2 21 polytope

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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
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 2 21 t2 E6.svg
Birectified 221
(Rectified 122)
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
orthogonal projections in E6 Coxeter plane

In 6-dimensional geometry, the 221 polytope is a uniform 6-polytope, constructed within the symmetry of the E6 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 6-ic semi-regular figure.[1]

Coxeter named it 221 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 2-node sequences. He also studied[2] its connection with the 27 lines on the cubic surface, which are naturally in correspondence with the vertices of 221.

The rectified 221 is constructed by points at the mid-edges of the 221. The birectified 221 is constructed by points at the triangle face centers of the 221, and is the same as the rectified 122.

These polytopes are a part of family of 39 convex uniform polytopes in 6-dimensions, made of uniform 5-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.

2_21 polytope[edit]

221 polytope
Type Uniform 6-polytope
Family k21 polytope
Schläfli symbol {3,3,32,1}
Coxeter symbol 221
Coxeter-Dynkin diagram CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
5-faces 99 total:
27 2115-orthoplex.svg
72 {34}5-simplex t0.svg
4-faces 648:
432 {33}4-simplex t0.svg
216 {33}4-simplex t0.svg
Cells 1080 {3,3}3-simplex t0.svg
Faces 720 {3}2-simplex t0.svg
Edges 216
Vertices 27
Vertex figure 121 (5-demicube)
Petrie polygon Dodecagon
Coxeter group E6, [32,2,1], order 51840
Properties convex

The 221 has 27 vertices, and 99 facets: 27 5-orthoplexes and 72 5-simplices. Its vertex figure is a 5-demicube.

For visualization this 6-dimensional polytope is often displayed in a special skewed orthographic projection direction that fits its 27 vertices within a 12-gonal regular polygon (called a Petrie polygon). Its 216 edges are drawn between 2 rings of 12 vertices, and 3 vertices projected into the center. Higher elements (faces, cells, etc.) can also be extracted and drawn on this projection.

Alternate names[edit]

  • E. L. Elte named it V27 (for its 27 vertices) in his 1912 listing of semiregular polytopes.[3]
  • Icosihepta-heptacontidi-peton - 27-72 facetted polypeton (acronym jak) (Jonathan Bowers)[4]

Coordinates[edit]

The 27 vertices can be expressed in 8-space as an edge-figure of the 421 polytope:

  • (-2,0,0,0,-2,0,0,0)(0,-2,0,0,-2,0,0,0)(0,0,-2,0,-2,0,0,0)(0,0,0,-2,-2,0,0,0)(0,0,0,0,-2,0,0,-2)(0,0,0,0,0,-2,-2,0)
  • ( 2,0,0,0,-2,0,0,0)(0, 2,0,0,-2,0,0,0)(0,0, 2,0,-2,0,0,0)(0,0,0, 2,-2,0,0,0)(0,0,0,0,-2,0,0, 2)
  • (-1,-1,-1,-1,-1,-1,-1,-1)
  • (-1,-1,-1, 1,-1,-1,-1, 1) (-1,-1, 1,-1,-1,-1,-1, 1) (-1,-1, 1, 1,-1,-1,-1,-1) (-1, 1,-1,-1,-1,-1,-1, 1) (-1, 1,-1, 1,-1,-1,-1,-1) (-1, 1, 1,-1,-1,-1,-1,-1) (1,-1,-1,-1,-1,-1,-1, 1) (1,-1, 1,-1,-1,-1,-1,-1) (1,-1,-1, 1,-1,-1,-1,-1) (1, 1,-1,-1,-1,-1,-1,-1)
  • (-1, 1, 1, 1,-1,-1,-1, 1) (1,-1, 1, 1,-1,-1,-1, 1) (1, 1,-1, 1,-1,-1,-1, 1) (1, 1, 1,-1,-1,-1,-1, 1) (1, 1, 1, 1,-1,-1,-1,-1)

Construction[edit]

Its construction is based on the E6 group.

The facet information can be extracted from its Coxeter-Dynkin diagram, CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png.

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

Removing the node on the end of the 2-length branch leaves the 5-orthoplex in its alternated form: (211), CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png.

Every simplex facet touches an 5-orthoplex facet, while alternate facets of the orthoplex touch either a simplex or another orthoplex.

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes 5-demicube (121 polytope), CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea 1.png.

Images[edit]

Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow. The number of vertices by color are given in parentheses.

Coxeter plane orthographic projections
E6
[12]
D5
[8]
D4 / A2
[6]
B6
[12/2]
Up 2 21 t0 E6.svg
(1,3)
Up 2 21 t0 D5.svg
(1,3)
Up 2 21 t0 D4.svg
(3,9)
Up 2 21 t0 B6.svg
(1,3)
A5
[6]
A4
[5]
A3 / D3
[4]
Up 2 21 t0 A5.svg
(1,3)
Up 2 21 t0 A4.svg
(1,2)
Up 2 21 t0 D3.svg
(1,4,7)

Geometric folding[edit]

The 221 is related to the 24-cell by a geometric folding of the E6/F4 Coxeter-Dynkin diagrams. 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 221.

