9-simplex

Regular decayotton (9-simplex)
File:9-simplex t0.svg Orthogonal projection inside Petrie polygon
Type	Regular 9-polytope
Family	simplex
Schläfli symbol	{3,3,3,3,3,3,3,3}
Coxeter-Dynkin diagram	File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
8-faces	10 8-simplex File:8-simplex t0.svg
7-faces	45 7-simplex File:7-simplex t0.svg
6-faces	120 6-simplex File:6-simplex t0.svg
5-faces	210 5-simplex File:5-simplex t0.svg
4-faces	252 5-cell File:4-simplex t0.svg
Cells	210 tetrahedron File:3-simplex t0.svg
Faces	120 triangle File:2-simplex t0.svg
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.
Jonathan Bowers gives it acronym day.

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

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

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

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

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

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

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

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

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

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

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.

Images

orthographic projections

Ak Coxeter plane	A9	A8	A7	A6
Graph	File:9-simplex t0.svg	File:9-simplex t0 A8.svg	File:9-simplex t0 A7.svg	File:9-simplex t0 A6.svg
Dihedral symmetry	[10]	[9]	[8]	[7]
Ak Coxeter plane	A5	A4	A3	A2
Graph	File:9-simplex t0 A5.svg	File:9-simplex t0 A4.svg	File:9-simplex t0 A3.svg	File:9-simplex t0 A2.svg
Dihedral symmetry	[6]	[5]	[4]	[3]

References

• Coxeter, H.S.M.:
  • • (Paper 22)
    • (Paper 23)
    • (Paper 24)

External links

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