1 32 polytope
{{Short description|Uniform polytope}}
{{DISPLAYTITLE:1 32 polytope}}
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! colspan="6" |Orthogonal projections in E7 Coxeter plane |
In 7-dimensional geometry, 132 is a uniform polytope, constructed from the E7 group.
Its Coxeter symbol is 132, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 1-node sequences.
The rectified 132 is constructed by points at the mid-edges of the 132.
These polytopes are part of a family of 127 (27-1) convex uniform polytopes in 7-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|3a|nodea}}.
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1<sub>32</sub> polytope
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!bgcolor=#e7dcc3 colspan=2|132 | |
bgcolor=#e7dcc3|Type | Uniform 7-polytope |
bgcolor=#e7dcc3|Family | 1k2 polytope |
bgcolor=#e7dcc3|Schläfli symbol | {3,33,2} |
bgcolor=#e7dcc3|Coxeter symbol | 132 |
bgcolor=#e7dcc3|Coxeter diagram | {{CDD|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea|3a|nodea}} |
bgcolor=#e7dcc3|6-faces | 182: 56 122 30px 126 131 30px |
bgcolor=#e7dcc3|5-faces | 4284: 756 121 25px 1512 121 25px 2016 {34} 25px |
bgcolor=#e7dcc3|4-faces | 23688: 4032 {33} 25px 7560 111 25px 12096 {33} 25px |
bgcolor=#e7dcc3|Cells | 50400: 20160 {32} 25px 30240 {32} 25px |
8
|bgcolor=#e7dcc3|Faces | 40320 {3}25px |
bgcolor=#e7dcc3|Edges | 10080 |
bgcolor=#e7dcc3|Vertices | 576 |
bgcolor=#e7dcc3|Vertex figure | t2{35} 25px |
bgcolor=#e7dcc3|Petrie polygon | Octadecagon |
bgcolor=#e7dcc3|Coxeter group | E7, [33,2,1], order 2903040 |
bgcolor=#e7dcc3|Properties | convex |
This polytope can tessellate 7-dimensional space, with symbol 133, and Coxeter-Dynkin diagram, {{CDD|nodea|3a|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea|3a|nodea}}. It is the Voronoi cell of the dual E7* 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 =
- Emanuel Lodewijk Elte named it V576 (for its 576 vertices) in his 1912 listing of semiregular polytopes.Elte, 1912
- Coxeter called it 132 for its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node branch.
- Pentacontahexa-hecatonicosihexa-exon (Acronym: lin) - 56-126 facetted polyexon (Jonathan Bowers)Klitzing, (o3o3o3x *c3o3o3o - lin)
= Images =
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|+ Coxeter plane projections |
E7
!E6 / F4 !B7 / A6 |
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|200px |200px |200px |
A5
!D7 / B6 !D6 / B5 |
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D5 / B4 / A4
!D4 / B3 / A2 / G2 !D3 / B2 / A3 |
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= Construction =
It is created by a Wythoff construction upon a set of 7 hyperplane mirrors in 7-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}}
Removing the node on the end of the 2-length branch leaves the 6-demicube, 131, {{CDD|nodea|3a|branch_01lr|3a|nodea|3a|nodea|3a|nodea}}
