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{{DISPLAYTITLE:4<sub><span style="display:none"> </span>21</sub> polytope}}
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{| class=wikitable width=540 align=right
|- valign=top
!colspan=3|[[Orthogonal projection]]s in E<sub>6</sub> [[Coxeter plane]]
|- align=center valign=top
|[[File:4 21 t0 E6.svg|180px]]<BR>4<sub>21</sub><BR>{{CDD|nodea_1|3a|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea}}
|[[File:1 42 polytope E6 Coxeter plane.svg|180px]]<BR>[[1 42 polytope|1<sub>42</sub>]]<BR>{{CDD|nodea|3a|nodea|3a|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea}}
|[[File:2 41 t0 E6.svg|180px]]<BR>[[2 41 polytope|2<sub>41</sub>]]<BR>{{CDD|nodea|3a|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea_1}}
|- align=center valign=top
|[[File:4 21 t1 E6.svg|180px]]<BR>Rectified 4<sub>21</sub><BR>{{CDD|nodea|3a|nodea_1|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea}}
|[[File:4 21 t4 E6.svg|180px]]<BR>[[Rectified 1 42 polytope|Rectified 1<sub>42</sub>]]<BR>{{CDD|nodea|3a|nodea|3a|nodea|3a|nodea|3a|branch_10|3a|nodea|3a|nodea}}
|[[File:2 41 t1 E6.svg|180px]]<BR>[[Rectified 2 41 polytope|Rectified 2<sub>41</sub>]]<BR>{{CDD|nodea|3a|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea_1|3a|nodea}}
|- align=center valign=top
|[[File:4 21 t2 E6.svg|180px]]<BR>Birectified 4<sub>21</sub><BR>{{CDD|nodea|3a|nodea|3a|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea}}
|[[File:4 21 t3 E6.svg|180px]]<BR>Trirectified 4<sub>21</sub><BR>{{CDD|nodea|3a|nodea|3a|nodea|3a|nodea_1|3a|branch|3a|nodea|3a|nodea}}
|}
In 8-dimensional [[geometry]], the '''4<sub>21</sub>''' is a semiregular [[uniform 8-polytope]], constructed within the symmetry of the [[E8 (mathematics)|E<sub>8</sub>]] group. It was discovered by [[Thorold Gosset]], published in his 1900 paper. He called it an ''8-ic semi-regular figure''.<ref name=gosset>Gosset, 1900</ref>


[[Coxeter]] named it '''4<sub>21</sub>''' by its bifurcating [[Coxeter-Dynkin diagram]], with a single ring on the end of the 4-node sequences, {{CDD|nodea_1|3a|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea}}.
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The '''rectified 4<sub>21</sub>''' is constructed by points at the mid-edges of the '''4<sub>21</sub>'''. The '''birectified 4<sub>21</sub>''' is constructed by points at the triangle face centers of the '''4<sub>21</sub>'''. The '''trirectified 4<sub>21</sub>''' is constructed by points at the tetrahedral centers of the '''4<sub>21</sub>''', and  is the same as the rectified 1<sub>42</sub>.
 
These polytopes are part of a family of 255 = 2<sup>8</sup>&nbsp;&minus;&nbsp;1 convex [[uniform 8-polytope]]s, made of [[uniform 7-polytope]] facets and [[vertex figure]]s, defined by all permutations of one or more rings in this Coxeter-Dynkin diagram: {{CDD|nodea|3a|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea}}.
 
