨{ }∨( )]] || E<sub>6</sub>/A<sub>1</sub>A<sub>1</sub>A<sub>1</sub> = 72*6!/2/2/2 = 6480
|- align=right
|rowspan=2|A<sub>2</sub>A<sub>1</sub> ||{{CDD|nodea_x|2|nodea|3a|nodes_1x|2|nodea_x|2|nodea}}|| rowspan=3|[[triangle|{3}]]
!rowspan=3|f<sub>2</sub>
||3||3||BGCOLOR="#ffffe0"|'''4320'''||BGCOLOR="#ffffe0"|*||BGCOLOR="#ffffe0"|*||1||2||1||0||0||2||1||1||2||0||1||2||1||rowspan=2|[[Mirrored sphenoid|Sphenoid]] || rowspan=2|E<sub>6</sub>/A<sub>2</sub>A<sub>1</sub> = 72*6!/3!/2 = 4320
|- align=right
||{{CDD|nodea|2|nodea_x|2|nodes_1x|3a|nodea|2|nodea_x}}
||3||3||BGCOLOR="#ffffe0"|*||BGCOLOR="#ffffe0"|'''4320'''||BGCOLOR="#ffffe0"|*||0||2||0||1||1||1||0||2||2||1||1||1||2
|- align=right
|A<sub>2</sub>A<sub>1</sub>A<sub>1</sub> ||{{CDD|nodea|2|nodea_x|2|branch_10|2|nodea_x|2|nodea}}
||3||3||BGCOLOR="#ffffe0"|*||BGCOLOR="#ffffe0"|*||BGCOLOR="#ffffe0"|'''2160'''||0||0||2||0||2||0||1||0||4||1||0||2||2||[[Digonal disphenoid|{ }∨{ }]] || E<sub>6</sub>/A<sub>2</sub>A<sub>1</sub>A<sub>1</sub> = 72*6!/3!/2/2 = 2160
|- align=right
|A<sub>2</sub>A<sub>1</sub> ||{{CDD|nodea|3a|nodea|3a|nodes_1x|2|nodea_x|2|nodea}}||[[tetrahedron|{3,3}]]
!rowspan=5|f<sub>3</sub>
||4||6||4||0||0||BGCOLOR="#e0ffe0"|'''1080'''||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|*||2||1||0||0||0||1||2||0||[[Isosceles triangle|{ }∨( )]] || E<sub>6</sub>/A<sub>2</sub>A<sub>1</sub> = 72*6!/3!/2 = 1080
|- align=right
|A<sub>3</sub> ||{{CDD|nodea_x|2|nodea|3a|nodes_1x|3a|nodea|2|nodea_x}}||rowspan=2|[[octahedron|r{3,3}]]
||6||12||4||4||0||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|'''2160'''||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|*||1||0||1||1||0||1||1||1||[[triangle|{3}]] || E<sub>6</sub>/A<sub>3</sub> = 72*6!/4! = 2160
|- align=right
|rowspan=3|A<sub>3</sub>A<sub>1</sub> ||{{CDD|nodea_x|2|nodea|3a|branch_10|2|nodea_x|2|nodea}}
||6||12||4||0||4||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|'''1080'''||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|*||0||1||0||2||0||0||2||1||rowspan=3|[[Isosceles triangle|{ }∨( )]] || rowspan=3|E<sub>6</sub>/A<sub>3</sub>A<sub>1</sub> = 72*6!/4!/2 = 1080
|- align=right
||{{CDD|nodea|2|nodea_x|2|nodes_1x|3a|nodea|3a|nodea}}|| [[tetrahedron|{3,3}]]
||4||6||0||4||0||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|'''1080'''||BGCOLOR="#e0ffe0"|*||0||0||2||0||1||1||0||2
|- align=right
||{{CDD|nodea|2|nodea_x|2|branch_10|3a|nodea|2|nodea_x}}|| [[octahedron|r{3,3}]]
||6||12||0||4||4||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|'''1080'''||0||0||0||2||1||0||1||2
|- align=right
|A<sub>4</sub> ||{{CDD|nodea|3a|nodea|3a|nodes_1x|3a|nodea|2|nodea_x}}|| rowspan=3| [[rectified 5-cell|r{3,3,3}]]
!rowspan=5|f<sub>4</sub>
||10||30||20||10||0||5||5||0||0||0||BGCOLOR="#e0ffff"|'''432'''||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|*||1||1||0||rowspan=5|{ } || E<sub>6</sub>/A<sub>4</sub> = 72*6!/5! = 432
|- align=right
|A<sub>4</sub>A<sub>1</sub> ||{{CDD|nodea|3a|nodea|3a|branch_10|2|nodea_x|2|nodea}}
||10||30||20||0||10||5||0||5||0||0||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|'''216'''||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|*||0||2||0|| E<sub>6</sub>/A<sub>4</sub>A<sub>1</sub> = 72*6!/5!/2 = 216
|- align=right
|A<sub>4</sub> ||{{CDD|nodea_x|2|nodea|3a|nodes_1x|3a|nodea|3a|nodea}}
||10||30||10||20||0||0||5||0||5||0||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|'''432'''||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|*||1||0||1 || E<sub>6</sub>/A<sub>4</sub> = 72*6!/5! = 432
|- align=right
|D<sub>4</sub> ||{{CDD|nodea_x|2|nodea|3a|branch_10|3a|nodea|2|nodea_x}}|| [[24-cell|{3,4,3}]]
||24||96||32||32||32||0||8||8||0||8||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|'''270'''||BGCOLOR="#e0ffff"|*||0||1||1|| E<sub>6</sub>/D<sub>4</sub> = 72*6!/8/4! = 270
|- align=right
|A<sub>4</sub>A<sub>1</sub> ||{{CDD|nodea|2|nodea_x|2|branch_10|3a|nodea|3a|nodea}}|| [[rectified 5-cell|r{3,3,3}]]
||10||30||0||20||10||0||0||0||5||5||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|'''216'''||0||0||2 || E<sub>6</sub>/A<sub>4</sub>A<sub>1</sub> = 72*6!/5!/2 = 216
|- align=right
|A<sub>5</sub> ||{{CDD|nodea|3a|nodea|3a|nodes_1x|3a|nodea|3a|nodea}}|| [[birectified 5-simplex|2r{3,3,3,3}]]
!rowspan=3|f<sub>5</sub>
||20||90||60||60||0||15||30||0||15||0||6||0||6||0||0||BGCOLOR="#e0e0ff"|'''72'''||BGCOLOR="#e0e0ff"|*||BGCOLOR="#e0e0ff"|*||rowspan=3|( ) || E<sub>6</sub>/A<sub>5</sub> = 72*6!/6! = 72
|- align=right
|rowspan=2|D<sub>5</sub> ||{{CDD|nodea|3a|nodea|3a|branch_10|3a|nodea|2|nodea_x}}|| rowspan=2|[[Birectified 5-cube|2r{4,3,3,3}]]
||80||480||320||160||160||80||80||80||0||40||16||16||0||10||0||BGCOLOR="#e0e0ff"|*||BGCOLOR="#e0e0ff"|'''27'''||BGCOLOR="#e0e0ff"|* || rowspan=2|E<sub>6</sub>/D<sub>5</sub> = 72*6!/16/5! = 27
|- align=right
||{{CDD|nodea_x|2|nodea|3a|branch_10|3a|nodea|3a|nodea}}
||80||480||160||320||160||0||80||40||80||80||0||0||16||10||16||BGCOLOR="#e0e0ff"|*||BGCOLOR="#e0e0ff"|*||BGCOLOR="#e0e0ff"|'''27'''
|}