Removing the node on the end of the 3-length branch leaves the 122, {{CDD|nodea|3a|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 6-simplex, 032, {{CDD|nodea|3a|nodea|3a|nodea_1|3a|nodea|3a|nodea|3a|nodea}}
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 Gossett figures in six, seven, and eight dimensions, p. 202-203
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!E7 | {{CDD|nodea|3a|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea}} | k-face | fk | f0 | f1 | f2 | colspan=2|f3 | colspan=3|f4 | colspan=3|f5 | colspan=2|f6 | k-figures | notes | |||||
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|A6 | width=65|{{CDD|nodea|3a|nodea|3a|nodea|3a|nodes_0x|3a|nodea|3a|nodea}} | ( )
!f0 |BGCOLOR="#ffe0ff"|576 | 35 | 210 | 140 | 210 | 35 | 105 | 105 | 21 | 42 | 21 | 7 | 7 | 2r{3,3,3,3,3} | E7/A6 = 72*8!/7! = 576 | |
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|A3A2A1 | {{CDD|nodea|3a|nodea|3a|nodea|2|nodes_x1|2|nodea|3a|nodea}} | { }
!f1 | 2 | BGCOLOR="#ffe0e0"|10080 | 12 | 12 | 18 | 4 | 12 | 12 | 6 | 12 | 3 | 4 | 3 | {3,3}x{3} | E7/A3A2A1 = 72*8!/4!/3!/2 = 10080 |
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|A2A2A1 | {{CDD|nodea|3a|nodea|2|nodea_x|2|branch_01|2|nodea_x|2|nodea}} | {3}
!f2 | 3 | 3 | BGCOLOR="#ffffe0"|40320 | 2 | 3 | 1 | 6 | 3 | 3 | 6 | 1 | 3 | 2 | { }∨{3} | E7/A2A2A1 = 72*8!/3!/3!/2 = 40320 |
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|A3A2 | {{CDD|nodea|3a|nodea|2|nodea_x|2|branch_01r|3a|nodea|2|nodea_x}} | rowspan=2|{3,3}
!rowspan=2|f3 | 4 | 6 | 4 | BGCOLOR="#e0ffe0"|20160 | BGCOLOR="#e0ffe0"|* | 1 | 3 | 0 | 3 | 3 | 0 | 3 | 1 | {3}∨( ) | E7/A3A2 = 72*8!/4!/3! = 20160 |
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|A3A1A1 | {{CDD|nodea|2|nodea_x|2|nodea|3a|branch_01l|2|nodea_x|2|nodea}} | 4 | 6 | 4 | BGCOLOR="#e0ffe0"|* | BGCOLOR="#e0ffe0"|30240 | 0 | 2 | 2 | 1 | 4 | 1 | 2 | 2 | Phyllic disphenoid | E7/A3A1A1 = 72*8!/4!/2/2 = 30240 | |
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|A4A2 | {{CDD|nodea|3a|nodea|2|nodea_x|2|branch_01r|3a|nodea|3a|nodea}} | {3,3,3}
!rowspan=3|f4 | 5 | 10 | 10 | 5 | 0 | BGCOLOR="#e0ffff"|4032 | BGCOLOR="#e0ffff"|* | BGCOLOR="#e0ffff"|* | 3 | 0 | 0 | 3 | 0 | {3} | E7/A4A2 = 72*8!/5!/3! = 4032 |
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|D4A1 | {{CDD|nodea|2|nodea_x|2|nodea|3a|branch_01lr|3a|nodea|2|nodea_x}} | {3,3,4} | 8 | 24 | 32 | 8 | 8 | BGCOLOR="#e0ffff"|* | BGCOLOR="#e0ffff"|7560 | BGCOLOR="#e0ffff"|* | 1 | 2 | 0 | 2 | 1 | rowspan=2|{ }∨( ) | E7/D4A1 = 72*8!/8/4!/2 = 7560 |
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|A4A1 | {{CDD|nodea_x|2|nodea|3a|nodea|3a|branch_01l|2|nodea_x|2|nodea}} | {3,3,3} | 5 | 10 | 10 | 0 | 5 | BGCOLOR="#e0ffff"|* | BGCOLOR="#e0ffff"|* | BGCOLOR="#e0ffff"|12096 | 0 | 2 | 1 | 1 | 2 | E7/A4A1 = 72*8!/5!/2 = 12096 | |
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|D5A1 | {{CDD|nodea|2|nodea_x|2|nodea|3a|branch_01lr|3a|nodea|3a|nodea}} | rowspan=2|h{4,3,3,3}
