# 142 polytope

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 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.

Coxeter named it 142 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequences.

The rectified 121 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: .

## 1_42 polytope

142
Type Uniform 8-polytope
Family 1k2 polytope
Schläfli symbol {3,34,2}
Coxeter symbol 142
Coxeter-Dynkin diagram
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 2√2 in this coordinate set, and the polytope radius is 4√2.

### 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, .

### 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]

### Related polytopes and honeycombs

1k2 figures in n dimensions
n 4 4 5 6 7 8 9 10
Coxeter
group
E3=A2×A1 E4=A4 E5=D5 E6 E7 E8 E9 = ${\tilde{E}}_{8}$ = E8+ E10 = E8++
Coxeter
diagram
Symmetry
(order)
[3-1,2,1]
(12)
[30,2,1]
(120)
[31,2,1]
(192)
[[32,2,1]]
(103,680)
[33,2,1]
(2,903,040)
[34,2,1]
(696,729,600)
[35,2,1]
(∞)
[36,2,1]
(∞)
Graph
Name 1-1,2 102 112 122 132 142 152 162

## Rectified 1_42 polytope

Rectified 142
Type Uniform 8-polytope
Schläfli symbol t1{3,34,2}
Coxeter symbol t1(142)
Coxeter-Dynkin diagram
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}x{3}x{}
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.

### Alternate names

• Birectified 241 polytope
• Quadrirectified 421 polytope
• Rectified diacositetracont-dischiliahectohexaconta-zetton as a rectified 240-2160 facetted polyzetton (Acronym buffy) (Jonathan Bowers)[3]

### 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, .

### 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, [20], [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]

## Notes

1. ^ Elte, E. L. (1912), The Semiregular Polytopes of the Hyperspaces, Groningen: University of Groningen
2. ^ Klitzing, (o3o3o3x *c3o3o3o3o - bif)
3. ^ Klitzing, (o3o3o3x *c3o3o3o3o - buffy)

## 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]
• Richard Klitzing, 8D, Uniform polyzetta o3o3o3x *c3o3o3o3o - bif, o3o3o3x *c3o3o3o3o - buffy