E6
Dyn-node.pngDyn-3.pngDyn-loop1.pngDyn-nodes.pngDyn-3s.pngDyn-nodes.png
F4
Dyn2-node.pngDyn2-3.pngDyn2-node.pngDyn2-4b.pngDyn2-node.pngDyn2-3.pngDyn2-node.png
E6 graph.svg
221
CDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 10l.png
4-cube t2 F4.svg
24-cell
CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png

This polytope can tessellate Euclidean 6-space, forming the 222 honeycomb with this Coxeter-Dynkin diagram: CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png.

Related polytopes[edit]

The 221 is fourth in a dimensional series of semiregular polytopes. Each progressive uniform polytope is constructed vertex figure of the previous polytope. Thorold Gosset identified this series in 1900 as containing all regular polytope facets, containing all simplexes and orthoplexes.

k21 figures in n dimensional
En 3 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 1.pngCDel 2.pngCDel node 1.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea 1.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png
Symmetry
(order)
[3-1,2,1]
(12)
[30,2,1]
(120)
[31,2,1]
(192)
[32,2,1]
(51,840)
[33,2,1]
(2,903,040)
[34,2,1]
(696,729,600)
[35,2,1]
(∞)
[36,2,1]
(∞)
Graph Triangular prism.png 4-simplex t1.svg Demipenteract graph ortho.svg E6 graph.svg E7 graph.svg E8 graph.svg
Name −121 021 121 221 321 421 521 621
2k1 figures in n dimensions
n 3 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 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.png CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.png CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.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]]
(384)
[32,2,1]
(51,840)
[33,2,1]
(2,903,040)
[34,2,1]
(696,729,600)
[35,2,1]
(∞)
[36,2,1]
(∞)
Graph Trigonal dihedron.png 4-simplex t0.svg 5-cube t4.svg Up 2 21 t0 E6.svg Up2 2 31 t0 E7.svg 2 41 t0 E8.svg
Name 2-1,1 201 211 221 231 241 251 261

Rectified 2_21 polytope[edit]

Rectified 221 polytope
Type Uniform 6-polytope
Schläfli symbol t1{3,3,32,1}
Coxeter symbol t1(221)
Coxeter-Dynkin diagram CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.png
5-faces 126 total:

72 t1{34} 5-simplex t1.svg
27 t1{33,4} 5-cube t3.svg
27 t1{3,32,1} 5-demicube t0 D5.svg

4-faces 1350
Cells 4320
Faces 5040
Edges 2160
Vertices 216
Vertex figure rectified 5-cell prism
Coxeter group E6, [32,2,1], order 51840
Properties convex

The rectified 221 has 216 vertices, and 126 facets: 72 rectified 5-simplices, and 27 rectified 5-orthoplexes and 27 5-demicubes . Its vertex figure is a rectified 5-cell prism.

Alternate names[edit]

  • Rectified icosihepta-heptacontidi-peton as a rectified 27-72 facetted polypeton (acronym rojak) (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.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.png.

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

Removing the ring on the end of the other 2-length branch leaves the rectified 5-orthoplex in its alternated form: t1(211), CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.png.

Removing the ring on the end of the same 2-length branch leaves the 5-demicube: (121), CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea 1.png.

The vertex figure is determined by removing the ringed ring and ringing the neighboring ring. This makes rectified 5-cell prism, t1{3,3,3}x{}, CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 2.pngCDel nodea 1.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 2 21 t1 E6.svg Up 2 21 t1 D5.svg Up 2 21 t1 D4.svg Up 2 21 t1 B6.svg
A5
[6]
A4
[5]
A3 / D3
[4]
Up 2 21 t1 A5.svg Up 2 21 t1 A4.svg Up 2 21 t1 D3.svg

See also[edit]

Notes[edit]

  1. ^ Gosset, 1900
  2. ^ Coxeter, H.S.M. (1940). "The Polytope 221 Whose Twenty-Seven Vertices Correspond to the Lines on the General Cubic Surface". Amer. J. Math. 62: 457–486. JSTOR 2371466. 
  3. ^ Elte, 1912
  4. ^ Klitzing, (x3o3o3o3o *c3o - jak)
  5. ^ Klitzing, (o3x3o3o3o *c3o - rojak)

References[edit]

  • T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
  • Elte, E. L. (1912), The Semiregular Polytopes of the Hyperspaces, Groningen: University of Groningen 
  • 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 17) Coxeter, The Evolution of Coxeter-Dynkin diagrams, [Nieuw Archief voor Wiskunde 9 (1991) 233-248] See figure 1: (p. 232) (Node-edge graph of polytope)
  • Richard Klitzing, 6D, uniform polytopes (polypeta) x3o3o3o3o *c3o - jak, o3x3o3o3o *c3o - rojak