View source for 3 21 polytope

{{DISPLAYTITLE:3<sub>&thinsp;21</sub> polytope}}
{| class="wikitable" align="right" width="360" style="margin-left:1em;"
|- align=center valign=top
|colspan=2|[[File:Up2 3 21 t0 E7.svg|120px]]<BR>'''3<sub>21</sub>'''<BR>{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea}}
|colspan=2|[[File:Up2 2 31 t0 E7.svg|120px]]<BR>[[2 31 polytope|2<sub>31</sub>]]<BR>{{CDD|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea_1}}
|colspan=2|[[File:Up2 1 32 t0 E7.svg|120px]]<BR>[[1 32 polytope|1<sub>32</sub>]]<BR>{{CDD|nodea|3a|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea}}

|- align=center valign=top
|colspan=3|[[File:Up2 3 21 t1 E7.svg|150px]]<BR>Rectified 3<sub>21</sub><BR>{{CDD|nodea|3a|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea}}
|colspan=3|[[File:Up2 3 21 t2 E7.svg|150px]]<BR>birectified 3<sub>21</sub><BR>{{CDD|nodea|3a|nodea|3a|nodea_1|3a|branch|3a|nodea|3a|nodea}}
|- align=center valign=top
|colspan=3|[[File:Up2 2 31 t1 E7.svg|150px]]<BR>Rectified 2<sub>31</sub><BR>{{CDD|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea_1|3a|nodea}}
|colspan=3 valign=center|[[File:Up2 1 32 t1 E7.svg|150px]]<BR>Rectified 1<sub>32</sub><BR>{{CDD|nodea|3a|nodea|3a|nodea|3a|branch_10|3a|nodea|3a|nodea}}
|- valign="top"
! colspan="6" |Orthogonal projections in E<sub>7</sub> Coxeter plane
|}
In 7-dimensional [[Geometry|geometry]], the '''3<sub>21</sub>''' polytope is a uniform 7-polytope, constructed within the symmetry of the [[E7 (mathematics)|E<sub>7</sub>]] group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an [[Semiregular polytope|7-ic semi-regular figure]].<ref name=gosset>Gosset, 1900</ref>

Its Coxeter symbol is '''3<sub>21</sub>''', describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 3-node sequences.

The '''rectified 3<sub>21</sub>''' is constructed by points at the mid-edges of the '''3<sub>21</sub>'''. The '''birectified 3<sub>21</sub>''' is constructed by points at the triangle face centers of the '''3<sub>21</sub>'''. The '''trirectified 3<sub>21</sub>''' is constructed by points at the tetrahedral centers of the '''3<sub>21</sub>''', and  is the same as the rectified 1<sub>32</sub>.

These polytopes are part of a family of 127 (2<sup>7</sup>-1) convex uniform polytopes in 7-dimensions, made of [[Uniform 6-polytope|uniform 6-polytope]] facets and [[Vertex figure|vertex figure]]s, defined by all permutations of rings in this Coxeter-Dynkin diagram: {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea}}.
{{clear}}

== 3<sub>21</sub> polytope==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|3<sub>21</sub> polytope
|-
|bgcolor=#e7dcc3|Type||Uniform 7-polytope
|-
|bgcolor=#e7dcc3|Family||k<sub>21</sub> polytope
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| {3,3,3,3<sup>2,1</sup>}
|-
|bgcolor=#e7dcc3|Coxeter symbol|| 3<sub>21</sub>
|-
|bgcolor=#e7dcc3|Coxeter diagram||{{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea_1}}
|-
|bgcolor=#e7dcc3|6-faces||702 total:<BR>126 3<sub>11</sub>[[Image:6-orthoplex.svg|25px]]<BR>576 [[6-simplex|{3<sup>5</sup>}]][[Image:6-simplex t0.svg|25px]]
|-
|bgcolor=#e7dcc3|5-faces||6048:<BR>4032 [[5-simplex|{3<sup>4</sup>}]][[Image:5-simplex t0.svg|25px]]<BR>2016 [[5-simplex|{3<sup>4</sup>}]][[Image:5-simplex t0.svg|25px]]
|-
|bgcolor=#e7dcc3|4-faces||12096 {3<sup>3</sup>}[[Image:4-simplex t0.svg|25px]]
|-
|bgcolor=#e7dcc3|Cells||10080 [[Tetrahedron|{3,3}]][[Image:3-simplex t0.svg|25px]]
|-
|bgcolor=#e7dcc3|Faces||4032 [[Triangle|{3}]][[Image:2-simplex t0.svg|25px]]
|-
|bgcolor=#e7dcc3|Edges||756
|-
|bgcolor=#e7dcc3|Vertices||56
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||2<sub>21</sub> polytope
|-
|bgcolor=#e7dcc3|[[Petrie polygon]]||[[Octadecagon|octadecagon]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||[[E7 (mathematics)|E<sub>7</sub>]], [3<sup>3,2,1</sup>], order 2903040
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}
In 7-dimensional [[Geometry|geometry]], the '''3<sub>21</sub>''' polytope is a uniform polytope. It has 56 vertices, and 702 facets: 126 3<sub>11</sub> and 576 [[6-simplex|6-simplex]]es.

For visualization this 7-dimensional polytope is often displayed in a special skewed orthographic projection direction that fits its 56 vertices within an 18-gonal regular polygon (called a [[Petrie polygon]]). Its 756 edges are drawn between 3 rings of 18 vertices, and 2 vertices in the center. Specific higher elements (faces, cells, etc.) can also be extracted and drawn on this projection.

The 1-[[N-skeleton|skeleton]] of the '''3<sub>21</sub>''' polytope is the [[Gosset graph]].

This polytope, along with the [[7-simplex|7-simplex]], can [[Honeycomb (geometry)|tessellate]] 7-dimensional space, represented by '''3<sub>31</sub>''' and Coxeter-Dynkin diagram: {{CDD||node|3|node|split1|nodes|3ab|nodes|3ab|nodes_10l}}.

=== Alternate names===
*It is also called the '''Hess polytope''' for Edmund Hess who first discovered it.
*It was enumerated by Thorold Gosset in his 1900 paper. He called it an ''7-ic semi-regular figure''.<ref name=gosset />
*E. L. Elte named it V<sub>56</sub> (for its 56 vertices) in his 1912 listing of semiregular polytopes.<ref>Elte, 1912</ref>
*H.S.M. Coxeter called it '''3<sub>21</sub>''' due to its bifurcating Coxeter-Dynkin diagram, having 3 branches of length 3, 2, and 1, and having a single ring on the final node of the 3 branch.
* ''Hecatonicosihexa-pentacosiheptacontihexa-exon'' (Acronym Naq) - 126-576 facetted polyexon (Jonathan Bowers)<ref>Klitzing, (o3o3o3o *c3o3o3x - naq)</ref>

=== Coordinates ===
The 56 vertices can be most simply represented in 8-dimensional space, obtained by the 28 permutations of the coordinates and their opposite:
: ± (-3, -3, 1, 1, 1, 1, 1, 1)

===Construction===

Its construction is based on the [[E7 (mathematics)|E7]] group. Coxeter named it as '''3<sub>21</sub>''' by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 3-node sequence.

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

Removing the node on the short branch leaves the [[6-simplex|6-simplex]], {{CDD|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea_1}}.

Removing the node on the end of the 2-length branch leaves the [[6-orthoplex|6-orthoplex]] in its alternated form: '''3<sub>11</sub>''', {{CDD|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea_1}}.

Every simplex facet touches a 6-orthoplex facet, while alternate facets of the orthoplex touch either a simplex or another orthoplex.

The [[Vertex figure|vertex figure]] is determined by removing the ringed node and ringing the neighboring node. This makes '''2<sub>21</sub>''' polytope, {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea_1}}.

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>Coxeter, Regular Polytopes, 11.8 Gossett figures in  six, seven, and eight dimensions, p. 202-203</ref>
{| class=wikitable
!E<sub>7</sub>||{{CDD|nodea_1|3a|nodea|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>||f<sub>4</sub>||colspan=2|f<sub>5</sub>||colspan=2|f<sub>6</sub>|| [[Vertex figure|''k''-figures]]|| notes
|- align=right
|E<sub>6</sub> ||{{CDD|nodea_x|2|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea}}||( )
!f<sub>0</sub>
|BGCOLOR="#ffe0ff"|'''56'''||27||216||720||1080||432||216||72||27||[[2 21 polytope|2<sub>21</sub>]]|| E<sub>7</sub>/E<sub>6</sub> =  72x8!/72x6! = 56
|- align=right
|D<sub>5</sub>A<sub>1</sub> ||{{CDD|nodea_1|2|nodea_x|2|nodea|3a|branch|3a|nodea|3a|nodea}}||{ }
!f<sub>1</sub>
||2||BGCOLOR="#ffe0e0"|'''756'''||16||80||160||80||40||16||10||[[5-demicube|5-demicube]] || E<sub>7</sub>/D<sub>5</sub>A<sub>1</sub> =  72x8!/16/5!/2 = 756
|- align=right
|A<sub>4</sub>A<sub>2</sub> ||{{CDD|nodea_1|3a|nodea|2|nodea_x|2|branch|3a|nodea|3a|nodea}}||[[Triangle|{3}]]
!f<sub>2</sub>
||3||3||BGCOLOR="#ffffe0"|'''4032'''||10||30||20||10||5||5||[[Rectified 5-cell|rectified 5-cell]] || E<sub>7</sub>/A<sub>4</sub>A<sub>2</sub> =  72x8!/5!/2 = 4032
|- align=right
|A<sub>3</sub>A<sub>2</sub>A<sub>1</sub> ||{{CDD|nodea_1|3a|nodea|3a|nodea|2|nodes_x0|2|nodea|3a|nodea}}||[[Tetrahedron|{3,3}]]
!f<sub>3</sub>
||4||6||4||BGCOLOR="#e0ffe0"|'''10080'''||6||6||3||2||3||[[Triangular prism|triangular prism]] || E<sub>7</sub>/A<sub>3</sub>A<sub>2</sub>A<sub>1</sub> = 72x8!/4!/3!/2 = 10080
|- align=right
|A<sub>4</sub>A<sub>1</sub> ||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|nodes_0x|2|nodea_x|2|nodea}}||[[5-cell|{3,3,3}]]
!f<sub>4</sub>
||5||10||10||5||BGCOLOR="#e0ffff"|'''12096'''||2||1||1||2||[[Isosceles triangle|isosceles triangle]] || E<sub>7</sub>/A<sub>4</sub>A<sub>1</sub> = 72x8!/5!/2 = 12096
|- align=right
|A<sub>5</sub>A<sub>1</sub> ||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch|2|nodea_x|2|nodea}}||rowspan=2|[[5-simplex|{3,3,3,3}]]
!rowspan=2|f<sub>5</sub>
||6||15||20||15||6||BGCOLOR="#e0e0ff"|'''4032'''||BGCOLOR="#e0e0ff"|*||1||1||rowspan=2|{ } || E<sub>7</sub>/A<sub>5</sub>A<sub>1</sub> = 72x8!/6!/2 = 4032
|- align=right
|A<sub>5</sub> ||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|nodes_0x|3a|nodea|2|nodea_x}}
||6||15||20||15||6||BGCOLOR="#e0e0ff"|*||BGCOLOR="#e0e0ff"|'''2016'''||0||2 || E<sub>7</sub>/A<sub>5</sub> = 72x8!/6! = 2016
|- align=right
|A<sub>6</sub> ||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|nodes_0x|3a|nodea|3a|nodea}}||[[6-simplex|{3,3,3,3,3}]]
!rowspan=2|f<sub>6</sub>
||7||21||35||35||21||10||0||BGCOLOR="#ffe0ff"|'''576'''||BGCOLOR="#ffe0ff"|*||rowspan=2|( ) || E<sub>7</sub>/A<sub>6</sub> = 72x8!/7! = 576
|- align=right
|D<sub>6</sub> ||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch|3a|nodea|2|nodea_x}}||[[6-orthoplex|{3,3,3,3,4}]]
||12||60||160||240||192||32||32||BGCOLOR="#ffe0ff"|*||BGCOLOR="#ffe0ff"|'''126''' || E<sub>7</sub>/D<sub>6</sub> = 72x8!/32/6! = 126
|}

===Images===

{| class=wikitable width=600
|+ Coxeter plane projections
|-
!E7
!E6 / F4
!B7 / A6
|- valign=top align=center
|[[File:up2 3 21 t0 E7.svg|200px]]<BR>[18]
|[[File:up2 3 21 t0 E6.svg|200px]]<BR>[12]
|[[File:up2 3 21 t0 A6.svg|200px]]<BR>[7x2]
|-
!A5
!D7 / B6
!D6 / B5
|- valign=top align=center
|[[File:up2 3 21 t0 A5.svg|200px]]<BR>[6]
|[[File:up2 3 21 t0 D7.svg|200px]]<BR>[12/2]
|[[File:up2 3 21 t0 D6.svg|200px]]<BR>[10]
|-
!D5 / B4 / A4
!D4 / B3 / A2 / G2
!D3 / B2 / A3
|- valign=top align=center
|[[File:up2 3 21 t0 D5.svg|200px]]<BR>[8]
|[[File:up2 3 21 t0 D4.svg|200px]]<BR>[6]
|[[File:up2 3 21 t0 D3.svg|200px]]<BR>[4]
|}

=== Related polytopes===
The 3<sub>21</sub> is fifth in a dimensional series of semiregular polytopes. Each progressive uniform polytope is constructed [[Vertex figure|vertex figure]] of the previous polytope. Thorold Gosset identified this series in 1900 as containing all [[Regular polytope|regular polytope]] facets, containing all [[Simplex|simplex]]es and orthoplexes.

It is in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 3<sub>k1</sub> series. (A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral [[Hosohedron|hosohedron]].)

== Rectified 3<sub>21</sub> polytope==

{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Rectified 3<sub>21</sub> polytope
|-
|bgcolor=#e7dcc3|Type||Uniform 7-polytope
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| t<sub>1</sub>{3,3,3,3<sup>2,1</sup>}
|-
|bgcolor=#e7dcc3|Coxeter symbol|| t<sub>1</sub>(3<sub>21</sub>)
|-
|bgcolor=#e7dcc3|Coxeter diagram||{{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea_1|3a|nodea}}
|-
|bgcolor=#e7dcc3|6-faces||758
|-
|bgcolor=#e7dcc3|5-faces||44352
|-
|bgcolor=#e7dcc3|4-faces||70560
|-
|bgcolor=#e7dcc3|Cells||48384
|-
|bgcolor=#e7dcc3|Faces||11592
|-
|bgcolor=#e7dcc3|Edges||12096
|-
|bgcolor=#e7dcc3|Vertices||756
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[5-demicube|5-demicube]] prism
|-
|bgcolor=#e7dcc3|[[Petrie polygon]]||[[Octadecagon|octadecagon]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||[[E7 (mathematics)|E<sub>7</sub>]], [3<sup>3,2,1</sup>], order 2903040
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}

=== Alternate names===
* Rectified hecatonicosihexa-pentacosiheptacontihexa-exon as a rectified 126-576 facetted polyexon (acronym ranq) (Jonathan Bowers)<ref>Klitzing. (o3o3o3o *c3o3x3o - ranq)</ref>

===Construction===

Its construction is based on the [[E7 (mathematics)|E7]] group. Coxeter named it as '''3<sub>21</sub>''' by its bifurcating Coxeter-Dynkin diagram, with a single node on the end of the 3-node sequence.

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

Removing the node on the short branch leaves the [[6-simplex|6-simplex]], {{CDD|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea_1|3a|nodea}}.

Removing the node on the end of the 2-length branch leaves the rectified 6-orthoplex in its alternated form: '''t<sub>1</sub>3<sub>11</sub>''', {{CDD|nodea|3a|branch|3a|nodea|3a|nodea_1|3a|nodea}}.

Removing the node on the end of the 3-length branch leaves the [[2 21 polytope|2<sub>21</sub>]], {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea_1}}.

The [[Vertex figure|vertex figure]] is determined by removing the ringed node and ringing the neighboring node. This makes [[5-demicube|5-demicube]] prism, {{CDD|nodea|3a|nodea|3a|branch|3a|nodea_1|2|nodea_1}}.

===Images===

{| class=wikitable width=600
|+ Coxeter plane projections
|-
!E7
!E6 / F4
!B7 / A6
|- valign=top align=center
|[[File:up2 3 21 t1 E7.svg|200px]]<BR>[18]
|[[File:up2 3 21 t1 E6.svg|200px]]<BR>[12]
|[[File:up2 3 21 t1 A6.svg|200px]]<BR>[7x2]
|-
!A5
!D7 / B6
!D6 / B5
|- valign=top align=center
|[[File:up2 3 21 t1 A5.svg|200px]]<BR>[6]
|[[File:up2 3 21 t1 D7.svg|200px]]<BR>[12/2]
|[[File:up2 3 21 t1 D6.svg|200px]]<BR>[10]
|-
!D5 / B4 / A4
!D4 / B3 / A2 / G2
!D3 / B2 / A3
|- valign=top align=center
|[[File:up2 3 21 t1 D5.svg|200px]]<BR>[8]
|[[File:up2 3 21 t1 D4.svg|200px]]<BR>[6]
|[[File:up2 3 21 t1 D3.svg|200px]]<BR>[4]
|}

== Birectified 3<sub>21</sub> polytope==

{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Birectified 3<sub>21</sub> polytope
|-
|bgcolor=#e7dcc3|Type||Uniform 7-polytope
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| t<sub>2</sub>{3,3,3,3<sup>2,1</sup>}
|-
|bgcolor=#e7dcc3|Coxeter symbol|| t<sub>2</sub>(3<sub>21</sub>)
|-
|bgcolor=#e7dcc3|Coxeter diagram||{{CDD|nodea|3a|nodea|3a|branch|3a|nodea_1|3a|nodea|3a|nodea}}
|-
|bgcolor=#e7dcc3|6-faces||758
|-
|bgcolor=#e7dcc3|5-faces||12348
|-
|bgcolor=#e7dcc3|4-faces||68040
|-
|bgcolor=#e7dcc3|Cells||161280
|-
|bgcolor=#e7dcc3|Faces||161280
|-
|bgcolor=#e7dcc3|Edges||60480
|-
|bgcolor=#e7dcc3|Vertices||4032
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[5-cell|5-cell]]-triangle duoprism
|-
|bgcolor=#e7dcc3|[[Petrie polygon]]||[[Octadecagon|octadecagon]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||[[E7 (mathematics)|E<sub>7</sub>]], [3<sup>3,2,1</sup>], order 2903040
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}

=== Alternate names===
* Birectified hecatonicosihexa-pentacosiheptacontihexa-exon as a birectified 126-576 facetted polyexon (acronym branq) (Jonathan Bowers)<ref>Klitzing, (o3o3o3o *c3x3o3o - branq)</ref>

===Construction===

Its construction is based on the [[E7 (mathematics)|E7]] group. Coxeter named it as '''3<sub>21</sub>''' by its bifurcating Coxeter-Dynkin diagram, with a single node on the end of the 3-node sequence.

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

Removing the node on the short branch leaves the birectified 6-simplex, {{CDD|nodea|3a|nodea|3a|nodea|3a|nodea_1|3a|nodea|3a|nodea}}.

Removing the node on the end of the 2-length branch leaves the birectified 6-orthoplex in its alternated form: '''t<sub>2</sub>(3<sub>11</sub>)''', {{CDD|nodea|3a|branch|3a|nodea_1|3a|nodea|3a|nodea}}.

Removing the node on the end of the 3-length branch leaves the rectified 2<sub>21</sub> polytope in its alternated form: '''t<sub>1</sub>(2<sub>21</sub>)''', {{CDD|nodea|3a|nodea|3a|branch|3a|nodea_1|3a|nodea}}.

The [[Vertex figure|vertex figure]] is determined by removing the ringed node and ringing the neighboring node. This makes [[Rectified 5-cell|rectified 5-cell]]-triangle duoprism, {{CDD|nodea|3a|nodea|3a|branch_10|2|nodea_1|3a|nodea}}.

===Images===

{| class=wikitable width=600
|+ Coxeter plane projections
|-
!E7
!E6 / F4
!B7 / A6
|- valign=top align=center
|[[File:up2 3 21 t2 E7.svg|200px]]<BR>[18]
|[[File:up2 3 21 t2 E6.svg|200px]]<BR>[12]
|[[File:up2 3 21 t2 A6.svg|200px]]<BR>[7x2]
|-
!A5
!D7 / B6
!D6 / B5
|- valign=top align=center
|[[File:up2 3 21 t2 A5.svg|200px]]<BR>[6]
|[[File:up2 3 21 t2 D7.svg|200px]]<BR>[12/2]
|[[File:up2 3 21 t2 D6.svg|200px]]<BR>[10]
|-
!D5 / B4 / A4
!D4 / B3 / A2 / G2
!D3 / B2 / A3
|- valign=top align=center
|[[File:up2 3 21 t2 D5.svg|200px]]<BR>[8]
|[[File:up2 3 21 t2 D4.svg|200px]]<BR>[6]
|[[File:up2 3 21 t2 D3.svg|200px]]<BR>[4]
|}

== See also ==
* List of E7 polytopes

==Notes==
{{reflist}}

== References ==
* 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}}
* 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 p342 (figure 3.7c) by Peter mcMullen: (18-gonal node-edge graph of 3<sub>21</sub>)
* {{KlitzingPolytopes|polyexa.htm|7D|uniform polytopes (polyexa)}} o3o3o3o *c3o3o3x - naq, o3o3o3o *c3o3x3o - ranq, o3o3o3o *c3x3o3o - branq

==External links==
* [http://vzome.com/home/geometry/gossets-polytopes ''Gosset’s Polytopes''] in vZome

{{Polytopes}}

{{DEFAULTSORT:3 21 Polytope}}
[[Category:7-polytopes]]

{{Sourceattribution|3 21 polytope}}