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122
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Up 1 22 t1 E6.svg
Rectified 122
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Up 1 22 t2 E6.svg
Birectified 122
CDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.png
Up 2 21 t0 E6.svg
221
CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Up 2 21 t1 E6.svg
Rectified 221
CDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
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).[1]

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, construcated 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: CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png.

1_22 polytopeEdit

122 polytope
Type Uniform 6-polytope
Family 1k2 polytope
Schläfli symbol {3,32,2}
Coxeter symbol 122
Coxeter-Dynkin diagram           or        
5-faces 54:
27 121 
27 121 
4-faces 702:
270 111 
432 120 
Cells 2160:
1080 110 
1080 {3,3} 
Faces 2160 {3} 
Edges 720
Vertices 72
Vertex figure Birectified 5-simplex:
022  
Petrie polygon Dodecagon
Coxeter group E6, [[3,32,2]], order 103680
Properties convex, isotopic

The 1_22 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 namesEdit

  • Pentacontatetra-peton (Acronym Mo) - 54-facetted polypeton (Jonathan Bowers)[2]

ImagesEdit

Coxeter plane orthographic projections
E6
[12]
D5
[8]
D4 / A2
[6]
 
(1,2)
 
(1,3)
 
(1,9,12)
B6
[12/2]
A5
[6]
A4
[[5]] = [10]
A3 / D3
[4]
 
(1,2)
 
(2,3,6)
 
(1,2)
 
(1,6,8,12)

ConstructionEdit

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,          .

Removing the node on either of 2-length branches leaves the 5-demicube, 131,        .

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 5-simplex, 022,          .

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

E6           k-face fk f0 f1 f2 f3 f4 f5 k-figure notes
A5           ( ) f0 72 20 90 60 60 15 15 30 6 6 r{3,3,3} E6/A5 = 72*6!/6! = 72
A2A2A1           { } f1 2 720 9 9 9 3 3 9 3 3 {3}x{3} E6/A2A2A1 = 72*6!/3!/3!/2 = 720
A2A1A1           {3} f2 3 3 2160 2 2 1 1 4 2 2 s{2,4} E6/A2A1A1 = 72*6!/3!/2/2 = 2160
A3A1           {3,3} f3 4 6 4 1080 * 1 0 2 2 1 { }v( ) E6/A3A1 = 72*6!/4!/2 = 1080
          4 6 4 * 1080 0 1 2 1 2
A4A1           {3,3,3} f4 5 10 10 5 0 216 * * 2 0 { } E6/A4A1 = 72*6!/5!/2 = 216
          5 10 10 0 5 * 216 * 0 2
D4           h{4,3,3} 8 24 32 8 8 * * 270 1 1 E6/D4 = 72*6!/8/4! = 270
D5           h{4,3,3,3} f5 16 80 160 80 40 16 0 10 27 * ( ) E6/D5 = 72*6!/16/5! = 27
          16 80 160 40 80 0 16 10 * 27

Related complex polyhedronEdit

 
Orthographic projection in Aut(E6) Coxeter plane with 18-gonal symmetry for complex polyhedron, 3{3}3{4}2. It has 72 vertices, 216 3-edges, and 54 3{3}3 faces.

The regular complex polyhedron 3{3}3{4}2,      , 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      , as a rectification of the Hessian polyhedron,      .[4]

Related polytopes and honeycombEdit

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:          .

Geometric foldingEdit

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.

E6/F4 Coxeter planes
 
122
 
24-cell
D4/B4 Coxeter planes
 
122
 
24-cell

TessellationsEdit

This polytope is the vertex figure for a uniform tessellation of 6-dimensional space, 222,          .

