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2 31 polytope

January 01, 1970

In 7-dimensional geometry, 231 is a uniform polytope, constructed from the E7 group.

Its Coxeter symbol is 231, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node branch.

The rectified 231 is constructed by points at the mid-edges of the 231.

These polytopes are part of a family of 127 (or 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: .

2_31 polytope edit

Gosset 231 polytope
Type Uniform 7-polytope
Family 2k1 polytope
Schläfli symbol {3,3,33,1}
Coxeter symbol 231
Coxeter diagram            
6-faces 632:
56 221 
576 {35} 
5-faces 4788:
756 211 
4032 {34} 
4-faces 16128:
4032 201 
12096 {33} 
Cells 20160 {32} 
Faces 10080 {3} 
Edges 2016
Vertices 126
Vertex figure 131
 
Petrie polygon Octadecagon
Coxeter group E7, [33,2,1]
Properties convex

The 231 is composed of 126 vertices, 2016 edges, 10080 faces (Triangles), 20160 cells (tetrahedra), 16128 4-faces (3-simplexes), 4788 5-faces (756 pentacrosses, and 4032 5-simplexes), 632 6-faces (576 6-simplexes and 56 221). Its vertex figure is a 6-demicube. Its 126 vertices represent the root vectors of the simple Lie group E7.

This polytope is the vertex figure for a uniform tessellation of 7-dimensional space, 331.

Alternate names edit

  • E. L. Elte named it V126 (for its 126 vertices) in his 1912 listing of semiregular polytopes.[1]
  • It was called 231 by Coxeter for its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequence.
  • Pentacontihexa-pentacosiheptacontihexa-exon (Acronym laq) - 56-576 facetted polyexon (Jonathan Bowers)[2]

Construction edit

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,            .

Removing the node on the short branch leaves the 6-simplex. There are 576 of these facets. These facets are centered on the locations of the vertices of the 321 polytope,            .

Removing the node on the end of the 3-length branch leaves the 221. There are 56 of these facets. These facets are centered on the locations of the vertices of the 132 polytope,          .

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 6-demicube, 131,          .

Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.[3]

E7             k-face fk f0 f1 f2 f3 f4 f5 f6 k-figures notes
D6             ( ) f0 126 32 240 640 160 480 60 192 12 32 6-demicube E7/D6 = 72x8!/32/6! = 126
A5A1             { } f1 2 2016 15 60 20 60 15 30 6 6 rectified 5-simplex E7/A5A1 = 72x8!/6!/2 = 2016
A3A2A1             {3} f2 3 3 10080 8 4 12 6 8 4 2 tetrahedral prism E7/A3A2A1 = 72x8!/4!/3!/2 = 10080
A3A2             {3,3} f3 4 6 4 20160 1 3 3 3 3 1 tetrahedron E7/A3A2 = 72x8!/4!/3! = 20160
A4A2             {3,3,3} f4 5 10 10 5 4032 * 3 0 3 0 {3} E7/A4A2 = 72x8!/5!/3! = 4032
A4A1             5 10 10 5 * 12096 1 2 2 1 Isosceles triangle E7/A4A1 = 72x8!/5!/2 = 12096
D5A1             {3,3,3,4} f5 10 40 80 80 16 16 756 * 2 0 { } E7/D5A1 = 72x8!/32/5! = 756
A5             {3,3,3,3} 6 15 20 15 0 6 * 4032 1 1 E7/A5 = 72x8!/6! = 72*8*7 = 4032
E6             {3,3,32,1} f6 27 216 720 1080 216 432 27 72 56 * ( ) E7/E6 = 72x8!/72x6! = 8*7 = 56
A6             {3,3,3,3,3} 7 21 35 35 0 21 0 7 * 576 E7/A6 = 72x8!/7! = 72×8 = 576

Images edit

Coxeter plane projections
E7 E6 / F4 B6 / A6
 
[18] 		 
[12] 		 
[7x2]
A5 D7 / B6 D6 / B5
 
[6] 		 
[12/2] 		 
[10]
D5 / B4 / A4 D4 / B3 / A2 / G2 D3 / B2 / A3
 
[8] 		 
[6] 		 
[4]

