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 eight-dimensions. The name enneazetton is derived from ennea for nine facets in Greek and -zetta for having seven-dimensional facets, and -on.

As a configuration[]

This configuration matrix represents the 8-simplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces, 6-faces and 7-faces. The diagonal numbers say how many of each element occur in the whole 8-simplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element. This self-dual simplex's matrix is identical to its 180 degree rotation.^[1][2]

${\displaystyle {\begin{bmatrix}{\begin{matrix}9&8&28&56&70&56&28&8\\2&36&7&21&35&35&21&7\\3&3&84&6&15&20&15&6\\4&6&4&126&5&10&10&5\\5&10&10&5&126&4&6&4\\6&15&20&15&6&84&3&3\\7&21&35&35&21&7&36&2\\8&28&56&70&56&28&8&9\end{matrix}}\end{bmatrix}}}$

Coordinates[]

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

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

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

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

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

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

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

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

${\displaystyle \left(-4/3,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)}$

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.

Another origin-centered construction uses (1,1,1,1,1,1,1,1)/3 and permutations of (1,1,1,1,1,1,1,-11)/12 for edge length √2.

Images[]

orthographic projections

Ak Coxeter plane	A8	A7	A6	A5
Graph
Dihedral symmetry	[9]	[8]	[7]	[6]
Ak 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.

A8 polytopes
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
		t125		t135			t145
			t0123	t0124	t0134	t0234	t1234				t1235			t1245
	t1345	t2345
											t01234
	t12345
														t01234567

References[]

• Coxeter, H.S.M.:
  • — (1973). "Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)". Regular Polytopes (3rd ed.). Dover. pp. 296. ISBN 0-486-61480-8.
  • Sherk, F. Arthur; McMullen, Peter; Thompson, Anthony C.; Weiss, Asia Ivic, eds. (1995). Kaleidoscopes: Selected Writings of H.S.M. Coxeter. Wiley. ISBN 978-0-471-01003-6.
    • (Paper 22) — (1940). "Regular and Semi Regular Polytopes I". Math. Zeit. 46: 380–407. doi:10.1007/BF01181449.
    • (Paper 23) — (1985). "Regular and Semi-Regular Polytopes II". Math. Zeit. 188: 559–591. doi:10.1007/BF01161657.
    • (Paper 24) — (1988). "Regular and Semi-Regular Polytopes III". Math. Zeit. 200: 3–45. doi:10.1007/BF01161745.
• Conway, John H.; Burgiel, Heidi; Goodman-Strass, Chaim (2008). "26. Hemicubes: 1_n1". The Symmetries of Things. p. 409. ISBN 978-1-56881-220-5.
• Johnson, Norman (1991). "Uniform Polytopes" (Manuscript). Cite journal requires |journal= (help)
  • Johnson, N.W. (1966). The Theory of Uniform Polytopes and Honeycombs (PhD). University of Toronto. OCLC 258527038.
• Klitzing, Richard. "8D uniform polytopes (polyzetta) x3o3o3o3o3o3o3o — ene".

External links[]

• Glossary for hyperspace, George Olshevsky.
• Polytopes of Various Dimensions
• Multi-dimensional Glossary