== Truncated 1<sub>22</sub> polytope ==
{| class="wikitable" align="right" style="margin-left:10px" width="280"
!bgcolor=#e7dcc3 colspan=2|Truncated 1<sub>22</sub>
|-
|bgcolor=#e7dcc3|Type||[[Uniform 6-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| t{3,3<sup>2,2</sup>}
|-
|bgcolor=#e7dcc3|[[Coxeter symbol]]|| t(1<sub>22</sub>)
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]||{{CDD|nodea|3a|nodea|3a|branch_11|3a|nodea|3a|nodea}}<br>or {{CDD|node_1|3|node_1|split1|nodes|3ab|nodes}}
|-
|bgcolor=#e7dcc3|5-faces||72+27+27
|-
|bgcolor=#e7dcc3|4-faces||32+216+432+270+216
|-
|bgcolor=#e7dcc3|Cells||1080+2160+1080+1080+1080
|-
|bgcolor=#e7dcc3|Faces||4320+4320+2160
|-
|bgcolor=#e7dcc3|Edges||6480+720
|-
|bgcolor=#e7dcc3|Vertices||1440
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||( )v{3}x{3}
|-
|bgcolor=#e7dcc3|[[Petrie polygon]]||[[Dodecagon]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||[[E6 (mathematics)|E<sub>6</sub>]], [[3,3<sup>2,2</sup>]], order 103680
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}

