View source for 2 41 polytope

{{DISPLAYTITLE:2<sub>&thinsp;41</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 21 polytope|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<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 21 polytope|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<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 21 polytope|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 21 polytope|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 '''2<sub>41</sub>''' is a [[uniform 8-polytope]], constructed within the symmetry of the [[E8 (mathematics)|E<sub>8</sub>]] group.

Its [[Coxeter symbol]] is '''2<sub>41</sub>''', describing its bifurcating [[Coxeter-Dynkin diagram]], with a single ring on the end of the 2-node sequences.

The '''rectified 2<sub>41</sub>''' is constructed by points at the mid-edges of the '''2<sub>41</sub>'''. The '''birectified 2<sub>41</sub>''' is constructed by points at the triangle face centers of the '''2<sub>41</sub>''', and is the same as the [[Rectified 1 42 polytope|rectified 1<sub>42</sub>]].

These polytopes are part of a family of 255 (2<sup>8</sup>&nbsp;&minus;&nbsp;1) convex [[uniform polytope]]s in 8-dimensions, made of [[uniform polytope]] facets, defined by all permutations of rings in this [[Coxeter-Dynkin diagram]]: {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}.

== 2<sub>41</sub> polytope ==
{{clear}}
{| class="wikitable" align="right" style="margin-left:10px" width="280"
!bgcolor=#e7dcc3 colspan=2|2<sub>41</sub> polytope
|-
|bgcolor=#e7dcc3|Type||Uniform [[8-polytope]]
|-
|bgcolor=#e7dcc3|Family||[[Uniform 2 k1 polytope|2<sub>k1</sub> polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| {3,3,3<sup>4,1</sup>}
|-
|bgcolor=#e7dcc3|[[Coxeter symbol]]|| 2<sub>41</sub>
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}
|-
|bgcolor=#e7dcc3|7-faces||17520:<BR>240 [[Gosset 2 31 polytope|2<sub>31</sub>]][[Image:Gosset 2 31 polytope.svg|25px]]<BR>17280 [[7-simplex|{3<sup>6</sup>}]][[Image:7-simplex t0.svg|25px]]
|-
|bgcolor=#e7dcc3|6-faces||144960:<BR>6720 [[E6 polytope|2<sub>21</sub>]][[Image:E6 graph.svg|25px]]<BR>138240 [[6-simplex|{3<sup>5</sup>}]][[Image:6-simplex t0.svg|25px]]
|-
|bgcolor=#e7dcc3|5-faces||544320:<BR>60480 [[pentacross|2<sub>11</sub>]][[Image:Cross graph 5.svg|25px]]<BR>483840 [[5-simplex|{3<sup>4</sup>}]][[Image:5-simplex t0.svg|25px]]
|-
|bgcolor=#e7dcc3|4-faces||1209600:<BR>241920 [[pentachoron|2<sub>01</sub>]][[Image:4-simplex t0.svg|25px]]<BR>967680 [[pentachoron|{3<sup>3</sup>}]][[Image:4-simplex t0.svg|25px]]
|-
|bgcolor=#e7dcc3|Cells||1209600 [[tetrahedron|{3<sup>2</sup>}]][[Image:3-simplex t0.svg|25px]]
|-
|bgcolor=#e7dcc3|Faces||483840 [[triangle|{3}]][[Image:2-simplex t0.svg|25px]]
|-
|bgcolor=#e7dcc3|Edges||69120
|-
|bgcolor=#e7dcc3|Vertices||2160
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||'''[[7-demicube|1<sub>41</sub>]]'''
|-
|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 '''2<sub>41</sub>''' is composed of 17,520 [[Facet (geometry)|facets]] (240 [[Gosset 2 31 polytope|2<sub>31</sub>]] polytopes and 17,280 [[7-simplex|7-simplices]]), 144,960 ''6-faces'' (6,720 [[E6 polytope|2<sub>21</sub>]] polytopes and 138,240 [[6-simplex|6-simplices]]), 544,320 5-faces (60,480 [[pentacross|2<sub>11</sub>]] and 483,840 [[5-simplex|5-simplices]]), 1,209,600 ''4-faces'' ([[pentachoron|4-simplices]]), 1,209,600 cells ([[tetrahedron|tetrahedra]]), 483,840 [[Face (geometry)|faces]] ([[triangle]]s), 69,120 [[Edge (geometry)|edges]], and 2160 [[Vertex (geometry)|vertices]]. Its [[vertex figure]] is a [[7-demicube]].