Related polytopes and honeycombs edit

2k1 figures in n dimensions
Space Finite Euclidean Hyperbolic
n 3 4 5 6 7 8 9 10
Coxeter
group 		E3=A2A1 E4=A4 E5=D5 E6 E7 E8 E9 =   = E8+ E10 =   = E8++
Coxeter
diagram 		                                                                                         
Symmetry [3−1,2,1] [30,2,1] [[31,2,1]] [32,2,1] [33,2,1] [34,2,1] [35,2,1] [36,2,1]
Order 12 120 384 51,840 2,903,040 696,729,600 ∞
Graph             - -
Name 2−1,1 201 211 221 231 241 251 261

Rectified 2_31 polytope edit

Rectified 231 polytope
Type Uniform 7-polytope
Family 2k1 polytope
Schläfli symbol {3,3,33,1}
Coxeter symbol t1(231)
Coxeter diagram            
6-faces 758
5-faces 10332
4-faces 47880
Cells 100800
Faces 90720
Edges 30240
Vertices 2016
Vertex figure 6-demicube
Petrie polygon Octadecagon
Coxeter group E7, [33,2,1]
Properties convex

The rectified 231 is a rectification of the 231 polytope, creating new vertices on the center of edge of the 231.

Alternate names edit

  • Rectified pentacontihexa-pentacosiheptacontihexa-exon - as a rectified 56-576 facetted polyexon (acronym rolaq) (Jonathan Bowers)[4]

Construction edit

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,            .

Removing the node on the short branch leaves the rectified 6-simplex,            .

Removing the node on the end of the 2-length branch leaves the, 6-demicube,          .

Removing the node on the end of the 3-length branch leaves the rectified 221,          .

The vertex figure is determined by removing the ringed node and ringing the neighboring node.

         

Images edit

Coxeter plane projections
E7 E6 / F4 B6 / A6
 
[18] 		 
[12] 		 
[7x2]
A5 D7 / B6 D6 / B5
 
[6] 		 
[12/2] 		 
[10]
D5 / B4 / A4 D4 / B3 / A2 / G2 D3 / B2 / A3
 
[8] 		 
[6] 		 
[4]

See also edit

Notes edit

  1. ^ Elte, 1912
  2. ^ Klitzing, (x3o3o3o *c3o3o3o - laq)
  3. ^ Coxeter, Regular Polytopes, 11.8 Gossett figures in six, seven, and eight dimensions, p. 202-203
  4. ^ Klitzing, (o3x3o3o *c3o3o3o - rolaq)

References edit

  • Elte, E. L. (1912), The Semiregular Polytopes of the Hyperspaces, Groningen: University of Groningen
  • 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 [1]
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
  • Klitzing, Richard. "7D uniform polytopes (polyexa)". x3o3o3o *c3o3o3o - laq, o3x3o3o *c3o3o3o - rolaq
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

