View source for 2 21 polytope

{{DISPLAYTITLE:2<sub>&thinsp;21</sub> polytope}}
{| class=wikitable width=320 align=right style="margin-left:1em"
|- align=center valign=top
|[[File:Up 2 21 t0 E6.svg|160px]]<BR>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<sub>21</sub><BR>{{CDD|nodea|3a|nodea_1|3a|branch|3a|nodea|3a|nodea}}
|- align=center valign=top
|[[File:Up 1 22 t0 E6.svg|160px]]<BR>([[1 22 polytope|1<sub>22</sub>]])<BR>{{CDD|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea}}
|[[File:Up 2 21 t2 E6.svg|160px]]<BR>Birectified 2<sub>21</sub><BR>([[Rectified 1 22 polytope|Rectified 1<sub>22</sub>]])<BR>{{CDD|nodea|3a|nodea|3a|branch_10|3a|nodea|3a|nodea}}
|- valign=top
!colspan=3|[[orthogonal projection]]s in E<sub>6</sub> [[Coxeter plane]]
|}
In 6-dimensional [[geometry]], the '''2<sub>21</sub>''' polytope is a [[uniform 6-polytope]], constructed within the symmetry of the [[E6 (mathematics)|E<sub>6</sub>]] group. It was discovered by [[Thorold Gosset]], published in his 1900 paper. He called it an [[Semiregular polytope|6-ic semi-regular figure]].<ref>Gosset, 1900</ref> It is also called the [[Schläfli]] polytope.

Its [[Coxeter symbol]] is '''2<sub>21</sub>''', describing its bifurcating [[Coxeter-Dynkin diagram]], with a single ring on the end of one of the 2-node sequences.  He also studied<ref>{{cite journal | last=Coxeter | first=H.S.M. | author-link=Harold Scott MacDonald Coxeter | year=1940 | title=The Polytope 2<sub>21</sub> Whose Twenty-Seven Vertices Correspond to the Lines on the General Cubic Surface | journal=Amer. J. Math. | volume=62 | issue=1 | pages=457–486 | jstor=2371466 | doi=10.2307/2371466}}</ref> its connection with the 27 lines on the [[cubic surface]], which are naturally in correspondence with the vertices of 2<sub>21</sub>.

The '''rectified 2<sub>21</sub>''' is constructed by points at the mid-edges of the '''2<sub>21</sub>'''. The '''birectified 2<sub>21</sub>''' is constructed by points at the triangle face centers of the '''2<sub>21</sub>''', and is the same as the rectified 1<sub>22</sub>.

These polytopes are a part of family of 39 convex [[uniform 6-polytope|uniform polytopes in 6-dimensions]], made of [[uniform 5-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}}.

== 2_21 polytope==

{| class="wikitable" align="right" style="margin-left:10px" width="280"
!bgcolor=#e7dcc3 colspan=2|2<sub>21</sub> polytope
|-
|bgcolor=#e7dcc3|Type||[[Uniform 6-polytope]]
|-
|bgcolor=#e7dcc3|Family||[[Semiregular k 21 polytope|k<sub>21</sub> polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| {3,3,3<sup>2,1</sup>}
|-
|bgcolor=#e7dcc3|Coxeter symbol|| 2<sub>21</sub>
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]||{{CDD|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea}} or {{CDD|nodes_10r|3ab|nodes|split2|node|3|node}}
|-
|bgcolor=#e7dcc3|5-faces||99 total:<BR>27 [[pentacross|2<sub>11</sub>]][[File:5-orthoplex.svg|40px]]<BR>72 [[5-simplex|{3<sup>4</sup>}]][[File:5-simplex t0.svg|40px]]
|-
|bgcolor=#e7dcc3|4-faces||648:<BR>432 [[pentachoron|{3<sup>3</sup>}]][[File:4-simplex t0.svg|40px]]<BR>216 [[pentachoron|{3<sup>3</sup>}]][[File:4-simplex t0.svg|40px]]
|-
|bgcolor=#e7dcc3|Cells||1080 [[tetrahedron|{3,3}]][[File:3-simplex t0.svg|40px]]
|-
|bgcolor=#e7dcc3|Faces||720 [[triangle|{3}]][[File:2-simplex t0.svg|40px]]
|-
|bgcolor=#e7dcc3|Edges||216
|-
|bgcolor=#e7dcc3|Vertices||27
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||'''1<sub>21</sub>''' ([[5-demicube]])
|-
|bgcolor=#e7dcc3|[[Petrie polygon]]||[[Dodecagon]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||[[E6 (mathematics)|E<sub>6</sub>]], [3<sup>2,2,1</sup>], order 51840
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}
The '''2<sub>21</sub>''' has 27 vertices, and 99 facets: 27 [[5-orthoplex]]es and 72 [[5-simplex|5-simplices]]. Its [[vertex figure]] is a [[5-demicube]].

