1 42 polytope

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4 21 t0 E6.svg
421
CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
1 42 polytope E6 Coxeter plane.svg
142
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
2 41 t0 E6.svg
241
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png
4 21 t1 E6.svg
Rectified 421
CDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
4 21 t4 E6.svg
Rectified 142
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
2 41 t1 E6.svg
Rectified 241
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.png
4 21 t2 E6.svg
Birectified 421
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
4 21 t3 E6.svg
Trirectified 421
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
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.

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 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: CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png.

1_42 polytope[edit]

142
Type Uniform 8-polytope
Family 1k2 polytope
Schläfli symbol {3,34,2}
Coxeter symbol 142
Coxeter diagram CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
7-faces 2400:
240 132Gosset 1 32 petrie.svg
2160 141Demihepteract ortho petrie.svg
6-faces 106080:
6720 122Gosset 1 22 polytope.svg
30240 131Demihexeract ortho petrie.svg
69120 {35}6-simplex t0.svg
5-faces 725760:
60480 112Demipenteract graph ortho.svg
181440 121Demipenteract graph ortho.svg
483840 {34}5-simplex t0.svg
4-faces 2298240:
241920 1024-simplex t0.svg
604800 1114-cube t3.svg
1451520 {33}4-simplex t0.svg
Cells 3628800:
1209600 1013-simplex t0.svg
2419200 {32}3-simplex t0.svg
Faces 2419200 {3}2-simplex t0.svg
Edges 483840
Vertices 17280
Vertex figure t2{36} 7-simplex t2.svg
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: CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png.

Alternate names[edit]

  • 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[edit]

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[edit]

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: CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png.

Removing the node on the end of the 2-length branch leaves the 7-demicube, 141, CDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png.

Removing the node on the end of the 4-length branch leaves the 132, CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png.

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 7-simplex, 042, CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png.

Projections[edit]

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]
Gosset 1 42 polytope petrie.svg
(1)
1 42 t0 e7.svg
(1,3,6)
1 42 polytope E6 Coxeter plane.svg
(8,16,24,32,48,64,96)
[20] [24] [6]
1 42 t0 p20.svg 1 42 t0 p24.svg 1 42 t0 mox.svg
(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]
1 42 t0 B2.svg
(32,160,192,240,480,512,832,960)
1 42 t0 B3.svg
(72,216,432,720,864,1080)
1 42 t0 B4.svg
(8,16,24,32,48,64,96)
D6 / B5 / A4
[10]
D7 / B6
[12]
D8 / B7 / A6
[14]
1 42 t0 B5.svg 1 42 t0 B6.svg 1 42 t0 B7.svg
B8
[16/2]
A5
[6]
A7
[8]
1 42 t0 B8.svg 1 42 t0 A5.svg 1 42 t0 A7.svg

Related polytopes and honeycombs[edit]

1k2 figures in n dimensions
Space Finite Euclidean Hyperbolic
n 4 4 5 6 7 8 9 10
Coxeter
group
E3=A2A1 E4=A4 E5=D5 E6 E7 E8 E9 = {\tilde{E}}_{8} = E8+ E10 = {\bar{T}}_8 = E8++
Coxeter
diagram
CDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01l.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Symmetry
(order)
[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 12 120 192 103,680 2,903,040 696,729,600
Graph Trigonal hosohedron.png 4-simplex t0.svg Demipenteract graph ortho.svg Up 1 22 t0 E6.svg Up2 1 32 t0 E7.svg Gosset 1 42 polytope petrie.svg - -
Name 1-1,2 102 112 122 132 142 152 162

Rectified 1_42 polytope[edit]

Rectified 142
Type Uniform 8-polytope
Schläfli symbol t1{3,34,2}
Coxeter symbol 0421
Coxeter diagram CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
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[edit]

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

Construction[edit]

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: CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png.

Removing the node on the end of the 1-length branch leaves the birectified 7-simplex, CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png

Removing the node on the end of the 2-length branch leaves the 7-demicube, 141, CDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png.

Removing the node on the end of the 3-length branch leaves the 132, CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png.

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 5-cell-triangle duoprism prism, CDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 2.pngCDel nodea 1.pngCDel 2.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png.

Projections[edit]

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]
4 21 t4 B2.svg 4 21 t4 B3.svg 4 21 t4 B4.svg
D6 / B5 / A4
[10]
D7 / B6
[12]
[6]
4 21 t4 B5.svg 4 21 t4 B6.svg 4 21 t4 mox.svg
A5
[6]
A7
[8]
4 21 t4 A5.svg 4 21 t4 A7.svg

See also[edit]

Notes[edit]

  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[edit]

  • 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