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Regular decayotton (9-simplex)
Orthogonal projection inside
Petrie polygon
Type
Regular
9-polytope
Family
simplex
Schläfli symbol
{3,3,3,3,3,3,3,3}
Coxeter-Dynkin diagram
8-faces
10
8-simplex
7-faces
45
7-simplex
6-faces
120
6-simplex
5-faces
210
5-simplex
4-faces
252
5-cell
Cells
210
tetrahedron
Faces
120
triangle
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 .
Coordinates
The
Cartesian 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
,
−
2
1
/
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
,
−
2
2
/
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)}
(
−
3
1
/
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 the 9-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 one
Facet of the
10-orthoplex .
Images
References
External links