8-simplex

Regular enneazetton (8-simplex)
Orthogonal projection inside Petrie polygon
Type	Regular 8-polytope
Family	simplex
Schläfli symbol	{3,3,3,3,3,3,3}
Coxeter-Dynkin diagram
7-faces	9 7-simplex
6-faces	36 6-simplex
5-faces	84 5-simplex
4-faces	126 5-cell
Cells	126 tetrahedron
Faces	84 triangle
Edges	36
Vertices	9
Vertex figure	7-simplex
Petrie polygon	enneagon
Coxeter group	A₈ [3,3,3,3,3,3,3]
Dual	Self-dual
Properties	convex

In geometry, an 8-simplex is a self-dual regular 8-polytope. It has 9 vertices, 36 edges, 84 triangle faces, 126 tetrahedral cells, 126 5-cell 4-faces, 84 5-simplex 5-faces, 36 6-simplex 6-faces, and 9 7-simplex 7-faces. Its dihedral angle is cos⁻¹(1/8), or approximately 82.82°.

It can also be called an enneazetton, or ennea-8-tope, as a 9-facetted polytope in 8-dimensions.. The name enneazetton is derived from ennea for nine facets in Greek and -zetta for having seven-dimensional facets, and -on.

Coordinates

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

\left(1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ \sqrt{1/6},\ \sqrt{1/3},\ \pm1\right)

\left(1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ \sqrt{1/6},\ -2\sqrt{1/3},\ 0\right)

\left(1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ -\sqrt{3/2},\ 0,\ 0\right)

\left(1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ -2\sqrt{2/5},\ 0,\ 0,\ 0\right)

\left(1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ -\sqrt{5/3},\ 0,\ 0,\ 0,\ 0\right)

\left(1/6,\ \sqrt{1/28},\ -\sqrt{12/7},\ 0,\ 0,\ 0,\ 0,\ 0\right)

\left(1/6,\ -\sqrt{7/4},\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)

\left(-4/3,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)

More simply, the vertices of the 8-simplex can be positioned in 9-space as permutations of (0,0,0,0,0,0,0,0,1). This construction is based on facets of the 9-orthoplex.

Images

orthographic projections
A_k Coxeter plane	A₈	A₇	A₆	A₅
Graph
Dihedral symmetry	[9]	[8]	[7]	[6]
A_k Coxeter plane	A₄	A₃	A₂
Graph
Dihedral symmetry	[5]	[4]	[3]

Related polytopes and honeycombs

This polytope is a facet in the uniform tessellations: 2₅₁, and 5₂₁ with respective Coxeter-Dynkin diagrams:

This polytope is one of 135 uniform 8-polytopes with A₈ symmetry.

t₀	t₁	t₂	t₃	t₀₁	t₀₂	t₁₂	t₀₃	t₁₃	t₂₃	t₀₄	t₁₄	t₂₄	t₃₄	t₀₅
t₁₅	t₂₅	t₀₆	t₁₆	t₀₇	t₀₁₂	t₀₁₃	t₀₂₃	t₁₂₃	t₀₁₄	t₀₂₄	t₁₂₄	t₀₃₄	t₁₃₄	t₂₃₄
t₀₁₅	t₀₂₅	t₁₂₅	t₀₃₅	t₁₃₅	t₂₃₅	t₀₄₅	t₁₄₅	t₀₁₆	t₀₂₆	t₁₂₆	t₀₃₆	t₁₃₆	t₀₄₆	t₀₅₆
t₀₁₇	t₀₂₇	t₀₃₇	t₀₁₂₃	t₀₁₂₄	t₀₁₃₄	t₀₂₃₄	t₁₂₃₄	t₀₁₂₅	t₀₁₃₅	t₀₂₃₅	t₁₂₃₅	t₀₁₄₅	t₀₂₄₅	t₁₂₄₅
t₀₃₄₅	t₁₃₄₅	t₂₃₄₅	t₀₁₂₆	t₀₁₃₆	t₀₂₃₆	t₁₂₃₆	t₀₁₄₆	t₀₂₄₆	t₁₂₄₆	t₀₃₄₆	t₁₃₄₆	t₀₁₅₆	t₀₂₅₆	t₁₂₅₆
t₀₃₅₆	t₀₄₅₆	t₀₁₂₇	t₀₁₃₇	t₀₂₃₇	t₀₁₄₇	t₀₂₄₇	t₀₃₄₇	t₀₁₅₇	t₀₂₅₇	t₀₁₆₇	t₀₁₂₃₄	t₀₁₂₃₅	t₀₁₂₄₅	t₀₁₃₄₅
t₀₂₃₄₅	t₁₂₃₄₅	t₀₁₂₃₆	t₀₁₂₄₆	t₀₁₃₄₆	t₀₂₃₄₆	t₁₂₃₄₆	t₀₁₂₅₆	t₀₁₃₅₆	t₀₂₃₅₆	t₁₂₃₅₆	t₀₁₄₅₆	t₀₂₄₅₆	t₀₃₄₅₆	t₀₁₂₃₇
t₀₁₂₄₇	t₀₁₃₄₇	t₀₂₃₄₇	t₀₁₂₅₇	t₀₁₃₅₇	t₀₂₃₅₇	t₀₁₄₅₇	t₀₁₂₆₇	t₀₁₃₆₇	t₀₁₂₃₄₅	t₀₁₂₃₄₆	t₀₁₂₃₅₆	t₀₁₂₄₅₆	t₀₁₃₄₅₆	t₀₂₃₄₅₆
t₁₂₃₄₅₆	t₀₁₂₃₄₇	t₀₁₂₃₅₇	t₀₁₂₄₅₇	t₀₁₃₄₅₇	t₀₂₃₄₅₇	t₀₁₂₃₆₇	t₀₁₂₄₆₇	t₀₁₃₄₆₇	t₀₁₂₅₆₇	t_0123456	t_0123457	t_0123467	t_0123567	t_01234567

References

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)
Richard Klitzing, 8D uniform polytopes (polyzetta), x3o3o3o3o3o3o3o - ene

External links

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