View source for 4 21 polytope

{{Short description|Polytope in 8-dimensional geometry}}
{{DISPLAYTITLE:4<sub>&hairsp;21</sub> polytope}}
{| class=wikitable width=450 align=right style="margin-left:1em;"
|- align=center valign=top
|[[File:4 21 t0 E6.svg|150px]]<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|150px]]<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|150px]]<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|150px]]<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|150px]]<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|150px]]<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|150px]]<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|150px]]<BR>Trirectified 4<sub>21</sub><BR>{{CDD|nodea|3a|nodea|3a|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 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 (mathematics)|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>

Its [[Coxeter symbol]] is '''4<sub>21</sub>''', describing 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}}.

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>'''.

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 [[permutation]]s of one or more rings in this Coxeter-Dynkin diagram: {{CDD|nodea|3a|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea}}.
{{clear}}

== 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 diagram]]s||{{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea_1}}<BR>= {{CDD|nodes_10r|3ab|nodes|3ab|nodes|split5c|nodes}}
|-
|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>], order  696729600
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}
The '''4<sub>21</sub>''' polytope has 17,280 [[7-simplex]] and 2,160 [[7-orthoplex]] [[Facet (geometry)|facets]], and 240 vertices.  Its [[vertex figure]] is the '''[[3 21 polytope|3<sub>21</sub>]]''' polytope.   As its vertices represent the [[root system|root vectors]] of the [[simple Lie group]] [[E8 (mathematics)|E<sub>8</sub>]], this polytope is sometimes referred to as the '''E<sub>8</sub> root polytope'''.

The vertices of this polytope can also be obtained by taking the 240 [[Octonion#Integral_octonions|integral octonions]] of norm 1.  Because the [[octonions]] are a nonassociative [[normed division algebra]], these 240 points have a multiplication operation making them not into a group but rather a [[Quasigroup#Loop|loop]], in fact a [[Moufang loop]].

For visualization this 8-dimensional polytope is often displayed in a special skewed [[orthographic projection]] direction that fits its 240 vertices within a regular [[triacontagon]] (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.

=== Alternate names ===
*This polytope 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 ({{math|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 a multiple of 4).

Each vertex has 56 nearest neighbors; for example, the nearest neighbors of the vertex <math>(1,1,1,1,1,1,1,1)</math> are those whose coordinates sum to 4, namely the 28 obtained by permuting the coordinates of <math>(2, 2,0,0,0,0,0,0)\,</math> and the 28 obtained by permuting the coordinates of <math>(1,1,1,1,1,1,-1,-1)</math>.  These 56 points are the vertices of a [[3_21 polytope|3<sub>21</sub>]] polytope in 7 dimensions.

Each vertex has 126 second nearest neighbors:  for example, the nearest neighbors of the vertex <math>(1,1,1,1,1,1,1,1)</math> are those whose coordinates sum to 0, namely the 56 obtained by permuting the coordinates of <math>(2, -2,0,0,0,0,0,0)\,</math> and the 70 obtained by permuting the coordinates of <math>(1,1,1,1,-1,-1,-1,-1)</math>.   These 126 points are the vertices of a [[2_31 polytope|2<sub>31</sub>]] polytope in 7 dimensions.

Each vertex also has 56 third nearest neighbors, which are the negatives of its nearest neighbors, and one antipodal vertex, for a total of <math>1 + 56 + 126 + 56 + 1 = 240</math> vertices.


Another construction is by taking signed combination of 14 codewords of  [[Hamming code#Encoding|8-bit Extended Hamming code(8,4)]] that give 14 × 2<sup>4</sup> = 224 vertices and adding trivial signed axis <math>( \pm 2,0,0,0, 0,0,0,0)</math> for last 16 vertices. In this case, vertices are distance of <math>\sqrt{4}</math>  from origin rather than <math>\sqrt{8}</math>.

  Hamming 8-bit Code
  0  0 0 0 0 0 0 0 0
  1  1 1 1 1 0 0 0 0   ⇒   ± ± ± ± 0 0 0 0
  2  1 1 0 0 1 1 0 0   ⇒   ± ± 0 0 ± ± 0 0
  3  0 0 1 1 1 1 0 0   ⇒   0 0 ± ± ± ± 0 0
  4  1 0 1 0 1 0 1 0   ⇒   ± 0 ± 0 ± 0 ± 0        ±2 0 0 0 0 0 0 0
  5  0 1 0 1 1 0 1 0   ⇒   0 ± 0 ± ± 0 ± 0        0 ±2 0 0 0 0 0 0
  6  0 1 1 0 0 1 1 0   ⇒   0 ± ± 0 0 ± ± 0        0 0 ±2 0 0 0 0 0
  7  1 0 0 1 0 1 1 0   ⇒   ± 0 0 ± 0 ± ± 0        0 0 0 ±2 0 0 0 0
  8  0 1 1 0 1 0 0 1   ⇒   0 ± ± 0 ± 0 0 ±        0 0 0 0 ±2 0 0 0
  9  1 0 0 1 1 0 0 1   ⇒   ± 0 0 ± ± 0 0 ±        0 0 0 0 0 ±2 0 0
  A  1 0 1 0 0 1 0 1   ⇒   ± 0 ± 0 0 ± 0 ±        0 0 0 0 0 0 ±2 0
  B  0 1 0 1 0 1 0 1   ⇒   0 ± 0 ± 0 ± 0 ±        0 0 0 0 0 0 0 ±2
  C  1 1 0 0 0 0 1 1   ⇒   ± ± 0 0 0 0 ± ±
  D  0 0 1 1 0 0 1 1   ⇒   0 0 ± ± 0 0 ± ±
  E  0 0 0 0 1 1 1 1   ⇒   0 0 0 0 ± ± ± ±
  F  1 1 1 1 1 1 1 1
                           ( 224 vertices     +     16 vertices )

Another decomposition gives the 240 points in 9-dimensions as an [[Heptellated_8-simplex#Coordinates|expanded 8-simplex]], {{CDD|nodes_11|3ab|nodes|3ab|nodes|3ab|branch}} and two opposite [[Birectified_8-simplex#Coordinates_2|birectified 8-simplexes]], {{CDD|nodes|3ab|nodes|3ab|nodes_10lr|3ab|branch}} and {{CDD|nodes|3ab|nodes|3ab|nodes_01lr|3ab|branch}}. 
: {{tmath|(3,-3,0,0,0,0,0,0,0)}} : 72 vertices
: {{tmath|(-2,-2,-2,1,1,1,1,1,1)}} : 84 vertices
: {{tmath|(2,2,2,-1,-1,-1,-1,-1,-1)}} : 84 vertices
This arises similarly to the relation of the [[A8 lattice]] and [[E8 lattice]], sharing 8 mirrors of A8: [[File:A8-e8 lattice relation.png]].

{| class=wikitable
|+ A7 Coxeter plane projections
!Name
!4<sub>21</sub><BR>{{CDD|nodea|3a|nodes|split2|node|3|node|3|node|3|node|3|node_1}}
![[Heptellated_8-simplex#Coordinates|expanded 8-simplex]]<BR>{{CDD|nodes_11|3ab|nodes|3ab|nodes|3ab|branch}}
![[Birectified_8-simplex#Coordinates_2|birectified 8-simplex]]<BR>{{CDD|nodes|3ab|nodes|3ab|nodes_10lr|3ab|branch}}
![[Birectified_8-simplex#Coordinates_2|birectified 8-simplex]]<BR>{{CDD|nodes|3ab|nodes|3ab|nodes_01lr|3ab|branch}}
|-
!Vertices||240||72||84||84
|-
!Image
|[[File:4 21 t0 A7.svg|200px]]
|[[File:8-simplex t07 A7.svg|200px]]
|colspan=2 align=center|[[File:8-simplex t2 A7.svg|200px]]
|}

=== 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 and faces ===
The facet information of this polytope 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 7-simplex facet touches only 7-orthoplex facets, while alternate facets of an orthoplex facet touch either a simplex or another orthoplex. There are 17,280 simplex facets and 2160 orthoplex facets.

Since every 7-simplex has 7 6-simplex facets, each incident to no other 6-simplex, the 4<sub>21</sub> polytope has 120,960 (7×17,280) 6-simplex faces that are facets of 7-simplexes. Since every 7-orthoplex has 128 (2<sup>7</sup>) 6-simplex facets, half of which are not incident to 7-simplexes, the 4<sub>21</sub> polytope has 138,240 (2<sup>6</sup>×2160) <!-- maybe should be half as many, 2<sup>5</sup>×2160? --> 6-simplex faces that are not facets of 7-simplexes. The 4<sub>21</sub> polytope thus has two kinds of 6-simplex faces, not interchanged by symmetries of this polytope. The total number of 6-simplex faces is 259200 (120,960+138,240).

