# 2 41 polytope

Last updated

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.

## Contents

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: .

## 241 polytope

241 polytope
TypeUniform 8-polytope
Family 2k1 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 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 fkf0f1f2f3f4f5f6f7 k-figure notes
D7( )f0216064672224056022402801344844481464 h{4,3,3,3,3,3} E8/D7 = 192*10!/64/7! = 2160
A6A1{ }f1269120211053514035105214277 r{3,3,3,3,3} E8/A6A1 = 192*10!/7!/2 = 69120
A4A2A1 {3} f233483840105201020101052 {}x{3,3,3} E8/A4A2A1 = 192*10!/5!/3!/2 = 483840
A3A3 {3,3} f3464120960014466441 {3,3}V( ) E8/A3A3 = 192*10!/4!/4! = 1209600
A4A3 {3,3,3} f4510105241920*406040 {3,3} E8/A4A3 = 192*10!/5!/4! = 241920
A4A2510105*967680133331 {3}V( ) E8/A4A2 = 192*10!/5!/3! = 967680
D5A2 {3,3,31,1} f510408080161660480*3030 {3} E8/D5A2 = 192*10!/16/5!/2 = 40480
A5A1 {3,3,3,3} 615201506*4838401221 { }V( ) E8/A5A1 = 192*10!/6!/2 = 483840
E6A1 {3,3,32,1} f627216720108021643227726720*20{ }E8/E6A1 = 192*10!/72/6! = 6720
A6 {3,3,3,3,3} 721353502107*13824011E8/A6 = 192*10!/7! = 138240
E7 {3,3,33,1} f712620161008020160403212096756403256576240*( )E8/E7 = 192*10!/72!/8! = 240
A7 {3,3,3,3,3,3} 828567005602808*17280E8/A7 = 192*10!/8! = 17280

### 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]
2k1 figures in n dimensions
SpaceFiniteEuclideanHyperbolic
n 3 4 5 6 7 8 9 10
Coxeter
group
E3=A2A1E4=A4E5=D5 E6 E7 E8 E9 = ${\displaystyle {\tilde {E}}_{8}}$ = E8+E10 = ${\displaystyle {\bar {T}}_{8}}$ = E8++
Coxeter
diagram
Symmetry [3−1,2,1][30,2,1][[31,2,1]][32,2,1][33,2,1][34,2,1][35,2,1][36,2,1]
Order 1212038451,8402,903,040696,729,600
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 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)

## Related Research Articles

In six-dimensional geometry, a uniform polypeton is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform 5-polytopes.

In five-dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a 5-hypercube (penteract) with alternated vertices removed.

In five-dimensional geometry, a rectified 5-simplex is a convex uniform 5-polytope, being a rectification of the regular 5-simplex.

In 7-dimensional geometry, 231 is a uniform polytope, constructed from the E7 group.

In 6-dimensional geometry, the 122 polytope is a uniform polytope, constructed from the E6 group. It was first published in E. L. Elte's 1912 listing of semiregular polytopes, named as V72 (for its 72 vertices).

In 7-dimensional geometry, 132 is a uniform polytope, constructed from the E7 group.

In 8-dimensional geometry, the 142 is a uniform 8-polytope, constructed within the symmetry of the E8 group.

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. It is also called the Schläfli polytope.

In 7-dimensional geometry, the 321 polytope is a uniform 7-polytope, constructed within the symmetry of the E7 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 7-ic semi-regular figure.

In 8-dimensional geometry, the 421 is a semiregular uniform 8-polytope, constructed within the symmetry of the E8 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 8-ic semi-regular figure.

In 7-dimensional geometry, the 331 honeycomb is a uniform honeycomb, also given by Schläfli symbol {3,3,3,33,1} and is composed of 321 and 7-simplex facets, with 56 and 576 of them respectively around each vertex.

In 7-dimensional geometry, 133 is a uniform honeycomb, also given by Schläfli symbol {3,33,3}, and is composed of 132 facets.

In six-dimensional geometry, a pentellated 6-simplex is a convex uniform 6-polytope with 5th order truncations of the regular 6-simplex.

In six-dimensional geometry, a rectified 6-simplex is a convex uniform 6-polytope, being a rectification of the regular 6-simplex.

In geometry, the 521 honeycomb is a uniform tessellation of 8-dimensional Euclidean space. The symbol 521 is from Coxeter, named for the length of the 3 branches of its Coxeter-Dynkin diagram.

In seven-dimensional geometry, a rectified 7-simplex is a convex uniform 7-polytope, being a rectification of the regular 7-simplex.

In eight-dimensional geometry, a rectified 8-simplex is a convex uniform 8-polytope, being a rectification of the regular 8-simplex.

In eight-dimensional geometry, a heptellated 8-simplex is a convex uniform 8-polytope, including 7th-order truncations (heptellation) from the regular 8-simplex.

In geometry, an E9 honeycomb is a tessellation of uniform polytopes in hyperbolic 9-dimensional space. , also (E10) is a paracompact hyperbolic group, so either facets or vertex figures will not be bounded.

In seven-dimensional Euclidean geometry, the quarter 7-cubic honeycomb is a uniform space-filling tessellation. It has half the vertices of the 7-demicubic honeycomb, and a quarter of the vertices of a 7-cube honeycomb. Its facets are 7-demicubes, pentellated 7-demicubes, and {31,1,1}×{3,3} duoprisms.

## 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
• (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
Family An Bn I2(p) / Dn E6 / E7 / E8 / / Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 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