1 22 polytope#Related complex polyhedron

{{Short description|Uniform 6-polytope}}

{{DISPLAYTITLE:1 22 polytope}}

class=wikitable width=500 align=right style="margin-left:0.5em;"
align=center valign=top

|160px
122
{{CDD|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea}}

|160px
Rectified 122
{{CDD|nodea|3a|nodea|3a|branch_10|3a|nodea|3a|nodea}}

|160px
Birectified 122
{{CDD|nodea|3a|nodea_1|3a|branch|3a|nodea_1|3a|nodea}}

align=center valign=top

|{{clarify|reason=Image needed|date=June 2025}}
Trirectified 122
{{CDD|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea_1}}

|160px
Truncated 122
{{CDD|nodea|3a|nodea|3a|branch_11|3a|nodea|3a|nodea}}

align=center valign=top

|160px
221
{{CDD|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea}}

|160px
Rectified 221
{{CDD|nodea|3a|nodea_1|3a|branch|3a|nodea|3a|nodea}}

valign=top

!colspan=3|orthogonal projections in E6 Coxeter plane

In 6-dimensional geometry, the 122 polytope is a uniform polytope, constructed from the E6 group. It was first published in E. L. Elte's 1912 listing of semiregular polytopes, named as V72 (for its 72 vertices).Elte, 1912

Its Coxeter symbol is 122, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequence. There are two rectifications of the 122, constructed by positions points on the elements of 122. The rectified 122 is constructed by points at the mid-edges of the 122. The birectified 122 is constructed by points at the triangle face centers of the 122.

These polytopes are from a family of 39 convex uniform polytopes in 6-dimensions, made of uniform polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea}}.

{{-}}

1<sub>22</sub> polytope

class="wikitable" align="right" style="margin-left:10px" width="280"

!bgcolor=#e7dcc3 colspan=2|122 polytope

bgcolor=#e7dcc3|TypeUniform 6-polytope
bgcolor=#e7dcc3|Family1k2 polytope
bgcolor=#e7dcc3|Schläfli symbol{3,32,2}
bgcolor=#e7dcc3|Coxeter symbol122
bgcolor=#e7dcc3|Coxeter-Dynkin diagram{{CDD|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea}} or {{CDD|node_1|3|node|split1|nodes|3ab|nodes}}
bgcolor=#e7dcc3|5-faces54:
27 121 30px
27 121 30px
bgcolor=#e7dcc3|4-faces702:
270 111 25px
432 120 25px
bgcolor=#e7dcc3|Cells2160:
1080 110 25px
1080 {3,3} 25px
bgcolor=#e7dcc3|Faces2160 {3} 25px
bgcolor=#e7dcc3|Edges720
bgcolor=#e7dcc3|Vertices72
bgcolor=#e7dcc3|Vertex figureBirectified 5-simplex:
022 50px
bgcolor=#e7dcc3|Petrie polygonDodecagon
bgcolor=#e7dcc3|Coxeter groupE6, 3,32,2, order 103680
bgcolor=#e7dcc3|Propertiesconvex, isotopic

The 122 polytope contains 72 vertices, and 54 5-demicubic facets. It has a birectified 5-simplex vertex figure. Its 72 vertices represent the root vectors of the simple Lie group E6.

= Alternate names =

  • Pentacontatetrapeton (Acronym: mo) - 54-facetted polypeton (Jonathan Bowers)Klitzing, (o3o3o3o3o *c3x - [http://bendwavy.org/klitzing/incmats/mo.htm mo])

= Images =

class=wikitable width=480

|+ Coxeter plane orthographic projections

!E6
[12]

!D5
[8]

!D4 / A2
[6]

valign=top align=center

|120px
(1,2)

|120px
(1,3)

|120px
(1,9,12)

B6
[12/2]

!A5
[6]

!A4
{{brackets|5}} = [10]

!A3 / D3
[4]

valign=top align=center

|120px
(1,2)

|120px
(2,3,6)

|120px
(1,2)

|120px
(1,6,8,12)

= Construction =

It is created by a Wythoff construction upon a set of 6 hyperplane mirrors in 6-dimensional space.

The facet information can be extracted from its Coxeter-Dynkin diagram, {{CDD|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea}}.

Removing the node on either of 2-length branches leaves the 5-demicube, 131, {{CDD|nodea|3a|branch_01lr|3a|nodea|3a|nodea}}.

