Regular decayotton
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 CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
8-faces 10 8-simplex8-simplex t0.svg
7-faces 45 7-simplex7-simplex t0.svg
6-faces 120 6-simplex6-simplex t0.svg
5-faces 210 5-simplex5-simplex t0.svg
4-faces 252 5-cell4-simplex t0.svg
Cells 210 tetrahedron3-simplex t0.svg
Faces 120 triangle2-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.


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


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). This construction is based on facets of the 10-orthoplex.


orthographic projections
Ak Coxeter plane A9 A8 A7 A6
Dihedral symmetry [10] [9] [8] [7]
Ak Coxeter plane A5 A4 A3 A2
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.
  • Conway, John H.; Burgiel, Heidi; Goodman-Strass, Chaim (2008). "26. Hemicubes: 1n1". The Symmetries of Things. p. 409. ISBN 978-1-56881-220-5.
  • Johnson, Norman (1991). "Uniform Polytopes" (Manuscript). {{cite journal}}: Cite journal requires |journal= (help)
  • Klitzing, Richard. "9D uniform polytopes (polyyotta) x3o3o3o3o3o3o3o3o — day".

External linksEdit

Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds