E8 polytope

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Orthographic projections in the E8 Coxeter plane
E8 graph.svg
421
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2 41 t0 E8.svg
241
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png
Gosset 1 42 polytope petrie.svg
142
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In 8-dimensional geometry, there are 255 uniform polytopes with E8 symmetry. The three simplest forms are the 421, 241, and 142 polytopes, composed of 240, 2160 and 17280 vertices respectively.

Geometry branch of mathematics that measures the shape, size and position of objects

Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer.

Uniform 8-polytope vertex-transitive 8-polytope bounded by uniform facets

In eight-dimensional geometry, an eight-dimensional polytope or 8-polytope is a polytope contained by 7-polytope facets. Each 6-polytope ridge being shared by exactly two 7-polytope facets.

4<sub> 21</sub> polytope semiregular uniform 8-polytope

In 8-dimensional geometry, the 421 is a semiregular uniform 8-polytope, constructed within the symmetry of the E8 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 8-ic semi-regular figure.

Contents

These polytopes can be visualized as symmetric orthographic projections in Coxeter planes of the E8 Coxeter group, and other subgroups.

Orthographic projection form of parallel projection in which all the projection lines are orthogonal to the projection plane

Orthographic projection is a means of representing three-dimensional objects in two dimensions. It is a form of parallel projection, in which all the projection lines are orthogonal to the projection plane, resulting in every plane of the scene appearing in affine transformation on the viewing surface. The obverse of an orthographic projection is an oblique projection, which is a parallel projection in which the projection lines are not orthogonal to the projection plane.

Graphs

Symmetric orthographic projections of these 255 polytopes can be made in the E8, E7, E6, D7, D6, D5, D4, D3, A7, A5 Coxeter planes. Ak has [k+1] symmetry, Dk has [2(k-1)] symmetry, and E6, E7, E8 have [12], [18], [30] symmetry respectively. In addition there are two other degrees of fundamental invariants, order [20] and [24] for the E8 group that represent Coxeter planes.

11 of these 255 polytopes are each shown in 14 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position.

# Coxeter plane projections Coxeter-Dynkin diagram
Name
E8
[30]
E7
[18]
E6
[12]
[24][20]D4-E6
[6]
A3
D3
[4]
A2
D4
[6]
D5
[8]
A4
D6
[10]
D7
[12]
A6
B7
[14]
B8
[16/2]
A5
 