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 5-simplex, 022, {{CDD|node|3|node|3|node_1|3|node|3|node}}.

Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.Coxeter, Regular Polytopes, 11.8 Gosset figures in six, seven, and eight dimensions, pp. 202–203

class=wikitable

!E6

width=60|{{CDD|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea}}k-facefkf0f1f2colspan=2|f3colspan=3|f4colspan=2|f5k-figurenotes
align=right

|A5

{{CDD|nodea|3a|nodea|3a|nodes_0x|3a|nodea|3a|nodea}}( )

! f0

|BGCOLOR="#ffe0ff"|72

2090606015153066r{3,3,3}E6/A5 = 72*6!/6! = 72
align=right

|A2A2A1

{{CDD|nodea|3a|nodea|2|nodes_x1|2|nodea|3a|nodea}}{ }

! f1

2BGCOLOR="#ffe0e0"|72099933933{3}×{3}E6/A2A2A1 = 72*6!/3!/3!/2 = 720
align=right

|A2A1A1

{{CDD|nodea|2|nodea_x|2|branch_01|2|nodea_x|2|nodea}}{3}

! f2

33BGCOLOR="#ffffe0"|21602211422s{2,4}E6/A2A1A1 = 72*6!/3!/2/2 = 2160
align=right

|rowspan=2|A3A1

{{CDD|nodea_x|2|nodea|3a|branch_01l|2|nodea_x|2|nodea}}rowspan=2|{3,3}

! rowspan=2|f3

464BGCOLOR="#e0ffe0"|1080BGCOLOR="#e0ffe0"|*10221rowspan=2|{ }∨( )rowspan=2| E6/A3A1 = 72*6!/4!/2 = 1080
align=right

|{{CDD|nodea|2|nodea_x|2|branch_01r|3a|nodea|2|nodea_x}}

464BGCOLOR="#e0ffe0"|*BGCOLOR="#e0ffe0"|108001212
align=right

|rowspan=2|A4A1

{{CDD|nodea|2|nodea_x|2|branch_01r|3a|nodea|3a|nodea}}rowspan=2| {3,3,3}

!rowspan=3|f4

5101050BGCOLOR="#e0ffff"|216BGCOLOR="#e0ffff"|*BGCOLOR="#e0ffff"|*20rowspan=3|{ }rowspan=2| E6/A4A1 = 72*6!/5!/2 = 216
align=right

|{{CDD|nodea|3a|nodea|3a|branch_01l|2|nodea_x|2|nodea}}

5101005BGCOLOR="#e0ffff"|*BGCOLOR="#e0ffff"|216BGCOLOR="#e0ffff"|*02
align=right

|D4

{{CDD|nodea_x|2|nodea|3a|branch_01lr|3a|nodea|2|nodea_x}}h{4,3,3}8243288BGCOLOR="#e0ffff"|*BGCOLOR="#e0ffff"|*BGCOLOR="#e0ffff"|27011E6/D4 = 72*6!/8/4! = 270
align=right

|rowspan=2|D5

{{CDD|nodea|3a|nodea|3a|branch_01lr|3a|nodea|2|nodea_x}}rowspan=2|h{4,3,3,3}

!rowspan=2|f5

1680160804016010BGCOLOR="#e0e0ff"|27BGCOLOR="#e0e0ff"|*rowspan=2|( )rowspan=2| E6/D5 = 72*6!/16/5! = 27
align=right

|{{CDD|nodea_x|2|nodea|3a|branch_01lr|3a|nodea|3a|nodea}}

1680160408001610BGCOLOR="#e0e0ff"|*BGCOLOR="#e0e0ff"|27

= Related complex polyhedron =

File:Complex polyhedron 3-3-3-4-2.png

The regular complex polyhedron 3{3}3{4}2, {{CDD|3node_1|3|3node|4|node}}, in \mathbb{C}^2 has a real representation as the 122 polytope in 4-dimensional space. It has 72 vertices, 216 3-edges, and 54 3{3}3 faces. Its complex reflection group is 3[3]3[4]2, order 1296. It has a half-symmetry quasiregular construction as {{CDD|3node|3|3node_1|3|3node}}, as a rectification of the Hessian polyhedron, {{CDD|3node_1|3|3node|3|3node}}. Coxeter, H. S. M., Regular Complex Polytopes, second edition, Cambridge University Press, (1991). p.30 and p.47

= Related polytopes and honeycomb =

Along with the semiregular polytope, 221, it is also one of a family of 39 convex uniform polytopes in 6-dimensions, made of uniform polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea}}.