This polytope is a facet in the [[2 51 honeycomb|uniform tessellation, 2<sub>51</sub>]] with [[Coxeter-Dynkin diagram]]:
:{{CDD|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}

=== Alternate names ===
*[[E. L. Elte]] named it V<sub>2160</sub> (for its 2160 vertices) in his 1912 listing of semiregular polytopes.<ref>Elte, 1912</ref>
*It is named '''2<sub>41</sub>''' by [[Coxeter]] for its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequence.
* '''Diacositetracont-myriaheptachiliadiacosioctaconta-zetton''' (Acronym Bay) - 240-17280 facetted polyzetton (Jonathan Bowers)<ref>Klitzing, (x3o3o3o *c3o3o3o3o - bay)</ref>

=== Coordinates ===

The 2160 vertices can be defined as follows:
: 16 permutations of (±4,0,0,0,0,0,0,0) of ([[8-orthoplex]])
: 1120 permutations of (±2,±2,±2,±2,0,0,0,0) of ([[trirectified 8-orthoplex]])
: 1024 permutations of (±3,±1,±1,±1,±1,±1,±1,±1) ''with an odd number of minus-signs''

=== Construction ===
It is created by a [[Wythoff construction]] upon a set of 8 [[hyperplane]] mirrors in 8-dimensional space.

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

Removing the node on the short branch leaves the [[7-simplex]]: {{CDD|nodea_1|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}. There are 17280 of these facets

Removing the node on the end of the 4-length branch leaves the [[Gosset 2 31 polytope|2<sub>31</sub>]], {{CDD|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea}}. There are 240 of these facets. They are centered at the positions of the 240 vertices in the [[Gosset 4 21 polytope|4<sub>21</sub>]] polytope.

The [[vertex figure]] is determined by removing the ringed node and ringing the neighboring node. This makes the [[7-demicube]], 1<sub>41</sub>, {{CDD|nodea_1|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}.

