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1 42 polytope

October 17, 2023

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 non-empty combinations of rings in this Coxeter-Dynkin diagram: .

142 polytope Edit

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 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: .

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]

Projections Edit

The projection of 142 to the E8 Coxeter plane (aka. the Petrie projection) with polytope radius is shown below with 483,840 edges of length culled 53% on the interior to only 226,444:
Shown in 3D projection using the basis vectors [u,v,w] giving H3 symmetry:
  • u = (1, φ, 0, −1, φ, 0,0,0)
  • v = (φ, 0, 1, φ, 0, −1,0,0)
  • w = (0, 1, φ, 0, −1, φ,0,0)
The 17280 projected 142 polytope vertices are sorted and tallied by their 3D norm generating the increasingly transparent hulls for each set of tallied norms. Notice the last two outer hulls are a combination of two overlapped Dodecahedrons (40) and a Nonuniform Rhombicosidodecahedron (60).
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)

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.

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 Edit

1k2 figures in n dimensions
Space Finite Euclidean Hyperbolic
n 3 4 5 6 7 8 9 10
Coxeter
group 		E3=A2A1 E4=A4 E5=D5 E6 E7 E8 E9 = = E8+ E10 = = E8++
Coxeter
diagram
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 1,920 103,680 2,903,040 696,729,600 ∞
Graph - -
Name 1−1,2 102 112 122 132 142 152 162

Rectified 142 polytope Edit

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 Edit

  • 0421 polytope
  • Birectified 241 polytope
  • Quadrirectified 421 polytope
  • Rectified diacositetracont-dischiliahectohexaconta-zetton as a rectified 240-2160 facetted polyzetton (acronym buffy) (Jonathan Bowers)[4]

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: .

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]

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, [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]

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. ^ 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 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]
  • 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 Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

October 17, 2023
polytope, rectified, rectified, rectified, birectified, trirectified, orthogonal, projections, coxeter, plane, dimensional, geometry, uniform, polytope, constructed, within, symmetry, group, coxeter, symbol, describing, bifurcating, coxeter, dynkin, diagram, w. 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 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 28 160 8722 160 1 convex uniform polytopes in 8 dimensions made of uniform polytope facets and vertex figures defined by all non empty combinations of rings in this Coxeter Dynkin diagram Contents 1 142 polytope 1 1 Alternate names 1 2 Coordinates 1 3 Construction 1 4 Projections 1 5 Related polytopes and honeycombs 2 Rectified 142 polytope 2 1 Alternate names 2 2 Construction 2 3 Projections 3 See also 4 Notes 5 References 142 polytope Edit 142 Type Uniform 8 polytope Family 1k2 polytope Schlafli 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 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 91 1 93 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 91 2 93 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 8730 2 in this coordinate set and the polytope radius is 4 8730 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 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 91 3 93 Configuration matrix 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 Edit The projection of 142 to the E8 Coxeter plane aka the Petrie projection with polytope radius 4 2 displaystyle 4 sqrt 2 is shown below with 483 840 edges of length 2 2 displaystyle 2 sqrt 2 culled 53 on the interior to only 226 444 Shown in 3D projection using the basis vectors u v w giving H3 symmetry u 1 f 0 1 f 0 0 0 v f 0 1 f 0 1 0 0 w 0 1 f 0 1 f 0 0 The 17280 projected 142 polytope vertices are sorted and tallied by their 3D norm generating the increasingly transparent hulls for each set of tallied norms Notice the last two outer hulls are a combination of two overlapped Dodecahedrons 40 and a Nonuniform Rhombicosidodecahedron 60 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 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 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 Edit 1k2 figures in n dimensions Space Finite Euclidean Hyperbolic n 3 4 5 6 7 8 9 10 Coxetergroup E3 A2A1 E4 A4 E5 D5 E6 E7 E8 E9 E x007E 8 displaystyle tilde E 8 E8 E10 T x00AF 8 displaystyle bar T 8 E8 Coxeterdiagram 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 1 920 103 680 2 903 040 696 729 600 8734 Graph Name 1 1 2 102 112 122 132 142 152 162 Rectified 142 polytope Edit Rectified 142 Type Uniform 8 polytope Schlafli 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 Edit 0421 polytope Birectified 241 polytope Quadrirectified 421 polytope Rectified diacositetracont dischiliahectohexaconta zetton as a rectified 240 2160 facetted polyzetton acronym buffy Jonathan Bowers 91 4 93 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 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 91 5 93 Configuration matrix 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 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 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 8 160 20 See also Edit List of E8 polytopes Notes Edit Elte E L 1912 The Semiregular Polytopes of the Hyperspaces Groningen University of Groningen Klitzing o3o3o3x c3o3o3o3o bif Coxeter Regular Polytopes 11 8 Gossett figures in six seven and eight dimensions p 202 203 Klitzing o3o3o3x c3o3o3o3o buffy Coxeter Regular Polytopes 11 8 Gossett figures in six seven and eight dimensions p 202 203 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 160 978 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 vteFundamental 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 Octahedron Cube Demicube Dodecahedron Icosahedron Uniform polychoron Pentachoron 16 cell Tesseract Demitesseract 24 cell 120 cell 600 cell Uniform 5 polytope 5 simplex 5 orthoplex 5 cube 5 demicube Uniform 6 polytope 6 simplex 6 orthoplex 6 cube 6 demicube 122 221 Uniform 7 polytope 7 simplex 7 orthoplex 7 cube 7 demicube 132 231 321 Uniform 8 polytope 8 simplex 8 orthoplex 8 cube 8 demicube 142 241 421 Uniform 9 polytope 9 simplex 9 orthoplex 9 cube 9 demicube Uniform 10 polytope 10 simplex 10 orthoplex 10 cube 10 demicube Uniform n polytope n simplex n orthoplex n cube n demicube 1k2 2k1 k21 n pentagonal polytope Topics Polytope families Regular polytope List of regular polytopes and compounds Retrieved from https en wikipedia org w index php title 1 42 polytope amp oldid 1149263005, wikipedia, wiki, book, books, library,

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