{{1 k2 polytopes}}

== Geometric folding ==

The 122 is related to the 24-cell by a geometric folding E6 → F4 of Coxeter-Dynkin diagrams, E6 corresponding to 122 in 6 dimensions, F4 to the 24-cell in 4 dimensions. This can be seen in the Coxeter plane projections. The 24 vertices of the 24-cell are projected in the same two rings as seen in the 122.

class=wikitable
colspan=2|E6/F4 Coxeter planes
align=center valign=top

|160px
122

|160px
24-cell

colspan=2|D4/B4 Coxeter planes
align=center valign=top

|160px
122

|160px
24-cell

== Tessellations ==

This polytope is the vertex figure for a uniform tessellation of 6-dimensional space, 222, {{CDD|node_1|3|node|3|node|split1|nodes|3ab|nodes}}.

Rectified 1<sub>22</sub> polytope

class="wikitable" align="right" style="margin-left:10px" width="280"

!bgcolor=#e7dcc3 colspan=2|Rectified 122

bgcolor=#e7dcc3|TypeUniform 6-polytope
bgcolor=#e7dcc3|Schläfli symbol2r{3,3,32,1}
r{3,32,2}
bgcolor=#e7dcc3|Coxeter symbol0221
bgcolor=#e7dcc3|Coxeter-Dynkin diagram{{CDD|nodea|3a|nodea|3a|branch_10|3a|nodea|3a|nodea}}
or {{CDD|node|3|node_1|split1|nodes|3ab|nodes}}
bgcolor=#e7dcc3|5-faces126
bgcolor=#e7dcc3|4-faces1566
bgcolor=#e7dcc3|Cells6480
bgcolor=#e7dcc3|Faces6480
bgcolor=#e7dcc3|Edges6480
bgcolor=#e7dcc3|Vertices720
bgcolor=#e7dcc3|Vertex figure3-3 duoprism prism
bgcolor=#e7dcc3|Petrie polygonDodecagon
bgcolor=#e7dcc3|Coxeter groupE6, 3,32,2, order 103680
bgcolor=#e7dcc3|Propertiesconvex

The rectified 122 polytope (also called 0221) can tessellate 6-dimensional space as the Voronoi cell of the E6* honeycomb lattice (dual of E6 lattice).[http://home.digital.net/~pervin/publications/vermont.html The Voronoi Cells of the E6* and E7* Lattices] {{Webarchive|url=https://web.archive.org/web/20160130193811/http://home.digital.net/~pervin/publications/vermont.html |date=2016-01-30 }}, Edward Pervin

= Alternate names =

  • Birectified 221 polytope
  • Rectified pentacontatetrapeton (Acronym: ram) - rectified 54-facetted polypeton (Jonathan Bowers)Klitzing, (o3o3x3o3o *c3o - [http://bendwavy.org/klitzing/incmats/ram.htm ram])

= Images =

Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow.

class=wikitable width=480

|+ Coxeter plane orthographic projections

E6
[12]

!D5
[8]

!D4 / A2
[6]

!B6
[12/2]

valign=top align=center

|120px

|120px

|120px

|120px

A5
[6]

!A4
[5]

!A3 / D3
[4]

valign=top align=center

|120px

|120px

|120px

= Construction =

Its construction is based on the E6 group and information can be extracted from the ringed Coxeter-Dynkin diagram representing this polytope: {{CDD|nodea|3a|nodea|3a|branch_10|3a|nodea|3a|nodea}}.

Removing the ring on the short branch leaves the birectified 5-simplex, {{CDD|nodea|3a|nodea|3a|nodea_1|3a|nodea|3a|nodea}}.

Removing the ring on either of 2-length branches leaves the birectified 5-orthoplex in its alternated form: t2(211), {{CDD|nodea|3a|branch_10|3a|nodea|3a|nodea}}.

The vertex figure is determined by removing the ringed node and ringing the neighboring ring. This makes 3-3 duoprism prism, {3}×{3}×{}, {{CDD|node|3|node_1|2|node_1|2|node_1|3|node}}.

Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.{{r|ram}}

class=wikitable

!E6

{{CDD|nodea|3a|nodea|3a|branch_10|3a|nodea|3a|nodea}}k-facefkf0f1colspan=3|f2colspan=5|f3colspan=5|f4colspan=3|f5k-figurenotes
align=right

|A2A2A1

{{CDD|nodea|3a|nodea|2|nodes_x0|2|nodea|3a|nodea}}( )

!f0

|BGCOLOR="#ffe0ff"|720

181818961896963693233{3}×{3}×{ }E6/A2A2A1 = 72*6!/3!/3!/2 = 720
align=right

|A1A1A1

{{CDD|nodea|2|nodea_x|2|nodes_1x|2|nodea_x|2|nodea}}{ }

!f1

2BGCOLOR="#ffe0e0"|64802211421221241122{ }∨{ }∨( )E6/A1A1A1 = 72*6!/2/2/2 = 6480
align=right

|rowspan=2|A2A1

{{CDD|nodea_x|2|nodea|3a|nodes_1x|2|nodea_x|2|nodea}}rowspan=3|{3}

!rowspan=3|f2

33BGCOLOR="#ffffe0"|4320BGCOLOR="#ffffe0"|*BGCOLOR="#ffffe0"|*1210021120121rowspan=2|Sphenoidrowspan=2|E6/A2A1 = 72*6!/3!/2 = 4320
align=right{{CDD|nodea|2|nodea_x|2|nodes_1x|3a|nodea|2|nodea_x}}33BGCOLOR="#ffffe0"|*BGCOLOR="#ffffe0"|4320BGCOLOR="#ffffe0"|*0201110221112
align=right

|A2A1A1

{{CDD|nodea|2|nodea_x|2|branch_10|2|nodea_x|2|nodea}}33BGCOLOR="#ffffe0"|*BGCOLOR="#ffffe0"|*BGCOLOR="#ffffe0"|21600020201041022{ }∨{ }E6/A2A1A1 = 72*6!/3!/2/2 = 2160
align=right

|A2A1

{{CDD|nodea|3a|nodea|3a|nodes_1x|2|nodea_x|2|nodea}}{3,3}

!rowspan=5|f3

46400BGCOLOR="#e0ffe0"|1080BGCOLOR="#e0ffe0"|*BGCOLOR="#e0ffe0"|*BGCOLOR="#e0ffe0"|*BGCOLOR="#e0ffe0"|*21000120{ }∨( )E6/A2A1 = 72*6!/3!/2 = 1080
align=right

|A3

{{CDD|nodea_x|2|nodea|3a|nodes_1x|3a|nodea|2|nodea_x}}rowspan=2|r{3,3}612440BGCOLOR="#e0ffe0"|*BGCOLOR="#e0ffe0"|2160BGCOLOR="#e0ffe0"|*BGCOLOR="#e0ffe0"|*BGCOLOR="#e0ffe0"|*10110111{3}E6/A3 = 72*6!/4! = 2160
align=right

|rowspan=3|A3A1

{{CDD|nodea_x|2|nodea|3a|branch_10|2|nodea_x|2|nodea}}612404BGCOLOR="#e0ffe0"|*BGCOLOR="#e0ffe0"|*BGCOLOR="#e0ffe0"|1080BGCOLOR="#e0ffe0"|*BGCOLOR="#e0ffe0"|*01020021rowspan=3|{ }∨( )rowspan=3|E6/A3A1 = 72*6!/4!/2 = 1080
align=right{{CDD|nodea|2|nodea_x|2|nodes_1x|3a|nodea|3a|nodea}}{3,3}46040BGCOLOR="#e0ffe0"|*BGCOLOR="#e0ffe0"|*BGCOLOR="#e0ffe0"|*BGCOLOR="#e0ffe0"|1080BGCOLOR="#e0ffe0"|*00201102
align=right{{CDD|nodea|2|nodea_x|2|branch_10|3a|nodea|2|nodea_x}}r{3,3}612044BGCOLOR="#e0ffe0"|*BGCOLOR="#e0ffe0"|*BGCOLOR="#e0ffe0"|*BGCOLOR="#e0ffe0"|*BGCOLOR="#e0ffe0"|108000021012
align=right

|A4

{{CDD|nodea|3a|nodea|3a|nodes_1x|3a|nodea|2|nodea_x}}rowspan=3| r{3,3,3}

!rowspan=5|f4

10302010055000BGCOLOR="#e0ffff"|432BGCOLOR="#e0ffff"|*BGCOLOR="#e0ffff"|*BGCOLOR="#e0ffff"|*BGCOLOR="#e0ffff"|*110rowspan=5|{ }E6/A4 = 72*6!/5! = 432
align=right

