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1 22 Polytope

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1 22 Polytope


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, 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: .

1_22 polytope

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 hexateron:
022
Petrie polygon Dodecagon
Coxeter group E6, , 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 names

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

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, .

Images

Coxeter plane orthographic projections
E6
[12]
D5
[8]
D4 / A2
[6]
B6
[12/2]

(1,2)

(1,3)

(1,9,12)

(1,2)
A5
[6]
A4
= [10]
A3 / D3
[4]

(2,3,6)

(1,2)

(1,6,8,12)

Related polytopes and honeycomb

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 4 4 5 6 7 8 9 10
Coxeter
group
E3=A2×A1 E4=A4 E5=D5 E6 E7 E8 E9 = {\tilde{E}}_{8} = E8+ E10 = {\bar{T}}_8 = E8++
Coxeter
diagram
Symmetry
(order)
[3-1,2,1] [30,2,1] [31,2,1] [33,2,1] [34,2,1] [35,2,1] [36,2,1]
Order 12 120 192 103,680 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 1_22 polytope

Rectified 122
Type Uniform 6-polytope
Schläfli symbol t2{3,3,32,1}
t1{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, , 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).[3]

Alternate names

  • Birectified 221 polytope
  • Rectified pentacontatetrapeton (acronym Ram) - rectified 54-facetted polypeton (Jonathan Bowers)[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}×{}, .

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 1_22 polytope

Rectified 122 polytope
Type Uniform 6-polytope
Schläfli symbol t2{3,32,2}
Coxeter symbol t2(122)
Coxeter-Dynkin diagram
or
5-faces
4-faces
Cells
Faces
Edges 12960
Vertices 2160
Vertex figure
Coxeter group E6, , order 103680
Properties convex

Alternate names

  • Bicantellated 221
  • Birectified pentacontitetrapeton (barm) (Jonathan Bowers)[5]

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]

See also

Notes

  1. ^ Elte, 1912
  2. ^ Klitzing, (o3o3o3o3o *c3x - mo)
  3. ^ The Voronoi Cells of the E6* and E7* Lattices, Edward Pervin
  4. ^ Klitzing, (o3o3x3o3o *c3o - ram)
  5. ^ Klitzing, (o3x3o3x3o *c3o - barm)

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 [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)
  • Richard Klitzing, 6D, uniform polytopes (polypeta) o3o3o3o3o *c3x - mo, o3o3x3o3o *c3o - ram, o3x3o3x3o *c3o - barm
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