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9-simplex

From Wikipedia, the free encyclopedia
Convex regular 9-polytope
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Regular decayotton
(9-simplex)

Orthogonal projection
insidePetrie polygon
TypeRegular9-polytope
Familysimplex
Schläfli symbol{3,3,3,3,3,3,3,3}
Coxeter-Dynkin
diagram
8-faces108-simplex
7-faces457-simplex
6-faces1206-simplex
5-faces2105-simplex
4-faces2525-cell
Cells210tetrahedron
Faces120triangle
Edges45
Vertices10
Vertex figure8-simplex
Petrie polygondecagon
Coxeter groupA9 [3,3,3,3,3,3,3,3]
DualSelf-dual
Propertiesconvex

Ingeometry, a 9-simplex is a self-dualregular9-polytope. It has 10vertices, 45edges, 120 trianglefaces, 210 tetrahedralcells, 2525-cell 4-faces, 2105-simplex 5-faces, 1206-simplex 6-faces, 457-simplex 7-faces, and 108-simplex 8-faces. Itsdihedral angle is cos−1(1/9), or approximately 83.62°.

It can also be called adecayotton, ordeca-9-tope, as a 10-facetted polytope in 9-dimensions. Thenamedecayotton is derived fromdeca for tenfacets inGreek andyotta (a variation of "oct" for eight), having 8-dimensional facets, and-on.
Jonathan Bowers gives it acronymday.[1]

Coordinates

[edit]

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

(1/45, 1/6, 1/28, 1/21, 1/15, 1/10, 1/6, 1/3, ±1){\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)}
(1/45, 1/6, 1/28, 1/21, 1/15, 1/10, 1/6, −21/3, 0){\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)}
(1/45, 1/6, 1/28, 1/21, 1/15, 1/10, −3/2, 0, 0){\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)}
(1/45, 1/6, 1/28, 1/21, 1/15, −22/5, 0, 0, 0){\displaystyle \left({\sqrt {1/45}},\ 1/6,\ {\sqrt {1/28}},\ {\sqrt {1/21}},\ {\sqrt {1/15}},\ -2{\sqrt {2/5}},\ 0,\ 0,\ 0\right)}
(1/45, 1/6, 1/28, 1/21, −5/3, 0, 0, 0, 0){\displaystyle \left({\sqrt {1/45}},\ 1/6,\ {\sqrt {1/28}},\ {\sqrt {1/21}},\ -{\sqrt {5/3}},\ 0,\ 0,\ 0,\ 0\right)}
(1/45, 1/6, 1/28, −12/7, 0, 0, 0, 0, 0){\displaystyle \left({\sqrt {1/45}},\ 1/6,\ {\sqrt {1/28}},\ -{\sqrt {12/7}},\ 0,\ 0,\ 0,\ 0,\ 0\right)}
(1/45, 1/6, −7/4, 0, 0, 0, 0, 0, 0){\displaystyle \left({\sqrt {1/45}},\ 1/6,\ -{\sqrt {7/4}},\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)}
(1/45, −4/3, 0, 0, 0, 0, 0, 0, 0){\displaystyle \left({\sqrt {1/45}},\ -4/3,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)}
(−31/5, 0, 0, 0, 0, 0, 0, 0, 0){\displaystyle \left(-3{\sqrt {1/5}},\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)}

More simply, the vertices of the9-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 oneFacet of the10-orthoplex.

Images

[edit]
orthographic projections
AkCoxeter planeA9A8A7A6
Graph
Dihedral symmetry[10][9][8][7]
Ak Coxeter planeA5A4A3A2
Graph
Dihedral symmetry[6][5][4][3]

References

[edit]
  1. ^Klitzing,(x3o3o3o3o3o3o3o3o – day)

External links

[edit]
Fundamental convexregular anduniform polytopes in dimensions 2–10
FamilyAnBnI2(p) /DnE6 /E7 /E8 /F4 /G2Hn
Regular polygonTriangleSquarep-gonHexagonPentagon
Uniform polyhedronTetrahedronOctahedron •CubeDemicubeDodecahedron •Icosahedron
Uniform polychoronPentachoron16-cell •TesseractDemitesseract24-cell120-cell •600-cell
Uniform 5-polytope5-simplex5-orthoplex •5-cube5-demicube
Uniform 6-polytope6-simplex6-orthoplex •6-cube6-demicube122 •221
Uniform 7-polytope7-simplex7-orthoplex •7-cube7-demicube132 •231 •321
Uniform 8-polytope8-simplex8-orthoplex •8-cube8-demicube142 •241 •421
Uniform 9-polytope9-simplex9-orthoplex •9-cube9-demicube
Uniform 10-polytope10-simplex10-orthoplex •10-cube10-demicube
Uniformn-polytopen-simplexn-orthoplex •n-cuben-demicube1k2 •2k1 •k21n-pentagonal polytope
Topics:Polytope families •Regular polytope •List of regular polytopes and compounds •Polytope operations
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