122

Rectified 122

Birectified 122

221

Rectified 221
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, 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: .

122 polytope

122 polytope
TypeUniform 6-polytope
Family1k2 polytope
Schläfli symbol{3,32,2}
Coxeter symbol122
Coxeter-Dynkin diagram or
5-faces54:
27 121
27 121
4-faces702:
270 111
432 120
Cells2160:
1080 110
1080 {3,3}
Faces2160 {3}
Edges720
Vertices72
Vertex figureBirectified 5-simplex:
022
Petrie polygonDodecagon
Coxeter groupE6, [[3,32,2]], order 103680
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

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

Images

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)

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, .

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]

E6k-facefkf0f1f2f3f4f5k-figurenotes
A5( ) f0 722090606015153066r{3,3,3}E6/A5 = 72*6!/6! = 72
A2A2A1{ } f1 272099933933{3}×{3}E6/A2A2A1 = 72*6!/3!/3!/2 = 720
A2A1A1{3} f2 3321602211422s{2,4}E6/A2A1A1 = 72*6!/3!/2/2 = 2160
A3A1{3,3} f3 4641080*10221{ }∨( )E6/A3A1 = 72*6!/4!/2 = 1080
464*108001212
A4A1{3,3,3} f4 5101050216**20{ }E6/A4A1 = 72*6!/5!/2 = 216
5101005*216*02
D4h{4,3,3} 8243288**27011E6/D4 = 72*6!/8/4! = 270
D5h{4,3,3,3} f5 168016080401601027*( )E6/D5 = 72*6!/16/5! = 27
1680160408001610*27
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]

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: .

1k2 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
(order)
[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 1,920 103,680 2,903,040 696,729,600
Graph - -
Name 1−1,2 102 112 122 132 142 152 162

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.

E6/F4 Coxeter planes

122

24-cell
D4/B4 Coxeter planes

122

24-cell

Tessellations

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

Rectified 122 polytope

Rectified 122
TypeUniform 6-polytope
Schläfli symbol2r{3,3,32,1}
r{3,32,2}
Coxeter symbol0221
Coxeter-Dynkin diagram
or
5-faces126
4-faces1566
Cells6480
Faces6480
Edges6480
Vertices720
Vertex figure3-3 duoprism prism
Petrie polygonDodecagon
Coxeter groupE6, [[3,32,2]], order 103680
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).[5]

Alternate names

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

Images

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]

Construction

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]

E6k-facefkf0f1f2f3f4f5k-figurenotes
A2A2A1( ) f0 720181818961896963693233{3}×{3}×{ }E6/A2A2A1 = 72*6!/3!/3!/2 = 720
A1A1A1{ } f1 264802211421221241122{ }∨{ }∨( )E6/A1A1A1 = 72*6!/2/2/2 = 6480
A2A1{3} f2 334320**1210021120121SphenoidE6/A2A1 = 72*6!/3!/2 = 4320
33*4320*0201110221112
A2A1A1 33**21600020201041022{ }∨{ }E6/A2A1A1 = 72*6!/3!/2/2 = 2160
A2A1{3,3} f3 464001080****21000120{ }∨( )E6/A2A1 = 72*6!/3!/2 = 1080
A3r{3,3} 612440*2160***10110111{3}E6/A3 = 72*6!/4! = 2160
A3A1 612404**1080**01020021{ }∨( )E6/A3A1 = 72*6!/4!/2 = 1080
{3,3} 46040***1080*00201102
r{3,3} 612044****108000021012
A4r{3,3,3} f4 10302010055000432****110{ }E6/A4 = 72*6!/5! = 432
A4A1 10302001050500*216***020E6/A4A1 = 72*6!/5!/2 = 216
A4 10301020005050**432**101E6/A4 = 72*6!/5! = 432
D4{3,4,3} 249632323208808***270*011E6/D4 = 72*6!/8/4! = 270
A4A1r{3,3,3} 10300201000055****216002E6/A4A1 = 72*6!/5!/2 = 216
A52r{3,3,3,3} f5 209060600153001506060072**( )E6/A5 = 72*6!/6! = 72
D52r{4,3,3,3} 8048032016016080808004016160100*27*E6/D5 = 72*6!/16/5! = 27
8048016032016008040808000161016**27

Truncated 122 polytope

Truncated 122
TypeUniform 6-polytope
Schläfli symbolt{3,32,2}
Coxeter symbolt(122)
Coxeter-Dynkin diagram
or
5-faces72+27+27
4-faces32+216+432+270+216
Cells1080+2160+1080+1080+1080
Faces4320+4320+2160
Edges6480+720
Vertices1440
Vertex figure( )v{3}x{3}
Petrie polygonDodecagon
Coxeter groupE6, [[3,32,2]], order 103680
Propertiesconvex

Alternate names

  • Truncated 122 polytope

Construction

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

Images

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 122 polytope

Birectified 122 polytope
TypeUniform 6-polytope
Schläfli symbol2r{3,32,2}
Coxeter symbol2r(122)
Coxeter-Dynkin diagram
or
5-faces126
4-faces2286
Cells10800
Faces19440
Edges12960
Vertices2160
Vertex figure
Coxeter groupE6, [[3,32,2]], order 103680
Propertiesconvex

Alternate names

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

Images

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 122 polytope

Trirectified 122 polytope
TypeUniform 6-polytope
Schläfli symbol3r{3,32,2}
Coxeter symbol3r(122)
Coxeter-Dynkin diagram
or
5-faces558
4-faces4608
Cells8640
Faces6480
Edges2160
Vertices270
Vertex figure
Coxeter groupE6, [[3,32,2]], order 103680
Propertiesconvex

Alternate names

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


See also

Notes

  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 Archived 2016-01-30 at the Wayback Machine, 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 - cacam

References

  • 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
    • (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
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds
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