January 01, 1970
polytope, rectified, birectified, rectified, rectified, orthogonal, projections, coxeter, plane, dimensional, geometry, uniform, polytope, constructed, from, group, coxeter, symbol, describing, bifurcating, coxeter, dynkin, diagram, with, single, ring, node, b. 321 231 132 Rectified 321 birectified 321 Rectified 231 Rectified 132 Orthogonal projections in E7 Coxeter plane In 7 dimensional geometry 231 is a uniform polytope constructed from the E7 group Its Coxeter symbol is 231 describing its bifurcating Coxeter Dynkin diagram with a single ring on the end of the 2 node branch The rectified 231 is constructed by points at the mid edges of the 231 These polytopes are part of a family of 127 or 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 Contents 1 2 31 polytope 1 1 Alternate names 1 2 Construction 1 3 Images 1 4 Related polytopes and honeycombs 2 Rectified 2 31 polytope 2 1 Alternate names 2 2 Construction 2 3 Images 3 See also 4 Notes 5 References2 31 polytope editGosset 231 polytope Type Uniform 7 polytope Family 2k1 polytope Schlafli symbol 3 3 33 1 Coxeter symbol 231 Coxeter diagram nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 6 faces 632 56 221 nbsp 576 35 nbsp 5 faces 4788 756 211 nbsp 4032 34 nbsp 4 faces 16128 4032 201 nbsp 12096 33 nbsp Cells 20160 32 nbsp Faces 10080 3 nbsp Edges 2016 Vertices 126 Vertex figure 131 nbsp Petrie polygon Octadecagon Coxeter group E7 33 2 1 Properties convex The 231 is composed of 126 vertices 2016 edges 10080 faces Triangles 20160 cells tetrahedra 16128 4 faces 3 simplexes 4788 5 faces 756 pentacrosses and 4032 5 simplexes 632 6 faces 576 6 simplexes and 56 221 Its vertex figure is a 6 demicube Its 126 vertices represent the root vectors of the simple Lie group E7 This polytope is the vertex figure for a uniform tessellation of 7 dimensional space 331 Alternate names edit E L Elte named it V126 for its 126 vertices in his 1912 listing of semiregular polytopes 1 It was called 231 by Coxeter for its bifurcating Coxeter Dynkin diagram with a single ring on the end of the 2 node sequence Pentacontihexa pentacosiheptacontihexa exon Acronym laq 56 576 facetted polyexon Jonathan Bowers 2 Construction edit 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 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Removing the node on the short branch leaves the 6 simplex There are 576 of these facets These facets are centered on the locations of the vertices of the 321 polytope nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Removing the node on the end of the 3 length branch leaves the 221 There are 56 of these facets These facets are centered on the locations of the vertices of the 132 polytope nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp The vertex figure is determined by removing the ringed node and ringing the neighboring node This makes the 6 demicube 131 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Seen in a configuration matrix the element counts can be derived by mirror removal and ratios of Coxeter group orders 3 E7 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp k face fk f0 f1 f2 f3 f4 f5 f6 k figures notes D6 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp f0 126 32 240 640 160 480 60 192 12 32 6 demicube E7 D6 72x8 32 6 126 A5A1 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp f1 2 2016 15 60 20 60 15 30 6 6 rectified 5 simplex E7 A5A1 72x8 6 2 2016 A3A2A1 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 3 f2 3 3 10080 8 4 12 6 8 4 2 tetrahedral prism E7 A3A2A1 72x8 4 3 2 10080 A3A2 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 3 3 f3 4 6 4 20160 1 3 3 3 3 1 tetrahedron E7 A3A2 72x8 4 3 20160 A4A2 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 3 3 3 f4 5 10 10 5 4032 3 0 3 0 3 E7 A4A2 72x8 5 3 4032 A4A1 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 5 10 10 5 12096 1 2 2 1 Isosceles triangle E7 A4A1 72x8 5 2 12096 D5A1 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 3 3 3 4 f5 10 40 80 80 16 16 756 2 0 E7 D5A1 72x8 32 5 756 A5 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 