=== Alternate names ===
* Truncated 1<sub>22</sub> polytope

=== Construction ===

Its construction is based on the [[E6 (mathematics)|E<sub>6</sub>]] group and information can be extracted from the ringed [[Coxeter-Dynkin diagram]] representing this polytope: {{CDD|nodea|3a|nodea|3a|branch_11|3a|nodea|3a|nodea}}.

=== Images ===

Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow.

{| class=wikitable width=480
|+ [[Coxeter plane]] [[orthographic projection]]s
|-
!E6<br>[12]
!D5<br>[8]
!D4 / A2<br>[6]
!B6<br>[12/2]
|- valign=top align=center
|[[File:up 1 22 t01 E6.svg|120px]]
|[[File:up 1 22 t01 D5.svg|120px]]
|[[File:up 1 22 t01 D4.svg|120px]]
|[[File:up 1 22 t01 B6.svg|120px]]
|-
!A5<br>[6]
!A4<br>[5]
!A3 / D3<br>[4]
|- valign=top align=center
|[[File:up 1 22 t01 A5.svg|120px]]
|[[File:up 1 22 t01 A4.svg|120px]]
|[[File:up 1 22 t01 D3.svg|120px]]
|}

== Birectified 1<sub>22</sub> polytope ==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Birectified 1<sub>22</sub> polytope
|-
|bgcolor=#e7dcc3|Type||[[Uniform 6-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| 2r{3,3<sup>2,2</sup>}
|-
|bgcolor=#e7dcc3|Coxeter symbol|| 2r(1<sub>22</sub>)
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]||{{CDD|nodea|3a|nodea_1|3a|branch|3a|nodea_1|3a|nodea}}<br>or {{CDD|node|3|node|split1|nodes_11|3ab|nodes}}
|-
|bgcolor=#e7dcc3|5-faces||126
|-
|bgcolor=#e7dcc3|4-faces||2286
|-
|bgcolor=#e7dcc3|Cells||10800
|-
|bgcolor=#e7dcc3|Faces||19440
|-
|bgcolor=#e7dcc3|Edges||12960
|-
|bgcolor=#e7dcc3|Vertices||2160
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||[[E6 (mathematics)|E<sub>6</sub>]], [[3,3<sup>2,2</sup>]], order 103680
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}

=== Alternate names ===
* Bicantellated 2<sub>21</sub>
* Birectified pentacontitetrapeton (barm) (Jonathan Bowers)<ref>Klitzing, (o3x3o3x3o *c3o - [http://bendwavy.org/klitzing/incmats/scram.htm barm])</ref>

=== Images ===
Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow.

{| class=wikitable width=480
|+ [[Coxeter plane]] [[orthographic projection]]s
|-
!E6<br>[12]
!D5<br>[8]
!D4 / A2<br>[6]
!B6<br>[12/2]
|- valign=top align=center
|[[File:up 1 22 t2 E6.svg|120px]]
|[[File:up 1 22 t2 D5.svg|120px]]
|[[File:up 1 22 t2 D4.svg|120px]]
|[[File:up 1 22 t2 B6.svg|120px]]
|-
!A5<br>[6]
!A4<br>[5]
!A3 / D3<br>[4]
|- valign=top align=center
|[[File:up 1 22 t2 A5.svg|120px]]
|[[File:up 1 22 t2 A4.svg|120px]]
|[[File:up 1 22 t2 D3.svg|120px]]
|}

== Trirectified 1<sub>22</sub> polytope ==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Trirectified 1<sub>22</sub> polytope
|-
|bgcolor=#e7dcc3|Type||[[Uniform 6-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| 3r{3,3<sup>2,2</sup>}
|-
|bgcolor=#e7dcc3|Coxeter symbol|| 3r(1<sub>22</sub>)
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]||{{CDD|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea_1}}<br>or {{CDD|node|3|node|split1|nodes|3ab|nodes_11}}
|-
|bgcolor=#e7dcc3|5-faces||558
|-
|bgcolor=#e7dcc3|4-faces||4608
|-
|bgcolor=#e7dcc3|Cells||8640
|-
|bgcolor=#e7dcc3|Faces||6480
|-
|bgcolor=#e7dcc3|Edges||2160
|-
|bgcolor=#e7dcc3|Vertices||270
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||[[E6 (mathematics)|E<sub>6</sub>]], [[3,3<sup>2,2</sup>]], order 103680
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}

=== Alternate names ===
* Tricantellated 2<sub>21</sub>
* Trirectified pentacontitetrapeton (trim or cacam) (Jonathan Bowers)<ref>Klitzing, (x3o3o3o3x *c3o - [http://bendwavy.org/klitzing/incmats/cacam.htm cacam]</ref>

<!--=== Images ===
Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow.

{| class=wikitable width=480
|+ [[Coxeter plane]] [[orthographic projection]]s
|-
!E6<br>[12]
!D5<br>[8]
!D4 / A2<br>[6]
!B6<br>[12/2]
|- valign=top align=center
|[[File:up 1 22 t3 E6.svg|120px]]
|[[File:up 1 22 t3 D5.svg|120px]]
|[[File:up 1 22 t3 D4.svg|120px]]
|[[File:up 1 22 t3 B6.svg|120px]]
|-
!A5<br>[6]
!A4<br>[5]
!A3 / D3<br>[4]
|- valign=top align=center
|[[File:up 1 22 t3 A5.svg|120px]]
|[[File:up 1 22 t3 A4.svg|120px]]
|[[File:up 1 22 t3 D3.svg|120px]]
|}-->

== See also ==
* [[List of E6 polytopes]]

== Notes ==
{{reflist}}

== References ==
* {{citation | last = Elte | first = E. L. | title = The Semiregular Polytopes of the Hyperspaces | publisher = University of Groningen | location = Groningen | year = 1912}}
* [[Harold Scott MacDonald Coxeter|H. S. M. Coxeter]], ''Regular Polytopes'', 3rd Edition, Dover New York, 1973
* '''Kaleidoscopes: Selected Writings of H.S.M. Coxeter''', edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, {{ISBN|978-0-471-01003-6}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45] See p334 (figure 3.6a) by Peter mcMullen: (12-gonal node-edge graph of 1<sub>22</sub>)
* {{KlitzingPolytopes|polypeta.htm|6D|uniform polytopes (polypeta)}} o3o3o3o3o *c3x - mo, o3o3x3o3o *c3o - ram, o3x3o3x3o *c3o - barm

{{Polytopes}}

[[Category:6-polytopes]]