|A4A1

{{CDD|nodea|3a|nodea|3a|branch_10|2|nodea_x|2|nodea}}10302001050500BGCOLOR="#e0ffff"|*BGCOLOR="#e0ffff"|216BGCOLOR="#e0ffff"|*BGCOLOR="#e0ffff"|*BGCOLOR="#e0ffff"|*020E6/A4A1 = 72*6!/5!/2 = 216
align=right

|A4

{{CDD|nodea_x|2|nodea|3a|nodes_1x|3a|nodea|3a|nodea}}10301020005050BGCOLOR="#e0ffff"|*BGCOLOR="#e0ffff"|*BGCOLOR="#e0ffff"|432BGCOLOR="#e0ffff"|*BGCOLOR="#e0ffff"|*101E6/A4 = 72*6!/5! = 432
align=right

|D4

{{CDD|nodea_x|2|nodea|3a|branch_10|3a|nodea|2|nodea_x}}{3,4,3}249632323208808BGCOLOR="#e0ffff"|*BGCOLOR="#e0ffff"|*BGCOLOR="#e0ffff"|*BGCOLOR="#e0ffff"|270BGCOLOR="#e0ffff"|*011E6/D4 = 72*6!/8/4! = 270
align=right

|A4A1

{{CDD|nodea|2|nodea_x|2|branch_10|3a|nodea|3a|nodea}}r{3,3,3}10300201000055BGCOLOR="#e0ffff"|*BGCOLOR="#e0ffff"|*BGCOLOR="#e0ffff"|*BGCOLOR="#e0ffff"|*BGCOLOR="#e0ffff"|216002E6/A4A1 = 72*6!/5!/2 = 216
align=right

|A5

{{CDD|nodea|3a|nodea|3a|nodes_1x|3a|nodea|3a|nodea}}2r{3,3,3,3}

!rowspan=3|f5

2090606001530015060600BGCOLOR="#e0e0ff"|72BGCOLOR="#e0e0ff"|*BGCOLOR="#e0e0ff"|*rowspan=3|( )E6/A5 = 72*6!/6! = 72
align=right

|rowspan=2|D5

{{CDD|nodea|3a|nodea|3a|branch_10|3a|nodea|2|nodea_x}}rowspan=2|2r{4,3,3,3}8048032016016080808004016160100BGCOLOR="#e0e0ff"|*BGCOLOR="#e0e0ff"|27BGCOLOR="#e0e0ff"|*rowspan=2|E6/D5 = 72*6!/16/5! = 27
align=right{{CDD|nodea_x|2|nodea|3a|branch_10|3a|nodea|3a|nodea}}8048016032016008040808000161016BGCOLOR="#e0e0ff"|*BGCOLOR="#e0e0ff"|*BGCOLOR="#e0e0ff"|27

Truncated 1<sub>22</sub> polytope

class="wikitable" align="right" style="margin-left:10px" width="280"

!bgcolor=#e7dcc3 colspan=2|Truncated 122

bgcolor=#e7dcc3|TypeUniform 6-polytope
bgcolor=#e7dcc3|Schläfli symbolt{3,32,2}
bgcolor=#e7dcc3|Coxeter symbolt(122)
bgcolor=#e7dcc3|Coxeter-Dynkin diagram{{CDD|nodea|3a|nodea|3a|branch_11|3a|nodea|3a|nodea}}
or {{CDD|node_1|3|node_1|split1|nodes|3ab|nodes}}
bgcolor=#e7dcc3|5-faces72+27+27
bgcolor=#e7dcc3|4-faces32+216+432+270+216
bgcolor=#e7dcc3|Cells1080+2160+1080+1080+1080
bgcolor=#e7dcc3|Faces4320+4320+2160
bgcolor=#e7dcc3|Edges6480+720
bgcolor=#e7dcc3|Vertices1440
bgcolor=#e7dcc3|Vertex figure( )v{3}x{3}
bgcolor=#e7dcc3|Petrie polygonDodecagon
bgcolor=#e7dcc3|Coxeter groupE6, 3,32,2, order 103680
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Truncated 122 polytope (Acronym: tim)Klitzing, (o3o3x3o3o *c3x - [http://bendwavy.org/klitzing/incmats/tim.htm tim])

= Construction =

Its construction is based on the E6 group and information can be extracted from the ringed Coxeter-Dynkin diagram representing this polytope: {{CDD|nodea|3a|nodea|3a|branch_11|3a|nodea|3a|nodea}}.

