12-simplex
Rank 12 Type Regular Notation Coxeter diagram x3o3o3o3o3o3o3o3o3o3o3o ( ) Schläfli symbol {3,3,3,3,3,3,3,3,3,3,3} Elements Henda 13 11-simplices Daka 78 10-simplices Xenna 286 9-simplices Yotta 715 8-simplices Zetta 1287 7-simplices Exa 1716 6-simplices Peta 1716 5-simplices Tera 1287 5-cells Cells 715 tetrahedra Faces 286 triangles Edges 78 Vertices 13 Vertex figure 11-simplex , edge length 1Measures (edge length 1) Circumradius
78
13
≈
0.67937
{\displaystyle {\frac {\sqrt {78}}{13}}\approx 0.67937}
Inradius
78
156
≈
0.056614
{\displaystyle {\frac {\sqrt {78}}{156}}\approx 0.056614}
Hypervolume
13
30656102400
≈
1.1761
×
10
−
10
{\displaystyle {\frac {\sqrt {13}}{30656102400}}\approx 1.1761\times 10^{-10}}
Dihedral angle
arccos
(
1
12
)
≈
85.21981
∘
{\displaystyle \arccos \left({\frac {1}{12}}\right)\approx 85.21981^{\circ }}
Height
78
12
≈
0.73598
{\displaystyle {\frac {\sqrt {78}}{12}}\approx 0.73598}
Central density 1 Number of external pieces 13 Level of complexity 1 Related polytopes Dual 12-simplex Conjugate None Convex hull 12-simplex Abstract & topological properties Flag count6227020800 Euler characteristic 0 Orientable Yes Properties Symmetry A12 , order 6227020800Flag orbits 1 Convex Yes Nature Tame
The 12-simplex (also called tridecahendon or tridecahendakon ) is the simplest possible non-degenerate 12-polytope . The full symmetry version has 13 regular dodecadaka as facets , joining 3 to a ridge and 12 to a vertex, and is one of the 3 convex regular 12-polytopes . It is the 12-dimensional simplex .
The vertices of a regular tridecahendon of edge length 1, centered at the origin, are given by:
(
±
1
2
,
−
3
6
,
−
6
12
,
−
10
20
,
−
15
30
,
−
21
42
,
−
7
28
,
−
1
12
,
−
5
30
,
−
55
110
,
−
66
132
,
−
78
156
)
{\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}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}}\right)}
,
(
0
,
3
3
,
−
6
12
,
−
10
20
,
−
15
30
,
−
21
42
,
−
7
28
,
−
1
12
,
−
5
30
,
−
55
110
,
−
66
132
,
−
78
156
)
{\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}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}}\right)}
,
(
0
,
0
,
6
4
,
−
10
20
,
−
15
30
,
−
21
42
,
−
7
28
,
−
1
12
,
−
5
30
,
−
55
110
,
−
66
132
,
−
78
156
)
{\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}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}}\right)}
,
(
0
,
0
,
0
,
10
5
,
−
15
30
,
−
21
42
,
−
7
28
,
−
1
12
,
−
5
30
,
−
55
110
,
−
66
132
,
−
78
156
)
{\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}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}}\right)}
,
(
0
,
0
,
0
,
0
,
15
6
,
−
21
42
,
−
7
28
,
−
1
12
,
−
5
30
,
−
55
110
,
−
66
132
,
−
78
156
)
{\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}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}}\right)}
,
(
0
,
0
,
0
,
0
,
0
,
21
7
,
−
7
28
,
−
1
12
,
−
5
30
,
−
55
110
,
−
66
132
,
−
78
156
)
{\displaystyle \left(0,\,0,\,0,\,0,\,0,\,{\frac {\sqrt {21}}{7}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}}\right)}
,
(
0
,
0
,
0
,
0
,
0
,
0
,
7
4
,
−
1
12
,
−
5
30
,
−
55
110
,
−
66
132
,
−
78
156
)
{\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,{\frac {\sqrt {7}}{4}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}}\right)}
,
(
0
,
0
,
0
,
0
,
0
,
0
,
0
,
2
3
,
−
5
30
,
−
55
110
,
−
66
132
,
−
78
156
)
{\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,{\frac {2}{3}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}}\right)}
,
(
0
,
0
,
0
,
0
,
0
,
0
,
0
,
0
,
3
5
10
,
−
55
110
,
−
66
132
,
−
78
156
)
{\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,{\frac {3{\sqrt {5}}}{10}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}}\right)}
,
(
0
,
0
,
0
,
0
,
0
,
0
,
0
,
0
,
0
,
55
11
,
−
66
132
,
−
78
156
)
{\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,{\frac {\sqrt {55}}{11}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}}\right)}
,
(
0
,
0
,
0
,
0
,
0
,
0
,
0
,
0
,
0
,
0
,
66
12
,
−
78
156
)
{\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,{\frac {\sqrt {66}}{12}},\,-{\frac {\sqrt {78}}{156}}\right)}
,
(
0
,
0
,
0
,
0
,
0
,
0
,
0
,
0
,
0
,
0
,
0
,
78
13
)
{\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,{\frac {\sqrt {78}}{13}}\right)}
.
Much simpler coordinates can be given in 13 dimensions , as all permutations of:
(
2
2
,
0
,
0
,
0
,
0
,
0
,
0
,
0
,
0
,
0
,
0
,
0
,
0
)
{\displaystyle \left({\frac {\sqrt {2}}{2}},\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0\right)}
.