Английская Википедия:2 41 polytope

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Файл:4 21 t0 E6.svg
421
Шаблон:CDD
Файл:1 42 polytope E6 Coxeter plane.svg
142
Шаблон:CDD
Файл:2 41 t0 E6.svg
241
Шаблон:CDD
Файл:4 21 t1 E6.svg
Rectified 421
Шаблон:CDD
Файл:4 21 t4 E6.svg
Rectified 142
Шаблон:CDD
Файл:2 41 t1 E6.svg
Rectified 241
Шаблон:CDD
Файл:4 21 t2 E6.svg
Birectified 421
Шаблон:CDD
Файл:4 21 t3 E6.svg
Trirectified 421
Шаблон:CDD
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.

Its Coxeter symbol is 241, describing 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, defined by all permutations of rings in this Coxeter-Dynkin diagram: Шаблон:CDD.

241 polytope

241 polytope
Type Uniform 8-polytope
Family 2k1 polytope
Schläfli symbol {3,3,34,1}
Coxeter symbol 241
Coxeter diagram Шаблон:CDD
7-faces 17520:
240 231Файл:Gosset 2 31 polytope.svg
17280 {36}Файл:7-simplex t0.svg
6-faces 144960:
6720 221Файл:E6 graph.svg
138240 {35}Файл:6-simplex t0.svg
5-faces 544320:
60480 211Файл:Cross graph 5.svg
483840 {34}Файл:5-simplex t0.svg
4-faces 1209600:
241920 {201Файл:4-simplex t0.svg
967680 {33}Файл:4-simplex t0.svg
Cells 1209600 {32}Файл:3-simplex t0.svg
Faces 483840 {3}Файл: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 and 17,280 7-simplices), 144,960 6-faces (6,720 221 polytopes and 138,240 6-simplices), 544,320 5-faces (60,480 211 and 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:

Шаблон:CDD

Alternate names

  • 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

The 2160 vertices can be defined as follows:

16 permutations of (±4,0,0,0,0,0,0,0) of (8-orthoplex)
1120 permutations of (±2,±2,±2,±2,0,0,0,0) of (trirectified 8-orthoplex)
1024 permutations of (±3,±1,±1,±1,±1,±1,±1,±1) with an odd number of minus-signs

Construction

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: Шаблон:CDD.

Removing the node on the short branch leaves the 7-simplex: Шаблон:CDD. There are 17280 of these facets

Removing the node on the end of the 4-length branch leaves the 231, Шаблон:CDD. 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, Шаблон:CDD.

Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.[3]

Visualizations

Файл:E8 241 Petrie Projection.png
The projection of 241 to the E8 Coxeter plane (aka. the Petrie projection) with polytope radius <math>2\sqrt{2}</math> and 69120 edges of length <math>2\sqrt{2}</math>
Файл:E8 241-3D.png
Shown in 3D projection using the basis vectors [u,v,w] giving H3 symmetry: Шаблон:Ubl The 2160 projected 241 polytope vertices are sorted and tallied by their 3D norm generating the increasingly transparent hulls for each set of tallied norms. The overlapping vertices are color coded by overlap count. Also shown is a list of each hull group, the normed distance from the origin, and the number of vertices in the group.
Файл:E8 241-3D Concentric Hulls List.png
The 2160 projected 241 polytope projected to 3D (as above) with each normed hull group listed individually with vertex counts. Notice the last two outer hulls are a combination of two overlapped Icosahedrons (24) and a Icosidodecahedron (30).
E8
[30]
[20] [24]
Файл:2 41 t0 E8.svg
(1)
Файл:2 41 t0 p20.svg Файл:2 41 t0 p24.svg
E7
[18]
E6
[12]
[6]
Файл:2 41 t0 E7.svg Файл:2 41 t0 E6.svg
(1,8,24,32)
Файл:2 41 t0 mox.svg

Petrie polygon projections are 12, 18, or 30-sided based on the E6, E7, and E8 symmetries (respectively). 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.

D3 / B2 / A3
[4]
D4 / B3 / A2
[6]
D5 / B4
[8]
Файл:2 41 t0 B2.svg Файл:2 41 t0 B3.svg Файл:2 41 t0 B4.svg
D6 / B5 / A4
[10]
D7 / B6
[12]
D8 / B7 / A6
[14]
Файл:2 41 t0 B5.svg Файл:2 41 t0 B6.svg
(1,3,9,12,18,21,36)
Файл:2 41 t0 B7.svg
B8
[16/2]
A5
[6]
A7
[8]
Файл:2 41 t0 B8.svg Файл:2 41 t0 A5.svg Файл:2 41 t0 A7.svg

Related polytopes and honeycombs

Шаблон:2 k1 polytopes

Rectified 2_41 polytope

Rectified 241 polytope
Type Uniform 8-polytope
Schläfli symbol t1{3,3,34,1}
Coxeter symbol t1(241)
Coxeter diagram Шаблон:CDD
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

  • Rectified Diacositetracont-myriaheptachiliadiacosioctaconta-zetton for rectified 240-17280 facetted polyzetton (known as robay for short)[4][5]

Construction

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: Шаблон:CDD.

Removing the node on the short branch leaves the rectified 7-simplex: Шаблон:CDD.

Removing the node on the end of the 4-length branch leaves the rectified 231, Шаблон:CDD.

Removing the node on the end of the 2-length branch leaves the 7-demicube, 141Шаблон:CDD.

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the rectified 6-simplex prism, Шаблон:CDD.

Visualizations

Petrie polygon projections are 12, 18, or 30-sided based on the E6, E7, and E8 symmetries (respectively). 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]
Файл:2 41 t1 E8.svg
(1)
Файл:2 41 t1 p20.svg Файл:2 41 t1 p24.svg
E7
[18]
E6
[12]
[6]
Файл:2 41 t1 E7.svg Файл:2 41 t1 E6.svg
(1,8,24,32)
Файл:2 41 t1 mox.svg
D3 / B2 / A3
[4]
D4 / B3 / A2
[6]
D5 / B4
[8]
Файл:2 41 t1 B2.svg Файл:2 41 t1 B3.svg Файл:2 41 t1 B4.svg
D6 / B5 / A4
[10]
D7 / B6
[12]
D8 / B7 / A6
[14]
Файл:2 41 t1 B5.svg Файл:2 41 t1 B6.svg
(1,3,9,12,18,21,36)
Файл:2 41 t1 B7.svg
B8
[16/2]
A5
[6]
A7
[8]
Файл:2 41 t1 B8.svg Файл:2 41 t1 A5.svg Файл:2 41 t1 A7.svg

See also

Notes

Шаблон:Reflist

References

  • Шаблон:Citation
  • 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, Шаблон:ISBN [1]
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
  • Шаблон:KlitzingPolytopes x3o3o3o *c3o3o3o3o - bay, o3x3o3o *c3o3o3o3o - robay

Шаблон:Polytopes

  1. Elte, 1912
  2. Klitzing, (x3o3o3o *c3o3o3o3o - bay)
  3. Coxeter, Regular Polytopes, 11.8 Gossett figures in six, seven, and eight dimensions, p. 202-203
  4. Jonathan Bowers
  5. Klitzing, (o3x3o3o *c3o3o3o3o - robay)