View source for 1 42 polytope

{{DISPLAYTITLE:1{{hair space}}<sub>42</sub> polytope}}<!--Space after "1" needed for title equivalence-->
{| 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<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 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<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 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 '''1<sub>42</sub>''' is a [[uniform 8-polytope]], constructed within the symmetry of the [[E8 (mathematics)|E<sub>8</sub>]] group.

Its [[Coxeter symbol]] is '''1<sub>42</sub>''', describing its bifurcating [[Coxeter-Dynkin diagram]], with a single ring on the end of the 1-node sequences.

The '''rectified 1<sub>42</sub>''' is constructed by points at the mid-edges of the '''1<sub>42</sub>''' and is the same as the birectified 2<sub>41</sub>, and the quadrirectified 4<sub>21</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 and [[vertex figure]]s, defined by all non-empty combinations of rings in this [[Coxeter-Dynkin diagram]]: {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}.

== 1<sub>42</sub> polytope ==
{{clear}}

{| class="wikitable" align="right" style="margin-left:10px" width="280"
!bgcolor=#e7dcc3 colspan=2|1<sub>42</sub>
|-
|bgcolor=#e7dcc3|Type||[[Uniform 8-polytope]]
|-
|bgcolor=#e7dcc3|Family||[[Uniform 1 k2 polytope|1<sub>k2</sub> polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| {3,3<sup>4,2</sup>}
|-
|bgcolor=#e7dcc3|[[Coxeter symbol]]|| 1<sub>42</sub>
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]s||{{CDD|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}<BR>{{CDD|node_1|3|node|split1|nodes|3ab|nodes|3a|nodea|3a|nodea}}
|-
|bgcolor=#e7dcc3|7-faces||2400:<BR>240 '''[[1 32 polytope|1<sub>32</sub>]]'''[[File:Gosset 1 32 petrie.svg|25px]]<BR>2160 '''[[7-demicube|1<sub>41</sub>]]'''[[Image:Demihepteract ortho petrie.svg|25px]]
|-
|bgcolor=#e7dcc3|6-faces||106080:<BR>6720 '''[[1 22 polytope|1<sub>22</sub>]]'''[[Image:Gosset 1 22 polytope.svg|25px]]<BR>30240 '''[[6-demicube|1<sub>31</sub>]]'''[[Image:Demihexeract ortho petrie.svg|25px]]<BR>69120 [[6-simplex|{3<sup>5</sup>}]][[Image:6-simplex t0.svg|25px]]
|-
|bgcolor=#e7dcc3|5-faces||725760:<BR>60480 '''[[5-demicube|1<sub>12</sub>]]'''[[Image:Demipenteract graph ortho.svg|25px]]<BR>181440 '''[[5-demicube|1<sub>21</sub>]]'''[[Image:Demipenteract graph ortho.svg|25px]]<BR>483840 [[5-simplex|{3<sup>4</sup>}]][[Image:5-simplex t0.svg|25px]]
|-
|bgcolor=#e7dcc3|4-faces||2298240:<BR>241920 '''[[5-cell|1<sub>02</sub>]]'''[[Image:4-simplex t0.svg|25px]]<BR>604800 '''[[16-cell|1<sub>11</sub>]]'''[[Image:4-cube t3.svg|25px]]<BR>1451520 [[5-cell|{3<sup>3</sup>}]][[Image:4-simplex t0.svg|25px]]
|-
|bgcolor=#e7dcc3|Cells||3628800:<BR>1209600 '''[[tetrahedron|1<sub>01</sub>]]'''[[Image:3-simplex t0.svg|25px]]<BR>2419200 [[tetrahedron|{3<sup>2</sup>}]][[Image:3-simplex t0.svg|25px]]
|-
|bgcolor=#e7dcc3|Faces||2419200 [[triangle|{3}]][[Image:2-simplex t0.svg|25px]]
|-
|bgcolor=#e7dcc3|Edges||483840
|-
|bgcolor=#e7dcc3|Vertices||17280
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[birectified 7-simplex|t<sub>2</sub>{3<sup>6</sup>}]] [[File:7-simplex t2.svg|25px]]
|-
|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 '''1<sub>42</sub>''' is composed of 2400 facets: 240 '''[[Gosset 1 32 polytope|1<sub>32</sub>]]''' polytopes, and 2160 [[7-demicube]]s ('''1<sub>41</sub>'''). Its [[vertex figure]] is a [[birectified 7-simplex]].

This polytope, along with the [[demiocteract]], can [[Honeycomb (geometry)|tessellate]] 8-dimensional space, represented by the symbol '''1<sub>52</sub>''', and Coxeter-Dynkin diagram: {{CDD|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}.