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>||{{CDD|nodea|3a|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea_1}}||[[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>||colspan=2|f<sub>4</sub>||colspan=2|f<sub>5</sub>||colspan=2|f<sub>6</sub>||colspan=2|f<sub>7</sub>|| [[vertex figure|''k''-figure]]||notes
|- align=right
|D<sub>7</sub> ||{{CDD|nodea|3a|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea|2|nodea_x}}||( )
!f<sub>0</sub>
|BGCOLOR="#e0e0ff" |'''2160'''||64||672||2240||560||2240||280||1344||84||448||14||64||[[7-demicube|h{4,3,3,3,3,3}]] || E<sub>8</sub>/D<sub>7</sub> = 192*10!/64/7! = 2160
|- align=right
|A<sub>6</sub>A<sub>1</sub> ||{{CDD|nodea|3a|nodea|3a|nodea|3a|nodea|3a|branch|2|nodea_x|2|nodea_1}}|| { }
!f<sub>1</sub>
||2||BGCOLOR="#ffe0ff"|'''69120'''||21||105||35||140||35||105||21||42||7||7||[[Rectified 6-simplex|r{3,3,3,3,3}]] || E<sub>8</sub>/A<sub>6</sub>A<sub>1</sub> = 192*10!/7!/2 = 69120
|- align=right
|A<sub>4</sub>A<sub>2</sub>A<sub>1</sub> ||{{CDD|nodea|3a|nodea|3a|nodea|3a|nodea|2|nodes_x0|2|nodea|3a|nodea_1}}|| [[triangle|{3}]]
!f<sub>2</sub>
||3||3||BGCOLOR="#ffe0e0"|'''483840'''||10||5||20||10||20||10||10||5||2||[[5-cell prism|{}x{3,3,3}]] || 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>3</sub>A<sub>3</sub> ||{{CDD|nodea|3a|nodea|3a|nodea|2|nodea_x|2|nodes_0x|3a|nodea|3a|nodea_1}}|| [[tetrahedron|{3,3}]]
!f<sub>3</sub>
||4||6||4||BGCOLOR="#ffffe0"|'''1209600'''||1||4||4||6||6||4||4||1||[[tetrahedral pyramid|{3,3}V( )]] || E<sub>8</sub>/A<sub>3</sub>A<sub>3</sub> = 192*10!/4!/4! = 1209600
|- align=right
|A<sub>4</sub>A<sub>3</sub> ||{{CDD|nodea|3a|nodea|3a|nodea|2|nodea_x|2|branch|3a|nodea|3a|nodea_1}}|| rowspan=2|[[5-cell|{3,3,3}]]
!rowspan=2|f<sub>4</sub>
||5||10||10||5||BGCOLOR="#e0ffe0"|'''241920'''||BGCOLOR="#e0ffe0"|*||4||0||6||0||4||0||[[Tetrahedron|{3,3}]] || E<sub>8</sub>/A<sub>4</sub>A<sub>3</sub> = 192*10!/5!/4! = 241920
|- align=right
|A<sub>4</sub>A<sub>2</sub> ||{{CDD|nodea|3a|nodea|2|nodea_x|2|nodea|3a|nodes_0x|3a|nodea|3a|nodea_1}}
||5||10||10||5||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|'''967680'''||1||3||3||3||3||1||[[triangular pyramid|{3}V( )]] || E<sub>8</sub>/A<sub>4</sub>A<sub>2</sub> = 192*10!/5!/3! = 967680
|- align=right
|D<sub>5</sub>A<sub>2</sub> ||{{CDD|nodea|3a|nodea|2|nodea_x|2|nodea|3a|branch|3a|nodea|3a|nodea_1}}||[[5-orthoplex|{3,3,3<sup>1,1</sup>}]] 
!rowspan=2|f<sub>5</sub>
||10||40||80||80||16||16||BGCOLOR="#e0ffff"|'''60480'''||BGCOLOR="#e0ffff"|*||3||0||3||0||[[Equilateral triangle|{3}]] || E<sub>8</sub>/D<sub>5</sub>A<sub>2</sub> = 192*10!/16/5!/2 = 40480
|- align=right
|A<sub>5</sub>A<sub>1</sub> ||{{CDD|nodea|2|nodea_x|2|nodea|3a|nodea|3a|nodes_0x|3a|nodea|3a|nodea_1}}|| [[5-simplex|{3,3,3,3}]]
||6||15||20||15||0||6||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|'''483840'''||1||2||2||1||[[Isosceles triangle|{ }V( )]] || E<sub>8</sub>/A<sub>5</sub>A<sub>1</sub> = 192*10!/6!/2 = 483840
|- align=right
|E<sub>6</sub>A<sub>1</sub> ||{{CDD|nodea|2|nodea_x|2|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea_1}}|| [[2_21 polytope|{3,3,3<sup>2,1</sup>}]]
!rowspan=2|f<sub>6</sub>
||27||216||720||1080||216||432||27||72||BGCOLOR="#e0e0ff"|'''6720'''||BGCOLOR="#e0e0ff"|*||2||0||rowspan=2|{ } || E<sub>8</sub>/E<sub>6</sub>A<sub>1</sub> = 192*10!/72/6! = 6720
|- align=right
|A<sub>6</sub> ||{{CDD|nodea_x|2|nodea|3a|nodea|3a|nodea|3a|nodes_0x|3a|nodea|3a|nodea_1}}|| [[6-simplex|{3,3,3,3,3}]]
||7||21||35||35||0||21||0||7||BGCOLOR="#e0e0ff"|*||BGCOLOR="#e0e0ff"|'''138240'''||1||1|| E<sub>8</sub>/A<sub>6</sub> = 192*10!/7! = 138240
|- align=right
|E<sub>7</sub> ||{{CDD|nodea_x|2|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea_1}}|| [[2_31 polytope|{3,3,3<sup>3,1</sup>}]]
!rowspan=2|f<sub>7</sub>
||126||2016||10080||20160||4032||12096||756||4032||56||576||BGCOLOR="#ffe0ff"|'''240'''||BGCOLOR="#ffe0ff"|*||rowspan=2|( ) ||E<sub>8</sub>/E<sub>7</sub> = 192*10!/72!/8! = 240
|- align=right
|A<sub>7</sub> ||{{CDD|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodes_0x|3a|nodea|3a|nodea_1}}|| [[7-simplex|{3,3,3,3,3,3}]]
||8||28||56||70||0||56||0||28||0||8||BGCOLOR="#ffe0ff"|*||BGCOLOR="#ffe0ff"|'''17280'''|| E<sub>8</sub>/A<sub>7</sub> = 192*10!/8! = 17280
|}

=== Visualizations ===

[[File:E8 241 Petrie Projection.png|right|thumb|290px|
The projection of '''2<sub>41</sub>''' to the '''E<sub>8</sub>''' Coxeter plane (aka. the Petrie projection) with polytope radius <math>2\sqrt{2}</math> and 69120 edges of length <math>2\sqrt{2}</math>]]

