1 22 polytope#Related complex polyhedron
{{Short description|Uniform 6-polytope}}
{{DISPLAYTITLE:1 22 polytope}}
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!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
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!bgcolor=#e7dcc3 colspan=2|122 polytope | |
bgcolor=#e7dcc3|Type | Uniform 6-polytope |
bgcolor=#e7dcc3|Family | 1k2 polytope |
bgcolor=#e7dcc3|Schläfli symbol | {3,32,2} |
bgcolor=#e7dcc3|Coxeter symbol | 122 |
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-faces | 54: 27 121 30px 27 121 30px |
bgcolor=#e7dcc3|4-faces | 702: 270 111 25px 432 120 25px |
bgcolor=#e7dcc3|Cells | 2160: 1080 110 25px 1080 {3,3} 25px |
bgcolor=#e7dcc3|Faces | 2160 {3} 25px |
bgcolor=#e7dcc3|Edges | 720 |
bgcolor=#e7dcc3|Vertices | 72 |
bgcolor=#e7dcc3|Vertex figure | Birectified 5-simplex: 022 50px |
bgcolor=#e7dcc3|Petrie polygon | Dodecagon |
bgcolor=#e7dcc3|Coxeter group | E6, 3,32,2, order 103680 |
bgcolor=#e7dcc3|Properties | convex, 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 =
= 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
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!E6 | width=60|{{CDD|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea}} | k-face | fk | f0 | f1 | f2 | colspan=2|f3 | colspan=3|f4 | colspan=2|f5 | k-figure | notes | |||
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|A5 | {{CDD|nodea|3a|nodea|3a|nodes_0x|3a|nodea|3a|nodea}} | ( )
! f0 |BGCOLOR="#ffe0ff"|72 | 20 | 90 | 60 | 60 | 15 | 15 | 30 | 6 | 6 | r{3,3,3} | E6/A5 = 72*6!/6! = 72 | |
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|A2A2A1 | {{CDD|nodea|3a|nodea|2|nodes_x1|2|nodea|3a|nodea}} | { }
! f1 | 2 | BGCOLOR="#ffe0e0"|720 | 9 | 9 | 9 | 3 | 3 | 9 | 3 | 3 | {3}×{3} | E6/A2A2A1 = 72*6!/3!/3!/2 = 720 |
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|A2A1A1 | {{CDD|nodea|2|nodea_x|2|branch_01|2|nodea_x|2|nodea}} | {3}
! f2 | 3 | 3 | BGCOLOR="#ffffe0"|2160 | 2 | 2 | 1 | 1 | 4 | 2 | 2 | s{2,4} | E6/A2A1A1 = 72*6!/3!/2/2 = 2160 |
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|rowspan=2|A3A1 | {{CDD|nodea_x|2|nodea|3a|branch_01l|2|nodea_x|2|nodea}} | rowspan=2|{3,3}
! rowspan=2|f3 | 4 | 6 | 4 | BGCOLOR="#e0ffe0"|1080 | BGCOLOR="#e0ffe0"|* | 1 | 0 | 2 | 2 | 1 | rowspan=2|{ }∨( ) | rowspan=2| E6/A3A1 = 72*6!/4!/2 = 1080 |
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|{{CDD|nodea|2|nodea_x|2|branch_01r|3a|nodea|2|nodea_x}} | 4 | 6 | 4 | BGCOLOR="#e0ffe0"|* | BGCOLOR="#e0ffe0"|1080 | 0 | 1 | 2 | 1 | 2 | ||||
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|rowspan=2|A4A1 | {{CDD|nodea|2|nodea_x|2|branch_01r|3a|nodea|3a|nodea}} | rowspan=2| {3,3,3}