=== Alternate names ===
*[[E. L. Elte]] (1912) excluded this polytope from his listing of semiregular polytopes, because it has more than two types of 6-faces, but under his naming scheme it would be called V<sub>17280</sub> for its 17280 vertices.<ref>{{citation | last = Elte | first = E. L. | title = The Semiregular Polytopes of the Hyperspaces | publisher = University of Groningen | location = Groningen | year = 1912}}</ref>
*[[Coxeter]] named it '''1<sub>42</sub>''' for its bifurcating [[Coxeter-Dynkin diagram]], with a single ring on the end of the 1-node branch.
* '''Diacositetracont-dischiliahectohexaconta-zetton''' (acronym ''bif'') - 240-2160 facetted polyzetton (Jonathan Bowers)<ref>Klitzing, (o3o3o3x *c3o3o3o3o - bif)</ref>

=== Coordinates ===
The 17280 vertices can be defined as sign and location permutations of:

All sign combinations (32): (280×32=8960 vertices)
: (4, 2, 2, 2, 2, 0, 0, 0)
Half of the sign combinations (128): ((1+8+56)×128=8320 vertices)
: (2, 2, 2, 2, 2, 2, 2, 2)
: (5, 1, 1, 1, 1, 1, 1, 1)
: (3, 3, 3, 1, 1, 1, 1, 1)

The edge length is 2{{radic|2}} in this coordinate set, and the polytope radius is 4{{radic|2}}.

=== 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|3a|nodea|3a|branch_01lr|3a|nodea|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|3a|branch_01lr|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}.

Removing the node on the end of the 4-length branch leaves the [[1 32 polytope|1<sub>32</sub>]], {{CDD|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea|3a|nodea}}.