For visualization this 6-dimensional polytope is often displayed in a special skewed orthographic projection direction that fits its 27 vertices within a 12-gonal regular polygon (called a [[Petrie polygon]]). Its 216 edges are drawn between 2 rings of 12 vertices, and 3 vertices projected into the center. Higher elements (faces, cells, etc.) can also be extracted and drawn on this projection.

The [[Schläfli graph]] is the [[1-skeleton]] of this polytope.

=== Alternate names ===
*[[E. L. Elte]] named it V<sub>27</sub> (for its 27 vertices) in his 1912 listing of semiregular polytopes.<ref>Elte, 1912</ref>
* '''Icosihepta-heptacontidi-peton''' - 27-72 facetted polypeton (acronym jak) (Jonathan Bowers)<ref>Klitzing, (x3o3o3o3o *c3o - jak)</ref>

=== Coordinates ===

The 27 vertices can be expressed in 8-space as an edge-figure of the [[4 21 polytope|4<sub>21</sub>]] polytope:

 (-2, 0, 0, 0,-2, 0, 0, 0), 
 ( 0,-2, 0, 0,-2, 0, 0, 0), 
 ( 0, 0,-2, 0,-2, 0, 0, 0), 
 ( 0, 0, 0,-2,-2, 0, 0, 0), 
 ( 0, 0, 0, 0,-2, 0, 0,-2), 
 ( 0, 0, 0, 0, 0,-2,-2, 0)

 ( 2, 0, 0, 0,-2, 0, 0, 0), 
 ( 0, 2, 0, 0,-2, 0, 0, 0), 
 ( 0, 0, 2, 0,-2, 0, 0, 0), 
 ( 0, 0, 0, 2,-2, 0, 0, 0), 
 ( 0, 0, 0, 0,-2, 0, 0, 2)

 (-1,-1,-1,-1,-1,-1,-1,-1),
 (-1,-1,-1, 1,-1,-1,-1, 1), 
 (-1,-1, 1,-1,-1,-1,-1, 1), 
 (-1,-1, 1, 1,-1,-1,-1,-1), 
 (-1, 1,-1,-1,-1,-1,-1, 1), 
 (-1, 1,-1, 1,-1,-1,-1,-1), 
 (-1, 1, 1,-1,-1,-1,-1,-1), 
 ( 1,-1,-1,-1,-1,-1,-1, 1), 
 ( 1,-1, 1,-1,-1,-1,-1,-1), 
 ( 1,-1,-1, 1,-1,-1,-1,-1), 
 ( 1, 1,-1,-1,-1,-1,-1,-1), 
 (-1, 1, 1, 1,-1,-1,-1, 1),
 ( 1,-1, 1, 1,-1,-1,-1, 1),
 ( 1, 1,-1, 1,-1,-1,-1, 1),
 ( 1, 1, 1,-1,-1,-1,-1, 1),
 ( 1, 1, 1, 1,-1,-1,-1,-1)

===Construction===

Its construction is based on the [[E6 (mathematics)|E<sub>6</sub>]] group.

