# Decayotton

(Redirected from 9-simplex)
Decayotton
Rank9
TypeRegular
SpaceSpherical
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 figureEnneazetton, edge length 1
Measures (edge length 1)
Circumradius${\displaystyle \frac{3\sqrt5}{10} ≈ 0.67082}$
Inradius${\displaystyle \frac{\sqrt5}{30} ≈ 0.074536}$
Hypervolume${\displaystyle \frac{\sqrt5}{5806080} ≈ 3.8513×10^{-7}}$
Diyottal angle${\displaystyle \arccos\left(\frac19\right) ≈ 83.62063°}$
Height${\displaystyle \frac{\sqrt5}{3} ≈ 0.74536}$
Central density1
Number of pieces10
Level of complexity1
Related polytopes
ArmyDay
RegimentDay
DualDecayotton
ConjugateNone
Abstract properties
Euler characteristic2
Topological properties
OrientableYes
Properties
SymmetryA9, order 3628800
ConvexYes
NatureTame
Discovered by{{{discoverer}}}

The decayotton, or day, also commonly called the 9-simplex, is the simplest possible non-degenerate polyyotton. The full symmetry version has 10 regular enneazetta as facets, joining 3 to a heptapeton peak and 9 to a vertex, and is one of the 3 regular polyyotta. It is the 9-dimensional simplex.

## Vertex coordinates

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

• ${\displaystyle \left(±\frac{1}{2},\,-\frac{\sqrt{3}}{6},\,-\frac{\sqrt{6}}{12},\,-\frac{\sqrt{10}}{20},\,-\frac{\sqrt{15}}{30},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28},\,-\frac{1}{12},\,-\frac{\sqrt5}{30}\right),}$
• ${\displaystyle \left(0,\,\frac{\sqrt{3}}{3},\,-\frac{\sqrt{6}}{12},\,-\frac{\sqrt{10}}{20},\,-\frac{\sqrt{15}}{30},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28},\,-\frac{1}{12},\,-\frac{\sqrt5}{30}\right),}$
• ${\displaystyle \left(0,\,0,\,\frac{\sqrt{6}}{4},\,-\frac{\sqrt{10}}{20},\,-\frac{\sqrt{15}}{30},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28},\,-\frac{1}{12},\,-\frac{\sqrt5}{30}\right),}$
• ${\displaystyle \left(0,\,0,\,0,\,\frac{\sqrt{10}}{5},\,-\frac{\sqrt{15}}{30},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28},\,-\frac{1}{12},\,-\frac{\sqrt5}{30}\right),}$
• ${\displaystyle \left(0,\,0,\,0,\,0,\,\frac{\sqrt{15}}{6},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28},\,-\frac{1}{12},\,-\frac{\sqrt5}{30}\right),}$
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,\frac{\sqrt{21}}{7},\,-\frac{\sqrt7}{28},\,-\frac{1}{12},\,-\frac{\sqrt5}{30}\right),}$
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,\frac{\sqrt7}{4},\,-\frac{1}{12},\,-\frac{\sqrt5}{30}\right),}$
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,\frac23,\,-\frac{\sqrt5}{30}\right),}$
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,\frac{3\sqrt5}{10}\right).}$

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

• ${\displaystyle \left(\frac{\sqrt2}{2},\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0\right).}$