The [[vertex figure]] is determined by removing the ringed node and ringing the neighboring node. This makes the [[birectified 7-simplex]], 0<sub>42</sub>, {{CDD|nodea|3a|nodea|3a|nodea_1|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 name="auto">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_01lr|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>||colspan=2|f<sub>3</sub>||colspan=3|f<sub>4</sub>||colspan=3|f<sub>5</sub>||colspan=3|f<sub>6</sub>||colspan=2|f<sub>7</sub>|| [[vertex figure|''k''-figure]]||notes
|- align=right
|A<sub>7</sub> ||{{CDD|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodes_0x|3a|nodea|3a|nodea}}|| ( )
!f<sub>0</sub>
|BGCOLOR="#e0e0ff"|'''17280'''||56||420||280||560||70||280||420||56||168||168||28||56||28||8||8||[[Birectified 7-simplex|2r{3<sup>6</sup>}]] || E<sub>8</sub>/A<sub>7</sub> = 192*10!/8! = 17280
|- align=right
|A<sub>4</sub>A<sub>2</sub>A<sub>1</sub> ||{{CDD|nodea|3a|nodea|3a|nodea|3a|nodea|2|nodes_x1|2|nodea|3a|nodea}}|| { }
!f<sub>1</sub>
||2||BGCOLOR="#ffe0ff"|'''483840'''||15||15||30||5||30||30||10||30||15||10||15||3||5||3||{3}x{3,3,3} || E<sub>8</sub>/A<sub>4</sub>A<sub>2</sub>A<sub>1</sub> = 192*10!/5!/2/2 = 483840
|- align=right
|A<sub>3</sub>A<sub>2</sub>A<sub>1</sub> ||{{CDD|nodea|3a|nodea|3a|nodea|2|nodea_x|2|branch_01|2|nodea_x|2|nodea}}|| [[triangle|{3}]]
!f<sub>2</sub>
||3||3||BGCOLOR="#ffe0e0"|'''2419200'''||2||4||1||8||6||4||12||4||6||8||1||4||2||{3.3}v{ }|| E<sub>8</sub>/A<sub>3</sub>A<sub>2</sub>A<sub>1</sub> = 192*10!/4!/3!/2 = 2419200
|- align=right
|A<sub>3</sub>A<sub>3</sub> ||{{CDD|nodea|3a|nodea|3a|nodea|2|nodea_x|2|branch_01r|3a|nodea|2|nodea_x}}|| rowspan=2|[[tetrahedron|1<sub>10</sub>]]
!rowspan=2|f<sub>3</sub>
||4||6||4||BGCOLOR="#ffffe0"|'''1209600'''||BGCOLOR="#ffffe0"|*||1||4||0||4||6||0||6||4||0||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>3</sub>A<sub>2</sub>A<sub>1</sub> ||{{CDD|nodea|3a|nodea|2|nodea_x|2|nodea|3a|branch_01l|2|nodea_x|2|nodea}}
||4||6||4||BGCOLOR="#ffffe0"|*||BGCOLOR="#ffffe0"|'''2419200'''||0||2||3||1||6||3||3||6||1||3||2||{3}v{ } || E<sub>8</sub>/A<sub>3</sub>A<sub>2</sub>A<sub>1</sub> = 192*10!/4!/3!/2 = 2419200
|- align=right
|A<sub>4</sub>A<sub>3</sub> ||{{CDD|nodea|3a|nodea|3a|nodea|2|nodea_x|2|branch_01r|3a|nodea|3a|nodea}}|| [[5-cell|1<sub>20</sub>]]
!rowspan=3|f<sub>4</sub>
||5||10||10||5||0||BGCOLOR="#e0ffe0"|'''241920'''||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|*||4||0||0||6||0||0||4||0||[[tetrahedron|{3,3}]] || E<sub>8</sub>/A<sub>4</sub>A<sub>3</sub> = 192*10!/4!/4! = 241920
|- align=right
|D<sub>4</sub>A<sub>2</sub> ||{{CDD|nodea|3a|nodea|2|nodea_x|2|nodea|3a|branch_01lr|3a|nodea|2|nodea_x}}|| [[16-cell|1<sub>11</sub>]]
||8||24||32||8||8||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|'''604800'''||BGCOLOR="#e0ffe0"|*||1||3||0||3||3||0||3||1||[[Triangular pyramid|{3}v( )]] || E<sub>8</sub>/D<sub>4</sub>A<sub>2</sub> = 192*10!/8/4!/3! = 604800
|- align=right
|A<sub>4</sub>A<sub>1</sub>A<sub>1</sub> ||{{CDD|nodea|2|nodea_x|2|nodea|3a|nodea|3a|branch_01l|2|nodea_x|2|nodea}}|| [[5-cell|1<sub>20</sub>]] 
||5||10||10||0||5||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|'''1451520'''||0||2||2||1||4||1||2||2||[[Digonal disphenoid|{ }v{ }]] || E<sub>8</sub>/A<sub>4</sub>A<sub>1</sub>A<sub>1</sub> = 192*10!/5!/2/2 = 1451520
|- align=right
|D<sub>5</sub>A<sub>2</sub> ||{{CDD|nodea|3a|nodea|2|nodea_x|2|nodea|3a|branch_01lr|3a|nodea|3a|nodea}}||rowspan=2| [[5-demicube|1<sub>21</sub>]]
!rowspan=3|f<sub>5</sub>
||16||80||160||80||40||16||10||0||BGCOLOR="#e0ffff"|'''60480'''||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|*||3||0||0||3||0||[[Equilateral triangle|{3}]] || E<sub>8</sub>/D<sub>5</sub>A<sub>2</sub> = 192*10!/16/5!/3! = 40480
|- align=right
|D<sub>5</sub>A<sub>1</sub> ||{{CDD|nodea|2|nodea_x|2|nodea|3a|nodea|3a|branch_01lr|3a|nodea|2|nodea_x}}
||16||80||160||40||80||0||10||16||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|'''181440'''||BGCOLOR="#e0ffff"|*||1||2||0||2||1||rowspan=2|[[isosceles triangle|{ }v( )]] || E<sub>8</sub>/D<sub>5</sub>A<sub>1</sub> = 192*10!/16/5!/2 = 181440
|- align=right
|A<sub>5</sub>A<sub>1</sub> ||{{CDD|nodea_x|2|nodea|3a|nodea|3a|nodea|3a|branch_01l|2|nodea_x|2|nodea}}|| [[5-simplex|1<sub>30</sub>]]
||6||15||20||0||15||0||0||6||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|'''483840'''||0||2||1||1||2 || 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_01lr|3a|nodea|3a|nodea}}|| [[1_22 polytope|1<sub>22</sub>]]
!rowspan=3|f<sub>6</sub>
 ||72||720||2160||1080||1080||216||270||216||27||27||0||BGCOLOR="#e0e0ff"|'''6720'''||BGCOLOR="#e0e0ff"|*||BGCOLOR="#e0e0ff"|*||2||0||rowspan=3|{ } || E<sub>8</sub>/E<sub>6</sub>A<sub>1</sub> = 192*10!/72/6!/2 = 6720
|- align=right
|D<sub>6</sub> ||{{CDD|nodea_x|2|nodea|3a|nodea|3a|nodea|3a|branch_01lr|3a|nodea|2|nodea_x}}|| [[6-demicube|1<sub>31</sub>]]
||32||240||640||160||480||0||60||192||0||12||32||BGCOLOR="#e0e0ff"|*||BGCOLOR="#e0e0ff"|'''30240'''||BGCOLOR="#e0e0ff"|*||1||1|| E<sub>8</sub>/D<sub>6</sub> = 192*10!/32/6! = 30240
|- align=right
|A<sub>6</sub>A<sub>1</sub> ||{{CDD|nodea|3a|nodea|3a|nodea|3a|nodea|3a|branch_01l|2|nodea_x|2|nodea}}|| [[6-simplex|1<sub>40</sub>]]
||7||21||35||0||35||0||0||21||0||0||7||BGCOLOR="#e0e0ff"|*||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
|E<sub>7</sub> ||{{CDD|nodea_x|2|nodea|3a|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea}}|| [[1_32 polytope|1<sub>32</sub>]]
!rowspan=2|f<sub>7</sub>
||576||10080||40320||20160||30240||4032||7560||12096||756||1512||2016||56||126||0||BGCOLOR="#ffe0ff"|'''240'''||BGCOLOR="#ffe0ff"|*||rowspan=2|( ) || E<sub>8</sub>/E<sub>7</sub>  = 192*10!/72/8! =  240
|- align=right
|D<sub>7</sub> ||{{CDD|nodea|3a|nodea|3a|nodea|3a|nodea|3a|branch_01lr|3a|nodea|2|nodea_x}}|| [[7-demicube|1<sub>41</sub>]]
||64||672||2240||560||2240||0||280||1344||0||84||448||0||14||64||BGCOLOR="#ffe0ff"|*||BGCOLOR="#ffe0ff"|'''2160''' || E<sub>8</sub>/D<sub>7</sub> = 192*10!/64/7! = 2160
|}

=== Projections ===
[[File:E8 142 Petrie Projection.png|250px|thumb|The projection of '''1<sub>42</sub>''' to the '''E<sub>8</sub>''' Coxeter plane (aka. the Petrie projection) with polytope radius <math>4\sqrt{2}</math> is shown below with  483,840 edges of length <math>2\sqrt{2}</math> culled 53% on the interior to only 226,444:]]

[[File:E8_142-3D_Concentric_Hulls.png|thumb|250px|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 17280 projected 1<sub>42</sub> polytope vertices are sorted and tallied by their 3D norm generating the increasingly transparent hulls for each set of tallied norms.