[6]
A7
 
[8]
1 4 21 t0 E8.svg 4 21 t0 E7.svg 4 21 t0 E6.svg 4 21 t0 p20.svg 4 21 t0 p24.svg 4 21 t0 mox.svg 4 21 t0 B2.svg 4 21 t0 B3.svg 4 21 t0 B4.svg 4 21 t0 B5.svg 4 21 t0 B6.svg 4 21 t0 B7.svg 4 21 t0 B8.svg 4 21 t0 A5.svg 4 21 t0 A7.svg 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
421 (fy)
2 4 21 t1 E8.svg 4 21 t1 E7.svg 4 21 t1 E6.svg 4 21 t1 p20.svg 4 21 t1 p24.svg 4 21 t1 mox.svg 4 21 t1 B2.svg 4 21 t1 B3.svg 4 21 t1 B4.svg 4 21 t1 B5.svg 4 21 t1 B6.svg 4 21 t1 B7.svg 4 21 t1 B8.svg 4 21 t1 A5.svg 4 21 t1 A7.svg 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
Rectified 421 (riffy)
3 4 21 t2 E8.svg 4 21 t2 E7.svg 4 21 t2 E6.svg 4 21 t2 p20.svg 4 21 t2 p24.svg 4 21 t2 mox.svg 4 21 t2 B2.svg 4 21 t2 B3.svg 4 21 t2 B4.svg 4 21 t2 B5.svg 4 21 t2 B6.svg 4 21 t2 B7.svg 4 21 t2 B8.svg 4 21 t2 A5.svg 4 21 t2 A7.svg 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
Birectified 421 (borfy)
4 4 21 t3 E7.svg 4 21 t3 E6.svg 4 21 t3 mox.svg 4 21 t3 B2.svg 4 21 t3 B3.svg 4 21 t3 B4.svg 4 21 t3 B5.svg 4 21 t3 B6.svg 4 21 t3 B7.svg 4 21 t3 A5.svg 4 21 t3 A7.svg 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
Trirectified 421 (torfy)
5 4 21 t4 E7.svg 4 21 t4 E6.svg 4 21 t4 mox.svg 4 21 t4 B2.svg 4 21 t4 B3.svg 4 21 t4 B4.svg 4 21 t4 B5.svg 4 21 t4 B6.svg 4 21 t4 A5.svg 4 21 t4 A7.svg 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
Rectified 142 (buffy)
6 2 41 t1 E8.svg 2 41 t1 E7.svg 2 41 t1 E6.svg 2 41 t1 p20.svg 2 41 t1 p24.svg 2 41 t1 mox.svg 2 41 t1 B2.svg 2 41 t1 B3.svg 2 41 t1 B4.svg 2 41 t1 B5.svg 2 41 t1 B6.svg 2 41 t1 B7.svg 2 41 t1 B8.svg 2 41 t1 A5.svg 2 41 t1 A7.svg 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
Rectified 241 (robay)
7 2 41 t0 E8.svg 2 41 t0 E7.svg 2 41 t0 E6.svg 2 41 t0 p20.svg 2 41 t0 p24.svg 2 41 t0 mox.svg 2 41 t0 B2.svg 2 41 t0 B3.svg 2 41 t0 B4.svg 2 41 t0 B5.svg 2 41 t0 B6.svg 2 41 t0 B7.svg 2 41 t0 B8.svg 2 41 t0 A5.svg 2 41 t0 A7.svg 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
241 (bay)
8 2 41 t01 E7.svg 2 41 t01 E6.svg 2 41 t01 B2.svg 2 41 t01 B3.svg 2 41 t01 B4.svg 2 41 t01 B5.svg 2 41 t01 B6.svg 2 41 t01 B7.svg 2 41 t01 A5.svg 2 41 t01 A7.svg 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 1.png
Truncated 241
9 4 21 t01 E8.svg 4 21 t01 E7.svg 4 21 t01 E6.svg 4 21 t01 p20.svg 4 21 t01 p24.svg 4 21 t01 B2.svg 4 21 t01 B3.svg 4 21 t01 B4.svg 4 21 t01 B5.svg 4 21 t01 B6.svg 4 21 t01 B7.svg 4 21 t01 B8.svg 4 21 t01 A5.svg 4 21 t01 A7.svg CDel nodea 1.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
Truncated 421 (tiffy)
10 Gosset 1 42 polytope petrie.svg 1 42 t0 e7.svg 1 42 polytope E6 Coxeter plane.svg 1 42 t0 p20.svg 1 42 t0 p24.svg 1 42 t0 mox.svg 1 42 t0 B2.svg 1 42 t0 B3.svg 1 42 t0 B4.svg 1 42 t0 B5.svg 1 42 t0 B6.svg 1 42 t0 B7.svg 1 42 t0 B8.svg 1 42 t0 A5.svg 1 42 t0 A7.svg 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
142 (bif)
11 1 42 t01 E6.svg 1 42 t01 B2.svg 1 42 t01 B3.svg 1 42 t01 B4.svg 1 42 t01 B5.svg 1 42 t01 B6.svg 1 42 t01 A5.svg 1 42 t01 A7.svg CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 11.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Truncated 142

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References

Harold Scott MacDonald Coxeter Canadian mathematician

Harold Scott MacDonald "Donald" Coxeter, FRS, FRSC, was a British-born Canadian geometer. Coxeter is regarded as one of the greatest geometers of the 20th century. He was born in London, received his BA (1929) and PhD (1931) from Cambridge, but lived in Canada from age 29. He was always called Donald, from his third name MacDonald. He was most noted for his work on regular polytopes and higher-dimensional geometries. He was a champion of the classical approach to geometry, in a period when the tendency was to approach geometry more and more via algebra.

International Standard Book Number Unique numeric book identifier

The International Standard Book Number (ISBN) is a numeric commercial book identifier which is intended to be unique. Publishers purchase ISBNs from an affiliate of the International ISBN Agency.

Notes

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