# 2 41 polytope

 Orthogonal projections in E6 Coxeter plane 421 142 241 Rectified 421 Rectified 142 Rectified 241 Birectified 421 Trirectified 421

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
Type Uniform 8-polytope
Family 2k1 polytope
Schläfli symbol {3,3,34,1}
Coxeter symbol 241
Coxeter diagram
7-faces 17520:
240 231
17280 {36}
6-faces 144960:
6720 221
138240 {35}
5-faces 544320:
60480 211
483840 {34}
4-faces 1209600:
241920 {201
967680 {33}
Cells 1209600 {32}
Faces 483840 {3}
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:

### 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:              .

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,            .

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

E8               k-face fk f0 f1 f2 f3 f4 f5 f6 f7 k-figure notes
D7               ( ) f0 2160 64 672 2240 560 2240 280 1344 84 448 14 64 h{4,3,3,3,3,3} E8/D7 = 192*10!/64/7! = 2160
A6A1               { } f1 2 69120 21 105 35 140 35 105 21 42 7 7 r{3,3,3,3,3} E8/A6A1 = 192*10!/7!/2 = 69120
A4A2A1               {3} f2 3 3 483840 10 5 20 10 20 10 10 5 2 {}x{3,3,3} E8/A4A2A1 = 192*10!/5!/3!/2 = 483840
A3A3               {3,3} f3 4 6 4 1209600 1 4 4 6 6 4 4 1 {3,3}V( ) E8/A3A3 = 192*10!/4!/4! = 1209600
A4A3               {3,3,3} f4 5 10 10 5 241920 * 4 0 6 0 4 0 {3,3} E8/A4A3 = 192*10!/5!/4! = 241920
A4A2               5 10 10 5 * 967680 1 3 3 3 3 1 {3}V( ) E8/A4A2 = 192*10!/5!/3! = 967680
D5A2               {3,3,31,1} f5 10 40 80 80 16 16 60480 * 3 0 3 0 {3} E8/D5A2 = 192*10!/16/5!/2 = 40480
A5A1               {3,3,3,3} 6 15 20 15 0 6 * 483840 1 2 2 1 { }V( ) E8/A5A1 = 192*10!/6!/2 = 483840
E6A1               {3,3,32,1} f6 27 216 720 1080 216 432 27 72 6720 * 2 0 { } E8/E6A1 = 192*10!/72/6! = 6720
A6               {3,3,3,3,3} 7 21 35 35 0 21 0 7 * 138240 1 1 E8/A6 = 192*10!/7! = 138240
E7               {3,3,33,1} f7 126 2016 10080 20160 4032 12096 756 4032 56 576 240 * ( ) E8/E7 = 192*10!/72!/8! = 240
A7               {3,3,3,3,3,3} 8 28 56 70 0 56 0 28 0 8 * 17280 E8/A7 = 192*10!/8! = 17280

### Images

Shown in 3D projection using the basis vectors [u,v,w] giving H3 symmetry:
• u = (1, φ, 0, −1, φ, 0,0,0)
• v = (φ, 0, 1, φ, 0, −1,0,0)
• w = (0, 1, φ, 0, −1, φ,0,0)
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 Norm'd distance from the origin, and the number of vertices in the goup.

The 2160 projected 241 polytope projected to 3D (as above) with each Norm'd 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).

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]

## Rectified 2_41 polytope

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

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,            .

### Visualizations

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]

## Notes

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)

## References

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