Notice the last two outer hulls are a combination of two overlapped Dodecahedrons (40) and a Nonuniform Rhombicosidodecahedron (60).
]]

{| class=wikitable width=500
!E8<BR>[30]
!E7<BR>[18]
!E6<BR>[12]
|- align=center
|[[File:Gosset 1 42 polytope petrie.svg|168px]]<BR>(1)
|[[File:1 42 t0 e7.svg|168px]]<BR>(1,3,6)
|[[File:1 42 polytope E6 Coxeter plane.svg|168px]]<BR>(8,16,24,32,48,64,96)
|- align=center
![20]
![24]
![6]
|- align=center
|[[File:1 42 t0 p20.svg|168px]]
|[[File:1 42 t0 p24.svg|168px]]
|[[File:1 42 t0 mox.svg|168px]]<BR>(1,2,3,4,5,6,7,8,10,11,12,14,16,18,19,20)
|}

[[Orthographic projection]]s are shown for the sub-symmetries of E<sub>8</sub>: E<sub>7</sub>, E<sub>6</sub>, B<sub>8</sub>, B<sub>7</sub>, B<sub>6</sub>, B<sub>5</sub>, B<sub>4</sub>, B<sub>3</sub>, B<sub>2</sub>, A<sub>7</sub>, and A<sub>5</sub> [[Coxeter plane]]s, as well as two more symmetry planes of order 20 and 24. Vertices are shown as circles, colored by their order of overlap in each projective plane.

{| class=wikitable width=570
|- align=center
!D3 / B2 / A3<BR>[4]
!D4 / B3 / A2<BR>[6]
!D5 / B4<BR>[8]
|- align=center
|[[File:1 42 t0 B2.svg|190px]]<BR>(32,160,192,240,480,512,832,960)
|[[File:1 42 t0 B3.svg|190px]]<BR>(72,216,432,720,864,1080)
|[[File:1 42 t0 B4.svg|190px]]<BR>(8,16,24,32,48,64,96)
|- align=center
!D6 / B5 / A4<BR>[10]
!D7 / B6<BR>[12]
!D8 / B7 / A6<BR>[14]
|- align=center
|[[File:1 42 t0 B5.svg|190px]]
|[[File:1 42 t0 B6.svg|190px]]
|[[File:1 42 t0 B7.svg|190px]]
|- align=center
!B8<BR>[16/2]
!A5<BR>[6]
!A7<BR>[8]
|- align=center
|[[File:1 42 t0 B8.svg|190px]]
|[[File:1 42 t0 A5.svg|190px]]
|[[File:1 42 t0 A7.svg|190px]]
|}

=== Related polytopes and honeycombs ===
{{1 k2 polytopes}}

== Rectified 1<sub>42</sub> polytope ==
{| class="wikitable" align="right" style="margin-left:10px" width="280"
!bgcolor=#e7dcc3 colspan=2|Rectified 1<sub>42</sub>
|-
|bgcolor=#e7dcc3|Type||[[Uniform 8-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| t<sub>1</sub>{3,3<sup>4,2</sup>}
|-
|bgcolor=#e7dcc3|[[Coxeter symbol]]|| 0<sub>421</sub>
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]s||{{CDD|nodea|3a|nodea|3a|branch_10|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}<BR>{{CDD|node|3|node_1|split1|nodes|3ab|nodes|3a|nodea|3a|nodea}}
|-
|bgcolor=#e7dcc3|7-faces||19680
|-
|bgcolor=#e7dcc3|6-faces||382560
|-
|bgcolor=#e7dcc3|5-faces||2661120
|-
|bgcolor=#e7dcc3|4-faces||9072000
|-
|bgcolor=#e7dcc3|Cells||16934400
|-
|bgcolor=#e7dcc3|Faces||16934400
|-
|bgcolor=#e7dcc3|Edges||7257600
|-
|bgcolor=#e7dcc3|Vertices||483840
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||{3,3,3}×{3}×{}
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||[[E8 (mathematics)|E<sub>8</sub>]], [3<sup>4,2,1</sup>]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}
The '''rectified 1<sub>42</sub>''' is named from being a [[Rectification (geometry)|rectification]] of the 1<sub>42</sub> polytope, with vertices positioned at the mid-edges of the 1<sub>42</sub>. It can also be called a '''0<sub>421</sub>''' polytope with the ring at the center of 3 branches of length 4, 2, and 1.

=== Alternate names ===
* 0<sub>421</sub> polytope
* Birectified 2<sub>41</sub> polytope
* Quadrirectified 4<sub>21</sub> polytope
* Rectified diacositetracont-dischiliahectohexaconta-zetton as a rectified 240-2160 facetted polyzetton (acronym ''buffy'') (Jonathan Bowers)<ref>Klitzing, (o3o3o3x *c3o3o3o3o - buffy)</ref>

=== 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|3a|nodea|3a|branch_10|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}.