== 4<sub>21</sub> polytope==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|4<sub>21</sub>
|-
|bgcolor=#e7dcc3|Type||[[Uniform 8-polytope]]
|-
|bgcolor=#e7dcc3|Family||[[Semiregular k 21 polytope|k<sub>21</sub> polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| {3,3,3,3,3<sup>2,1</sup>}
|-
|bgcolor=#e7dcc3|Coxeter symbol|| 4<sub>21</sub>
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]||{{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea_1}}
|-
|bgcolor=#e7dcc3|7-faces||19440 total:<BR>2160 [[heptacross|4<sub>11</sub>]][[Image:7-orthoplex.svg|25px]]<BR>17280 [[7-simplex|{3<sup>6</sup>}]][[Image:7-simplex t0.svg|25px]]
|-
|bgcolor=#e7dcc3|6-faces||207360:<BR>138240 [[6-simplex|{3<sup>5</sup>}]][[Image:6-simplex t0.svg|25px]]<BR>69120 [[6-simplex|{3<sup>5</sup>}]][[Image:6-simplex t0.svg|25px]]
|-
|bgcolor=#e7dcc3|5-faces||483840 [[5-simplex|{3<sup>4</sup>}]][[Image:5-simplex t0.svg|25px]]
|-
|bgcolor=#e7dcc3|4-faces||483840 [[pentachoron|{3<sup>3</sup>}]][[Image:4-simplex t0.svg|25px]]
|-
|bgcolor=#e7dcc3|Cells||241920 [[tetrahedron|{3,3}]][[Image:3-simplex t0.svg|25px]]
|-
|bgcolor=#e7dcc3|Faces||60480 [[triangle|{3}]][[Image:2-simplex t0.svg|25px]]
|-
|bgcolor=#e7dcc3|Edges||6720
|-
|bgcolor=#e7dcc3|Vertices||240
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[Gosset 3 21 polytope|3<sub>21</sub> polytope]]
|-
|bgcolor=#e7dcc3|[[Petrie polygon]]||[[regular polygon|30-gon]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||[[E8 (mathematics)|E<sub>8</sub>]], [3<sup>4,2,1</sup>]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}
The '''4<sub>21</sub>''' is composed of 17,280 [[7-simplex]] and 2,160 [[7-orthoplex]] [[Facet (geometry)|facets]]. Its [[vertex figure]] is the '''[[3 21 polytope|3<sub>21</sub>]]''' polytope.
 
For visualization this 8-dimensional polytope is often displayed in a special skewed orthographic projection direction that fits its 240 vertices within a 30-gonal regular polygon (called a [[Petrie polygon]]). Its 6720 edges are drawn between the 240 vertices. Specific higher elements (faces, cells, etc.) can also be extracted and drawn on this projection.
 
As its 240 vertices represent the root vectors of the [[simple Lie group]] [[E8 (mathematics)|E<sub>8</sub>]], the polytope is sometimes referred to as the '''E<sub>8</sub> polytope'''.
 
===Alternate names ===
*It was discovered by [[Thorold Gosset]], who described it in his 1900 paper as an '''8-ic semi-regular figure'''.<ref name=gosset /> It is the [[Semiregular k 21 polytope|last finite semiregular figure]] in his enumeration, semiregular to him meaning that it contained only regular facets.
*[[E. L. Elte]] named it V<sub>240</sub> (for its 240 vertices) in his 1912 listing of semiregular polytopes.<ref name=elte>Elte, 1912</ref>
*[[H.S.M. Coxeter]] called it '''4<sub>21</sub>''' because its [[Coxeter-Dynkin diagram]] has three branches of length 4, 2, and 1, with a single node on the terminal node of the 4 branch.
* '''Dischiliahectohexaconta-myriaheptachiliadiacosioctaconta-zetton''' (Acronym Fy) - 2160-17280 facetted polyzetton (Jonathan Bowers)<ref>Klitzing, (o3o3o3o *c3o3o3o3x - fy)</ref>
 
=== Coordinates===
It is created by a [[Wythoff construction]] upon a set of 8 [[hyperplane]] mirrors in 8-dimensional space.
 
The 240 vertices of the '''4<sub>21</sub>''' polytope can be constructed in two sets: 112 (2<sup>2</sup>×<sup>8</sup>C<sub>2</sub>) with coordinates obtained from <math>(\pm 2,\pm 2,0,0,0,0,0,0)\,</math> by taking an arbitrary [[combination]] of signs and an arbitrary [[permutation]] of coordinates, and 128 roots (2<sup>7</sup>) with coordinates obtained from <math>(\pm 1,\pm 1,\pm 1,\pm 1,\pm 1,\pm 1,\pm 1,\pm 1)\,</math> by taking an even number of minus signs (or, equivalently, requiring that the sum of all the eight coordinates be even).
 
=== Tessellations ===
This polytope is the [[vertex figure]] for a uniform tessellation of 8-dimensional space, represented by symbol '''[[5 21 honeycomb|5<sub>21</sub>]]''' and Coxeter-Dynkin diagram:
:{{CDD|nodea_1|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea}}
 
===Construction===
The facet information can be extracted from its [[Coxeter-Dynkin diagram]].
: {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea_1}}
 
Removing the node on the short branch leaves the [[7-simplex]]:
: {{CDD|nodea|3a|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 [[7-orthoplex]] in its alternated form ('''4<sub>11</sub>'''):
: {{CDD|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea_1}}
 
Every simplex facet touches a 7-orthoplex facet, while alternate facets of the orthoplex touch either a simplex or another orthoplex.
 