3 3 3 3 6 15 20 15 0 6 4032 1 1 E7 A5 72x8 6 72 8 7 4032 E6 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 3 3 32 1 f6 27 216 720 1080 216 432 27 72 56 E7 E6 72x8 72x6 8 7 56 A6 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 3 3 3 3 3 7 21 35 35 0 21 0 7 576 E7 A6 72x8 7 72 8 576 Images edit Coxeter plane projections E7 E6 F4 B6 A6 nbsp 18 nbsp 12 nbsp 7x2 A5 D7 B6 D6 B5 nbsp 6 nbsp 12 2 nbsp 10 D5 B4 A4 D4 B3 A2 G2 D3 B2 A3 nbsp 8 nbsp 6 nbsp 4 Related polytopes and honeycombs edit 2k1 figures in n dimensions Space Finite Euclidean Hyperbolic n 3 4 5 6 7 8 9 10 Coxetergroup E3 A2A1 E4 A4 E5 D5 E6 E7 E8 E9 E 8 displaystyle tilde E 8 nbsp E8 E10 T 8 displaystyle bar T 8 nbsp E8 Coxeterdiagram nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Symmetry 3 1 2 1 30 2 1 31 2 1 32 2 1 33 2 1 34 2 1 35 2 1 36 2 1 Order 12 120 384 51 840 2 903 040 696 729 600 Graph nbsp nbsp nbsp nbsp nbsp nbsp Name 2 1 1 201 211 221 231 241 251 261Rectified 2 31 polytope editRectified 231 polytope Type Uniform 7 polytope Family 2k1 polytope Schlafli symbol 3 3 33 1 Coxeter symbol t1 231 Coxeter diagram nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 6 faces 758 5 faces 10332 4 faces 47880 Cells 100800 Faces 90720 Edges 30240 Vertices 2016 Vertex figure 6 demicube Petrie polygon Octadecagon Coxeter group E7 33 2 1 Properties convex The rectified 231 is a rectification of the 231 polytope creating new vertices on the center of edge of the 231 Alternate names edit Rectified pentacontihexa pentacosiheptacontihexa exon as a rectified 56 576 facetted polyexon acronym rolaq Jonathan Bowers 4 Construction edit 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 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Removing the node on the short branch leaves the rectified 6 simplex nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Removing the node on the end of the 2 length branch leaves the 6 demicube nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Removing the node on the end of the 3 length branch leaves the rectified 221 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp The vertex figure is determined by removing the ringed node and ringing the neighboring node nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Images edit Coxeter plane projections E7 E6 F4 B6 A6 nbsp 18 nbsp 12 nbsp 7x2 A5 D7 B6 D6 B5 nbsp 6 nbsp 12 2 nbsp 10 D5 B4 A4 D4 B3 A2 G2 D3 B2 A3 nbsp 8 nbsp 6 nbsp 4 See also editList of E7 polytopesNotes edit Elte 1912 Klitzing x3o3o3o c3o3o3o laq Coxeter Regular Polytopes 11 8 Gossett figures in six seven and eight dimensions p 202 203 Klitzing o3x3o3o c3o3o3o rolaq References editElte E L 1912 The Semiregular Polytopes of the Hyperspaces Groningen University of Groningen 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 1 Paper 24 H S M Coxeter Regular and Semi Regular Polytopes III Math Zeit 200 1988 3 45 Klitzing Richard 7D uniform polytopes polyexa x3o3o3o c3o3o3o laq o3x3o3o c3o3o3o rolaq vteFundamental convex regular and uniform polytopes in dimensions 2 10 Family An Bn I2 p Dn E6 E7 E8 F4 G2 Hn Regular polygon Triangle Square p gon Hexagon Pentagon Uniform polyhedron Tetrahedron Octahedron Cube Demicube Dodecahedron Icosahedron Uniform polychoron Pentachoron 16 cell Tesseract Demitesseract 24 cell 120 cell 600 cell Uniform 5 polytope 5 simplex 5 orthoplex 5 cube 5 demicube Uniform 6 polytope 6 simplex 6 orthoplex 6 cube 6 demicube 122 221 Uniform 7 polytope 7 simplex 7 orthoplex 7 cube 7 demicube 132 231 321 Uniform 8 polytope 8 simplex 8 orthoplex 8 cube 8 demicube 142 241 421 Uniform 9 polytope 9 simplex 9 orthoplex 9 cube 9 demicube Uniform 10 polytope 10 simplex 10 orthoplex 10 cube 10 demicube Uniform n polytope n simplex n orthoplex n cube n demicube 1k2 2k1 k21 n pentagonal polytope Topics Polytope families Regular polytope List of regular polytopes and compounds Retrieved from https en wikipedia org w index php title 2 31 polytope amp oldid 1218108392, wikipedia, wiki, book, books, library,

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