Rectified 1_22 polytopeEdit

Rectified 122
Type Uniform 6-polytope
Schläfli symbol 2r{3,3,32,1}
r{3,32,2}
Coxeter symbol 0221
Coxeter-Dynkin diagram          
or        
5-faces 126
4-faces 1566
Cells 6480
Faces 6480
Edges 6480
Vertices 720
Vertex figure 3-3 duoprism prism
Petrie polygon Dodecagon
Coxeter group E6, [[3,32,2]], order 103680
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).[5]

Alternate namesEdit

  • Birectified 221 polytope
  • Rectified pentacontatetrapeton (acronym Ram) - rectified 54-facetted polypeton (Jonathan Bowers)[6]

ImagesEdit

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

Coxeter plane orthographic projections
E6
[12]
D5
[8]
D4 / A2
[6]
B6
[12/2]
       
A5
[6]
A4
[5]
A3 / D3
[4]
     

ConstructionEdit

Its construction is based on the E6 group and information can be extracted from the ringed Coxeter-Dynkin diagram representing this polytope:          .

Removing the ring on the short branch leaves the birectified 5-simplex,          .

Removing the ring on the either 2-length branch leaves the birectified 5-orthoplex in its alternated form: t2(211),        .

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

Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.[7][8]

E6           k-face fk f0 f1 f2 f3 f4 f5 k-figure notes
A2A2A1           ( ) f0 720 18 18 18 9 6 18 9 6 9 6 3 6 9 3 2 3 3 {3}x{3}x{ } E6/A2A2A1 = 72*6!/3!/3!/2 = 720
A1A1A1           { } f1 2 6480 2 2 1 1 4 2 1 2 2 1 2 4 1 1 2 2 { }v{ }v( ) E6/A1A1A1 = 72*6!/2/2/2 = 6480
A2A1           {3} f2 3 3 4320 * * 1 2 1 0 0 2 1 1 2 0 1 2 1 Sphenoid E6/A2A1 = 72*6!/3!/2 = 4320
          3 3 * 4320 * 0 2 0 1 1 1 0 2 2 1 1 1 2
A2A1A1           3 3 * * 2160 0 0 2 0 2 0 1 0 4 1 0 2 2 { }∨{ } E6/A2A1A1 = 72*6!/3!/2/2 = 2160
A2A1           {3,3} f3 4 6 4 0 0 1080 * * * * 2 1 0 0 0 1 2 0 { }v( ) E6/A2A1 = 72*6!/3!/2 = 1080
A3           r{3,3} 6 12 4 4 0 * 2160 * * * 1 0 1 1 0 1 1 1 {3} E6/A3 = 72*6!/4! = 2160
A3A1           6 12 4 0 4 * * 1080 * * 0 1 0 2 0 0 2 1 { }v( ) E6/A3A1 = 72*6!/4!/2 = 1080
          {3,3} 4 6 0 4 0 * * * 1080 * 0 0 2 0 1 1 0 2
          r{3,3} 6 12 0 4 4 * * * * 1080 0 0 0 2 1 0 1 2
A4           r{3,3,3} f4 10 30 20 10 0 5 5 0 0 0 432 * * * * 1 1 0 { } E6/A4 = 72*6!/5! = 432
A4A1           10 30 20 0 10 5 0 5 0 0 * 216 * * * 0 2 0 E6/A4A1 = 72*6!/5!/2 = 216
A4           10 30 10 20 0 0 5 0 5 0 * * 432 * * 1 0 1 E6/A4 = 72*6!/5! = 432
D4           h{4,3,3} 24 96 32 32 32 0 8 8 0 8 * * * 270 * 0 1 1 E6/D4 = 72*6!/8/4! = 270
A4A1           r{3,3,3} 10 30 0 20 10 0 0 0 5 5 * * * * 216 0 0 2 E6/A4A1 = 72*6!/5!/2 = 216
A5           2r{3,3,3,3} f5 20 90 60 60 0 15 30 0 15 0 6 0 6 0 0 72 * * ( ) E6/A5 = 72*6!/6! = 72
D5           rh{4,3,3,3} 80 480 320 160 160 80 80 80 0 40 16 16 0 10 0 * 27 * E6/D5 = 72*6!/16/5! = 27
          80 480 160 320 160 0 80 40 80 80 0 0 16 10 16 * * 27

Truncated 1_22 polytopeEdit

Truncated 122
Type Uniform 6-polytope
Schläfli symbol t{3,32,2}
Coxeter symbol t(122)
Coxeter-Dynkin diagram          
or        
5-faces 72+27+27
4-faces 32+216+432+270+216
Cells 1080+2160+1080+1080+1080
Faces 4320+4320+2160
Edges 6480+720
Vertices 1440
Vertex figure ( )v{3}x{3}
Petrie polygon Dodecagon
Coxeter group E6, [[3,32,2]], order 103680
Properties convex

Alternate namesEdit

  • Truncated 122 polytope

ConstructionEdit

Its construction is based on the E6 group and information can be extracted from the ringed Coxeter-Dynkin diagram representing this polytope:          .