Removing the node on the end of the 1-length branch leaves the [[birectified 7-simplex]], {{CDD|nodea|3a|nodea|3a|nodea_1|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}

Removing the node on the end of the 2-length branch leaves the [[birectified 7-cube]], {{CDD|nodea|3a|branch_10|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}.

Removing the node on the end of the 3-length branch leaves the [[rectified 1 32 polytope|rectified 1<sub>32</sub>]], {{CDD|nodea|3a|nodea|3a|branch_10|3a|nodea|3a|nodea|3a|nodea}}.

The [[vertex figure]] is determined by removing the ringed node and ringing the neighboring node. This makes the [[5-cell]]-[[triangle]] duoprism prism, {{CDD|nodea|3a|nodea_1|2|nodea_1|2|nodea_1|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 name="auto"/>
{| 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_10|3a|nodea|3a|nodea}}||[[Face_(geometry)#k-face|<small>''k''-face</small>]]|| f<sub>k</sub> || f<sub>0</sub> || f<sub>1</sub>||colspan=3|f<sub>2</sub>||colspan=5|f<sub>3</sub>||colspan=6|f<sub>4</sub>||colspan=6|f<sub>5</sub>||colspan=5|f<sub>6</sub>||colspan=3|f<sub>7</sub>|| [[vertex figure|''k''-figure]]
|- 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}}|| ( )
!f<sub>0</sub>
|BGCOLOR="#e0e0ff"|<small>'''483840'''</small>||30||30||15||60||10||15||60||30||60||5||20||30||60||30||30||10||20||30||30||15||6||10||10||15||6||3||5||2||3||{3,3,3}x{3,3}x{} 
|- align=right
|A<sub>3</sub>A<sub>1</sub>A<sub>1</sub> ||{{CDD|nodea|3a|nodea|3a|nodea|2|nodea_x|2|nodes_1x|2|nodea_x|2|nodea}}|| { }
!f<sub>1</sub>
||2||BGCOLOR="#ffe0ff"|<small>'''7257600'''</small>||2||1||4||1||2||8||4||6||1||4||8||12||6||4||4||6||12||8||4||1||6||4||8||2||1||4||1||2|| 
|- align=right
|A<sub>3</sub>A<sub>2</sub> ||{{CDD|nodea|3a|nodea|3a|nodea|2|nodea_x|2|nodes_1x|3a|nodea|2|nodea_x}}||rowspan=3| [[triangle|{3}]]
!rowspan=3|f<sub>2</sub>
||3||3||BGCOLOR="#ffe0e0"|<small>'''4838400'''</small>||BGCOLOR="#ffe0e0"|*||BGCOLOR="#ffe0e0"|*||1||1||4||0||0||1||4||4||6||0||0||4||6||6||4||0||0||6||4||4||1||0||4||1||1|| 
|- align=right
|A<sub>3</sub>A<sub>2</sub>A<sub>1</sub> ||{{CDD|nodea|3a|nodea|3a|nodea|2|nodea_x|2|branch_10|2|nodea_x|2|nodea}}
||3||3||BGCOLOR="#ffe0e0"|*||BGCOLOR="#ffe0e0"|<small>'''2419200'''</small>||BGCOLOR="#ffe0e0"|*||0||2||0||4||0||1||0||8||0||6||0||4||0||12||0||4||0||6||0||8||0||1||4||0||2||
|- align=right
|A<sub>2</sub>A<sub>2</sub>A<sub>1</sub> ||{{CDD|nodea|3a|nodea|2|node_x|2|nodea|3a|nodes_1x|2|nodea_x|2|nodea}}
||3||3||BGCOLOR="#ffe0e0"|*||BGCOLOR="#ffe0e0"|*||BGCOLOR="#ffe0e0"|<small>'''9676800'''</small>||0||0||2||1||3||0||1||2||6||3||3||1||3||6||6||3||1||3||3||6||2||1||3||1||2|| 
|- align=right
|rowspan=2|A<sub>3</sub>A<sub>3</sub> ||{{CDD|nodea|3a|nodea|3a|nodea|2|nodea_x|2|nodes_1x|3a|nodea|3a|nodea}}|| [[tetrahedron|0<sub>200</sub>]]
!rowspan=5|f<sub>3</sub>
||4||6||4||0||0||BGCOLOR="#ffffe0"|<small>'''1209600'''</small>||BGCOLOR="#ffffe0"|*||BGCOLOR="#ffffe0"|*||BGCOLOR="#ffffe0"|*||BGCOLOR="#ffffe0"|*||1||4||0||0||0||0||4||6||0||0||0||0||6||4||0||0||0||4||1||0|| 
|- align=right
||{{CDD|nodea|3a|nodea|3a|nodea|2|nodea_x|2|branch_10|3a|nodea|2|nodea_x}}|| rowspan=3|[[octahedron|0<sub>110</sub>]]
||6||12||4||4||0||BGCOLOR="#ffffe0"|*||BGCOLOR="#ffffe0"|<small>'''1209600'''</small>||BGCOLOR="#ffffe0"|*||BGCOLOR="#ffffe0"|*||BGCOLOR="#ffffe0"|*||1||0||4||0||0||0||4||0||6||0||0||0||6||0||4||0||0||4||0||1||
|- align=right
|A<sub>3</sub>A<sub>2</sub> ||{{CDD|nodea|3a|nodea|2|nodea_x|2|nodea|3a|nodes_1x|3a|nodea|2|nodea_x}}
||6||12||4||0||4||BGCOLOR="#ffffe0"|*||BGCOLOR="#ffffe0"|*||BGCOLOR="#ffffe0"|<small>'''4838400'''</small>||BGCOLOR="#ffffe0"|*||BGCOLOR="#ffffe0"|*||0||1||1||3||0||0||1||3||3||3||0||0||3||3||3||1||0||3||1||1||
|- align=right
|A<sub>3</sub>A<sub>2</sub>A<sub>1</sub> ||{{CDD|nodea|3a|nodea|2|nodea_x|2|nodea|3a|branch_10|2|nodea_x|2|nodea}}
||6||12||0||4||4||BGCOLOR="#ffffe0"|*||BGCOLOR="#ffffe0"|*||BGCOLOR="#ffffe0"|*||BGCOLOR="#ffffe0"|<small>'''2419200'''</small>||BGCOLOR="#ffffe0"|*||0||0||2||0||3||0||1||0||6||0||3||0||3||0||6||0||1||3||0||2||
|- align=right
|A<sub>3</sub>A<sub>1</sub>A<sub>1</sub> ||{{CDD|nodea|2|nodea_x|2|nodea|3a|nodea|3a|nodes_1x|2|nodea_x|2|nodea}}|| [[tetrahedron|0<sub>200</sub>]]
||4||6||0||0||4||BGCOLOR="#ffffe0"|*||BGCOLOR="#ffffe0"|*||BGCOLOR="#ffffe0"|*||BGCOLOR="#ffffe0"|*||BGCOLOR="#ffffe0"|<small>'''7257600'''</small>||0||0||0||2||1||2||0||1||2||4||2||1||1||2||4||2||1||2||1||2||
|- align=right
|A<sub>4</sub>A<sub>3</sub> ||{{CDD|nodea|3a|nodea|3a|nodea|2|nodea_x|2|branch_10|3a|nodea|3a|nodea}}|| rowspan=2|[[rectified 5-cell|0<sub>210</sub>]]
!rowspan=6|f<sub>4</sub>
||10||30||20||10||0||5||5||0||0||0||BGCOLOR="#e0ffe0"|<small>'''241920'''</small>||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|*||4||0||0||0||0||0||6||0||0||0||0||4||0||0|| 
|- align=right
|A<sub>4</sub>A<sub>2</sub> ||{{CDD|nodea|3a|nodea|2|nodea_x|2|nodea|3a|nodes_1x|3a|nodea|3a|nodea}}
||10||30||20||0||10||5||0||5||0||0||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|<small>'''967680'''</small>||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|*||1||3||0||0||0||0||3||3||0||0||0||3||1||0|| 
|- align=right
|D<sub>4</sub>A<sub>2</sub> ||{{CDD|nodea|3a|nodea|2|nodea_x|2|nodea|3a|branch_10|3a|nodea|2|nodea_x}}||  [[16-cell|0<sub>111</sub>]]
||24||96||32||32||32||0||8||8||8||0||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|<small>'''604800'''</small>||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|*||1||0||3||0||0||0||3||0||3||0||0||3||0||1|| 
|- align=right
|A<sub>4</sub>A<sub>1</sub> ||{{CDD|nodea|2|nodea_x|2|nodea|3a|nodea|3a|nodes_1x|3a|nodea|2|nodea_x}}|| rowspan=2|[[rectified 5-cell|0<sub>210</sub>]]
||10||30||10||0||20||0||0||5||0||5||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|<small>'''2903040'''</small>||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|*||0||1||1||2||0||0||1||2||2||1||0||2||1||1|| 
|- align=right
|A<sub>4</sub>A<sub>1</sub>A<sub>1</sub>||{{CDD|nodea|2|nodea_x|2|nodea|3a|nodea|3a|branch_10|2|nodea_x|2|nodea}}
||10||30||0||10||20||0||0||0||5||5||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|<small>'''1451520'''</small>||BGCOLOR="#e0ffe0"|*||0||0||2||0||2||0||1||0||4||0||1||2||0||2|| 
|- align=right
|A<sub>4</sub>A<sub>1</sub>||{{CDD|nodea_x|2|nodea|3a|nodea|3a|nodea|3a|nodes_1x|2|nodea_x|2|nodea}}||[[5-cell|0<sub>300</sub> ]]
||5||10||0||0||10||0||0||0||0||5||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|*||BGCOLOR="#e0ffe0"|<small>'''2903040'''</small>||0||0||0||2||1||1||0||1||2||2||1||1||1||2|| 
|- align=right
|D<sub>5</sub>A<sub>2</sub> ||{{CDD|nodea|3a|nodea|2|nodea_x|2|nodea|3a|branch_10|3a|nodea|3a|nodea}}|| [[rectified 5-demicube|0<sub>211</sub>]]
!rowspan=6|f<sub>5</sub>
||80||480||320||160||160||80||80||80||40||0||16||16||10||0||0||0||BGCOLOR="#e0ffff"|<small>'''60480'''</small>||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|*||3||0||0||0||0||3||0||0|| [[Equilateral triangle|{3}]]
|- align=right
|A<sub>5</sub>A<sub>1</sub> ||{{CDD|nodea|2|nodea_x|2|nodea|3a|nodea|3a|nodes_1x|3a|nodea|3a|nodea}}|| [[birectified 5-simplex|0<sub>220</sub>]]
||20||90||60||0||60||15||0||30||0||15||0||6||0||6||0||0||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|<small>'''483840'''</small>||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|*||1||2||0||0||0||2||1||0|| rowspan=2|[[Isosceles triangle|{ }v()]]
|- align=right
|D<sub>5</sub>A<sub>1</sub> ||{{CDD|nodea|2|nodea_x|2|nodea|3a|nodea|3a|branch_10|3a|nodea|2|nodea_x}}|| [[rectified 5-demicube|0<sub>211</sub>]]
||80||480||160||160||320||0||40||80||80||80||0||0||10||16||16||0||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|<small>'''181440'''</small>||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|*||1||0||2||0||0||2||0||1
|- align=right
|A<sub>5</sub> ||{{CDD|nodea_x|2|nodea|3a|nodea|3a|nodea|3a|nodes_1x|3a|nodea|2|nodea_x}}|| rowspan=2|[[rectified 5-simplex|0<sub>310</sub>]]
||15||60||20||0||60||0||0||15||0||30||0||0||0||6||0||6||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|<small>'''967680'''</small>||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|*||0||1||1||1||0||1||1||1|| [[Scalene triangle|( )v( )v()]]
|- align=right
