5-simplex

5-simplex Hexateron (hix)
Type	uniform 5-polytope
Schläfli symbol	{3⁴}
Coxeter diagram
4-faces	6	6 {3,3,3}
Cells	15	15 {3,3}
Faces	20	20 {3}
Edges	15
Vertices	6
Vertex figure	5-cell
Coxeter group	A₅, [3⁴], order 720
Dual	self-dual
Base point	(0,0,0,0,0,1)
Circumradius	0.645497
Properties	convex, isogonal regular, self-dual

In five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope. It has six vertices, 15 edges, 20 triangle faces, 15 tetrahedral cells, and 6 5-cell facets. It has a dihedral angle of cos⁻¹(1/5), or approximately 78.46°.

Alternate names

It can also be called a hexateron, or hexa-5-tope, as a 6-facetted polytope in 5-dimensions. The name hexateron is derived from hexa- for having six facets and teron (with ter- being a corruption of tetra-) for having four-dimensional facets.

By Jonathan Bowers, a hexateron is given the acronym hix.^[1]

Regular hexateron cartesian coordinates

The hexateron can be constructed from a 5-cell by adding a 6th vertex such that it is equidistant from all the other vertices of the 5-cell.

The Cartesian coordinates for the vertices of an origin-centered regular hexateron having edge length 2 are:

\left({\sqrt {1/15}},\ {\sqrt {1/10}},\ {\sqrt {1/6}},\ {\sqrt {1/3}},\ \pm 1\right)

\left({\sqrt {1/15}},\ {\sqrt {1/10}},\ {\sqrt {1/6}},\ -2{\sqrt {1/3}},\ 0\right)

\left({\sqrt {1/15}},\ {\sqrt {1/10}},\ -{\sqrt {3/2}},\ 0,\ 0\right)

\left({\sqrt {1/15}},\ -2{\sqrt {2/5}},\ 0,\ 0,\ 0\right)

\left(-{\sqrt {5/3}},\ 0,\ 0,\ 0,\ 0\right)

The vertices of the 5-simplex can be more simply positioned on a hyperplane in 6-space as permutations of (0,0,0,0,0,1) or (0,1,1,1,1,1). These construction can be seen as facets of the 6-orthoplex or rectified 6-cube respectively.

Projected images

orthographic projections
A_k Coxeter plane	A₅	A₄
Graph
Dihedral symmetry	[6]	[5]
A_k Coxeter plane	A₃	A₂
Graph
Dihedral symmetry	[4]	[3]

Stereographic projection 4D to 3D of Schlegel diagram 5D to 4D of hexateron.

Related uniform 5-polytopes

It is first in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 1_3k series. A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral dihedron.

1_3k dimensional figures
Space	Finite				Euclidean	Hyperbolic
n	4	5	6	7	8	9
Coxeter group	A₃A₁	A₅	D₆	E₇	${\tilde {E}}_{7}$ =E₇⁺	${\bar {T}}_{8}$ =E₇⁺⁺
Coxeter diagram
Symmetry	[3^−1,3,1]	[3^0,3,1]	[3^1,3,1]	[3^2,3,1]	[[3<sup>3,3,1</sup>]]	[3^4,3,1]
Order	48	720	23,040	2,903,040	∞
Graph					-	-
Name	1_3,-1	1₃₀	1₃₁	1₃₂	1₃₃	1₃₄

It is first in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 3_k1 series. A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral hosohedron.

3_k1 dimensional figures
Space	Finite				Euclidean	Hyperbolic
n	4	5	6	7	8	9
Coxeter group	A₃A₁	A₅	D₆	E₇	${\tilde {E}}_{7}$ =E₇⁺	${\bar {T}}_{8}$ =E₇⁺⁺
Coxeter diagram
Symmetry	[3^−1,3,1]	[3^0,3,1]	[[3<sup>1,3,1</sup>]]	[3^2,3,1]	[3^3,3,1]	[3^4,3,1]
Order	48	720	46,080	2,903,040	∞
Graph					-	-
Name	3_1,-1	3₁₀	3₁₁	3₂₁	3₃₁	3₄₁

The 5-simplex, as 2₂₀ polytope is first in dimensional series 2_2k.

2_2k figures of n dimensions
Space	Finite		Euclidean	Hyperbolic
n	5	6	7	8
Coxeter group	A₅	E₆	${\tilde {E}}_{6}$ =E₆⁺	E₆⁺⁺
Coxeter diagram
Graph			∞	∞
Name	2₂₀	2₂₁	2₂₂	2₂₃

The regular 5-simplex is one of 19 uniform polytera based on the [3,3,3,3] Coxeter group, all shown here in A₅ Coxeter plane orthographic projections. (Vertices are colored by projection overlap order, red, orange, yellow, green, cyan, blue, purple having progressively more vertices)

A5 polytopes
t₀	t₁	t₂	t_0,1	t_0,2	t_1,2	t_0,3
t_1,3	t_0,4	t_0,1,2	t_0,1,3	t_0,2,3	t_1,2,3	t_0,1,4
t_0,2,4	t_0,1,2,3	t_0,1,2,4	t_0,1,3,4	t_0,1,2,3,4

Other forms

The 5-simplex can also be considered a 5-cell pyramid, constructed as a 5-cell base in a 4-space hyperplane, and an apex point above the hyperplane. The five sides of the pyramid are made of 5-cell cells.

Notes

↑ Klitzing, (x3o3o3o3o - hix)

References

T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
H.S.M. Coxeter:
- Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, p.296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
- H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p.296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
- 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
  - (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]
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1_n1)
Norman Johnson Uniform Polytopes, Manuscript (1991)
- N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
Klitzing, Richard. "5D uniform polytopes (polytera) x3o3o3o3o - hix".

External links

Olshevsky, George. "Simplex". Glossary for Hyperspace. Archived from the original on 4 February 2007.
Polytopes of Various Dimensions, Jonathan Bowers
Multi-dimensional Glossary

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) / D_n	E₆ / E₇ / E₈ / E₉ / E₁₀ / 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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