[[File:E8_241-3D.png|right|thumb|290px|Shown in 3D projection using the basis vectors [u,v,w] giving H3 symmetry:
{{ubl
 | u {{=}} (1, ''φ'', 0, −1, ''φ'', 0,0,0)
 | v {{=}} (''φ'', 0, 1, ''φ'', 0, −1,0,0)
 | w {{=}} (0, 1, ''φ'', 0, −1, ''φ'',0,0)
}}

The 2160 projected 2<sub>41</sub> polytope vertices are sorted and tallied by their 3D norm generating the increasingly transparent hulls for each set of tallied norms.
The overlapping vertices are color coded by overlap count. Also shown is a list of each hull group, the normed distance from the origin, and the number of vertices in the group. 
]]

[[File:E8_241-3D_Concentric_Hulls_List.png|right|thumb|290px|The 2160 projected 2<sub>41</sub> polytope projected to 3D (as above) with each normed hull group listed individually with vertex counts. Notice the last two outer hulls are a combination of two overlapped Icosahedrons (24) and a Icosidodecahedron (30).
]]
{| class=wikitable width=600
!E8<BR>[30]
![20]
![24]
|- align=center
|[[File:2 41 t0 E8.svg|200px]]<BR>(1)
|[[File:2 41 t0 p20.svg|200px]]
|[[File:2 41 t0 p24.svg|200px]]
|- align=center
!E7<BR>[18]
!E6<BR>[12]
![6]
|- align=center
|[[File:2 41 t0 E7.svg|200px]]
|[[File:2 41 t0 E6.svg|200px]]<BR>(1,8,24,32)
|[[File:2 41 t0 mox.svg|200px]]
|}

[[Petrie polygon]] projections are 12, 18, or 30-sided based on the E6, E7, and E8 symmetries (respectively). The 2160 vertices are all displayed, but lower symmetry forms have projected positions overlapping, shown as different colored vertices. For comparison, a B6 coxeter group is also shown.

{| class=wikitable width=600
|- align=center
!D3 / B2 / A3<BR>[4]
!D4 / B3 / A2<BR>[6]
!D5 / B4<BR>[8]
|- align=center
|[[File:2 41 t0 B2.svg|200px]]
|[[File:2 41 t0 B3.svg|200px]]
|[[File:2 41 t0 B4.svg|200px]]
|- align=center
!D6 / B5 / A4<BR>[10]
!D7 / B6<BR>[12]
!D8 / B7 / A6<BR>[14]
|- align=center
|[[File:2 41 t0 B5.svg|200px]]
|[[File:2 41 t0 B6.svg|200px]]<BR>(1,3,9,12,18,21,36)
|[[File:2 41 t0 B7.svg|200px]]
|- align=center
!B8<BR>[16/2]
!A5<BR>[6]
!A7<BR>[8]
|- align=center
|[[File:2 41 t0 B8.svg|200px]]
|[[File:2 41 t0 A5.svg|200px]]
|[[File:2 41 t0 A7.svg|200px]]
|}

=== Related polytopes and honeycombs ===
{{2 k1 polytopes}}

== Rectified 2<sub>41</sub> polytope ==
{| class="wikitable" align="right" style="margin-left:10px" width="280"
!bgcolor=#e7dcc3 colspan=2|Rectified 2<sub>41</sub> polytope
|-
|bgcolor=#e7dcc3|Type||Uniform [[8-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| t<sub>1</sub>{3,3,3<sup>4,1</sup>}
|-
|bgcolor=#e7dcc3|[[Coxeter symbol]]|| t<sub>1</sub>(2<sub>41</sub>)
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|nodea|3a|nodea_1|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}
|-
|bgcolor=#e7dcc3|7-faces||19680 total:
240 [[Rectified 2 21 polytope|t<sub>1</sub>(2<sub>21</sub>)]]<BR>
17280 [[Rectified 7-simplex|t<sub>1</sub>{3<sup>6</sup>}]]<BR>
2160 [[7-demicube|1<sub>41</sub>]]
|-
|bgcolor=#e7dcc3|6-faces||313440
|-
|bgcolor=#e7dcc3|5-faces||1693440
|-
|bgcolor=#e7dcc3|4-faces||4717440
|-
|bgcolor=#e7dcc3|Cells||7257600
|-
|bgcolor=#e7dcc3|Faces||5322240
|-
|bgcolor=#e7dcc3|Edges||19680
|-
|bgcolor=#e7dcc3|Vertices||69120
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[rectified 6-simplex]] prism
|-
|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 '''rectified 2<sub>41</sub>''' is a [[Rectification (geometry)|rectification]] of the 2<sub>41</sub> polytope, with vertices positioned at the mid-edges of the 2<sub>41</sub>.