ImagesEdit

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

Coxeter plane orthographic projections
E6
[12]
D5
[8]
D4 / A2
[6]
B6
[12/2]
       
A5
[6]
A4
[5]
A3 / D3
[4]
     

Birectified 1_22 polytopeEdit

Birectified 122 polytope
Type Uniform 6-polytope
Schläfli symbol 2r{3,32,2}
Coxeter symbol 2r(122)
Coxeter-Dynkin diagram          
or        
5-faces 126
4-faces 2286
Cells 10800
Faces 19440
Edges 12960
Vertices 2160
Vertex figure
Coxeter group E6, [[3,32,2]], order 103680
Properties convex

Alternate namesEdit

  • Bicantellated 221
  • Birectified pentacontitetrapeton (barm) (Jonathan Bowers)[9]

ImagesEdit

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

Coxeter plane orthographic projections
E6
[12]
D5
[8]
D4 / A2
[6]
B6
[12/2]
       
A5
[6]
A4
[5]
A3 / D3
[4]
     

Trirectified 1_22 polytopeEdit

Trirectified 122 polytope
Type Uniform 6-polytope
Schläfli symbol 3r{3,32,2}
Coxeter symbol 3r(122)
Coxeter-Dynkin diagram          
or        
5-faces 558
4-faces 4608
Cells 8640
Faces 6480
Edges 2160
Vertices 270
Vertex figure
Coxeter group E6, [[3,32,2]], order 103680
Properties convex

Alternate namesEdit

  • Tricantellated 221
  • Trirectified pentacontitetrapeton (trim) (Jonathan Bowers)[10]


See alsoEdit

NotesEdit

  1. ^ Elte, 1912
  2. ^ Klitzing, (o3o3o3o3o *c3x - mo)
  3. ^ Coxeter, Regular Polytopes, 11.8 Gossett figures in six, seven, and eight dimensions, p. 202-203
  4. ^ Coxeter, H. S. M., Regular Complex Polytopes, second edition, Cambridge University Press, (1991). p.30 and p.47
  5. ^ The Voronoi Cells of the E6* and E7* Lattices, Edward Pervin
  6. ^ Klitzing, (o3o3x3o3o *c3o - ram)
  7. ^ Coxeter, Regular Polytopes, 11.8 Gossett figures in six, seven, and eight dimensions, p. 202-203
  8. ^ Klitzing, Richard. "6D convex uniform polypeta o3o3x3o3o *c3o - ram".
  9. ^ Klitzing, (o3x3o3x3o *c3o - barm)
  10. ^ Klitzing, (x3o3o3o3x *c3o - trim[permanent dead link])

ReferencesEdit

  • 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] See p334 (figure 3.6a) by Peter mcMullen: (12-gonal node-edge graph of 122)
  • Klitzing, Richard. "6D uniform polytopes (polypeta)". o3o3o3o3o *c3x - mo, o3o3x3o3o *c3o - ram, o3x3o3x3o *c3o - barm
Fundamental convex regular and uniform polytopes in dimensions 2–10
An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Triangle Square p-gon Hexagon Pentagon
Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
5-simplex 5-orthoplex5-cube 5-demicube
6-simplex 6-orthoplex6-cube 6-demicube 122221
7-simplex 7-orthoplex7-cube 7-demicube 132231321
8-simplex 8-orthoplex8-cube 8-demicube 142241421
9-simplex 9-orthoplex9-cube 9-demicube
10-simplex 10-orthoplex10-cube 10-demicube
n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds