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 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: 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.

142 polytope[edit]

142
Type Uniform 8-polytope
Family 1k2 polytope
Schläfli symbol {3,34,2}
Coxeter symbol 142
Coxeter diagrams 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 node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.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 22 in this coordinate set, and the polytope radius is 42.

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.

Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.[3]

E8 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 k-face fk f0 f1 f2 f3 f4 f5 f6 f7 k-figure notes
A7 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodes 0x.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png ( ) 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 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodes x1.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.png { } 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 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel branch 01.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png {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 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel branch 01r.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.png 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 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel branch 01l.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png 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 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel branch 01r.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png 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 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.png 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 CDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01l.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png 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 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png 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 CDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.png 16 80 160 40 80 0 10 16 * 181440 * 1 2 0 2 1 { }v( ) E8/D5A1 = 192*10!/16/5!/2 = 181440
A5A1 CDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01l.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png 130 6 15 20 0 15 0 0 6 * * 483840 0 2 1 1 2 E8/A5A1 = 192*10!/6!/2 = 483840
E6A1 CDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png 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 CDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.png 131 32 240 640 160 480 0 60 192 0 12 32 * 30240 * 1 1 E8/D6 = 192*10!/32/6! = 30240
A6A1 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01l.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png 140 7 21 35 0 35 0 0 21 0 0 7 * * 69120 0 2 E8/A6A1 = 192*10!/7!/2 = 69120
E7 CDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png 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 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.png 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]

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]

Rectified 142 polytope[edit]

Rectified 142
Type Uniform 8-polytope
Schläfli symbol t1{3,34,2}
Coxeter symbol 0421
Coxeter diagrams 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
CDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.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}×{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: 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.

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

E8 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 k-face fk f0 f1 f2 f3 f4 f5 f6 f7 k-figure
A4A2A1 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodes x0.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.png ( ) 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 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodes 1x.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png { } 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 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodes 1x.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.png {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 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel branch 10.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png 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 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodes 1x.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png 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 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodes 1x.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png 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
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.png 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 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodes 1x.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.png 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 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png 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 CDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodes 1x.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png 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 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png 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 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodes 1x.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png 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 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.png 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 CDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodes 1x.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.png 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 CDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png 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 CDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodes 1x.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png 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 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png 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 CDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodes 1x.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png 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 CDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.png 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 CDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodes 1x.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.png 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 CDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png 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()
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodes 1x.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png 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 CDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png 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 CDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodes 1x.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png 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 CDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.png 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 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodes 1x.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.png 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 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png 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 CDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png 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 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodes 1x.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png 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 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.png 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]
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]
 
[20]
4 21 t4 A5.svg 4 21 t4 A7.svg 4 21 t4 p20.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. ^ 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 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