# 9-simplex

9-simplex
Rank9
TypeRegular
Notation
Bowers style acronymDay
Coxeter diagramx3o3o3o3o3o3o3o3o ()
Schläfli symbol{3,3,3,3,3,3,3,3}
Tapertopic notation18
Elements
Yotta10 enneazetta
Zetta45 octaexa
Exa120 heptapeta
Peta210 hexatera
Tera252 pentachora
Cells210 tetrahedra
Faces120 triangles
Edges45
Vertices10
Vertex figure8-simplex, edge length 1
Measures (edge length 1)
Circumradius${\displaystyle {\frac {3{\sqrt {5}}}{10}}\approx 0.67082}$
Inradius${\displaystyle {\frac {\sqrt {5}}{30}}\approx 0.074536}$
Hypervolume${\displaystyle {\frac {\sqrt {5}}{5806080}}\approx 3.8513\times 10^{-7}}$
Diyottal angle${\displaystyle \arccos \left({\frac {1}{9}}\right)\approx 83.62063^{\circ }}$
Height${\displaystyle {\frac {\sqrt {5}}{3}}\approx 0.74536}$
Central density1
Number of external pieces10
Level of complexity1
Related polytopes
ArmyDay
RegimentDay
Dual9-simplex
ConjugateNone
Abstract & topological properties
Flag count3628800
Euler characteristic2
OrientableYes
Properties
SymmetryA9, order 3628800
Flag orbits1
ConvexYes
NatureTame

The 9-simplex, also called the decayotton, or day, is the simplest possible non-degenerate 9-polytope. The full symmetry version has 10 regular 8-simplices as facets, joining 3 to a 6-simplex peak and 9 to a vertex, and is one of the 3 regular 9-polytopes. It is the 9-dimensional simplex.

## Vertex coordinates

The vertices of a regular 9-simplex of edge length 1, centered at the origin, are given by:

• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\frac {\sqrt {3}}{6}},\,-{\frac {\sqrt {6}}{12}},\,-{\frac {\sqrt {10}}{20}},\,-{\frac {\sqrt {15}}{30}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}}\right)}$,
• ${\displaystyle \left(0,\,{\frac {\sqrt {3}}{3}},\,-{\frac {\sqrt {6}}{12}},\,-{\frac {\sqrt {10}}{20}},\,-{\frac {\sqrt {15}}{30}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}}\right)}$,
• ${\displaystyle \left(0,\,0,\,{\frac {\sqrt {6}}{4}},\,-{\frac {\sqrt {10}}{20}},\,-{\frac {\sqrt {15}}{30}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}}\right)}$,
• ${\displaystyle \left(0,\,0,\,0,\,{\frac {\sqrt {10}}{5}},\,-{\frac {\sqrt {15}}{30}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}}\right)}$,
• ${\displaystyle \left(0,\,0,\,0,\,0,\,{\frac {\sqrt {15}}{6}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}}\right)}$,
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,{\frac {\sqrt {21}}{7}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}}\right)}$,
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,{\frac {\sqrt {7}}{4}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}}\right)}$,
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,{\frac {2}{3}},\,-{\frac {\sqrt {5}}{30}}\right)}$,
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,{\frac {3{\sqrt {5}}}{10}}\right)}$.

Much simpler coordinates can be given in ten dimensions, as all permutations of:

• ${\displaystyle \left({\frac {\sqrt {2}}{2}},\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0\right)}$.