!rowspan=3|f4 | 5 | 10 | 10 | 5 | 0 | BGCOLOR="#e0ffff"|216 | BGCOLOR="#e0ffff"|* | BGCOLOR="#e0ffff"|* | 2 | 0 | rowspan=3|{ } | rowspan=2| E6/A4A1 = 72*6!/5!/2 = 216 |
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|{{CDD|nodea|3a|nodea|3a|branch_01l|2|nodea_x|2|nodea}} | 5 | 10 | 10 | 0 | 5 | BGCOLOR="#e0ffff"|* | BGCOLOR="#e0ffff"|216 | BGCOLOR="#e0ffff"|* | 0 | 2 | ||||
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|D4 | {{CDD|nodea_x|2|nodea|3a|branch_01lr|3a|nodea|2|nodea_x}} | h{4,3,3} | 8 | 24 | 32 | 8 | 8 | BGCOLOR="#e0ffff"|* | BGCOLOR="#e0ffff"|* | BGCOLOR="#e0ffff"|270 | 1 | 1 | E6/D4 = 72*6!/8/4! = 270 | |
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|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 | 16 | 80 | 160 | 80 | 40 | 16 | 0 | 10 | BGCOLOR="#e0e0ff"|27 | BGCOLOR="#e0e0ff"|* | rowspan=2|( ) | rowspan=2| E6/D5 = 72*6!/16/5! = 27 |
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|{{CDD|nodea_x|2|nodea|3a|branch_01lr|3a|nodea|3a|nodea}} | 16 | 80 | 160 | 40 | 80 | 0 | 16 | 10 | BGCOLOR="#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 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.
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== 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
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!bgcolor=#e7dcc3 colspan=2|Rectified 122 | |
bgcolor=#e7dcc3|Type | Uniform 6-polytope |
bgcolor=#e7dcc3|Schläfli symbol | 2r{3,3,32,1} r{3,32,2} |
bgcolor=#e7dcc3|Coxeter symbol | 0221 |
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-faces | 126 |
bgcolor=#e7dcc3|4-faces | 1566 |
bgcolor=#e7dcc3|Cells | 6480 |
bgcolor=#e7dcc3|Faces | 6480 |
bgcolor=#e7dcc3|Edges | 6480 |
bgcolor=#e7dcc3|Vertices | 720 |
bgcolor=#e7dcc3|Vertex figure | 3-3 duoprism prism |
bgcolor=#e7dcc3|Petrie polygon | Dodecagon |
bgcolor=#e7dcc3|Coxeter group | E6, 3,32,2, order 103680 |
bgcolor=#e7dcc3|Properties | convex |
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.
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E6 [12] !D5 !D4 / A2 !B6 |
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A5 [6] !A4 !A3 / D3 |
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= 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}}
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!E6 | {{CDD|nodea|3a|nodea|3a|branch_10|3a|nodea|3a|nodea}} | k-face | fk | f0 | f1 | colspan=3|f2 | colspan=5|f3 | colspan=5|f4 | colspan=3|f5 | k-figure | notes | |||||||||||
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|A2A2A1 | {{CDD|nodea|3a|nodea|2|nodes_x0|2|nodea|3a|nodea}} | ( )
