1 22 polytope

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Up 1 22 t0 E6.svg
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 polytope[edit]

122 polytope
Type Uniform 6-polytope
Family 1k2 polytope
Schläfli symbol {3,32,2}
Coxeter symbol 122
Coxeter-Dynkin diagram CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png or CDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png
5-faces 54:
27 121Demipenteract graph ortho.svg
27 121Demipenteract graph ortho.svg
4-faces 702:
270 111Cross graph 4.svg
432 1204-simplex t0.svg
Cells 2160:
1080 1103-simplex t0.svg
1080 {3,3}3-simplex t0.svg
Faces 2160 {3}2-simplex t0.svg
Edges 720
Vertices 72
Vertex figure Birectified hexateron:
022 5-simplex t2.svg
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 names[edit]

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

Construction[edit]

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, CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png.

Removing the node on either of 2-length branches leaves the 5-demicube, 131, CDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png.

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 5-simplex, 022, CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png.

Images[edit]

Coxeter plane orthographic projections
E6
[12]
D5
[8]
D4 / A2
[6]
B6
[12/2]
Up 1 22 t0 E6.svg
(1,2)
Up 1 22 t0 D5.svg
(1,3)
Up 1 22 t0 D4.svg
(1,9,12)
Up 1 22 t0 B6.svg
(1,2)
A5
[6]
A4
[[5]] = [10]
A3 / D3
[4]
Up 1 22 t0 A5.svg
(2,3,6)
Up 1 22 t0 A4.svg
(1,2)
Up 1 22 t0 D3.svg
(1,6,8,12)

Related polytopes and honeycomb[edit]

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

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 = E8++
Coxeter
diagram
CDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01l.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Symmetry
(order)
[3-1,2,1]
(12)
[30,2,1]
(120)
[31,2,1]
(192)
[[32,2,1]]
(103,680)
[33,2,1]
(2,903,040)
[34,2,1]
(696,729,600)
[35,2,1]
(∞)
[36,2,1]
(∞)
Graph Trigonal hosohedron.png 4-simplex t0.svg Demipenteract graph ortho.svg Up 1 22 t0 E6.svg Up2 1 32 t0 E7.svg Gosset 1 42 polytope petrie.svg
Name 1-1,2 102 112 122 132 142 152 162

Geometric folding[edit]

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
Up 1 22 t0 E6.svg
122
4-cube t2 F4.svg
24-cell
D4/B4 Coxeter planes
Up 1 22 t0 D4.svg
122
4-cube t2 B3.svg
24-cell

Tessellations[edit]

This polytope is the vertex figure for a uniform tessellation of 6-dimensional space, 222, CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png.

Rectified 1_22 polytope[edit]

Rectified 122
Type Uniform 6-polytope
Schläfli symbol t2{3,3,32,1}
t1{3,32,2}
Coxeter symbol 0221
Coxeter-Dynkin diagram CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
or CDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png
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).[3]

Alternate names[edit]

  • Birectified 221 polytope
  • Rectified pentacontatetrapeton (acronym Ram) - rectified 54-facetted polypeton (Jonathan Bowers)[4]

Construction[edit]

Its construction is based on the E6 group and information can be extracted from the ringed Coxeter-Dynkin diagram representing this polytope: CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png.

Removing the ring on the short branch leaves the birectified 5-simplex, CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png.

Removing the ring on the either 2-length branch leaves the birectified 5-orthoplex in its alternated form: t2(211), CDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png.

The vertex figure is determined by removing the ringed node and ringing the neighboring ring. This makes 3-3 duoprism prism, {3}×{3}×{}, CDel node.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 3.pngCDel node.png.

Images[edit]

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]
Up 1 22 t1 E6.svg Up 1 22 t1 D5.svg Up 1 22 t1 D4.svg Up 1 22 t1 B6.svg
A5
[6]
A4
[5]
A3 / D3
[4]
Up 1 22 t1 A5.svg Up 1 22 t1 A4.svg Up 1 22 t1 D3.svg

Birectified 1_22 polytope[edit]

Rectified 122 polytope
Type Uniform 6-polytope
Schläfli symbol t2{3,32,2}
Coxeter symbol t2(122)
Coxeter-Dynkin diagram CDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.png
or CDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.png
5-faces
4-faces
Cells
Faces
Edges 12960
Vertices 2160
Vertex figure
Coxeter group E6, [[3,32,2]], order 103680
Properties convex

Alternate names[edit]

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

Images[edit]

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]
Up 1 22 t2 E6.svg Up 1 22 t2 D5.svg Up 1 22 t2 D4.svg Up 1 22 t2 B6.svg
A5
[6]
A4
[5]
A3 / D3
[4]
Up 1 22 t2 A5.svg Up 1 22 t2 A4.svg Up 1 22 t2 D3.svg

See also[edit]

Notes[edit]

  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[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] 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