The facet information can be extracted from its [[Coxeter-Dynkin diagram]], {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea_1}}.

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

Removing the node on the end of the 2-length branch leaves the [[5-orthoplex]] in its alternated form: ('''2<sub>11</sub>'''), {{CDD|nodea|3a|branch|3a|nodea|3a|nodea_1}}.

Every simplex facet touches a 5-orthoplex facet, while alternate facets of the orthoplex touch either a simplex or another orthoplex.

The [[vertex figure]] is determined by removing the ringed node and ringing the neighboring node. This makes [[5-demicube]]  (1<sub>21</sub> polytope), {{CDD|nodea|3a|nodea|3a|branch|3a|nodea_1}}. The edge-figure is the vertex figure of the vertex figure, a [[rectified 5-cell]], (0<sub>21</sub> polytope), {{CDD|nodea|3a|nodea|3a|branch_10}}.

Seen in a [[configuration (polytope)|configuration matrix]], the element counts can be derived from the [[Coxeter group#Properties|Coxeter group]] orders.<ref>Coxeter, Regular Polytopes, 11.8 Gossett figures in  six, seven, and eight dimensions, p. 202-203</ref>
{| class=wikitable
!E<sub>6</sub>||{{CDD|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea}}||k-face|| f<sub>k</sub> || f<sub>0</sub> || f<sub>1</sub>||f<sub>2</sub>||f<sub>3</sub>||colspan=2|f<sub>4</sub>||colspan=2|f<sub>5</sub>|| ''k''-figure|| notes
|- align=right
|D<sub>5</sub>||{{CDD|nodea_x|2|nodea|3a|branch|3a|nodea|3a|nodea}}||( )
!f<sub>0</sub>
|BGCOLOR="#ffe0ff"|'''27'''||16||80||160||80||40||16||10||[[5-demicube|h{4,3,3,3}]] ||E<sub>6</sub>/D<sub>5</sub> = 51840/1920 = 27
|- align=right
|A<sub>4</sub>A<sub>1</sub>||{{CDD|nodea_1|2|nodea_x|2|branch|3a|nodea|3a|nodea}}||{ }
!f<sub>1</sub>
||2||BGCOLOR="#ffe0e0" |'''216'''||10||30||20||10||5||5||[[rectified 5-cell|r{3,3,3}]] ||E<sub>6</sub>/A<sub>4</sub>A<sub>1</sub> = 51840/120/2 = 216
|- align=right
|A<sub>2</sub>A<sub>2</sub>A<sub>1</sub>||{{CDD|nodea_1|3a|nodea|2|nodes_x0|2|nodea|3a|nodea}}||[[Triangle|{3}]]
!f<sub>2</sub>
||3||3||BGCOLOR="#ffffe0" |'''720'''||6||6||3||2||3||[[triangular prism|{3}x{ }]] ||E<sub>6</sub>/A<sub>2</sub>A<sub>2</sub>A<sub>1</sub> = 51840/6/6/2 = 720
|- align=right
|A<sub>3</sub>A<sub>1</sub>||{{CDD|nodea_1|3a|nodea|3a|nodes_0x|2|nodea_x|2|nodea}}||[[tetrahedron|{3,3}]]
!f<sub>3</sub>
||4||6||4||BGCOLOR="#e0ffe0" |'''1080'''||2||1||1||2||[[Isosceles triangle|{ }v( )]] ||E<sub>6</sub>/A<sub>3</sub>A<sub>1</sub> = 51840/24/2 = 1080
|- align=right
|A<sub>4</sub>||{{CDD|nodea_1|3a|nodea|3a|nodes_0x|3a|nodea|2|nodea_x}}||rowspan=2|[[5-cell|{3,3,3}]]
!rowspan=2|f<sub>4</sub>
||5||10||10||5||BGCOLOR="#e0ffff" |'''432'''||BGCOLOR="#e0ffff" |*||1||1||rowspan=2|{ } ||E<sub>6</sub>/A<sub>4</sub> = 51840/120 = 432
|- align=right
|A<sub>4</sub>A<sub>1</sub>||{{CDD|nodea_1|3a|nodea|3a|branch|2|nodea_x|2|nodea}}
||5||10||10||5||BGCOLOR="#e0ffff" |*||BGCOLOR="#e0ffff" |'''216'''||0||2 ||E<sub>6</sub>/A<sub>4</sub>A<sub>1</sub> = 51840/120/2 = 216
|- align=right
|A<sub>5</sub>||{{CDD|nodea_1|3a|nodea|3a|nodes_0x|3a|nodea|3a|nodea}}||[[5-simplex|{3,3,3,3}]]
!rowspan=2|f<sub>5</sub>
||6||15||20||15||6||0||BGCOLOR="#e0e0ff" |'''72'''||BGCOLOR="#e0e0ff"|*||rowspan=2|( ) ||E<sub>6</sub>/A<sub>5</sub> = 51840/720 = 72
|- align=right
|D<sub>5</sub>||{{CDD|nodea_1|3a|nodea|3a|branch|3a|nodea|2|nodea_x}}||[[5-orthoplex|{3,3,3,4}]]
||10||40||80||80||16||16||BGCOLOR="#e0e0ff"|*||BGCOLOR="#e0e0ff"|'''27''' ||E<sub>6</sub>/D<sub>5</sub> = 51840/1920 = 27
|}

=== Images ===

Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow. The number of vertices by color are given in parentheses.

{| 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 2 21 t0 E6.svg|120px]]<BR>(1,3)
|[[File:up 2 21 t0 D5.svg|120px]]<BR>(1,3)
|[[File:up 2 21 t0 D4.svg|120px]]<BR>(3,9)
|[[File:up 2 21 t0 B6.svg|120px]]<BR>(1,3)
|-
!A5<BR>[6]
!A4<BR>[5]
!A3 / D3<BR>[4]
|- valign=top align=center
|[[File:up 2 21 t0 A5.svg|120px]]<BR>(1,3)
|[[File:up 2 21 t0 A4.svg|120px]]<BR>(1,2)
|[[File:up 2 21 t0 D3.svg|120px]]<BR>(1,4,7)
|}

=== Geometric folding ===

The '''2<sub>21</sub>''' is related to the [[24-cell]] by a geometric [[folding (Dynkin diagram)|folding]] of the E6/F4 [[Coxeter-Dynkin diagram]]s. 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 2<sub>21</sub>. 
{| class=wikitable
|- align=center valign=top
|E<sub>6</sub><BR>{{Dynkin|node|3|loop1|nodes|3s|nodes}}
|F<sub>4</sub><BR>{{Dynkin2|node|3|node|4b|node|3|node}}
|- align=center valign=top
|[[File:E6 graph.svg|160px]]<BR>'''2<sub>21</sub>'''<BR>{{CDD|node|3|node|split1|nodes|3ab|nodes_10l}}
|[[File:24-cell t3 F4.svg|160px]]<BR>24-cell<BR>{{CDD|node|3|node|4|node|3|node_1}}
|}

This polytope can tessellate Euclidean 6-space, forming the '''[[Gosset 2 22 honeycomb|2<sub>22</sub>]]''' honeycomb with this Coxeter-Dynkin diagram: {{CDD|node_1|3|node|3|node|split1|nodes|3ab|nodes}}.

=== Related complex polyhedra ===
The [[regular complex polytope|regular complex polygon]] <sub>3</sub>{3}<sub>3</sub>{3}<sub>3</sub>, {{CDD|3node_1|3|3node|3|3node}}, in <math>\mathbb{C}^2</math> has a real representation as the ''2<sub>21</sub>'' polytope, {{CDD|nodes_10r|3ab|nodes|split2|node|3|node}}, in 4-dimensional space. It is called a [[Hessian polyhedron]] after [[Edmund Hess]]. It has 27 vertices, 72 3-edges, and 27 3{3}3 faces. Its [[complex reflection group]] is <sub>3</sub>[3]<sub>3</sub>[3]<sub>3</sub>, order 648.

=== Related polytopes===
The 2<sub>21</sub> is fourth in a dimensional series of [[Uniform k21 polytope|semiregular polytope]]s. Each progressive [[uniform polytope]] is constructed [[vertex figure]] of the previous polytope. [[Thorold Gosset]] identified this series in 1900 as containing all [[regular polytope]] facets, containing all [[simplex]]es and [[orthoplex]]es.
{{k 21 polytopes}}

The 2<sub>21</sub> polytope is fourth in dimensional series 2<sub>k2</sub>.
{{2 k1 polytopes}}

The 2<sub>21</sub> polytope is second in dimensional series 2<sub>2k</sub>.
{{2 2k polytopes}}

== Rectified 2_21 polytope==
{| class="wikitable" align="right" style="margin-left:10px" width="280"
!bgcolor=#e7dcc3 colspan=2|Rectified 2<sub>21</sub> polytope
|-
|bgcolor=#e7dcc3|Type||[[Uniform 6-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| t<sub>1</sub>{3,3,3<sup>2,1</sup>}
|-
|bgcolor=#e7dcc3|Coxeter symbol|| t<sub>1</sub>(2<sub>21</sub>)
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]||{{CDD|nodea|3a|nodea_1|3a|branch|3a|nodea|3a|nodea}} or {{CDD|nodes|3ab|nodes_10lru|split2|node|3|node}}
|-
|bgcolor=#e7dcc3|5-faces||126 total:
72 [[rectified 5-simplex|t<sub>1</sub>{3<sup>4</sup>}]] [[File:5-simplex t1.svg|40px]]<BR>
27 [[rectified 5-orthoplex|t<sub>1</sub>{3<sup>3</sup>,4}]] [[File:5-cube t3.svg|40px]]<BR>
27 [[5-demicube|t<sub>1</sub>{3,3<sup>2,1</sup>}]] [[File:5-demicube t0 D5.svg|40px]]
|-
|bgcolor=#e7dcc3|4-faces||1350
|-
|bgcolor=#e7dcc3|Cells||4320 
|-
|bgcolor=#e7dcc3|Faces||5040
|-
|bgcolor=#e7dcc3|Edges||2160
|-
|bgcolor=#e7dcc3|Vertices||216
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[rectified 5-cell]] prism
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||[[E6 (mathematics)|E<sub>6</sub>]], [3<sup>2,2,1</sup>], order 51840
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}
The '''rectified 2<sub>21</sub>''' has 216 vertices, and 126 facets: 72 [[rectified 5-simplex|rectified 5-simplices]], and 27 [[rectified 5-orthoplex]]es and 27 [[5-demicube]]s . Its [[vertex figure]] is a [[rectified 5-cell]] prism.

=== Alternate names ===
* Rectified icosihepta-heptacontidi-peton as a rectified 27-72 facetted polypeton (acronym rojak) (Jonathan Bowers)<ref>Klitzing, (o3x3o3o3o *c3o - rojak)</ref>

===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|3a|nodea_1|3a|nodea}}.

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

Removing the ring on the end of the other 2-length branch leaves the [[rectified 5-orthoplex]] in its alternated form: '''t<sub>1</sub>(2<sub>11</sub>)''', {{CDD|nodea|3a|branch|3a|nodea_1|3a|nodea}}.

Removing the ring on the end of the same 2-length branch leaves the [[5-demicube]]: '''(1<sub>21</sub>)''', {{CDD|nodea|3a|nodea|3a|branch|3a|nodea_1}}.

The [[vertex figure]] is determined by removing the ringed ring and ringing the neighboring ring. This makes [[rectified 5-cell]] prism, t<sub>1</sub>{3,3,3}x{}, {{CDD|nodea|3a|nodea|3a|branch_10|2|nodea_1}}.

=== Images ===

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

{| class=wikitable width=600
|+ [[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 2 21 t1 E6.svg|150px]]
|[[File:up 2 21 t1 D5.svg|150px]]
|[[File:up 2 21 t1 D4.svg|150px]]
|[[File:up 2 21 t1 B6.svg|150px]]
|-
!A5<BR>[6]
!A4<BR>[5]
!A3 / D3<BR>[4]
|- valign=top align=center
|[[File:up 2 21 t1 A5.svg|150px]]
|[[File:up 2 21 t1 A4.svg|150px]]
|[[File:up 2 21 t1 D3.svg|150px]]
|}

== Truncated 2_21 polytope==
{| class="wikitable" align="right" style="margin-left:10px" width="280"
!bgcolor=#e7dcc3 colspan=2|Truncated 2<sub>21</sub> polytope
|-
|bgcolor=#e7dcc3|Type||[[Uniform 6-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| t{3,3,3<sup>2,1</sup>}
|-
|bgcolor=#e7dcc3|Coxeter symbol|| t(2<sub>21</sub>)
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]||{{CDD|nodea_1|3a|nodea_1|3a|branch|3a|nodea|3a|nodea}} or {{CDD|nodes_10r|3ab|nodes_10lru|split2|node|3|node}}
|-
|bgcolor=#e7dcc3|5-faces||72+27+27
|-
|bgcolor=#e7dcc3|4-faces||432+216+432+270 
|-
|bgcolor=#e7dcc3|Cells||1080+2160+1080
|-
|bgcolor=#e7dcc3|Faces||720+4320
|-
|bgcolor=#e7dcc3|Edges||216+2160
|-
|bgcolor=#e7dcc3|Vertices||432 
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||( ) v r{3,3,3}
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||[[E6 (mathematics)|E<sub>6</sub>]], [3<sup>2,2,1</sup>], order 51840
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}
The '''truncated 2<sub>21</sub>''' has 432 vertices, 5040 edges, 4320 faces, 1350 cells, and 126 4-faces. Its [[vertex figure]] is a [[rectified 5-cell]] pyramid.

=== Images ===

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

{| class=wikitable width=600
|+ [[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 2 21 t01 E6.svg|150px]]
|[[File:up 2 21 t01 D5.svg|150px]]
|[[File:up 2 21 t01 D4.svg|150px]]
|[[File:up 2 21 t01 B6.svg|150px]]
|-
!A5<BR>[6]
!A4<BR>[5]
!A3 / D3<BR>[4]
|- valign=top align=center
|[[File:up 2 21 t01 A5.svg|150px]]
|[[File:up 2 21 t01 A4.svg|150px]]
|[[File:up 2 21 t01 D3.svg|150px]]
|}

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

==Notes==
{{reflist}}

== References ==
* [[Thorold Gosset|T. Gosset]]: ''On the Regular and Semi-Regular Figures in Space of n Dimensions'', Messenger of Mathematics, Macmillan, 1900
* {{citation | last = Elte | first = E. L. | title = The Semiregular Polytopes of the Hyperspaces | publisher = University of Groningen | location = Groningen | year = 1912}}
* '''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 17) [[Harold Scott MacDonald Coxeter|Coxeter]], ''The Evolution of Coxeter-Dynkin diagrams'', [Nieuw Archief voor Wiskunde 9 (1991) 233-248] See figure 1: (p.&nbsp;232) (Node-edge graph of polytope)
* {{KlitzingPolytopes|polypeta.htm|6D|uniform polytopes (polypeta)}} x3o3o3o3o *c3o - jak, o3x3o3o3o *c3o - rojak

{{Polytopes}}

[[Category:6-polytopes]]