10-simplex

Regular hendecaxennon (10-simplex)
File:10-simplex t0.svg Orthogonal projection inside Petrie polygon
Type	Regular 10-polytope
Family	simplex
Schläfli symbol	{3,3,3,3,3,3,3,3,3}
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.pngFile:CDel 3.pngFile:CDel node.png
9-faces	11 9-simplex File:9-simplex t0.svg
8-faces	55 8-simplex File:8-simplex t0.svg
7-faces	165 7-simplex File:7-simplex t0.svg
6-faces	330 6-simplex File:6-simplex t0.svg
5-faces	462 5-simplex File:5-simplex t0.svg
4-faces	462 5-cell File:4-simplex t0.svg
Cells	330 tetrahedron File:3-simplex t0.svg
Faces	165 triangle File:2-simplex t0.svg
Edges	55
Vertices	11
Vertex figure	9-simplex
Petrie polygon	hendecagon
Coxeter group	A10 [3,3,3,3,3,3,3,3,3]
Dual	Self-dual
Properties	convex

In geometry, a 10-simplex is a self-dual regular 10-polytope. It has 11 vertices, 55 edges, 165 triangle faces, 330 tetrahedral cells, 462 5-cell 4-faces, 462 5-simplex 5-faces, 330 6-simplex 6-faces, 165 7-simplex 7-faces, 55 8-simplex 8-faces, and 11 9-simplex 9-faces. Its dihedral angle is cos⁻¹(1/10), or approximately 84.26°.

It can also be called a hendecaxennon, or hendeca-10-tope, as an 11-facetted polytope in 10-dimensions. Acronym: ux

The name hendecaxennon is derived from hendeca for 11 facets in Greek and -xenn (variation of ennea for nine), having 9-dimensional facets, and -on.

The Cartesian coordinates of the vertices of an origin-centered regular 10-simplex having edge length 2 are:

$(\sqrt{1/55},\sqrt{1/45},1/6,\sqrt{1/28},\sqrt{1/21},\sqrt{1/15},\sqrt{1/10},\sqrt{1/6},\sqrt{1/3},\pm 1)$

$(\sqrt{1/55},\sqrt{1/45},1/6,\sqrt{1/28},\sqrt{1/21},\sqrt{1/15},\sqrt{1/10},\sqrt{1/6},-2\sqrt{1/3},0)$

$(\sqrt{1/55},\sqrt{1/45},1/6,\sqrt{1/28},\sqrt{1/21},\sqrt{1/15},\sqrt{1/10},-\sqrt{3/2},0,0)$

$(\sqrt{1/55},\sqrt{1/45},1/6,\sqrt{1/28},\sqrt{1/21},\sqrt{1/15},-2\sqrt{2/5},0,0,0)$

$(\sqrt{1/55},\sqrt{1/45},1/6,\sqrt{1/28},\sqrt{1/21},-\sqrt{5/3},0,0,0,0)$

$(\sqrt{1/55},\sqrt{1/45},1/6,\sqrt{1/28},-\sqrt{12/7},0,0,0,0,0)$

$(\sqrt{1/55},\sqrt{1/45},1/6,-\sqrt{7/4},0,0,0,0,0,0)$

$(\sqrt{1/55},\sqrt{1/45},-4/3,0,0,0,0,0,0,0)$

$(\sqrt{1/55},-3\sqrt{1/5},0,0,0,0,0,0,0,0)$

$(-\sqrt{20/11},0,0,0,0,0,0,0,0,0)$

More simply, the vertices of the 10-simplex can be positioned in 11-space as permutations of (0,0,0,0,0,0,0,0,0,0,1). This construction is based on facets of the 11-orthoplex.

Images

orthographic projections

Ak Coxeter plane	A10	A9	A8
Graph	File:10-simplex t0.svg	File:10-simplex t0 A9.svg	File:10-simplex t0 A8.svg
Dihedral symmetry	[11]	[10]	[9]
Ak Coxeter plane	A7	A6	A5
Graph	File:10-simplex t0 A7.svg	File:10-simplex t0 A6.svg	File:10-simplex t0 A5.svg
Dihedral symmetry	[8]	[7]	[6]
Ak Coxeter plane	A4	A3	A2
Graph	File:10-simplex t0 A4.svg	File:10-simplex t0 A3.svg	File:10-simplex t0 A2.svg
Dihedral symmetry	[5]	[4]	[3]

Related polytopes

The 2-skeleton of the 10-simplex is topologically related to the 11-cell abstract regular polychoron which has the same 11 vertices, 55 edges, but only 1/3 the faces (55).

References

• Coxeter, H.S.M.:
  • • (Paper 22)
    • (Paper 23)
    • (Paper 24)
• x3o3o3o3o3o3o3o3o3o – ux

External links

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