The [[vertex figure]] of a single-ring polytope is obtained by removing the ringed node and ringing its neighbor(s). This makes the '''[[Gosset 3 21 polytope|3<sub>21</sub>]]''' polytope.
: {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea_1}}
 
=== Projections ===
 
{| class=wikitable
|[[File:E8-with-thread.jpg|360px]]<BR>The 4<sub>21</sub> graph created as [[string art]].
|[[File:E8Petrie.svg|320px]]<BR>E<sub>8</sub> Coxeter plane projection
|}
 
==== 3D====
{| class=wikitable width=540
|- valign=top
|[[File:Zome-like.png|240px]]<BR>Mathematical representation of the physical Zome model isomorphic (?) to E8. This is constructed from [http://theoryofeverything.org/TOE/JGM/VisibLie_E8.nbp VisibLie_E8] pictured with all 3360 edges of length √2(√5-1) from two concentric [[600-cell]]s (at the golden ratio) with orthogonal projections to perspective 3-space
|[[File:E8 3D.png|240px]]<BR>The actual split real even E8 '''4<sub>21</sub>''' polytope projected into perspective 3-space pictured with all 6720 edges of length √2<ref>[http://theoryofeverything.org/TOE/JGM/e8Flyer.nbp e8Flyer.nb]</ref>
|}
 
====2D====
These graphs represent orthographic projections in the E<sub>8</sub>,E<sub>7</sub>,E<sub>6</sub>, and B<sub>8</sub>,D<sub>8</sub>,D<sub>7</sub>,D<sub>6</sub>,D<sub>5</sub>,D<sub>4</sub>,D<sub>3</sub>,A<sub>7</sub>,A<sub>5</sub> [[Coxeter plane]]s. The vertex colors are by overlapping multiplicity in the projection: colored by increasing order of multiplicities as red, orange, yellow, green.
 
{| class=wikitable width=720
!E<sub>8</sub> / H<sub>4</sub><BR>[30]
![20]
![24]
|- align=center
|[[File:4 21 t0 E8.svg|240px]]<BR>(Colors: 1)
|[[File:4 21 t0 p20.svg|240px]]<BR>(Colors: 1)
|[[File:4 21 t0 p24.svg|240px]]<BR>(Colors: 1)
|- align=center
!E<sub>7</sub><BR>[18]
!E<sub>6</sub> / F<sub>4</sub><BR>[12]
![6]
|- align=center
|[[File:4 21 t0 E7.svg|240px]]<BR>(Colors: 1,3,6)
|[[File:4 21 t0 E6.svg|240px]]<BR>(Colors: 1,8,24)
|[[File:4 21 t0 mox.svg|240px]]<BR>(Colors: 1,2,3)
|}
 
{| class=wikitable width=720
|- align=center
!D<sub>3</sub> / B<sub>2</sub> / A<sub>3</sub><BR>[4]
!D<sub>4</sub> / B<sub>3</sub> / A<sub>2</sub> / G<sub>2</sub><BR>[6]
!D<sub>5</sub> / B<sub>4</sub><BR>[8]
|- align=center
|[[File:4 21 t0 B2.svg|240px]]<BR>(Colors: 1,12,32,60)
|[[File:4 21 t0 B3.svg|240px]]<BR>(Colors: 1,27,72)
|[[File:4 21 t0 B4.svg|240px]]<BR>(Colors: 1,8,24)
|- align=center
!D<sub>6</sub> / B<sub>5</sub> / A<sub>4</sub><BR>[10]
!D<sub>7</sub> / B<sub>6</sub><BR>[12]
!D<sub>8</sub> / B<sub>7</sub> / A<sub>6</sub><BR>[14]
|- align=center
|[[File:4 21 t0 B5.svg|240px]]<BR>(Colors: 1,5,10,20)
|[[File:4 21 t0 B6.svg|240px]]<BR>(Colors: 1,3,9,12)
|[[File:4 21 t0 B7.svg|240px]]<BR>(Colors: 1,2,3)
|- align=center
!B<sub>8</sub><BR>[16/2]
!A<sub>5</sub><BR>[6]
!A<sub>7</sub><BR>[8]
|- align=center
|[[File:4 21 t0 B8.svg|240px]]<BR>(Colors: 1)
|[[File:4 21 t0 A5.svg|240px]]<BR>(Colors: 3,8,24,30)
|[[File:4 21 t0 A7.svg|240px]]<BR>(Colors: 1,2,4,8)
|}
 
