2 41 polytope

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4 21 t0 E6.svg
421
CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
1 42 polytope E6 Coxeter plane.svg
142
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
2 41 t0 E6.svg
241
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png
4 21 t1 E6.svg
Rectified 421
CDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
4 21 t4 E6.svg
Rectified 142
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
2 41 t1 E6.svg
Rectified 241
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.png
4 21 t2 E6.svg
Birectified 421
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
4 21 t3 E6.svg
Trirectified 421
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel 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 8-dimensional geometry, the 241 is a uniform 8-polytope, constructed within the symmetry of the E8 group.

Coxeter named it 241 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequences.

The rectified 241 is constructed by points at the mid-edges of the 241. The birectified 241 is constructed by points at the triangle face centers of the 241, and is the same as the rectified 142.

These polytopes are part of a family of 255 (28 − 1) convex uniform polytopes in 8-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.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png.


2_41 polytope[edit]

241 polytope
Type Uniform 8-polytope
Family 2k1 polytope
Schläfli symbol {3,3,34,1}
Coxeter symbol 241
Coxeter-Dynkin diagram 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
7-faces 17520:
240 231Gosset 2 31 polytope.svg
17280 {36}7-simplex t0.svg
6-faces 144960:
6720 221E6 graph.svg
138240 {35}6-simplex t0.svg
5-faces 544320:
60480 211Cross graph 5.svg
483840 {34}5-simplex t0.svg
4-faces 1209600:
241920 {2014-simplex t0.svg
967680 {33}4-simplex t0.svg
Cells 1209600 {32}3-simplex t0.svg
Faces 483840 {3}2-simplex t0.svg
Edges 69120
Vertices 2160
Vertex figure 141
Petrie polygon 30-gon
Coxeter group E8, [34,2,1]
Properties convex

The 241 is composed of 17,520 facets (240 231 polytopes, 17,280 7-simplices), 144,960 6-faces (6,720 221 polytopes, 138,240 6-simplices), 544,320 5-faces (60,480 211, 483,840 5-simplices, 1,209,600 4-faces (4-simplices), 1,209,600 cells (tetrahedra), 483,840 faces (triangles), 69,120 edges, and 2160 vertices. Its vertex figure is a 7-demicube.

This polytope is a facet in the uniform tessellation, 251 with Coxeter-Dynkin diagram:

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

Alternate names[edit]

  • E. L. Elte named it V2160 (for its 2160 vertices) in his 1912 listing of semiregular polytopes.[1]
  • It is named 241 by Coxeter for its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequence.
  • Diacositetracont-myriaheptachiliadiacosioctaconta-zetton (Acronym Bay) - 240-17280 facetted polyzetton (Jonathan Bowers)[2]

Coordinates[edit]

The 2160 vertices can be defined as follows:

16 permutations of (±4,0,0,0,0,0,0,0)
1120 permutations of (±2,±2,±2,±2,0,0,0,0)
1024 permutations of (±3,±1,±1,±1,±1,±1,±1,±1) with an even number of minus-signs

Construction[edit]

It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space.

The facet information can be extracted from its Coxeter-Dynkin diagram: 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.

Removing the node on the short branch leaves the 7-simplex: CDel nodea 1.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. There are 17280 of these facets

Removing the node on the end of the 4-length branch leaves the 231, CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png. There are 240 of these facets. They are centered at the positions of the 240 vertices in the 421 polytope.

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 7-demicube, 141, CDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png.

Images[edit]

Petrie polygon projections can be 12, 18, or 30-sided based on the E6, E7, and E8 symmetries. The 2160 vertices are all displayed, but lower symmetry forms have projected positions overlapping, shown as different colored vertices. For comparison, a B6 coxeter group is also shown.

E8
[30]
[20] [24]
2 41 t0 E8.svg
(1)
2 41 t0 p20.svg 2 41 t0 p24.svg
E7
[18]
E6
[12]
[6]
2 41 t0 E7.svg 2 41 t0 E6.svg
(1,8,24,32)
2 41 t0 mox.svg
D3 / B2 / A3
[4]
D4 / B3 / A2
[6]
D5 / B4
[8]
2 41 t0 B2.svg 2 41 t0 B3.svg 2 41 t0 B4.svg
D6 / B5 / A4
[10]
D7 / B6
[12]
D8 / B7 / A6
[14]
2 41 t0 B5.svg 2 41 t0 B6.svg
(1,3,9,12,18,21,36)
2 41 t0 B7.svg
B8
[16/2]
A5
[6]
A7
[8]
2 41 t0 B8.svg 2 41 t0 A5.svg 2 41 t0 A7.svg

Related polytopes and honeycombs[edit]

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_41 polytope[edit]

Rectified 241 polytope
Type Uniform 8-polytope
Schläfli symbol t1{3,3,34,1}
Coxeter symbol t1(241)
Coxeter-Dynkin diagram CDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
7-faces 19680 total:

240 t1(221)
17280 t1{36}
2160 141

6-faces 313440
5-faces 1693440
4-faces 4717440
Cells 7257600
Faces 5322240
Edges 19680
Vertices 69120
Vertex figure rectified 6-simplex prism
Petrie polygon 30-gon
Coxeter group E8, [34,2,1]
Properties convex

The rectified 241 is a rectification of the 241 polytope, with vertices positioned at the mid-edges of the 241.

Alternate names[edit]

  • Rectified Diacositetracont-myriaheptachiliadiacosioctaconta-zetton for rectified 240-17280 facetted polyzetton (acronym robay) (Jonathan Bowers)[3]

Construction[edit]

It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space, defined by root vectors of the E8 Coxeter group.

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

Removing the node on the short branch leaves the rectified 7-simplex: CDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png.

Removing the node on the end of the 4-length branch leaves the rectified 231, CDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png.

Removing the node on the end of the 2-length branch leaves the 7-demicube, 141CDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.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 rectified 6-simplex prism, CDel nodea 1.pngCDel 2.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png.

Images[edit]

Petrie polygon projections can be 12, 18, or 30-sided based on the E6, E7, and E8 symmetries. The 2160 vertices are all displayed, but lower symmetry forms have projected positions overlapping, shown as different colored vertices. For comparison, a B6 coxeter group is also shown.

E8
[30]
[20] [24]
2 41 t1 E8.svg
(1)
2 41 t1 p20.svg 2 41 t1 p24.svg
E7
[18]
E6
[12]
[6]
2 41 t1 E7.svg 2 41 t1 E6.svg
(1,8,24,32)
2 41 t1 mox.svg
D3 / B2 / A3
[4]
D4 / B3 / A2
[6]
D5 / B4
[8]
2 41 t1 B2.svg 2 41 t1 B3.svg 2 41 t1 B4.svg
D6 / B5 / A4
[10]
D7 / B6
[12]
D8 / B7 / A6
[14]
2 41 t1 B5.svg 2 41 t1 B6.svg
(1,3,9,12,18,21,36)
2 41 t1 B7.svg
B8
[16/2]
A5
[6]
A7
[8]
2 41 t1 B8.svg 2 41 t1 A5.svg 2 41 t1 A7.svg

See also[edit]

Notes[edit]

  1. ^ Elte, 1912
  2. ^ Klitzing, (x3o3o3o *c3o3o3o3o - bay)
  3. ^ Klitzing, (o3x3o3o *c3o3o3o3o - robay)

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]
  • Richard Klitzing, 8D, Uniform polyzetta x3o3o3o *c3o3o3o3o - bay, o3x3o3o *c3o3o3o3o - robay