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}}

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 collapsible collapsed" style="text-align: center;"
!colspan="2"|
! [[configuration (polytope)|Configuration matrix]]
|- valign=top
!E<sub>8</sub>||width=80|{{CDD|nodea_1|3a|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea}}||[[Face_(geometry)#k-face|''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>||f<sub>5</sub>||colspan=2|f<sub>6</sub>||colspan=2|f<sub>7</sub>|| [[vertex figure|''k''-figure]]||notes
|- align=right
|E<sub>7</sub> ||{{CDD|nodea_x|2|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea}}|| ( )
! f<sub>0</sub>
|BGCOLOR="#e0e0ff" |'''240'''||56||756||4032||10080||12096||4032||2016||576||126||[[3_21 polytope]] || E<sub>8</sub>/E<sub>7</sub> = 192×10!/(72×8!) = 240
|- align=right
|A<sub>1</sub>E<sub>6</sub> ||{{CDD|nodea_1|2|nodea_x|2|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea}}|| { }
! f<sub>1</sub>
||2||BGCOLOR="#ffe0ff" |'''6720'''||27||216||720||1080||432||216||72||27||[[2_21 polytope]] || E<sub>8</sub>/A<sub>1</sub>E<sub>6</sub> = 192×10!/(2×72×6!) = 6720
|- align=right
|A<sub>2</sub>D<sub>5</sub> ||{{CDD|nodea_1|3a|nodea|2|nodea_x|2|nodea|3a|branch|3a|nodea|3a|nodea}}|| [[triangle|{3}]]
! f<sub>2</sub>
||3||3||BGCOLOR="#ffe0e0" |'''60480'''||16||80||160||80||40||16||10||[[5-demicube]] || E<sub>8</sub>/A<sub>2</sub>D<sub>5</sub> = 192×10!/(6×2<sup>4</sup>×5!) = 60480
|- align=right
|A<sub>3</sub>A<sub>4</sub> ||{{CDD|nodea_1|3a|nodea|3a|nodea|2|nodea_x|2|branch|3a|nodea|3a|nodea}}|| [[tetrahedron|{3,3} ]]
! f<sub>3</sub>
||4||6||4||BGCOLOR="#ffffe0" |'''241920'''||10||30||20||10||5||5||[[Rectified 5-cell]] || E<sub>8</sub>/A<sub>3</sub>A<sub>4</sub> = 192×10!/(4!×5!) = 241920
|- align=right
|A<sub>4</sub>A<sub>2</sub>A<sub>1</sub> ||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|nodea|2|nodes_x0|2|nodea|3a|nodea}}|| [[5-cell|{3,3,3} ]]
! f<sub>4</sub>
||5||10||10||5||BGCOLOR="#e0ffe0" |'''483840'''||6||6||3||2||3||[[Triangular prism]] ||E<sub>8</sub>/A<sub>4</sub>A<sub>2</sub>A<sub>1</sub> = 192×10!/(5!×3!×2) = 483840
|- align=right
|A<sub>5</sub>A<sub>1</sub> ||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|nodea|3a|nodes_0x|2|nodea_x|2|nodea}}|| [[5-simplex|{3,3,3,3} ]]
! f<sub>5</sub>
||6||15||20||15||6||BGCOLOR="#e0ffff" |'''483840'''||2||1||1||2||[[Isosceles triangle]] ||E<sub>8</sub>/A<sub>5</sub>A<sub>1</sub> = 192×10!/(6!×2) = 483840
|- align=right
|A<sub>6</sub> ||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|nodea|3a|nodes_0x|3a|nodea|2|nodea_x}}||rowspan=2| [[6-simplex|{3,3,3,3,3}]]
!rowspan=2| f<sub>6</sub>
||7||21||35||35||21||7||BGCOLOR="#e0e0ff"|'''138240'''||BGCOLOR="#e0e0ff"|*||1||1||rowspan=2|{ } ||E<sub>8</sub>/A<sub>6</sub> = 192×(10!×7!) = 138240
|- align=right
|A<sub>6</sub>A<sub>1</sub> ||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|nodea|3a|branch|2|nodea_x|2|nodea}}
||7||21||35||35||21||7||BGCOLOR="#e0e0ff"|*||BGCOLOR="#e0e0ff"|'''69120'''||0||2 ||E<sub>8</sub>/A<sub>6</sub>A<sub>1</sub> = 192×10!/(7!×2) = 69120
|- align=right
|A<sub>7</sub> ||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|nodea|3a|nodes_0x|3a|nodea|3a|nodea}}|| [[7-simplex|{3,3,3,3,3,3} ]]
!rowspan=2| f<sub>7</sub>
||8||28||56||70||56||28||8||0||BGCOLOR="#ffe0ff"|'''17280'''||BGCOLOR="#ffe0ff"|*||rowspan=2| ( )||E<sub>8</sub>/A<sub>7</sub> = 192×10!/8! = 17280
|- align=right
|D<sub>7</sub> ||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea|2|nodea_x}}|| [[7-orthoplex|{3,3,3,3,3,4}]]
||14||84||280||560||672||448||64||64||BGCOLOR="#ffe0ff"|*||BGCOLOR="#ffe0ff"|'''2160''' ||E<sub>8</sub>/D<sub>7</sub> = 192×10!/(2<sup>6</sup>×7!) = 2160
|}

=== Projections ===
{| class=wikitable
|- align=center
|[[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 {{radic|2}}({{radic|5}}&minus;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 {{radic|2}} <ref>[http://theoryofeverything.org/TOE/JGM/e8Flyer.nbp e8Flyer.nb]</ref>
|[[File:E8-3Dprint-b.png|240px]] E8 rotated to H4+H4φ, projected to 3D, converted to STL, and printed in nylon plastic. Projection basis used: 
:x = {1, &phi;, 0, &minus;1, &phi;, 0,0,0}
:y = {&phi;, 0, 1, &phi;, 0, &minus;1,0,0}
:z = {0, 1, &phi;, 0, &minus;1, &phi;,0,0}
|}

==== 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 collapsible collapsed" width=720
!colspan=3| Orthogonal projections
|-
!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)
|- 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 {{radic|2}}({{radic|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 (scaled by the [[golden ratio]]) 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 collapsible collapsed"
!colspan=2|E8/H4 Coxeter plane foldings
|-
|- 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 ===
In 4-dimensional complex geometry, the  [[regular complex polytope]] <sub>3</sub>{3}<sub>3</sub>{3}<sub>3</sub>{3}<sub>3</sub>, and [[Coxeter diagram]] {{CDD|3node_1|3|3node|3|3node|3|3node}} exists with the same vertex arrangement as the 4<sub>21</sub> polytope. It is self-dual. Coxeter called it the [[Witting polytope]], after [[Alexander Witting]]. Coxeter expresses its [[Shephard group]] symmetry by <sub>3</sub>[3]<sub>3</sub>[3]<sub>3</sub>[3]<sub>3</sub>.<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<sub>21</sub> 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 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}}

=== Coordinates ===
The [[Cartesian coordinates]] of the 6720 vertices of the rectified 4<sub>21</sub> is given by all permutations of coordinates from three other uniform polytope:
* [[hexic 8-cube]] - odd negatives: ½(±1,±1,±1,±1,±1,±1,±3,±3) - 3584 vertices<ref>{{Cite web|url=https://bendwavy.org/klitzing/incmats/sotho.htm|title = Sotho}}</ref>
* [[birectified 8-cube]] - (0,0,±1,±1,±1,±1,±1,±1) - 1792 vertices<ref>{{Cite web|url=https://bendwavy.org/klitzing/incmats/bro.htm|title=Bro}}</ref>
* [[cantellated 8-orthoplex]] - (0,0,0,0,0,0,±1,±1,±2) - 1344 vertices<ref>{{Cite web|url=https://bendwavy.org/klitzing/incmats/srek.htm|title = Srek}}</ref>
{| class=wikitable
|+ D8 Coxeter plane projections
!Name
!Rectified 4<sub>21</sub><BR>{{CDD|nodea|3a|nodes|split2|node|3|node|3|node|3|node_1|3|node}}
![[birectified 8-cube]]<BR>{{CDD|nodes|split2|node_1|3|node|3|node|3|node|3|node|3|node}} = {{CDD|node|4|node|3|node_1|3|node|3|node|3|node|3|node|3|node}}
![[hexic 8-cube]]<BR>{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node|3|node|3|node|3|node|3|node_1|3|node}}
![[cantellated 8-orthoplex]]<BR>{{CDD|nodes|split2|node|3|node|3|node|3|node_1|3|node|3|node_1}} = {{CDD|node|4|node|3|node|3|node|3|node|3|node_1|3|node|3|node_1}}
|-
!Vertices||6720||1792||3584||1344
|-
!Image
|[[File:4 21 t1 B7.svg|200px]]
|[[File:8-cube t2 B7.svg|200px]]
|[[File:8-demicube t05 D8.svg|200px]]
|[[File:8-cube t57 B7.svg|200px]]
|}

=== 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 collapsible collapsed" width=720
!colspan=3|Orthogonal projections
|-
!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]]
|- 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<sub>21</sub> 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 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 collapsible collapsed" width=720
!colspan=3|Orthogonal projections
|-
!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]]
|- 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<sub>21</sub> 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 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>, 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>, and 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 collapsible collapsed" width=720
!colspan=3|Orthogonal projections
|-
<!--!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]]
|- 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, [https://www.wiley.com/en-us/Kaleidoscopes-p-9780471010036 wiley.com], {{ISBN|978-0-471-01003-6}}
** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45], p. 347 (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]]
[[Category:E8 (mathematics)]]