===k<sub>21</sub> family ===
The '''4<sub>21</sub>''' polytope is last in a family called the [[Semiregular k 21 polytope|k<sub>21</sub> polytopes]]. The first polytope in this family is the semiregular [[triangular prism]] which is constructed from three squares (2-orthoplexes) and two triangles (2-simplexes).
 
=== Geometric folding ===
[[File:E8 roots zome.jpg|thumb|right|The '''4<sub>21</sub>''' polytope can be projected into 3-space as a physical vertex-edge model. Pictured here as 2 concentric [[600-cell]]s (at the golden ratio) using [[Zome]] tools.<ref>David Richter: [http://homepages.wmich.edu/~drichter/gossetzome.htm Gosset's Figure in 8 Dimensions, A Zome Model]</ref> (Not all of the 3360 edges of length √2(√5-1) are represented.)]]
 
The '''4<sub>21</sub>''' is related to the [[600-cell]] by a geometric [[folding (Dynkin diagram)|folding]] of the [[Coxeter-Dynkin diagram]]s. This can be seen in the E8/H4 [[Coxeter plane]] projections. The 240 vertices of the '''4<sub>21</sub>''' polytope are projected into 4-space as two copies of the 120 vertices of the 600-cell, one copy smaller than the other with the same orientation. Seen as a 2D orthographic projection in the E8/H4 Coxeter plane, the 120 vertices of the 600-cell are projected in the same four rings as seen in the 4<sub>21</sub>. The other 4 rings of the 4<sub>21</sub> graph also match a smaller copy of the four rings of the 600-cell.
{| class=wikitable
|-
!colspan=2|E8/H4 Coxeter planes
|- align=center valign=top
|'''E<sub>8</sub>''' {{CDD|nodes_10r|3ab|nodes|3ab|nodes|split5c|nodes}}
|'''H<sub>4</sub>''' {{CDD|node_1|3|node|3|node|5|node}}
|- align=center valign=top
|[[File:4 21 t0 E8.svg|240px]]<BR>'''4<sub>21</sub>'''
|[[File:600-cell graph H4.svg|240px]]<BR>600-cell
|-
!colspan=2|[20] symmetry planes
|- align=center valign=top
|[[File:4 21 t0 p20.svg|240px]]<BR>'''4<sub>21</sub>'''
|[[File:600-cell t0 p20.svg|240px]]<BR>600-cell
|}
 
=== Related polytopes===
Using a [[complex number]] coordinate system, it can also be constructed as a 4-dimensional [[regular complex polytope]], named as: 3{3}3{3}3{3}3. Coxeter called it the '''Witting polytope''', after [[Alexander Witting]].<ref>Coxeter Regular Convex Polytopes, 12.5 The Witting polytope</ref>
 
The 4<sub>21</sub> is sixth 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.
{{Gosset_semiregular_polytopes}}
 
== Rectified 4_21 polytope==
 
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Rectified 4<sub>21</sub>
|-
|bgcolor=#e7dcc3|Type||[[Uniform 8-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| t<sub>1</sub>{3,3,3,3,3<sup>2,1</sup>}
|-
|bgcolor=#e7dcc3|Coxeter symbol|| t<sub>1</sub>(4<sub>21</sub>)
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]||{{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea_1|3a|nodea}}
|-
|bgcolor=#e7dcc3|7-faces||19680 total:<BR>
240 [[3 21 polytope|3<sub>21</sub>]]<BR>
17280 [[Rectified 7-simplex|t<sub>1</sub>{3<sup>6</sup>}]]<BR>
2160 [[Rectified 7-orthoplex|t<sub>1</sub>{3<sup>5</sup>,4}]]
|-
|bgcolor=#e7dcc3|6-faces||375840
|-
|bgcolor=#e7dcc3|5-faces||1935360
|-
|bgcolor=#e7dcc3|4-faces||3386880
|-
|bgcolor=#e7dcc3|Cells||2661120
|-
|bgcolor=#e7dcc3|Faces||1028160
|-
|bgcolor=#e7dcc3|Edges||181440
|-
|bgcolor=#e7dcc3|Vertices||6720
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||2<sub>21</sub> prism
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||[[E8 (mathematics)|E<sub>8</sub>]], [3<sup>4,2,1</sup>]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}
The '''rectified 4<sub>21</sub>''' can be seen as a [[Rectification (geometry)|rectification]] of the 4<sub>21</sub> polytope, creating new vertices on the center of edges of the 4<sub>21</sub>.
 
=== Alternative names ===
* Rectified dischiliahectohexaconta-myriaheptachiliadiacosioctaconta-zetton for rectified 2160-17280 polyzetton (Acronym riffy) (Jonathan Bowers)<ref>Klitzing, (o3o3o3o *c3o3o3x3o - riffy)</ref>
 
===Construction===
It is created by a [[Wythoff construction]] upon a set of 8 [[hyperplane]] mirrors in 8-dimensional space. It is named for being a [[Rectification (geometry)|rectification]] of the 4<sub>21</sub>. Vertices are positioned at the midpoint of all the edges of 4<sub>21</sub>, and new edges connecting them.
 
The facet information can be extracted from its [[Coxeter-Dynkin diagram]].
: {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea_1|3a|nodea}}
 
Removing the node on the short branch leaves the [[rectified 7-simplex]]:
: {{CDD|nodea|3a|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 7-orthoplex]] in its alternated form:
: {{CDD|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea_1|3a|nodea}}
 
Removing the node on the end of the 4-length branch leaves the [[3 21 polytope|3<sub>21</sub>]]:
: {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea_1}}
 
The [[vertex figure]] is determined by removing the ringed node and adding a ring to the neighboring node. This makes a '''[[Gosset 2 21 polytope|2<sub>21</sub>]]''' prism.
: {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea_1|2|nodea_1}}
 
=== Projections ===
 
====2D====
These graphs represent orthographic projections in the E<sub>8</sub>,E<sub>7</sub>,E<sub>6</sub>, and B<sub>8</sub>,D<sub>8</sub>,D<sub>7</sub>,D<sub>6</sub>,D<sub>5</sub>,D<sub>4</sub>,D<sub>3</sub>,A<sub>7</sub>,A<sub>5</sub> [[Coxeter plane]]s. The vertex colors are by overlapping multiplicity in the projection: colored by increasing order of multiplicities as red, orange, yellow, green.
 
{| class=wikitable width=720
!E<sub>8</sub> / H<sub>4</sub><BR>[30]
![20]
![24]
|- align=center
|[[File:4 21 t1 E8.svg|200px]]
|[[File:4 21 t1 p20.svg|200px]]
|[[File:4 21 t1 p24.svg|200px]]
|- align=center
!E<sub>7</sub><BR>[18]
!E<sub>6</sub> / F<sub>4</sub><BR>[12]
![6]
|- align=center
|[[File:4 21 t1 E7.svg|200px]]
|[[File:4 21 t1 E6.svg|200px]]
|[[File:4 21 t1 mox.svg|200px]]
|}
 
{| class=wikitable width=720
|- align=center
!D<sub>3</sub> / B<sub>2</sub> / A<sub>3</sub><BR>[4]
!D<sub>4</sub> / B<sub>3</sub> / A<sub>2</sub> / G<sub>2</sub><BR>[6]
!D<sub>5</sub> / B<sub>4</sub><BR>[8]
|- align=center
|[[File:4 21 t1 B2.svg|200px]]
|[[File:4 21 t1 B3.svg|200px]]
|[[File:4 21 t1 B4.svg|200px]]
|- align=center
!D<sub>6</sub> / B<sub>5</sub> / A<sub>4</sub><BR>[10]
!D<sub>7</sub> / B<sub>6</sub><BR>[12]
!D<sub>8</sub> / B<sub>7</sub> / A<sub>6</sub><BR>[14]
|- align=center
|[[File:4 21 t1 B5.svg|200px]]
|[[File:4 21 t1 B6.svg|200px]]
|[[File:4 21 t1 B7.svg|200px]]
|- align=center
!B<sub>8</sub><BR>[16/2]
!A<sub>5</sub><BR>[6]
!A<sub>7</sub><BR>[8]
|- align=center
|[[File:4 21 t1 B8.svg|200px]]
|[[File:4 21 t1 A5.svg|200px]]
|[[File:4 21 t1 A7.svg|200px]]
|}
 
== Birectified 4_21 polytope ==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Birectified 4<sub>21</sub> polytope
|-
|bgcolor=#e7dcc3|Type||[[Uniform 8-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| t<sub>2</sub>{3,3,3,3,3<sup>2,1</sup>}
|-
|bgcolor=#e7dcc3|Coxeter symbol|| t<sub>2</sub>(4<sub>21</sub>)
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]||{{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea_1|3a|nodea|3a|nodea}}
|-
|bgcolor=#e7dcc3|7-faces||19680 total:<BR>
17280 [[Birectified 7-simplex|t<sub>2</sub>{3<sup>6</sup>}]] [[File:7-simplex t2.svg|40px]]<BR>
2160 [[birectified 7-orthoplex|t<sub>2</sub>{3<sup>5</sup>,4}]] [[File:7-cube t4.svg|40px]]<BR>
240 [[Rectified 3 21 polytope|t<sub>1</sub>(3<sub>21</sub>)]] [[File:Up2 3 21 t1 E7.svg|40px]]
|-
|bgcolor=#e7dcc3|6-faces||382560
|-
|bgcolor=#e7dcc3|5-faces||2600640
|-
|bgcolor=#e7dcc3|4-faces||7741440
|-
|bgcolor=#e7dcc3|Cells||9918720
|-
|bgcolor=#e7dcc3|Faces||5806080
|-
|bgcolor=#e7dcc3|Edges||1451520
|-
|bgcolor=#e7dcc3|Vertices||60480
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[5-demicube]]-triangular duoprism
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||[[E8 (mathematics)|E<sub>8</sub>]], [3<sup>4,2,1</sup>]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}
The '''birectified 4<sub>21</sub>'''can be seen as a second [[Rectification (geometry)|rectification]] of the uniform 4<sub>21</sub> polytope. Vertices of this polytope are positioned at the centers of all the 60480 triangular faces of the 4<sub>21</sub>.
 
=== Alternative names ===
* Birectified dischiliahectohexaconta-myriaheptachiliadiacosioctaconta-zetton for birectified 2160-17280 polyzetton (acronym borfy) (Jonathan Bowers)<ref>Klitzing, (o3o3o3o *c3o3x3o3o - borfy)</ref>
 
===Construction===
It is created by a [[Wythoff construction]] upon a set of 8 [[hyperplane]] mirrors in 8-dimensional space. It is named for being a [[Rectification (geometry)|birectification]] of the 4<sub>21</sub>. Vertices are positioned at the center of all the triangle faces of 4<sub>21</sub>.
 
The facet information can be extracted from its [[Coxeter-Dynkin diagram]].
: {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea_1|3a|nodea|3a|nodea}}
 
Removing the node on the short branch leaves the [[birectified 7-simplex]]. There are 17280 of these facets.
: {{CDD|nodea|3a|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 7-orthoplex]] in its alternated form. There are 2160 of these facets.
: {{CDD|nodea|3a|branch|3a|nodea|3a|nodea_1|3a|nodea|3a|nodea}}
 
Removing the node on the end of the 4-length branch leaves the [[Rectified 3 21 polytope|rectified 3<sub>21</sub>]]. There are 240 of these facets.
: {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea_1|3a|nodea}}
 
The [[vertex figure]] is determined by removing the ringed node and adding rings to the neighboring nodes. This makes a [[5-demicube]]-triangular duoprism.
: {{CDD|nodea|3a|nodea|3a|branch|3a|nodea_1|2|nodea_1|3a|nodea}}
 
=== Projections ===
 
====2D====
These graphs represent orthographic projections in the E<sub>8</sub>,E<sub>7</sub>,E<sub>6</sub>, and B<sub>8</sub>,D<sub>8</sub>,D<sub>7</sub>,D<sub>6</sub>,D<sub>5</sub>,D<sub>4</sub>,D<sub>3</sub>,A<sub>7</sub>,A<sub>5</sub> [[Coxeter plane]]s. Edges are not drawn. The vertex colors are by overlapping multiplicity in the projection: colored by increasing order of multiplicities as red, orange, yellow, green, etc.
 
{| class=wikitable width=720
!E<sub>8</sub> / H<sub>4</sub><BR>[30]
![20]
![24]
|- align=center
|[[File:4 21 t2 E8.svg|200px]]
|[[File:4 21 t2 p20.svg|200px]]
|[[File:4 21 t2 p24.svg|200px]]
|- align=center
!E<sub>7</sub><BR>[18]
!E<sub>6</sub> / F<sub>4</sub><BR>[12]
![6]
|- align=center
|[[File:4 21 t2 E7.svg|200px]]
|[[File:4 21 t2 E6.svg|200px]]
|[[File:4 21 t2 mox.svg|200px]]
|}
 
{| class=wikitable width=720
|- align=center
!D<sub>3</sub> / B<sub>2</sub> / A<sub>3</sub><BR>[4]
!D<sub>4</sub> / B<sub>3</sub> / A<sub>2</sub> / G<sub>2</sub><BR>[6]
!D<sub>5</sub> / B<sub>4</sub><BR>[8]
|- align=center
|[[File:4 21 t2 B2.svg|200px]]
|[[File:4 21 t2 B3.svg|200px]]
|[[File:4 21 t2 B4.svg|200px]]
|- align=center
!D<sub>6</sub> / B<sub>5</sub> / A<sub>4</sub><BR>[10]
!D<sub>7</sub> / B<sub>6</sub><BR>[12]
!D<sub>8</sub> / B<sub>7</sub> / A<sub>6</sub><BR>[14]
|- align=center
|[[File:4 21 t2 B5.svg|200px]]
|[[File:4 21 t2 B6.svg|200px]]
|[[File:4 21 t2 B7.svg|200px]]
|- align=center
!B<sub>8</sub><BR>[16/2]
!A<sub>5</sub><BR>[6]
!A<sub>7</sub><BR>[8]
|- align=center
|[[File:4 21 t2 B8.svg|200px]]
|[[File:4 21 t2 A5.svg|200px]]
|[[File:4 21 t2 A7.svg|200px]]
|}
 
== Trirectified 4_21 polytope==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Trirectified 4<sub>21</sub> polytope
|-
|bgcolor=#e7dcc3|Type||[[Uniform 8-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| t<sub>3</sub>{3,3,3,3,3<sup>2,1</sup>}
|-
|bgcolor=#e7dcc3|Coxeter symbol|| t<sub>3</sub>(4<sub>21</sub>)
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]||{{CDD|nodea|3a|nodea|3a|branch|3a|nodea_1|3a|nodea|3a|nodea|3a|nodea}}
|-
|bgcolor=#e7dcc3|7-faces||19680
|-
|bgcolor=#e7dcc3|6-faces||382560
|-
|bgcolor=#e7dcc3|5-faces||2661120
|-
|bgcolor=#e7dcc3|4-faces||9313920
|-
|bgcolor=#e7dcc3|Cells||16934400
|-
|bgcolor=#e7dcc3|Faces||14515200
|-
|bgcolor=#e7dcc3|Edges||4838400
|-
|bgcolor=#e7dcc3|Vertices||241920
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[tetrahedron]]-[[rectified 5-cell]] duoprism
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||[[E8 (mathematics)|E<sub>8</sub>]], [3<sup>4,2,1</sup>]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}
 
=== Alternative names ===
* Trirectified dischiliahectohexaconta-myriaheptachiliadiacosioctaconta-zetton for trirectified 2160-17280 polyzetton (acronym torfy) (Jonathan Bowers)<ref>Klitzing, (o3o3o3o *c3x3o3o3o - torfy)</ref>
 
===Construction===
It is created by a [[Wythoff construction]] upon a set of 8 [[hyperplane]] mirrors in 8-dimensional space. It is named for being a [[Rectification (geometry)|birectification]] of the 4<sub>21</sub>. Vertices are positioned at the center of all the triangle faces of 4<sub>21</sub>.
 