!f0 |BGCOLOR="#ffe0ff"|720 | 18 | 18 | 18 | 9 | 6 | 18 | 9 | 6 | 9 | 6 | 3 | 6 | 9 | 3 | 2 | 3 | 3 | {3}×{3}×{ } | E6/A2A2A1 = 72*6!/3!/3!/2 = 720 | |
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|A1A1A1 | {{CDD|nodea|2|nodea_x|2|nodes_1x|2|nodea_x|2|nodea}} | { }
!f1 | 2 | BGCOLOR="#ffe0e0"|6480 | 2 | 2 | 1 | 1 | 4 | 2 | 1 | 2 | 2 | 1 | 2 | 4 | 1 | 1 | 2 | 2 | { }∨{ }∨( ) | E6/A1A1A1 = 72*6!/2/2/2 = 6480 |
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|rowspan=2|A2A1 | {{CDD|nodea_x|2|nodea|3a|nodes_1x|2|nodea_x|2|nodea}} | rowspan=3|{3}
!rowspan=3|f2 | 3 | 3 | BGCOLOR="#ffffe0"|4320 | BGCOLOR="#ffffe0"|* | BGCOLOR="#ffffe0"|* | 1 | 2 | 1 | 0 | 0 | 2 | 1 | 1 | 2 | 0 | 1 | 2 | 1 | rowspan=2|Sphenoid | rowspan=2|E6/A2A1 = 72*6!/3!/2 = 4320 |
align=right | {{CDD|nodea|2|nodea_x|2|nodes_1x|3a|nodea|2|nodea_x}} | 3 | 3 | BGCOLOR="#ffffe0"|* | BGCOLOR="#ffffe0"|4320 | BGCOLOR="#ffffe0"|* | 0 | 2 | 0 | 1 | 1 | 1 | 0 | 2 | 2 | 1 | 1 | 1 | 2 | |||
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|A2A1A1 | {{CDD|nodea|2|nodea_x|2|branch_10|2|nodea_x|2|nodea}} | 3 | 3 | BGCOLOR="#ffffe0"|* | BGCOLOR="#ffffe0"|* | BGCOLOR="#ffffe0"|2160 | 0 | 0 | 2 | 0 | 2 | 0 | 1 | 0 | 4 | 1 | 0 | 2 | 2 | { }∨{ } | E6/A2A1A1 = 72*6!/3!/2/2 = 2160 | |
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|A2A1 | {{CDD|nodea|3a|nodea|3a|nodes_1x|2|nodea_x|2|nodea}} | {3,3}
!rowspan=5|f3 | 4 | 6 | 4 | 0 | 0 | BGCOLOR="#e0ffe0"|1080 | BGCOLOR="#e0ffe0"|* | BGCOLOR="#e0ffe0"|* | BGCOLOR="#e0ffe0"|* | BGCOLOR="#e0ffe0"|* | 2 | 1 | 0 | 0 | 0 | 1 | 2 | 0 | { }∨( ) | E6/A2A1 = 72*6!/3!/2 = 1080 |
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|A3 | {{CDD|nodea_x|2|nodea|3a|nodes_1x|3a|nodea|2|nodea_x}} | rowspan=2|r{3,3} | 6 | 12 | 4 | 4 | 0 | BGCOLOR="#e0ffe0"|* | BGCOLOR="#e0ffe0"|2160 | BGCOLOR="#e0ffe0"|* | BGCOLOR="#e0ffe0"|* | BGCOLOR="#e0ffe0"|* | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | {3} | E6/A3 = 72*6!/4! = 2160 |
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|rowspan=3|A3A1 | {{CDD|nodea_x|2|nodea|3a|branch_10|2|nodea_x|2|nodea}} | 6 | 12 | 4 | 0 | 4 | BGCOLOR="#e0ffe0"|* | BGCOLOR="#e0ffe0"|* | BGCOLOR="#e0ffe0"|1080 | BGCOLOR="#e0ffe0"|* | BGCOLOR="#e0ffe0"|* | 0 | 1 | 0 | 2 | 0 | 0 | 2 | 1 | rowspan=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} | 4 | 6 | 0 | 4 | 0 | BGCOLOR="#e0ffe0"|* | BGCOLOR="#e0ffe0"|* | BGCOLOR="#e0ffe0"|* | BGCOLOR="#e0ffe0"|1080 | BGCOLOR="#e0ffe0"|* | 0 | 0 | 2 | 0 | 1 | 1 | 0 | 2 | ||