!rowspan=3|f5 | 16 | 80 | 160 | 80 | 40 | 16 | 10 | 0 | BGCOLOR="#e0e0ff"|756 | BGCOLOR="#e0e0ff"|* | BGCOLOR="#e0e0ff"|* | 2 | 0 | rowspan=3|{ } | E7/D5A1 = 72*8!/16/5!/2 = 756 |
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|D5 | {{CDD|nodea_x|2|nodea|3a|nodea|3a|branch_01lr|3a|nodea|2|nodea_x}} | 16 | 80 | 160 | 40 | 80 | 0 | 10 | 16 | BGCOLOR="#e0e0ff"|* | BGCOLOR="#e0e0ff"|1512 | BGCOLOR="#e0e0ff"|* | 1 | 1 | E7/D5 = 72*8!/16/5! = 1512 | ||
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|A5A1 | {{CDD|nodea|3a|nodea|3a|nodea|3a|branch_01l|2|nodea_x|2|nodea}} | {3,3,3,3,3} | 6 | 15 | 20 | 0 | 15 | 0 | 0 | 6 | BGCOLOR="#e0e0ff"|* | BGCOLOR="#e0e0ff"|* | BGCOLOR="#e0e0ff"|2016 | 0 | 2 | E7/A5A1 = 72*8!/6!/2 = 2016 | |
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|E6 | {{CDD|nodea_x|2|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea}} | {3,32,2}
!rowspan=2|f6 | 72 | 720 | 2160 | 1080 | 1080 | 216 | 270 | 216 | 27 | 27 | 0 | BGCOLOR="#ffe0ff"|56 | BGCOLOR="#ffe0ff"|* | rowspan=2|( ) | E7/E6 = 72*8!/72/6! = 56 |
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|D6 | {{CDD|nodea|3a|nodea|3a|nodea|3a|branch_01lr|3a|nodea|2|nodea_x}} | h{4,3,3,3,3} | 32 | 240 | 640 | 160 | 480 | 0 | 60 | 192 | 0 | 12 | 32 | BGCOLOR="#ffe0ff"|* | BGCOLOR="#ffe0ff"|126 | E7/D6 = 72*8!/32/6! = 126 |
= Related polytopes and honeycombs =
Rectified 1<sub>32</sub> polytope
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!bgcolor=#e7dcc3 colspan=2|Rectified 132 | |
bgcolor=#e7dcc3|Type | Uniform 7-polytope |
bgcolor=#e7dcc3|Schläfli symbol | t1{3,33,2} |
bgcolor=#e7dcc3|Coxeter symbol | 0321 |
bgcolor=#e7dcc3|Coxeter-Dynkin diagram | {{CDD|nodea|3a|nodea|3a|branch_10|3a|nodea|3a|nodea|3a|nodea}} |
bgcolor=#e7dcc3|6-faces | 758 |
bgcolor=#e7dcc3|5-faces | 12348 |
bgcolor=#e7dcc3|4-faces | 72072 |
bgcolor=#e7dcc3|Cells | 191520 |
bgcolor=#e7dcc3|Faces | 241920 |
bgcolor=#e7dcc3|Edges | 120960 |
bgcolor=#e7dcc3|Vertices | 10080 |
bgcolor=#e7dcc3|Vertex figure | {3,3}×{3}×{} |
bgcolor=#e7dcc3|Coxeter group | E7, [33,2,1], order 2903040 |
bgcolor=#e7dcc3|Properties | convex |
The rectified 132 (also called 0321) is a rectification of the 132 polytope, creating new vertices on the center of edge of the 132. Its vertex figure is a duoprism prism, the product of a regular tetrahedra and triangle, doubled into a prism: {3,3}×{3}×{}.
= Alternate names =
- Rectified pentacontahexa-hecatonicosihexa-exon for rectified 56-126 facetted polyexon (Acronym: rolin) (Jonathan Bowers)Klitzing, (o3o3x3o *c3o3o3o - rolin)
= Construction =
It is created by a Wythoff construction upon a set of 7 hyperplane mirrors in 7-dimensional space. These mirrors are represented by its Coxeter-Dynkin diagram, {{CDD|nodea|3a|nodea|3a|branch_10|3a|nodea|3a|nodea|3a|nodea}}, and the ring represents the position of the active mirror(s).
Removing the node on the end of the 3-length branch leaves the rectified 122 polytope, {{CDD|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea}}
Removing the node on the end of the 2-length branch leaves the demihexeract, 131, {{CDD|nodea|3a|branch_01lr|3a|nodea|3a|nodea|3a|nodea}}
Removing the node on the end of the 1-length branch leaves the birectified 6-simplex, {{CDD|nodea|3a|nodea|3a|nodea_1|3a|nodea|3a|nodea|3a|nodea}}
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the tetrahedron-triangle duoprism prism, {3,3}×{3}×{}, {{CDD|nodea|3a|nodea_1|2|nodea_1|2|nodea_1|3a|nodea|3a|nodea}}
Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.
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!E7 | {{CDD|nodea|3a|nodea|3a|nodea|3a|branch_10|3a|nodea|3a|nodea}} | k-face | fk | f0 | f1 | colspan=3|f2 | colspan=5|f3 | colspan=6|f4 | colspan=5|f5 | colspan=3|f6 | k-figures | notes | ||||||||||||||||
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|A3A2A1 | {{CDD|nodea|3a|nodea|3a|nodea|2|nodes_x0|2|nodea|3a|nodea}} | ( )
!f0 |BGCOLOR="#ffe0ff"|10080 | 24 | 24 | 12 | 36 | 8 | 12 | 36 | 18 | 24 | 4 | 12 | 18 | 24 | 12 | 6 | 6 | 8 | 12 | 6 | 3 | 4 | 2 | 3 | {3,3}x{3}x{ } | E7/A3A2A1 = 72*8!/4!/3!/2 = 10080 | |
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|A2A1A1 | {{CDD|nodea|3a|nodea|2|nodea_x|2|nodes_1x|2|nodea_x|2|nodea}} | { }
!f1 | 2 | BGCOLOR="#ffe0e0"|120960 | 2 | 1 | 3 | 1 | 2 | 6 | 3 | 3 | 1 | 3 | 6 | 6 | 3 | 1 | 3 | 3 | 6 | 2 | 1 | 3 | 1 | 2 | ( )v{3}v{ } | E7/A2A1A1 = 72*8!/3!/2/2 = 120960 |
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|A2A2 | {{CDD|nodea|3a|nodea|2|nodea_x|2|nodes_1x|3a|nodea|2|nodea_x}} | rowspan=3|01
!rowspan=3|f2 | 3 | 3 | BGCOLOR="#ffffe0"|80640 | BGCOLOR="#ffffe0"|* | BGCOLOR="#ffffe0"|* | 1 | 1 | 3 | 0 | 0 | 1 | 3 | 3 | 3 | 0 | 0 | 3 | 3 | 3 | 1 | 0 | 3 | 1 | 1 | {3}v( )v( ) | E7/A2A2 = 72*8!/3!/3! = 80640 |
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|A2A2A1 | {{CDD|nodea|3a|nodea|2|nodea_x|2|branch_10|2|nodea_x|2|nodea}} | 3 | 3 | BGCOLOR="#ffffe0"|* | BGCOLOR="#ffffe0"|40320 | BGCOLOR="#ffffe0"|* | 0 | 2 | 0 | 3 | 0 | 1 | 0 | 6 | 0 | 3 | 0 | 3 | 0 | 6 | 0 | 1 | 3 | 0 | 2 | {3}v{ } | E7/A2A2A1 = 72*8!/3!/3!/2 = 40320 | |
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|A2A1A1 | {{CDD|nodea|2|nodea_x|2|nodea|3a|nodes_1x|2|nodea_x|2|nodea}} | 3 | 3 | BGCOLOR="#ffffe0"|* | BGCOLOR="#ffffe0"|* | BGCOLOR="#ffffe0"|120960 | 0 | 0 | 2 | 1 | 2 | 0 | 1 | 2 | 4 | 2 | 1 | 1 | 2 | 4 | 2 | 1 | 2 | 1 | 2 | { }v{ }v( ) | E7/A2A1A1 = 72*8!/3!/2/2 = 120960 | |
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|rowspan=2|A3A2 | {{CDD|nodea|3a|nodea|2|nodea_x|2|nodes_1x|3a|nodea|3a|nodea}} | 02
!rowspan=5|f3 | 4 | 6 | 4 | 0 | 0 | BGCOLOR="#e0ffe0"|20160 | BGCOLOR="#e0ffe0"|* | BGCOLOR="#e0ffe0"|* | BGCOLOR="#e0ffe0"|* | BGCOLOR="#e0ffe0"|* | 1 | 3 | 0 | 0 | 0 | 0 | 3 | 3 | 0 | 0 | 0 | 3 | 1 | 0 | rowspan=2|{3}v( ) | rowspan=2|E7/A3A2 = 72*8!/4!/3! = 20160 |
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|{{CDD|nodea|3a|nodea|2|nodea_x|2|branch_10|3a|nodea|2|nodea_x}} | rowspan=3| 011 | 6 | 12 | 4 | 4 | 0 | BGCOLOR="#e0ffe0"|* | BGCOLOR="#e0ffe0"|20160 | BGCOLOR="#e0ffe0"|* | BGCOLOR="#e0ffe0"|* | BGCOLOR="#e0ffe0"|* | 1 | 0 | 3 | 0 | 0 | 0 | 3 | 0 | 3 | 0 | 0 | 3 | 0 | 1 | |||
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|A3A1 | {{CDD|nodea|2|nodea_x|2|nodea|3a|nodes_1x|3a|nodea|2|nodea_x}} | 6 | 12 | 4 | 0 | 4 | BGCOLOR="#e0ffe0"|* | BGCOLOR="#e0ffe0"|* | BGCOLOR="#e0ffe0"|60480 | BGCOLOR="#e0ffe0"|* | BGCOLOR="#e0ffe0"|* | 0 | 1 | 1 | 2 | 0 | 0 | 1 | 2 | 2 | 1 | 0 | 2 | 1 | 1 | Sphenoid | E7/A3A1 = 72*8!/4!/2 = 60480 | |
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|A3A1A1 | {{CDD|nodea|2|nodea_x|2|nodea|3a|branch_10|2|nodea_x|2|nodea}} | 6 | 12 | 0 | 4 | 4 | BGCOLOR="#e0ffe0"|* | BGCOLOR="#e0ffe0"|* | BGCOLOR="#e0ffe0"|* | BGCOLOR="#e0ffe0"|30240 | BGCOLOR="#e0ffe0"|* | 0 | 0 | 2 | 0 | 2 | 0 | 1 | 0 | 4 | 0 | 1 | 2 | 0 | 2 | { }v{ } | E7/A3A1A1 = 72*8!/4!/2/2 = 30240 | |
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|A3A1 | {{CDD|nodea_x|2|nodea|3a|nodea|3a|nodes_1x|2|nodea_x|2|nodea}} | 02 | 4 | 6 | 0 | 0 | 4 | BGCOLOR="#e0ffe0"|* | BGCOLOR="#e0ffe0"|* | BGCOLOR="#e0ffe0"|* | BGCOLOR="#e0ffe0"|* | BGCOLOR="#e0ffe0"|60480 | 0 | 0 | 0 | 2 | 1 | 1 | 0 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | Sphenoid | E7/A3A1 = 72*8!/4!/2 = 60480 |
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|A4A2 | {{CDD|nodea|3a|nodea|2|nodea_x|2|branch_10|3a|nodea|3a|nodea}} | rowspan=2|021
!rowspan=6|f4 | 10 | 30 | 20 | 10 | 0 | 5 | 5 | 0 | 0 | 0 | BGCOLOR="#e0ffff"|4032 | BGCOLOR="#e0ffff"|* | BGCOLOR="#e0ffff"|* | BGCOLOR="#e0ffff"|* | BGCOLOR="#e0ffff"|* | BGCOLOR="#e0ffff"|* | 3 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | {3} | E7/A4A2 = 72*8!/5!/3! = 4032 |
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|A4A1 | {{CDD|nodea|2|nodea_x|2|nodea|3a|nodes_1x|3a|nodea|3a|nodea}} | 10 | 30 | 20 | 0 | 10 | 5 | 0 | 5 | 0 | 0 | BGCOLOR="#e0ffff"|* | BGCOLOR="#e0ffff"|12096 | BGCOLOR="#e0ffff"|* | BGCOLOR="#e0ffff"|* | BGCOLOR="#e0ffff"|* | BGCOLOR="#e0ffff"|* | 1 | 2 | 0 | 0 | 0 | 2 | 1 | 0 | rowspan=2|{ }v() | E7/A4A1 = 72*8!/5!/2 = 12096 | |
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|D4A1 | {{CDD|nodea|2|nodea_x|2|nodea|3a|branch_10|3a|nodea|2|nodea_x}} | 0111 | 24 | 96 | 32 | 32 | 32 | 0 | 8 | 8 | 8 | 0 | BGCOLOR="#e0ffff"|* | BGCOLOR="#e0ffff"|* | BGCOLOR="#e0ffff"|7560 | BGCOLOR="#e0ffff"|* | BGCOLOR="#e0ffff"|* | BGCOLOR="#e0ffff"|* | 1 | 0 | 2 | 0 | 0 | 2 | 0 | 1 | E7/D4A1 = 72*8!/8/4!/2 = 7560 | |
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|A4 | {{CDD|nodea_x|2|nodea|3a|nodea|3a|nodes_1x|3a|nodea|2|nodea_x}} | rowspan=2|021 | 10 | 30 | 10 | 0 | 20 | 0 | 0 | 5 | 0 | 5 | BGCOLOR="#e0ffff"|* | BGCOLOR="#e0ffff"|* | BGCOLOR="#e0ffff"|* | BGCOLOR="#e0ffff"|24192 | BGCOLOR="#e0ffff"|* | BGCOLOR="#e0ffff"|* | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | ( )v( )v( ) | E7/A4 = 72*8!/5! = 34192 |
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|rowspan=2|A4A1 | {{CDD|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="#e0ffff"|* | BGCOLOR="#e0ffff"|* | BGCOLOR="#e0ffff"|* | BGCOLOR="#e0ffff"|* | BGCOLOR="#e0ffff"|12096 | BGCOLOR="#e0ffff"|* | 0 | 0 | 2 | 0 | 1 | 1 | 0 | 2 | rowspan=2|{ }v() | rowspan=2|E7/A4A1 = 72*8!/5!/2 = 12096 | |
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|{{CDD|nodea|3a|nodea|3a|nodea|3a|nodes_1x|2|nodea_x|2|nodea}} | 03 | 5 | 10 | 0 | 0 | 10 | 0 | 0 | 0 | 0 | 5 | BGCOLOR="#e0ffff"|* | BGCOLOR="#e0ffff"|* | BGCOLOR="#e0ffff"|* | BGCOLOR="#e0ffff"|* | BGCOLOR="#e0ffff"|* | BGCOLOR="#e0ffff"|12096 | 0 | 0 | 0 | 2 | 1 | 0 | 1 | 2 | |||
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|D5A1 | {{CDD|nodea|2|nodea_x|2|nodea|3a|branch_10|3a|nodea|3a|nodea}} | 0211
!rowspan=5|f5 | 80 | 480 | 320 | 160 | 160 | 80 | 80 | 80 | 40 | 0 | 16 | 16 | 10 | 0 | 0 | 0 | BGCOLOR="#e0e0ff"|756 | BGCOLOR="#e0e0ff"|* | BGCOLOR="#e0e0ff"|* | BGCOLOR="#e0e0ff"|* | BGCOLOR="#e0e0ff"|* | 2 | 0 | 0 | rowspan=5|{ } | E7/D5A1 = 72*8!/16/5!/2 = 756 |
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|A5 | {{CDD|nodea_x|2|nodea|3a|nodea|3a|nodes_1x|3a|nodea|3a|nodea}} | 022 | 20 | 90 | 60 | 0 | 60 | 15 | 0 | 30 | 0 | 15 | 0 | 6 | 0 | 6 | 0 | 0 | BGCOLOR="#e0e0ff"|* | BGCOLOR="#e0e0ff"|4032 | BGCOLOR="#e0e0ff"|* | BGCOLOR="#e0e0ff"|* | BGCOLOR="#e0e0ff"|* | 1 | 1 | 0 | E7/A5 = 72*8!/6! = 4032 | |
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|D5 | {{CDD|nodea_x|2|nodea|3a|nodea|3a|branch_10|3a|nodea|2|nodea_x}} | 0211 | 80 | 480 | 160 | 160 | 320 | 0 | 40 | 80 | 80 | 80 | 0 | 0 | 10 | 16 | 16 | 0 | BGCOLOR="#e0e0ff"|* | BGCOLOR="#e0e0ff"|* | BGCOLOR="#e0e0ff"|1512 | BGCOLOR="#e0e0ff"|* | BGCOLOR="#e0e0ff"|* | 1 | 0 | 1 | E7/D5 = 72*8!/16/5! = 1512 | |
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|A5 | {{CDD|nodea|3a|nodea|3a|nodea|3a|nodes_1x|3a|nodea|2|nodea_x}} | rowspan=2|031 | 15 | 60 | 20 | 0 | 60 | 0 | 0 | 15 | 0 | 30 | 0 | 0 | 0 | 6 | 0 | 6 | BGCOLOR="#e0e0ff"|* | BGCOLOR="#e0e0ff"|* | BGCOLOR="#e0e0ff"|* | BGCOLOR="#e0e0ff"|4032 | BGCOLOR="#e0e0ff"|* | 0 | 1 | 1 | E7/A5 = 72*8!/6! = 4032 | |
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|A5A1 | {{CDD|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="#e0e0ff"|* | BGCOLOR="#e0e0ff"|* | BGCOLOR="#e0e0ff"|* | BGCOLOR="#e0e0ff"|* | BGCOLOR="#e0e0ff"|2016 | 0 | 0 | 2 | E7/A5A1 = 72*8!/6!/2 = 2016 | ||
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|E6 | {{CDD|nodea_x|2|nodea|3a|nodea|3a|branch_10|3a|nodea|3a|nodea}} | 0221
!rowspan=3|f6 | 720 | 6480 | 4320 | 2160 | 4320 | 1080 | 1080 | 2160 | 1080 | 1080 | 216 | 432 | 270 | 432 | 216 | 0 | 27 | 72 | 27 | 0 | 0 | BGCOLOR="#ffe0ff"|56 | BGCOLOR="#ffe0ff"|* | BGCOLOR="#ffe0ff"|* | rowspan=3|( ) | E7/E6 = 72*8!/72/6! = 56 |
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|A6 | {{CDD|nodea|3a|nodea|3a|nodea|3a|nodes_1x|3a|nodea|3a|nodea}} | 032 | 35 | 210 | 140 | 0 | 210 | 35 | 0 | 105 | 0 | 105 | 0 | 21 | 0 | 42 | 0 | 21 | 0 | 7 | 0 | 7 | 0 | BGCOLOR="#ffe0ff"|* | BGCOLOR="#ffe0ff"|576 | BGCOLOR="#ffe0ff"|* | E7/A6 = 72*8!/7! = 576 | |
align=right
|D6 | {{CDD|nodea|3a|nodea|3a|nodea|3a|branch_10|3a|nodea|2|nodea_x}} | 0311 | 240 | 1920 | 640 | 640 | 1920 | 0 | 160 | 480 | 480 | 960 | 0 | 0 | 60 | 192 | 192 | 192 | 0 | 0 | 12 | 32 | 32 | BGCOLOR="#ffe0ff"|* | BGCOLOR="#ffe0ff"|* | BGCOLOR="#ffe0ff"|126 | E7/D6 = 72*8!/32/6! = 126 |
= Images =
class=wikitable width=600
|+ Coxeter plane projections |
E7
!E6 / F4 !B7 / A6 |
---|
valign=top align=center
|200px |200px |200px |
A5
!D7 / B6 !D6 / B5 |
valign=top align=center
|200px |200px |200px |
D5 / B4 / A4
!D4 / B3 / A2 / G2 !D3 / B2 / A3 |
valign=top align=center
|200px |200px |200px |
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]
- {{KlitzingPolytopes|polyexa.htm|7D uniform polytopes (polyexa)}} o3o3o3x *c3o3o3o - lin, o3o3x3o *c3o3o3o - rolin
{{Polytopes}}