The facet information can be extracted from its [[Coxeter-Dynkin diagram]].
: {{CDD|nodea|3a|nodea|3a|branch|3a|nodea_1|3a|nodea|3a|nodea|3a|nodea}}
 
Removing the node on the short branch leaves the [[trirectified 7-simplex]]:
: {{CDD|nodea|3a|nodea|3a|nodea|3a|nodea_1|3a|nodea|3a|nodea|3a|nodea}}
 
Removing the node on the end of the 2-length branch leaves the [[trirectified 7-orthoplex]] in its alternated form:
: {{CDD|nodea|3a|branch|3a|nodea_1|3a|nodea|3a|nodea|3a|nodea}}
 
Removing the node on the end of the 4-length branch leaves the [[Birectified 3 21 polytope|birectified 3<sub>21</sub>]]:
: {{CDD|nodea|3a|nodea|3a|branch|3a|nodea_1|3a|nodea|3a|nodea}}
 
The [[vertex figure]] is determined by removing the ringed node and ring the neighbor nodes. This makes a [[tetrahedron]]-[[rectified 5-cell]] duoprism.
: {{CDD|nodea|3a|nodea|3a|branch_10|2|nodea_1|3a|nodea|3a|nodea}}
 
=== Projections ===
 
====2D====
These graphs represent orthographic projections in the E<sub>7</sub>,E<sub>6</sub>, and B<sub>8</sub>,D<sub>8</sub>,D<sub>7</sub>,D<sub>6</sub>,D<sub>5</sub>,D<sub>4</sub>,D<sub>3</sub>,A<sub>7</sub>,A<sub>5</sub> [[Coxeter plane]]s. The vertex colors are by overlapping multiplicity in the projection: colored by increasing order of multiplicities as red, orange, yellow, green.
 
(E<sub>8</sub> and B<sub>8</sub> were too large to display)
 
{| class=wikitable width=600
<!--!E<sub>8</sub> / H<sub>4</sub><BR>[30]
![20]
![24]
|- align=center
|[[File:4 21 t3 E8.svg|200px]]
|[[File:4 21 t3 p20.svg|200px]]
|[[File:4_21_t3_p24.svg|200px]]-->
|- align=center
!E<sub>7</sub><BR>[18]
!E<sub>6</sub> / F<sub>4</sub><BR>[12]
!D<sub>4</sub> - E<sub>6</sub><BR>[6]
|- align=center
|[[File:4 21 t3 E7.svg|200px]]
|[[File:4 21 t3 E6.svg|200px]]
|[[File:4 21 t3 mox.svg|200px]]
|}
 
{| class=wikitable width=600
|- align=center
!D<sub>3</sub> / B<sub>2</sub> / A<sub>3</sub><BR>[4]
!D<sub>4</sub> / B<sub>3</sub> / A<sub>2</sub> / G<sub>2</sub><BR>[6]
!D<sub>5</sub> / B<sub>4</sub><BR>[8]
|- align=center
|[[File:4 21 t3 B2.svg|200px]]
|[[File:4 21 t3 B3.svg|200px]]
|[[File:4 21 t3 B4.svg|200px]]
|- align=center
!D<sub>6</sub> / B<sub>5</sub> / A<sub>4</sub><BR>[10]
!D<sub>7</sub> / B<sub>6</sub><BR>[12]
!D<sub>8</sub> / B<sub>7</sub> / A<sub>6</sub><BR>[14]
|- align=center
<!--|[[File:4 21 t3 B5.svg|200px]]-->
|[[File:4 21 t3 B6.svg|200px]]
|[[File:4 21 t3 B7.svg|200px]]
|- align=center
<!--!B<sub>8</sub><BR>[16/2]-->
!A<sub>5</sub><BR>[6]
!A<sub>7</sub><BR>[8]
|- align=center
<!--|[[File:4 21 t3 B8.svg|200px]]-->
|[[File:4 21 t3 A5.svg|200px]]
|[[File:4 21 t3 A7.svg|200px]]
|}
 
== See also==
* [[List of E8 polytopes]]
 
== Notes ==
{{reflist|2}}
 
== 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}}
* [[Harold Scott MacDonald Coxeter|Coxeter, H. S. M.]], ''Regular Complex Polytopes'', Cambridge University Press, (1974).
* '''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 p347 (figure 3.8c) by [[Peter McMullen]]: (30-gonal node-edge graph of 4<sub>21</sub>)
* {{KlitzingPolytopes|polyzetta.htm|8D|uniform polytopes (polyzetta)}} o3o3o3o *c3o3o3o3x - fy, o3o3o3o *c3o3o3x3o - riffy, o3o3o3o *c3o3x3o3o - borfy, o3o3o3o *c3x3o3o3o - torfy
 
{{Polytopes}}
 
{{DEFAULTSORT:4 21 Polytope}}
[[Category:8-polytopes]]

Latest revision as of 14:33, 21 July 2014

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