|rowspan=2|A<sub>5</sub>A<sub>1</sub> ||{{CDD|nodea_x|2|nodea|3a|nodea|3a|nodea|3a|branch_10|2|nodea_x|2|nodea}}
||15||60||0||20||60||0||0||0||15||30||0||0||0||0||6||6||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|<small>'''483840'''</small>||BGCOLOR="#e0ffff"|*||0||0||2||0||1||1||0||2|| rowspan=2|[[Isosceles triangle|{ }v()]]
|- align=right
||{{CDD|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodes_1x|2|nodea_x|2|nodea}}|| [[5-simplex|0<sub>400</sub>]]
||6||15||0||0||20||0||0||0||0||15||0||0||0||0||0||6||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|*||BGCOLOR="#e0ffff"|<small>'''483840'''</small>||0||0||0||2||1||0||1||2
|- align=right
|E<sub>6</sub>A<sub>1</sub>||{{CDD|nodea|2|nodea_x|2|nodea|3a|nodea|3a|branch_10|3a|nodea|3a|nodea}}|| [[rectified 1_22 polytope|0<sub>221</sub>]]
!rowspan=5|f<sub>6</sub>
||720||6480||4320||2160||4320||1080||1080||2160||1080||1080||216||432||270||432||216||0||27||72||27||0||0||0||BGCOLOR="#e0e0ff"|<small>'''6720'''</small>||BGCOLOR="#e0e0ff"|*||BGCOLOR="#e0e0ff"|*||BGCOLOR="#e0e0ff"|*||BGCOLOR="#e0e0ff"|*||2||0||0||rowspan=5| { }
|- align=right
|A<sub>6</sub> ||{{CDD|nodea_x|2|nodea|3a|nodea|3a|nodea|3a|nodes_1x|3a|nodea|3a|nodea}}|| [[birectified 6-simplex|0<sub>320</sub>]]
||35||210||140||0||210||35||0||105||0||105||0||21||0||42||0||21||0||7||0||7||0||0||BGCOLOR="#e0e0ff"|*||BGCOLOR="#e0e0ff"|<small>'''138240'''</small>||BGCOLOR="#e0e0ff"|*||BGCOLOR="#e0e0ff"|*||BGCOLOR="#e0e0ff"|*||1||1||0
|- align=right
|D<sub>6</sub> ||{{CDD|nodea_x|2|nodea|3a|nodea|3a|nodea|3a|branch_10|3a|nodea|2|nodea_x}}|| [[rectified 6-demicube|0<sub>311</sub>]]
||240||1920||640||640||1920||0||160||480||480||960||0||0||60||192||192||192||0||0||12||32||32||0||BGCOLOR="#e0e0ff"|*||BGCOLOR="#e0e0ff"|*||BGCOLOR="#e0e0ff"|<small>'''30240'''</small>||BGCOLOR="#e0e0ff"|*||BGCOLOR="#e0e0ff"|*||1||0||1
|- align=right
|A<sub>6</sub> ||{{CDD|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodes_1x|3a|nodea|2|nodea_x}}|| rowspan=2|[[rectified 6-simplex|0<sub>410</sub>]]
||21||105||35||0||140||0||0||35||0||105||0||0||0||21||0||42||0||0||0||7||0||7||BGCOLOR="#e0e0ff"|*||BGCOLOR="#e0e0ff"|*||BGCOLOR="#e0e0ff"|*||BGCOLOR="#e0e0ff"|<small>'''138240'''</small>||BGCOLOR="#e0e0ff"|*||0||1||1
|- align=right
|A<sub>6</sub>A<sub>1</sub> ||{{CDD|nodea|3a|nodea|3a|nodea|3a|nodea|3a|branch_10|2|nodea_x|2|nodea}}
||21||105||0||35||140||0||0||0||35||105||0||0||0||0||21||42||0||0||0||0||7||7||BGCOLOR="#e0e0ff"|*||BGCOLOR="#e0e0ff"|*||BGCOLOR="#e0e0ff"|*||BGCOLOR="#e0e0ff"|*||BGCOLOR="#e0e0ff"|<small>'''69120'''</small>||0||0||2
|- align=right
|E<sub>7</sub> ||{{CDD|nodea_x|2|nodea|3a|nodea|3a|nodea|3a|branch_10|3a|nodea|3a|nodea}}|| [[rectified 1_32 polytope|0<sub>321</sub>]]
!rowspan=3|f<sub>7</sub>
||<small>10080</small>||<small>120960</small>||<small>80640</small>||<small>40320</small>||<small>120960</small>||<small>20160</small>||<small>20160</small>||<small>60480</small>||<small>30240</small>||<small>60480</small>||<small>4032</small>||<small>12096</small>||<small>7560</small>||<small>24192</small>||<small>12096</small>||<small>12096</small>||<small>756</small>||<small>4032</small>||<small>1512</small>||<small>4032</small>||<small>2016</small>||<small>0</small>||<small>56</small>||<small>576</small>||<small>126</small>||<small>0</small>||<small>0</small>||BGCOLOR="#ffe0ff"|<small>'''240'''</small>||BGCOLOR="#ffe0ff"|*||BGCOLOR="#ffe0ff"|*|| rowspan=3|( )
|- align=right
|A<sub>7</sub> ||{{CDD|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodes_1x|3a|nodea|3a|nodea}}|| [[birectified 7-simplex|0<sub>420</sub>]]
||56||420||280||0||560||70||0||280||0||420||0||56||0||168||0||168||0||28||0||56||0||28||0||8||0||8||0||BGCOLOR="#ffe0ff"|*||BGCOLOR="#ffe0ff"|<small>'''17280'''</small>||BGCOLOR="#ffe0ff"|*
|- align=right
|D<sub>7</sub> ||{{CDD|nodea|3a|nodea|3a|nodea|3a|nodea|3a|branch_10|3a|nodea|2|nodea_x}}|| [[rectified 7-demicube|0<sub>411</sub>]]
||672||6720||2240||2240||8960||0||560||2240||2240||6720||0||0||280||1344||1344||2688||0||0||84||448||448||448||0||0||14||64||64||BGCOLOR="#ffe0ff"|*||BGCOLOR="#ffe0ff"|*||BGCOLOR="#ffe0ff"|<small>'''2160'''</small>
|}

=== Projections ===
[[Orthographic projection]]s are shown for the sub-symmetries of B<sub>6</sub>, B<sub>5</sub>, B<sub>4</sub>, B<sub>3</sub>, B<sub>2</sub>, A<sub>7</sub>, and A<sub>5</sub> [[Coxeter plane]]s. Vertices are shown as circles, colored by their order of overlap in each projective plane.

(Planes for E<sub>8</sub>: E<sub>7</sub>, E<sub>6</sub>, B<sub>8</sub>, B<sub>7</sub>, [24] are not shown for being too large to display.)

<!--{| class=wikitable width=600
!E7<BR>[18]
!E6<BR>[12]
|- align=center
|[[File:4 21 t4 e7.svg|200px]]
|[[File:4 21 t4 e6.svg|200px]]
|- align=center
![20]
![24]
|- align=center
|[[File:4 21 t4 p20.svg|200px]]
|[[File:1_42_t1_p24.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:4 21 t4 B2.svg|200px]]
|[[File:4 21 t4 B3.svg|200px]]
|[[File:4 21 t4 B4.svg|200px]]
|- align=center
!D6 / B5 / A4<BR>[10]
!D7 / B6<BR>[12]
![6]
<!--!D8 / B7 / A6<BR>[14]-->
|- align=center
|[[File:4 21 t4 B5.svg|200px]]
|[[File:4 21 t4 B6.svg|200px]]
|[[File:4 21 t4 mox.svg|200px]]
<!--|[[File:4 21 t4 B7.svg|240px]]-->
|- align=center
<!--!B8<BR>[16/2]-->
!A5<BR>[6]
!A7<BR>[8]
!&nbsp;<BR>[20]
|- align=center
<!--|[[File:4 21 t4 B8.svg|240px]]-->
|[[File:4 21 t4 A5.svg|200px]]
|[[File:4 21 t4 A7.svg|200px]]
|[[File:4 21 t4 p20.svg|200px]]
|}

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

== Notes ==
{{reflist}}

== References ==
* [[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 Kaleidoscopes: Selected Writings of H.S.M. Coxeter | Wiley]
** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45]
* {{KlitzingPolytopes|polyzetta.htm|8D|Uniform polyzetta}} o3o3o3x *c3o3o3o3o - bif, o3o3o3x *c3o3o3o3o - buffy

{{Polytopes}}

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