# 1 42 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 142 is a uniform 8-polytope, constructed within the symmetry of the E8 group.

## Contents

Its Coxeter symbol is 142, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequences.

The rectified 142 is constructed by points at the mid-edges of the 142 and is the same as the birectified 241, and the quadrirectified 421.

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: .

## 142 polytope

142
Type Uniform 8-polytope
Family 1k2 polytope
Schläfli symbol {3,34,2}
Coxeter symbol 142
Coxeter diagrams
7-faces2400:
240 132
2160 141
6-faces106080:
6720 122
30240 131
69120 {35}
5-faces725760:
60480 112
181440 121
483840 {34}
4-faces2298240:
241920 102
604800 111
1451520 {33}
Cells3628800:
1209600 101
2419200 {32}
Faces2419200 {3}
Edges483840
Vertices17280
Vertex figure t2{36}
Petrie polygon 30-gon
Coxeter group E8, [34,2,1]
Properties convex

The 142 is composed of 2400 facets: 240 132 polytopes, and 2160 7-demicubes (141). Its vertex figure is a birectified 7-simplex.

This polytope, along with the demiocteract, can tessellate 8-dimensional space, represented by the symbol 152, and Coxeter-Dynkin diagram: .

### Alternate names

• E. L. Elte (1912) excluded this polytope from his listing of semiregular polytopes, because it has more than two types of 6-faces, but under his naming scheme it would be called V17280 for its 17280 vertices. [1]
• Coxeter named it 142 for its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node branch.
• Diacositetracont-dischiliahectohexaconta-zetton (acronym bif) - 240-2160 facetted polyzetton (Jonathan Bowers) [2]

### Coordinates

The 17280 vertices can be defined as sign and location permutations of:

All sign combinations (32): (280×32=8960 vertices)

(4, 2, 2, 2, 2, 0, 0, 0)

Half of the sign combinations (128): ((1+8+56)×128=8320 vertices)

(2, 2, 2, 2, 2, 2, 2, 2)
(5, 1, 1, 1, 1, 1, 1, 1)
(3, 3, 3, 1, 1, 1, 1, 1)

The edge length is 22 in this coordinate set, and the polytope radius is 42.

### 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 end of the 2-length branch leaves the 7-demicube, 141, .

Removing the node on the end of the 4-length branch leaves the 132, .

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 7-simplex, 042, .

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
A7( )f01728056420280560702804205616816828562888 2r{36} E8/A7 = 192*10!/8! = 17280
A4A2A1{ }f12483840151530530301030151015353{3}x{3,3,3}E8/A4A2A1 = 192*10!/5!/2/2 = 483840
A3A2A1 {3} f233241920024186412468142{3.3}v{ }E8/A3A2A1 = 192*10!/4!/3!/2 = 2419200
A3A3 110 f34641209600*14046064041 {3,3}v( ) E8/A3A3 = 192*10!/4!/4! = 1209600
A3A2A1464*241920002316336132{3}v{ }E8/A3A2A1 = 192*10!/4!/3!/2 = 2419200
A4A3 120 f45101050241920**40060040 {3,3} E8/A4A3 = 192*10!/4!/4! = 241920
D4A2 111 8243288*604800*13033031 {3}v( ) E8/D4A2 = 192*10!/8/4!/3! = 604800
A4A1A1 120 5101005**145152002214122 { }v{ } E8/A4A1A1 = 192*10!/5!/2/2 = 1451520
D5A2 121 f5168016080401610060480**30030 {3} E8/D5A2 = 192*10!/16/5!/3! = 40480
D5A11680160408001016*181440*12021 { }v( ) E8/D5A1 = 192*10!/16/5!/2 = 181440
A5A1 130 61520015006**48384002112E8/A5A1 = 192*10!/6!/2 = 483840
E6A1 122 f672720216010801080216270216272706720**20{ }E8/E6A1 = 192*10!/72/6!/2 = 6720
D6 131 3224064016048006019201232*30240*11E8/D6 = 192*10!/32/6! = 30240
A6A1 140 721350350021007**6912002E8/A6A1 = 192*10!/7!/2 = 69120
E7 132 f757610080403202016030240403275601209675615122016561260240*( )E8/E7 = 192*10!/72/8! = 240
D7 141 64672224056022400280134408444801464*2160E8/D7 = 192*10!/64/7! = 2160

### Projections

Orthographic projections are shown for the sub-symmetries of E8: E7, E6, B8, B7, B6, B5, B4, B3, B2, A7, and A5 Coxeter planes, as well as two more symmetry planes of order 20 and 24. Vertices are shown as circles, colored by their order of overlap in each projective plane.

E8
[30]
E7
[18]
E6
[12]

(1)

(1,3,6)

(8,16,24,32,48,64,96)
[20][24][6]

(1,2,3,4,5,6,7,8,10,11,12,14,16,18,19,20)
D3 / B2 / A3
[4]
D4 / B3 / A2
[6]
D5 / B4
[8]

(32,160,192,240,480,512,832,960)

(72,216,432,720,864,1080)

(8,16,24,32,48,64,96)
D6 / B5 / A4
[10]
D7 / B6
[12]
D8 / B7 / A6
[14]
B8
[16/2]
A5
[6]
A7
[8]
1k2 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
(order)
[3−1,2,1][30,2,1][31,2,1][[3<sup>2,2,1</sup>]][33,2,1][34,2,1][35,2,1][36,2,1]
Order 121201,920103,6802,903,040696,729,600
Graph --
Name 1−1,2 102 112 122 132 142 152 162

## Rectified 142 polytope

Rectified 142
Type Uniform 8-polytope
Schläfli symbol t1{3,34,2}
Coxeter symbol 0421
Coxeter diagrams
7-faces19680
6-faces382560
5-faces2661120
4-faces9072000
Cells16934400
Faces16934400
Edges7257600
Vertices483840
Vertex figure {3,3,3}×{3}×{}
Coxeter group E8, [34,2,1]
Properties convex

The rectified 142 is named from being a rectification of the 142 polytope, with vertices positioned at the mid-edges of the 142. It can also be called a 0421 polytope with the ring at the center of 3 branches of length 4, 2, and 1.

### Alternate names

• 0421 polytope
• Birectified 241 polytope
• Rectified diacositetracont-dischiliahectohexaconta-zetton as a rectified 240-2160 facetted polyzetton (acronym buffy) (Jonathan Bowers) [4]

### 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 end of the 1-length branch leaves the birectified 7-simplex,

Removing the node on the end of the 2-length branch leaves the birectified 7-cube, .

Removing the node on the end of the 3-length branch leaves the rectified 132, .

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 5-cell-triangle duoprism prism, .

Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders. [5]

E8 k-face fkf0f1f2f3f4f5f6f7 k-figure
A4A2A1( )f0483840303015601015603060520306030301020303015610101563523{3,3,3}x{3,3}x{}
A3A1A1{ }f127257600214128461481264461284164821412
A3A2 {3} f2334838400**1140014460046640064410411
A3A2A133*2419200*02040108060401204060801402
A2A2A133**96768000021301263313663133621312
A3A3 0200 f3464001209600****14000046000064000410
0110 612440*1209600***10400040600060400401
A3A2612404**4838400**01130013330033310311
A3A2A1612044***2419200*00203010603030601302
A3A1A1 0200 46004****725760000021201242112421212
A4A3 0210 f410302010055000241920*****40000060000400
A4A210302001050500*967680****13000033000310
D4A2 0111 249632323208880**604800***10300030300301
A4A1 0210 10301002000505***2903040**01120012210211
A4A1A110300102000055****1451520*00202010401202
A4A1 0300 510001000005*****290304000021101221112
D5A2 0211 f58048032016016080808040016161000060480*****30000300 {3}
A5A1 0220 20906006015030015060600*483840****12000210 { }v()
D5A1 0211 80480160160320040808080001016160**181440***10200201
A5 0310 1560200600015030000606***967680**01110111 ( )v( )v()
A5A11560020600001530000066****483840*00201102 { }v()
0400 6150020000015000006*****48384000021012
E6A1 0221 f672064804320216043201080108021601080108021643227043221602772270006720****200{ }
A6 0320 3521014002103501050105021042021070700*138240***110
D6 0311 2401920640640192001604804809600060192192192001232320**30240**101
A6 0410 211053501400035010500021042000707***138240*011
A6A1211050351400003510500002142000077****69120002
E7 0321 f710080120960806404032012096020160201606048030240604804032120967560241921209612096756403215124032201605657612600240**( )
A7 0420 56420280056070028004200560168016802805602808080*17280*
D7 0411 6726720224022408960056022402240672000280134413442688008444844844800146464**2160

### Projections

Orthographic projections are shown for the sub-symmetries of B6, B5, B4, B3, B2, A7, and A5 Coxeter planes. Vertices are shown as circles, colored by their order of overlap in each projective plane.

(Planes for E8: E7, E6, B8, B7, [24] are not shown for being too large to display.)

D3 / B2 / A3
[4]
D4 / B3 / A2
[6]
D5 / B4
[8]
D6 / B5 / A4
[10]
D7 / B6
[12]
[6]
A5
[6]
A7
[8]

[20]

## Notes

1. Elte, E. L. (1912), The Semiregular Polytopes of the Hyperspaces, Groningen: University of Groningen
2. Klitzing, (o3o3o3x *c3o3o3o3o - bif)
3. Coxeter, Regular Polytopes, 11.8 Gossett figures in six, seven, and eight dimensions, p. 202-203
4. Klitzing, (o3o3o3x *c3o3o3o3o - buffy)
5. Coxeter, Regular Polytopes, 11.8 Gossett figures in six, seven, and eight dimensions, p. 202-203

## Related Research Articles

In geometry, the Gosset–Elte figures, named by Coxeter after Thorold Gosset and E. L. Elte, are a group of uniform polytopes which are not regular, generated by a Wythoff construction with mirrors all related by order-2 and order-3 dihedral angles. They can be seen as one-end-ringed Coxeter–Dynkin diagrams.

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 241 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 geometry, the 152 honeycomb is a uniform tessellation of 8-dimensional Euclidean space. It contains 142 and 151 facets, in a birectified 8-simplex vertex figure. It is the final figure in the 1k2 polytope family.

In geometry, the 222 honeycomb is a uniform tessellation of the six-dimensional Euclidean space. It can be represented by the Schläfli symbol {3,3,32,2}. It is constructed from 221 facets and has a 122 vertex figure, with 54 221 polytopes around every vertex.

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 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 geometry, the simplectic honeycomb is a dimensional infinite series of honeycombs, based on the affine Coxeter group symmetry. It is given a Schläfli symbol {3[n+1]}, and is represented by a Coxeter-Dynkin diagram as a cyclic graph of n+1 nodes with one node ringed. It is composed of n-simplex facets, along with all rectified n-simplices. It can be thought of as an n-dimensional hypercubic honeycomb that has been subdivided along all hyperplanes , then stretched along its main diagonal until the simplices on the ends of the hypercubes become regular. The vertex figure of an n-simplex honeycomb is an expanded n-simplex.

## References

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