align=right | {{CDD|nodea|2|nodea_x|2|branch_10|3a|nodea|2|nodea_x}} | r{3,3} | 6 | 12 | 0 | 4 | 4 | BGCOLOR="#e0ffe0"|* | BGCOLOR="#e0ffe0"|* | BGCOLOR="#e0ffe0"|* | BGCOLOR="#e0ffe0"|* | BGCOLOR="#e0ffe0"|1080 | 0 | 0 | 0 | 2 | 1 | 0 | 1 | 2 | ||
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|A4 | {{CDD|nodea|3a|nodea|3a|nodes_1x|3a|nodea|2|nodea_x}} | rowspan=3| r{3,3,3}
!rowspan=5|f4 | 10 | 30 | 20 | 10 | 0 | 5 | 5 | 0 | 0 | 0 | BGCOLOR="#e0ffff"|432 | BGCOLOR="#e0ffff"|* | BGCOLOR="#e0ffff"|* | BGCOLOR="#e0ffff"|* | BGCOLOR="#e0ffff"|* | 1 | 1 | 0 | rowspan=5|{ } | E6/A4 = 72*6!/5! = 432 |
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|A4A1 | {{CDD|nodea|3a|nodea|3a|branch_10|2|nodea_x|2|nodea}} | 10 | 30 | 20 | 0 | 10 | 5 | 0 | 5 | 0 | 0 | BGCOLOR="#e0ffff"|* | BGCOLOR="#e0ffff"|216 | BGCOLOR="#e0ffff"|* | BGCOLOR="#e0ffff"|* | BGCOLOR="#e0ffff"|* | 0 | 2 | 0 | E6/A4A1 = 72*6!/5!/2 = 216 | ||
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|A4 | {{CDD|nodea_x|2|nodea|3a|nodes_1x|3a|nodea|3a|nodea}} | 10 | 30 | 10 | 20 | 0 | 0 | 5 | 0 | 5 | 0 | BGCOLOR="#e0ffff"|* | BGCOLOR="#e0ffff"|* | BGCOLOR="#e0ffff"|432 | BGCOLOR="#e0ffff"|* | BGCOLOR="#e0ffff"|* | 1 | 0 | 1 | E6/A4 = 72*6!/5! = 432 | ||
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|D4 | {{CDD|nodea_x|2|nodea|3a|branch_10|3a|nodea|2|nodea_x}} | {3,4,3} | 24 | 96 | 32 | 32 | 32 | 0 | 8 | 8 | 0 | 8 | BGCOLOR="#e0ffff"|* | BGCOLOR="#e0ffff"|* | BGCOLOR="#e0ffff"|* | BGCOLOR="#e0ffff"|270 | BGCOLOR="#e0ffff"|* | 0 | 1 | 1 | E6/D4 = 72*6!/8/4! = 270 | |
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|A4A1 | {{CDD|nodea|2|nodea_x|2|branch_10|3a|nodea|3a|nodea}} | r{3,3,3} | 10 | 30 | 0 | 20 | 10 | 0 | 0 | 0 | 5 | 5 | BGCOLOR="#e0ffff"|* | BGCOLOR="#e0ffff"|* | BGCOLOR="#e0ffff"|* | BGCOLOR="#e0ffff"|* | BGCOLOR="#e0ffff"|216 | 0 | 0 | 2 | E6/A4A1 = 72*6!/5!/2 = 216 | |
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|A5 | {{CDD|nodea|3a|nodea|3a|nodes_1x|3a|nodea|3a|nodea}} | 2r{3,3,3,3}
!rowspan=3|f5 | 20 | 90 | 60 | 60 | 0 | 15 | 30 | 0 | 15 | 0 | 6 | 0 | 6 | 0 | 0 | BGCOLOR="#e0e0ff"|72 | BGCOLOR="#e0e0ff"|* | BGCOLOR="#e0e0ff"|* | rowspan=3|( ) | E6/A5 = 72*6!/6! = 72 |
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|rowspan=2|D5 | {{CDD|nodea|3a|nodea|3a|branch_10|3a|nodea|2|nodea_x}} | rowspan=2|2r{4,3,3,3} | 80 | 480 | 320 | 160 | 160 | 80 | 80 | 80 | 0 | 40 | 16 | 16 | 0 | 10 | 0 | BGCOLOR="#e0e0ff"|* | BGCOLOR="#e0e0ff"|27 | BGCOLOR="#e0e0ff"|* | rowspan=2|E6/D5 = 72*6!/16/5! = 27 | |
align=right | {{CDD|nodea_x|2|nodea|3a|branch_10|3a|nodea|3a|nodea}} | 80 | 480 | 160 | 320 | 160 | 0 | 80 | 40 | 80 | 80 | 0 | 0 | 16 | 10 | 16 | BGCOLOR="#e0e0ff"|* | BGCOLOR="#e0e0ff"|* | BGCOLOR="#e0e0ff"|27 |
Truncated 1<sub>22</sub> polytope
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!bgcolor=#e7dcc3 colspan=2|Truncated 122 | |
bgcolor=#e7dcc3|Type | Uniform 6-polytope |
bgcolor=#e7dcc3|Schläfli symbol | t{3,32,2} |
bgcolor=#e7dcc3|Coxeter symbol | t(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-faces | 72+27+27 |
bgcolor=#e7dcc3|4-faces | 32+216+432+270+216 |
bgcolor=#e7dcc3|Cells | 1080+2160+1080+1080+1080 |
bgcolor=#e7dcc3|Faces | 4320+4320+2160 |
bgcolor=#e7dcc3|Edges | 6480+720 |
bgcolor=#e7dcc3|Vertices | 1440 |
bgcolor=#e7dcc3|Vertex figure | ( )v{3}x{3} |
bgcolor=#e7dcc3|Petrie polygon | Dodecagon |
bgcolor=#e7dcc3|Coxeter group | E6, 3,32,2, order 103680 |
bgcolor=#e7dcc3|Properties | convex |
= 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.
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E6 [12] !D5 !D4 / A2 !B6 |
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A5 [6] !A4 !A3 / D3 |
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Birectified 1<sub>22</sub> polytope
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!bgcolor=#e7dcc3 colspan=2|Birectified 122 polytope | |
bgcolor=#e7dcc3|Type | Uniform 6-polytope |
bgcolor=#e7dcc3|Schläfli symbol | 2r{3,32,2} |
bgcolor=#e7dcc3|Coxeter symbol | 2r(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-faces | 126 |
bgcolor=#e7dcc3|4-faces | 2286 |
bgcolor=#e7dcc3|Cells | 10800 |
bgcolor=#e7dcc3|Faces | 19440 |
bgcolor=#e7dcc3|Edges | 12960 |
bgcolor=#e7dcc3|Vertices | 2160 |
bgcolor=#e7dcc3|Vertex figure | |
bgcolor=#e7dcc3|Coxeter group | E6, 3,32,2, order 103680 |
bgcolor=#e7dcc3|Properties | convex |
= 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.
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E6 [12] !D5 !D4 / A2 !B6 |
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A5 [6] !A4 !A3 / D3 |
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Trirectified 1<sub>22</sub> polytope
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!bgcolor=#e7dcc3 colspan=2|Trirectified 122 polytope | |
bgcolor=#e7dcc3|Type | Uniform 6-polytope |
bgcolor=#e7dcc3|Schläfli symbol | 3r{3,32,2} |
bgcolor=#e7dcc3|Coxeter symbol | 3r(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-faces | 558 |
bgcolor=#e7dcc3|4-faces | 4608 |
bgcolor=#e7dcc3|Cells | 8640 |
bgcolor=#e7dcc3|Faces | 6480 |
bgcolor=#e7dcc3|Edges | 2160 |
bgcolor=#e7dcc3|Vertices | 270 |
bgcolor=#e7dcc3|Vertex figure | |
bgcolor=#e7dcc3|Coxeter group | E6, 3,32,2, order 103680 |
bgcolor=#e7dcc3|Properties | convex |
= 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}}