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1 22 polytope

Uniform 6-polytope

1 22 polytope

Uniform 6-polytope

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).

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
Type
Family
Schläfli symbol
Coxeter symbol
Coxeter-Dynkin diagram
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Petrie polygon
Coxeter group
Properties

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)

Images

E6
[12]D5
[8]D4 / A2
[6]B6
[12/2]A5
[6]A4
= [10]A3 / D3
[4]
[[File:up 1 22 t0 E6.svg120px]]
(1,2)[[File:up 1 22 t0 D5.svg120px]]
(1,3)[[File:up 1 22 t0 D4.svg120px]]
(1,9,12)
[[File:up 1 22 t0 B6.svg120px]]
(1,2)[[File:up 1 22 t0 A5.svg120px]]
(2,3,6)[[File:up 1 22 t0 A4.svg120px]]
(1,2)[[File:up 1 22 t0 D3.svg120px]]
(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, 121, .

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.

E6*k*-facef*k*f0f1f2colspan=2f3colspan=3f4colspan=2f5*k*-figureNotesf0f1f2f3f4f5
A5( )**72**209060
A2A2A1{ }2**720**99
A2A1A1{3}33**2160**2
A3A1{3,3}464**1080**
464***1080**0
A4A1[{3,3,3}](5-cell)510105
5101005*
D4[h{4,3,3}](16-cell)824328
D5[h{4,3,3,3}](5-demicube)168016080
168016040800
Orthographic projection in Aut(E6) Coxeter plane with 18-gonal symmetry for complex polyhedron, <sub>3</sub>{3}<sub>3</sub>{4}<sub>2</sub>. It has 72 vertices, 216 3-edges, and 54 3{3}3 faces.

The regular complex polyhedron 3{3}3{4}2, , in \mathbb{C}^2 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, .

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 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 planesD4/B4 Coxeter planes
[[File:Up 1 22 t0 E6.svg160px]]
**122**[[File:24-cell t3 F4.svg160px]]
24-cell
[[File:up 1 22 t0 D4.svg160px]]
**122**[[File:24-cell t3 B3.svg160px]]
24-cell

Tessellations

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

Rectified 122 polytope

Rectified 122
Type
Schläfli symbol
Coxeter symbol
Coxeter-Dynkin diagram
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Petrie polygon
Coxeter group
Properties

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).

Alternate names

  • Birectified 221 polytope
  • Rectified pentacontatetrapeton (Acronym: ram) - rectified 54-facetted polypeton (Jonathan Bowers)

Images

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

E6
[12]D5
[8]D4 / A2
[6]B6
[12/2]A5
[6]A4
[5]A3 / D3
[4]
[[File:up 1 22 t1 E6.svg120px]][[File:up 1 22 t1 D5.svg120px]][[File:up 1 22 t1 D4.svg120px]][[File:up 1 22 t1 B6.svg120px]]
[[File:up 1 22 t1 A5.svg120px]][[File:up 1 22 t1 A4.svg120px]][[File:up 1 22 t1 D3.svg120px]]

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 either of 2-length branches 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.

E6*k*-facef*k*f0f1colspan=3f2colspan=5f3colspan=5f4colspan=3f5*k*-figureNotesf0f1f2f3f4f5
A2A2A1( )**720**181818
A1A1A1{ }2**6480**22
A2A1{3}33**4320***
33***4320***0
A2A1A133****2160**
A2A1{3,3}4640
A3r{3,3}61244
A3A1612404
{3,3}46040
r{3,3}612044
A4r{3,3,3}10302010
A4A1103020010
A4103010200
D4[{3,4,3}](24-cell)24963232
A4A1r{3,3,3}1030020
A52r{3,3,3,3}20906060
D52r{4,3,3,3}80480320160
804801603201600

Truncated 122 polytope

Truncated 122
Type
Schläfli symbol
Coxeter symbol
Coxeter-Dynkin diagram
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Petrie polygon
Coxeter group
Properties

Alternate names

  • Truncated 122 polytope (Acronym: tim)

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.

E6
[12]D5
[8]D4 / A2
[6]B6
[12/2]A5
[6]A4
[5]A3 / D3
[4]
[[File:up 1 22 t01 E6.svg120px]][[File:up 1 22 t01 D5.svg120px]][[File:up 1 22 t01 D4.svg120px]][[File:up 1 22 t01 B6.svg120px]]
[[File:up 1 22 t01 A5.svg120px]][[File:up 1 22 t01 A4.svg120px]][[File:up 1 22 t01 D3.svg120px]]

Birectified 122 polytope

Birectified 122 polytope
Type
Schläfli symbol
Coxeter symbol
Coxeter-Dynkin diagram
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter group
Properties

Alternate names

  • Bicantellated 221
  • Birectified pentacontatetrapeton (barm) (Jonathan Bowers)

Images

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

E6
[12]D5
[8]D4 / A2
[6]B6
[12/2]A5
[6]A4
[5]A3 / D3
[4]
[[File:up 1 22 t2 E6.svg120px]][[File:up 1 22 t2 D5.svg120px]][[File:up 1 22 t2 D4.svg120px]][[File:up 1 22 t2 B6.svg120px]]
[[File:up 1 22 t2 A5.svg120px]][[File:up 1 22 t2 A4.svg120px]][[File:up 1 22 t2 D3.svg120px]]

Trirectified 122 polytope

Trirectified 122 polytope
Type
Schläfli symbol
Coxeter symbol
Coxeter-Dynkin diagram
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter group
Properties

Alternate names

  • Tricantellated 221
  • Trirectified pentacontatetrapeton (Acronym: trim, old: cacam, tram, mak) (Jonathan Bowers)

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

E6
[12]D5
[8]D4 / A2
[6]B6
[12/2]A5
[6]A4
[5]A3 / D3
[4]
[[File:up 1 22 t3 E6.svg120px]][[File:up 1 22 t3 D5.svg120px]][[File:up 1 22 t3 D4.svg120px]][[File:up 1 22 t3 B6.svg120px]]
[[File:up 1 22 t3 A5.svg120px]][[File:up 1 22 t3 A4.svg120px]][[File:up 1 22 t3 D3.svg120px]]

Notes

References

  • 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, wiley.com,
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45], p. 334 (figure 3.6a) by Peter mcMullen: (12-gonal node-edge graph of 122)
  • o3o3o3o3o *c3x - mo, o3o3x3o3o *c3o - ram, o3o3x3o3o *c3x - tim, o3x3o3x3o *c3o - barm, x3o3o3o3x *c3o - trim

References

  1. Elte, 1912
  2. Klitzing, (o3o3o3o3o *c3x - [http://bendwavy.org/klitzing/incmats/mo.htm mo])
  3. Coxeter, Regular Polytopes, 11.8 Gosset figures in six, seven, and eight dimensions, pp. 202–203
  4. Coxeter, H. S. M., ''Regular Complex Polytopes'', second edition, Cambridge University Press, (1991). p.30 and p.47
  5. [http://home.digital.net/~pervin/publications/vermont.html The Voronoi Cells of the E6* and E7* Lattices] {{Webarchive. link. (2016-01-30 , Edward Pervin)
  6. Klitzing, (o3o3x3o3o *c3o - [http://bendwavy.org/klitzing/incmats/ram.htm ram])
  7. Klitzing, (o3o3x3o3o *c3x - [http://bendwavy.org/klitzing/incmats/tim.htm tim])
  8. Klitzing, (o3x3o3x3o *c3o - [http://bendwavy.org/klitzing/incmats/scram.htm barm])
  9. Klitzing, (x3o3o3o3x *c3o - [http://bendwavy.org/klitzing/incmats/cacam.htm trim])
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