= Images =

Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow.

class=wikitable width=480

|+ Coxeter plane orthographic projections

E6
[12]

!D5
[8]

!D4 / A2
[6]

!B6
[12/2]

valign=top align=center

|120px

|120px

|120px

|120px

A5
[6]

!A4
[5]

!A3 / D3
[4]

valign=top align=center

|120px

|120px

|120px

Birectified 1<sub>22</sub> polytope

class="wikitable" align="right" style="margin-left:10px" width="250"

!bgcolor=#e7dcc3 colspan=2|Birectified 122 polytope

bgcolor=#e7dcc3|TypeUniform 6-polytope
bgcolor=#e7dcc3|Schläfli symbol2r{3,32,2}
bgcolor=#e7dcc3|Coxeter symbol2r(122)
bgcolor=#e7dcc3|Coxeter-Dynkin diagram{{CDD|nodea|3a|nodea_1|3a|branch|3a|nodea_1|3a|nodea}}
or {{CDD|node|3|node|split1|nodes_11|3ab|nodes}}
bgcolor=#e7dcc3|5-faces126
bgcolor=#e7dcc3|4-faces2286
bgcolor=#e7dcc3|Cells10800
bgcolor=#e7dcc3|Faces19440
bgcolor=#e7dcc3|Edges12960
bgcolor=#e7dcc3|Vertices2160
bgcolor=#e7dcc3|Vertex figure
bgcolor=#e7dcc3|Coxeter groupE6, 3,32,2, order 103680
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Bicantellated 221
  • Birectified pentacontatetrapeton (barm) (Jonathan Bowers)Klitzing, (o3x3o3x3o *c3o - [http://bendwavy.org/klitzing/incmats/scram.htm barm])

= Images =

Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow.

class=wikitable width=480

|+ Coxeter plane orthographic projections

E6
[12]

!D5
[8]

!D4 / A2
[6]

!B6
[12/2]

valign=top align=center

|120px

|120px

|120px

|120px

A5
[6]

!A4
[5]

!A3 / D3
[4]

valign=top align=center

|120px

|120px

|120px

Trirectified 1<sub>22</sub> polytope

class="wikitable" align="right" style="margin-left:10px" width="250"

!bgcolor=#e7dcc3 colspan=2|Trirectified 122 polytope

bgcolor=#e7dcc3|TypeUniform 6-polytope
bgcolor=#e7dcc3|Schläfli symbol3r{3,32,2}
bgcolor=#e7dcc3|Coxeter symbol3r(122)
bgcolor=#e7dcc3|Coxeter-Dynkin diagram{{CDD|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea_1}}
or {{CDD|node|3|node|split1|nodes|3ab|nodes_11}}
bgcolor=#e7dcc3|5-faces558
bgcolor=#e7dcc3|4-faces4608
bgcolor=#e7dcc3|Cells8640
bgcolor=#e7dcc3|Faces6480
bgcolor=#e7dcc3|Edges2160
bgcolor=#e7dcc3|Vertices270
bgcolor=#e7dcc3|Vertex figure
bgcolor=#e7dcc3|Coxeter groupE6, 3,32,2, order 103680
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Tricantellated 221
  • Trirectified pentacontatetrapeton (Acronym: trim, old: cacam, tram, mak) (Jonathan Bowers)Klitzing, (x3o3o3o3x *c3o - [http://bendwavy.org/klitzing/incmats/cacam.htm trim])

See also

Notes

{{reflist}}

References

  • {{citation | last = Elte | first = E. L. | title = The Semiregular Polytopes of the Hyperspaces | publisher = University of Groningen | location = Groningen | year = 1912}}
  • H. S. M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
  • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, [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] See p334 (figure 3.6a) by Peter mcMullen: (12-gonal node-edge graph of 122)
  • {{KlitzingPolytopes|polypeta.htm|6D uniform polytopes (polypeta) with acronyms}} o3o3o3o3o *c3x - mo, o3o3x3o3o *c3o - ram, o3o3x3o3o *c3x - tim, o3x3o3x3o *c3o - barm, x3o3o3o3x *c3o - trim

{{Polytopes}}

Category:6-polytopes