9-simplex

Regular decayotton (9-simplex)
Orthogonal projection inside Petrie polygon
Type	Regular 9-polytope
Family	simplex
Schläfli symbol	{3,3,3,3,3,3,3,3}
Coxeter-Dynkin diagram
8-faces	10 8-simplex
7-faces	45 7-simplex
6-faces	120 6-simplex
5-faces	210 5-simplex
4-faces	252 5-cell
Cells	210 tetrahedron
Faces	120 triangle
Edges	45
Vertices	10
Vertex figure	8-simplex
Petrie polygon	decagon
Coxeter group	A9 [3,3,3,3,3,3,3,3]
Dual	Self-dual
Properties	convex

In geometry, a 9-simplex is a self-dual regular 9-polytope. It has 10 vertices, 45 edges, 120 triangle faces, 210 tetrahedral cells, 252 5-cell 4-faces, 210 5-simplex 5-faces, 120 6-simplex 6-faces, 45 7-simplex 7-faces, and 10 8-simplex 8-faces. Its dihedral angle is cos⁻¹(1/9), or approximately 83.62°.

It can also be called a decayotton, or deca-9-tope, as a 10-facetted polytope in 9-dimensions.. The name decayotton is derived from deca for ten facets in Greek and yotta (a variation of "oct" for eight), having 8-dimensional facets, and -on.

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

${\displaystyle \left({\sqrt {1/45}},\ 1/6,\ {\sqrt {1/28}},\ {\sqrt {1/21}},\ {\sqrt {1/15}},\ {\sqrt {1/10}},\ {\sqrt {1/6}},\ {\sqrt {1/3}},\ \pm 1\right)}$

${\displaystyle \left({\sqrt {1/45}},\ 1/6,\ {\sqrt {1/28}},\ {\sqrt {1/21}},\ {\sqrt {1/15}},\ {\sqrt {1/10}},\ {\sqrt {1/6}},\ -2{\sqrt {1/3}},\ 0\right)}$

${\displaystyle \left({\sqrt {1/45}},\ 1/6,\ {\sqrt {1/28}},\ {\sqrt {1/21}},\ {\sqrt {1/15}},\ {\sqrt {1/10}},\ -{\sqrt {3/2}},\ 0,\ 0\right)}$

${\displaystyle \left({\sqrt {1/45}},\ 1/6,\ {\sqrt {1/28}},\ {\sqrt {1/21}},\ {\sqrt {1/15}},\ -2{\sqrt {2/5}},\ 0,\ 0,\ 0\right)}$

${\displaystyle \left({\sqrt {1/45}},\ 1/6,\ {\sqrt {1/28}},\ {\sqrt {1/21}},\ -{\sqrt {5/3}},\ 0,\ 0,\ 0,\ 0\right)}$

${\displaystyle \left({\sqrt {1/45}},\ 1/6,\ {\sqrt {1/28}},\ -{\sqrt {12/7}},\ 0,\ 0,\ 0,\ 0,\ 0\right)}$

${\displaystyle \left({\sqrt {1/45}},\ 1/6,\ -{\sqrt {7/4}},\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)}$

${\displaystyle \left({\sqrt {1/45}},\ -4/3,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)}$

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

More simply, the vertices of the 9-simplex can be positioned in 10-space as permutations of (0,0,0,0,0,0,0,0,0,1). These are the vertices of one Facet of the 10-orthoplex.

orthographic projections

Ak Coxeter plane	A9	A8	A7	A6
Graph
Dihedral symmetry	[10]	[9]	[8]	[7]
Ak Coxeter plane	A5	A4	A3	A2
Graph
Dihedral symmetry	[6]	[5]	[4]	[3]

• Coxeter, H.S.M.:
  • — (1973). "Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)". Regular Polytopes (3rd ed.). Dover. p. 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. S2CID 186237114.
    • (Paper 23) — (1985). "Regular and Semi-Regular Polytopes II". Math. Zeit. 188 (4): 559–591. doi:10.1007/BF01161657. S2CID 120429557.
    • (Paper 24) — (1988). "Regular and Semi-Regular Polytopes III". Math. Zeit. 200: 3–45. doi:10.1007/BF01161745. S2CID 186237142.
• Conway, John H.; Burgiel, Heidi; Goodman-Strauss, Chaim (2008). "26. Hemicubes: 1_n1". The Symmetries of Things. p. 409. ISBN 978-1-56881-220-5.
• Johnson, Norman (1991), Uniform Polytopes (Manuscript)
  • Johnson, N.W. (1966). The Theory of Uniform Polytopes and Honeycombs (PhD). University of Toronto. OCLC 258527038.
• Klitzing, Richard. "9D uniform polytopes (polyyotta) x3o3o3o3o3o3o3o3o — day".

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