1 22 polytope

From Wikipedia, the free encyclopedia
Jump to: navigation, search
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 5-simplex:
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]
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)
B6
[12/2]
A5
[6]
A4
[[5]] = [10]
A3 / D3
[4]
Up 1 22 t0 B6.svg
(1,2)
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 complex polyhedron[edit]

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, CDel 3node 1.pngCDel 3.pngCDel 3node.pngCDel 4.pngCDel node.png, 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 CDel 3node.pngCDel 3.pngCDel 3node 1.pngCDel 3.pngCDel 3node.png, as a rectification of the Hessian polyhedron, CDel 3node 1.pngCDel 3.pngCDel 3node.pngCDel 3.pngCDel 3node.png.[3]

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.

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
24-cell t3 F4.svg
24-cell
D4/B4 Coxeter planes
Up 1 22 t0 D4.svg
122
24-cell t3 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 2r{3,3,32,1}
r{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).[4]

Alternate names[edit]

  • Birectified 221 polytope
  • Rectified pentacontatetrapeton (acronym Ram) - rectified 54-facetted polypeton (Jonathan Bowers)[5]

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]

Birectified 122 polytope
Type Uniform 6-polytope
Schläfli symbol 2r{3,32,2}
Coxeter symbol 2r(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 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 names[edit]

  • Bicantellated 221
  • Birectified pentacontitetrapeton (barm) (Jonathan Bowers)[6]

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

Trirectified 1_22 polytope[edit]

Trirectified 122 polytope
Type Uniform 6-polytope
Schläfli symbol 3r{3,32,2}
Coxeter symbol 3r(122)
Coxeter-Dynkin diagram CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png
or CDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.png
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 names[edit]

  • Tricantellated 221
  • Trirectified pentacontitetrapeton (trim) (Jonathan Bowers)[7]


See also[edit]

Notes[edit]

  1. ^ Elte, 1912
  2. ^ Klitzing, (o3o3o3o3o *c3x - mo)
  3. ^ Coxeter, H. S. M., Regular Complex Polytopes, second edition, Cambridge University Press, (1991). p.30 and p.47
  4. ^ The Voronoi Cells of the E6* and E7* Lattices, Edward Pervin
  5. ^ Klitzing, (o3o3x3o3o *c3o - ram)
  6. ^ Klitzing, (o3x3o3x3o *c3o - barm)
  7. ^ Klitzing, (x3o3o3o3x *c3o - trim)

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
Fundamental convex regular and uniform polytopes in dimensions 2–10
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 4-polytope 5-cell 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