E6 polytope

Last updated March 26, 2019

Orthographic projections in the E₆ Coxeter plane
2₂₁	1₂₂

In 6-dimensional geometry, there are 39 uniform polytopes with E₆ symmetry. The two simplest forms are the 2₂₁ and 1₂₂ polytopes, composed of 27 and 72 vertices respectively.

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.

In six-dimensional geometry, a uniform polypeton is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform 5-polytopes.

In 6-dimensional geometry, the 2₂₁ polytope is a uniform 6-polytope, constructed within the symmetry of the E₆ group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 6-ic semi-regular figure. It is also called the Schläfli polytope.

They can be visualized as symmetric orthographic projections in Coxeter planes of the E₆ Coxeter group, and other subgroups.

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 39 polytopes can be made in the E₆, D₅, D₄, D₂, A₅, A₄, A₃ Coxeter planes. A_k has k+1 symmetry, D_k has 2(k-1) symmetry, and E₆ has 12 symmetry.

Six symmetry planes graphs are shown for 9 of the 39 polytopes in the E₆ symmetry. The vertices and edges drawn with vertices colored by the number of overlapping vertices in each projective position.

#	Coxeter plane graphs						Coxeter diagram Names
#	Aut(E₆) [18/2]	E₆ [12]	D₅ [8]	D₄ / A₂ [6]	A₅ [6]	D₃ / A₃ [4]	Coxeter diagram Names
1							2₂₁ Icosihepta-heptacontidipeton (jak)
2							Rectified 2₂₁ Rectified icosihepta-heptacontidipeton (rojak)
3							Trirectified 2₂₁ Trirectified icosihepta-heptacontidipeton (harjak)
4							Truncated 2₂₁ Truncated icosihepta-heptacontidipeton (tojak)
5							Cantellated 2₂₁ Cantellated icosihepta-heptacontidipeton

#	Coxeter plane graphs							Coxeter diagram Names
#	Aut(E₆) [18]	E₆ [12]	D₅ [8]	D₄ / A₂ [6]	A₅ [6]	D₆ / A₄ [10]	D₃ / A₃ [4]	Coxeter diagram Names
6								1₂₂ Pentacontatetrapeton (mo)
7								Rectified 1₂₂ / Birectified 2₂₁ Rectified pentacontatetrapeton (ram)
8								Birectified 1₂₂ Birectified pentacontatetrapeton (barm)
9								Truncated 1₂₂ Truncated pentacontatetrapeton (tim)

Related Research Articles

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.

In 7-dimensional geometry, there are 127 uniform polytopes with E₇ symmetry. The three simplest forms are the 3₂₁, 2₃₁, and 1₃₂ polytopes, composed of 56, 126, and 576 vertices respectively.

In six-dimensional geometry, a truncated 6-simplex is a convex uniform 6-polytope, being a truncation of the regular 6-simplex.

In eight-dimensional geometry, a pentellated 8-simplex is a convex uniform 8-polytope with 5th order truncations of the regular 8-simplex.

In 8-dimensional geometry, there are 135 uniform polytopes with A₈ symmetry. There is one self-dual regular form, the 8-simplex with 9 vertices.

In 7-dimensional geometry, there are 128 uniform polytopes with B₇ symmetry. There are two regular forms, the 7-orthoplex, and 8-cube with 14 and 128 vertices respectively. The 7-demicube is added with half of the symmetry.

In 7-dimensional geometry, there are 71 uniform polytopes with A₇ symmetry. There is one self-dual regular form, the 7-simplex with 8 vertices.

In 6-dimensional geometry, there are 35 uniform polytopes with A₆ symmetry. There is one self-dual regular form, the 6-simplex with 7 vertices.

In 6-dimensional geometry, there are 64 uniform polytopes with B₆ symmetry. There are two regular forms, the 6-orthoplex, and 6-cube with 12 and 64 vertices respectively. The 6-demicube is added with half the symmetry.

In 6-dimensional geometry, there are 47 uniform polytopes with D₆ symmetry, 16 are unique, and 31 are shared with the B₆ symmetry. There are two regular forms, the 6-orthoplex, and 6-demicube with 12 and 32 vertices respectively.

In 7-dimensional geometry, there are 95 uniform polytopes with D₇ symmetry; 32 are unique, and 63 are shared with the B₇ symmetry. There are two regular forms, the 7-orthoplex, and 7-demicube with 14 and 64 vertices respectively.

In 5-dimensional geometry, there are 19 uniform polytopes with A₅ symmetry. There is one self-dual regular form, the 5-simplex with 6 vertices.

In 5-dimensional geometry, there are 31 uniform polytopes with B₅ symmetry. There are two regular forms, the 5-orthoplex, and 5-cube with 10 and 32 vertices respectively. The 5-demicube is added as an alternation of the 5-cube.

In 5-dimensional geometry, there are 23 uniform polytopes with D₅ symmetry, 8 are unique, and 15 are shared with the B₅ symmetry. There are two special forms, the 5-orthoplex, and 5-demicube with 10 and 16 vertices respectively.

In seven-dimensional geometry, a cantic 7-cube or truncated 7-demicube as a uniform 7-polytope, being a truncation of the 7-demicube.

In 4-dimensional geometry, there are 9 uniform polytopes with A₄ symmetry. There is one self-dual regular form, the 5-cell with 5 vertices.

In 4-dimensional geometry, there are 15 uniform 4-polytopes with B₄ symmetry. There are two regular forms, the tesseract, and 16-cell with 16 and 8 vertices respectively.

In 4-dimensional geometry, there are 15 uniform polytopes with H₄ symmetry. Two of these, the 120-cell and 600-cell, are regular.

In 4-dimensional geometry, there are 7 uniform 4-polytopes with reflections of D₄ symmetry, all are shared with higher symmetry constructions in the B₄ or F₄ symmetry families. there is also one half symmetry alternation, the snub 24-cell.

References

H.S.M. Coxeter:
- 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 Wiley::Kaleidoscopes: Selected Writings of H.S.M. Coxeter
- (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
- (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
Klitzing, Richard. "6D uniform polytopes (polypeta)".

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.

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) / D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform 4-polytope	5-cell	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	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

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