2 41 polytope


421

142

241

Rectified 421

Rectified 142

Rectified 241

Birectified 421

Trirectified 421
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 and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: .

241 polytope

241 polytope
TypeUniform 8-polytope
Family2k1 polytope
Schläfli symbol {3,3,34,1}
Coxeter symbol 241
Coxeter diagram
7-faces17520:
240 231
17280 {36}
6-faces144960:
6720 221
138240 {35}
5-faces544320:
60480 211
483840 {34}
4-faces1209600:
241920 {201
967680 {33}
Cells1209600 {32}
Faces483840 {3}
Edges69120
Vertices2160
Vertex figure141
Petrie polygon30-gon
Coxeter groupE8, [34,2,1]
Propertiesconvex

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:

Alternate names

Coordinates

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

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: .

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

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

Images

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]

(1)
E7
[18]
E6
[12]
[6]

(1,8,24,32)
D3 / B2 / A3
[4]
D4 / B3 / A2
[6]
D5 / B4
[8]
D6 / B5 / A4
[10]
D7 / B6
[12]
D8 / B7 / A6
[14]

(1,3,9,12,18,21,36)
B8
[16/2]
A5
[6]
A7
[8]

Related polytopes and honeycombs

2k1 figures in n dimensions
Space Finite Euclidean Hyperbolic
n 3 4 5 6 7 8 9 10
Coxeter
group
E3=A2A1 E4=A4 E5=D5 E6 E7 E8 E9 = {\tilde {E}}_{8} = E8+ E10 = {\bar {T}}_{8} = E8++
Coxeter
diagram
Symmetry [3−1,2,1] [30,2,1] [[3<sup>1,2,1</sup>]] [32,2,1] [33,2,1] [34,2,1] [35,2,1] [36,2,1]
Order 12 120 384 51,840 2,903,040 696,729,600
Graph - -
Name 2-1,1 201 211 221 231 241 251 261

Rectified 2_41 polytope

Rectified 241 polytope
TypeUniform 8-polytope
Schläfli symbol t1{3,3,34,1}
Coxeter symbol t1(241)
Coxeter diagram
7-faces19680 total:

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

6-faces313440
5-faces1693440
4-faces4717440
Cells7257600
Faces5322240
Edges19680
Vertices69120
Vertex figurerectified 6-simplex prism
Petrie polygon30-gon
Coxeter groupE8, [34,2,1]
Propertiesconvex

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

Alternate names

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: .

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

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

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

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

Images

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]

(1)
E7
[18]
E6
[12]
[6]

(1,8,24,32)
D3 / B2 / A3
[4]
D4 / B3 / A2
[6]
D5 / B4
[8]
D6 / B5 / A4
[10]
D7 / B6
[12]
D8 / B7 / A6
[14]

(1,3,9,12,18,21,36)
B8
[16/2]
A5
[6]
A7
[8]

See also

Notes

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

References

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