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

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-faces 2400:
240 132
2160 141
6-faces 106080:
6720 122
30240 131
69120 {35}
5-faces 725760:
60480 112
181440 121
483840 {34}
4-faces 2298240:
241920 102
604800 111
1451520 {33}
Cells 3628800:
1209600 101
2419200 {32}
Faces 2419200 {3}
Edges 483840
Vertices 17280
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 fk f0 f1 f2 f3 f4 f5 f6 f7 k-figure notes
A7 ( ) f0 17280 56 420 280 560 70 280 420 56 168 168 28 56 28 8 8 2r{36} E8/A7 = 192*10!/8! = 17280
A4A2A1 { } f1 2 483840 15 15 30 5 30 30 10 30 15 10 15 3 5 3 {3}x{3,3,3} E8/A4A2A1 = 192*10!/5!/2/2 = 483840
A3A2A1 {3} f2 3 3 2419200 2 4 1 8 6 4 12 4 6 8 1 4 2 {3.3}v{ } E8/A3A2A1 = 192*10!/4!/3!/2 = 2419200
A3A3 110 f3 4 6 4 1209600 * 1 4 0 4 6 0 6 4 0 4 1 {3,3}v( ) E8/A3A3 = 192*10!/4!/4! = 1209600
A3A2A1 4 6 4 * 2419200 0 2 3 1 6 3 3 6 1 3 2 {3}v{ } E8/A3A2A1 = 192*10!/4!/3!/2 = 2419200
A4A3 120 f4 5 10 10 5 0 241920 * * 4 0 0 6 0 0 4 0 {3,3} E8/A4A3 = 192*10!/4!/4! = 241920
D4A2 111 8 24 32 8 8 * 604800 * 1 3 0 3 3 0 3 1 {3}v( ) E8/D4A2 = 192*10!/8/4!/3! = 604800
A4A1A1 120 5 10 10 0 5 * * 1451520 0 2 2 1 4 1 2 2 { }v{ } E8/A4A1A1 = 192*10!/5!/2/2 = 1451520
D5A2 121 f5 16 80 160 80 40 16 10 0 60480 * * 3 0 0 3 0 {3} E8/D5A2 = 192*10!/16/5!/3! = 40480
D5A1 16 80 160 40 80 0 10 16 * 181440 * 1 2 0 2 1 { }v( ) E8/D5A1 = 192*10!/16/5!/2 = 181440
A5A1 130 6 15 20 0 15 0 0 6 * * 483840 0 2 1 1 2 E8/A5A1 = 192*10!/6!/2 = 483840
E6A1 122 f6 72 720 2160 1080 1080 216 270 216 27 27 0 6720 * * 2 0 { } E8/E6A1 = 192*10!/72/6!/2 = 6720
D6 131 32 240 640 160 480 0 60 192 0 12 32 * 30240 * 1 1 E8/D6 = 192*10!/32/6! = 30240
A6A1 140 7 21 35 0 35 0 0 21 0 0 7 * * 69120 0 2 E8/A6A1 = 192*10!/7!/2 = 69120
E7 132 f7 576 10080 40320 20160 30240 4032 7560 12096 756 1512 2016 56 126 0 240 * ( ) E8/E7 = 192*10!/72/8! = 240
D7 141 64 672 2240 560 2240 0 280 1344 0 84 448 0 14 64 * 2160 E8/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]

## Rectified 142 polytope

Rectified 142
Type Uniform 8-polytope
Schläfli symbol t1{3,34,2}
Coxeter symbol 0421
Coxeter diagrams
7-faces 19680
6-faces 382560
5-faces 2661120
4-faces 9072000
Cells 16934400
Faces 16934400
Edges 7257600
Vertices 483840
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 7-demicube, 141, .

Removing the node on the end of the 3-length branch leaves the 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 fk f0 f1 f2 f3 f4 f5 f6 f7 k-figure
A4A2A1 ( ) f0 483840 30 30 15 60 10 15 60 30 60 5 20 30 60 30 30 10 20 30 30 15 6 10 10 15 6 3 5 2 3 {3,3,3}x{3,3}x{}
A3A1A1 { } f1 2 7257600 2 1 4 1 2 8 4 6 1 4 8 12 6 4 4 6 12 8 4 1 6 4 8 2 1 4 1 2
A3A2 {3} f2 3 3 4838400 * * 1 1 4 0 0 1 4 4 6 0 0 4 6 6 4 0 0 6 4 4 1 0 4 1 1
A3A2A1 3 3 * 2419200 * 0 2 0 4 0 1 0 8 0 6 0 4 0 12 0 4 0 6 0 8 0 1 4 0 2
A2A2A1 3 3 * * 9676800 0 0 2 1 3 0 1 2 6 3 3 1 3 6 6 3 1 3 3 6 2 1 3 1 2
A3A3 0200 f3 4 6 4 0 0 1209600 * * * * 1 4 0 0 0 0 4 6 0 0 0 0 6 4 0 0 0 4 1 0
0110 6 12 4 4 0 * 1209600 * * * 1 0 4 0 0 0 4 0 6 0 0 0 6 0 4 0 0 4 0 1
A3A2 6 12 4 0 4 * * 4838400 * * 0 1 1 3 0 0 1 3 3 3 0 0 3 3 3 1 0 3 1 1
A3A2A1 6 12 0 4 4 * * * 2419200 * 0 0 2 0 3 0 1 0 6 0 3 0 3 0 6 0 1 3 0 2
A3A1A1 0200 4 6 0 0 4 * * * * 7257600 0 0 0 2 1 2 0 1 2 4 2 1 1 2 4 2 1 2 1 2
A4A3 0210 f4 10 30 20 10 0 5 5 0 0 0 241920 * * * * * 4 0 0 0 0 0 6 0 0 0 0 4 0 0
A4A2 10 30 20 0 10 5 0 5 0 0 * 967680 * * * * 1 3 0 0 0 0 3 3 0 0 0 3 1 0
D4A2 0111 24 96 32 32 32 0 8 8 8 0 * * 604800 * * * 1 0 3 0 0 0 3 0 3 0 0 3 0 1
A4A1 0210 10 30 10 0 20 0 0 5 0 5 * * * 2903040 * * 0 1 1 2 0 0 1 2 2 1 0 2 1 1
A4A1A1 10 30 0 10 20 0 0 0 5 5 * * * * 1451520 * 0 0 2 0 2 0 1 0 4 0 1 2 0 2
A4A1 0300 5 10 0 0 10 0 0 0 0 5 * * * * * 2903040 0 0 0 2 1 1 0 1 2 2 1 1 1 2
D5A2 0211 f5 80 480 320 160 160 80 80 80 40 0 16 16 10 0 0 0 60480 * * * * * 3 0 0 0 0 3 0 0 {3}
A5A1 0220 20 90 60 0 60 15 0 30 0 15 0 6 0 6 0 0 * 483840 * * * * 1 2 0 0 0 2 1 0 { }v()
D5A1 0211 80 480 160 160 320 0 40 80 80 80 0 0 10 16 16 0 * * 181440 * * * 1 0 2 0 0 2 0 1
A5 0310 15 60 20 0 60 0 0 15 0 30 0 0 0 6 0 6 * * * 967680 * * 0 1 1 1 0 1 1 1 ( )v( )v()
A5A1 15 60 0 20 60 0 0 0 15 30 0 0 0 0 6 6 * * * * 483840 * 0 0 2 0 1 1 0 2 { }v()
0400 6 15 0 0 20 0 0 0 0 15 0 0 0 0 0 6 * * * * * 483840 0 0 0 2 1 0 1 2
E6A1 0221 f6 720 6480 4320 2160 4320 1080 1080 2160 1080 1080 216 432 270 432 216 0 27 72 27 0 0 0 6720 * * * * 2 0 0 { }
A6 0320 35 210 140 0 210 35 0 105 0 105 0 21 0 42 0 21 0 7 0 7 0 0 * 138240 * * * 1 1 0
D6 0311 240 1920 640 640 1920 0 160 480 480 960 0 0 60 192 192 192 0 0 12 32 32 0 * * 30240 * * 1 0 1
A6 0410 21 105 35 0 140 0 0 35 0 105 0 0 0 21 0 42 0 0 0 7 0 7 * * * 138240 * 0 1 1
A6A1 21 105 0 35 140 0 0 0 35 105 0 0 0 0 21 42 0 0 0 0 7 7 * * * * 69120 0 0 2
E7 0321 f7 10080 120960 80640 40320 120960 20160 20160 60480 30240 60480 4032 12096 7560 24192 12096 12096 756 4032 1512 4032 2016 0 56 576 126 0 0 240 * * ( )
A7 0420 56 420 280 0 560 70 0 280 0 420 0 56 0 168 0 168 0 28 0 56 0 28 0 8 0 8 0 * 17280 *
D7 0411 672 6720 2240 2240 8960 0 560 2240 2240 6720 0 0 280 1344 1344 2688 0 0 84 448 448 448 0 0 14 64 64 * * 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

## References

• 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". o3o3o3x *c3o3o3o3o - bif, o3o3o3x *c3o3o3o3o - buffy
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