=== Alternate names ===
* Rectified Diacositetracont-myriaheptachiliadiacosioctaconta-zetton for rectified 240-17280 facetted polyzetton (known as robay for short)<ref>Jonathan Bowers</ref><ref>Klitzing, (o3x3o3o *c3o3o3o3o - robay)</ref>

=== Construction ===
It is created by a [[Wythoff construction]] upon a set of 8 [[hyperplane]] mirrors in 8-dimensional space, defined by root vectors of the [[E8 (mathematics)|E<sub>8</sub>]] [[Coxeter group]].

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

Removing the node on the short branch leaves the [[rectified 7-simplex]]: {{CDD|nodea|3a|nodea_1|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}.

Removing the node on the end of the 4-length branch leaves the [[rectified 2 31 polytope|rectified 2<sub>31</sub>]], {{CDD|nodea|3a|nodea_1|3a|branch|3a|nodea|3a|nodea|3a|nodea}}.

Removing the node on the end of the 2-length branch leaves the [[7-demicube]], 1<sub>41</sub>{{CDD|nodea_1|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}.

The [[vertex figure]] is determined by removing the ringed node and ringing the neighboring node. This makes the [[rectified 6-simplex]] prism, {{CDD|nodea_1|2|branch_10|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}.

=== Visualizations ===
[[Petrie polygon]] projections are 12, 18, or 30-sided based on the E6, E7, and E8 symmetries (respectively). The 2160 vertices are all displayed, but lower symmetry forms have projected positions overlapping, shown as different colored vertices. For comparison, a B6 coxeter group is also shown.

{| class=wikitable width=600
!E8<BR>[30]
![20]
![24]
|- align=center
|[[File:2 41 t1 E8.svg|200px]]<BR>(1)
|[[File:2 41 t1 p20.svg|200px]]
|[[File:2 41 t1 p24.svg|200px]]
|- align=center
!E7<BR>[18]
!E6<BR>[12]
![6]
|- align=center
|[[File:2 41 t1 E7.svg|200px]]
|[[File:2 41 t1 E6.svg|200px]]<BR>(1,8,24,32)
|[[File:2 41 t1 mox.svg|200px]]
|}

{| class=wikitable width=600
|- align=center
!D3 / B2 / A3<BR>[4]
!D4 / B3 / A2<BR>[6]
!D5 / B4<BR>[8]
|- align=center
|[[File:2 41 t1 B2.svg|200px]]
|[[File:2 41 t1 B3.svg|200px]]
|[[File:2 41 t1 B4.svg|200px]]
|- align=center
!D6 / B5 / A4<BR>[10]
!D7 / B6<BR>[12]
!D8 / B7 / A6<BR>[14]
|- align=center
|[[File:2 41 t1 B5.svg|200px]]
|[[File:2 41 t1 B6.svg|200px]]<BR>(1,3,9,12,18,21,36)
|[[File:2 41 t1 B7.svg|200px]]
|- align=center
!B8<BR>[16/2]
!A5<BR>[6]
!A7<BR>[8]
|- align=center
|[[File:2 41 t1 B8.svg|200px]]
|[[File:2 41 t1 A5.svg|200px]]
|[[File:2 41 t1 A7.svg|200px]]
|}

== See also ==
* [[List of E8 polytopes]]

== Notes ==
{{reflist}}

== References ==
* {{citation | last = Elte | first = E. L. | title = The Semiregular Polytopes of the Hyperspaces | publisher = University of Groningen | location = Groningen | year = 1912}}
* [[Harold Scott MacDonald Coxeter|H. S. M. Coxeter]], ''Regular Polytopes'', 3rd Edition, Dover New York, 1973
* '''Kaleidoscopes: Selected Writings of H.S.M. Coxeter''', edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, {{ISBN|978-0-471-01003-6}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45]
* {{KlitzingPolytopes|polyzetta.htm|8D|Uniform polyzetta}} x3o3o3o *c3o3o3o3o - bay, o3x3o3o *c3o3o3o3o - robay

{{Polytopes}}

[[Category:8-polytopes]]
[[Category:E8 (mathematics)]]