Hendecagonal-truncated dodecahedral duoprism Rank 5 Type Uniform Notation Bowers style acronym Hentid Coxeter diagram x11o x5x3o Elements Tera 20 triangular-hendecagonal duoprisms , 12 decagonal-hendecagonal duoprisms Cells 220 triangular prisms , 132 decagonal prisms , 30+60 hendecagonal prisms , 11 truncated dodecahedra Faces 220 triangles , 330+660 squares , 132 decagons , 60 hendecagons Edges 330+660+660 Vertices 660 Vertex figure Digonal disphenoidal pyramid , edge lengths 1, √(5+√5 )/2 , √(5+√5 )/2 (base triangle), cos(π/11) (top), √2 (side edges)Measures (edge length 1) Circumradius
37
+
15
5
+
2
sin
2
π
11
8
≈
3.45938
{\displaystyle {\sqrt {\frac {37+15{\sqrt {5}}+{\frac {2}{\sin ^{2}{\frac {\pi }{11}}}}}{8}}}\approx 3.45938}
Hypervolume
55
99
+
47
5
48
tan
π
11
≈
796.45088
{\displaystyle 55{\frac {99+47{\sqrt {5}}}{48\tan {\frac {\pi }{11}}}}\approx 796.45088}
Diteral angles Tiddip–tid–tiddip:
9
π
11
≈
147.27273
∘
{\displaystyle {\frac {9\pi }{11}}\approx 147.27273^{\circ }}
Thendip–henp–dahendip:
arccos
(
−
5
+
2
5
15
)
≈
142.62263
∘
{\displaystyle \arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }}
Dahendip–henp–dahendip:
arccos
(
−
5
5
)
≈
116.56505
∘
{\displaystyle \arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }}
Thendip–trip–tiddip: 90° Dahendip–dip–tiddip: 90° Central density 1 Number of external pieces 43 Level of complexity 30 Related polytopes Army Hentid Regiment Hentid Dual Hendecagonal-triakis icosahedral duotegum Conjugates Small hendecagrammic-truncated dodecahedral duoprism , Hendecagrammic-truncated dodecahedral duoprism , Great hendecagrammic-truncated dodecahedral duoprism , Grand hendecagrammic-truncated dodecahedral duoprism , Hendecagonal-quasitruncated great stellated dodecahedral duoprism , Small hendecagrammic-quasitruncated great stellated dodecahedral duoprism , Hendecagrammic-quasitruncated great stellated dodecahedral duoprism , Great hendecagrammic-quasitruncated great stellated dodecahedral duoprism , Grand hendecagrammic-quasitruncated great stellated dodecahedral duoprism Abstract & topological properties Euler characteristic 2 Orientable Yes Properties Symmetry H3 ×I2(11) , order 2640Convex Yes Nature Tame
The hendecagonal-truncated dodecahedral duoprism or hentid is a convex uniform duoprism that consists of 11 truncated dodecahedral prisms , 12 decagonal-hendecagonal duoprisms , and 20 triangular-hendecagonal duoprisms . Each vertex joins 2 truncated dodecahedral prisms, 1 triangular-hendecagonal duoprism, and 2 decagonal-hendecagonal duoprisms.
The vertices of a hendecagonal-truncated dodecahedral duoprism of edge length 2sin(π/11) are given by all even permutations of the last three coordinates of:
(
1
,
0
,
0
,
±
sin
π
11
,
±
(
5
+
3
5
)
sin
π
11
2
)
,
{\displaystyle \left(1,\,0,\,0,\,\pm \sin {\frac {\pi }{11}},\,\pm {\frac {(5+3{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}}\right),}
(
1
,
0
,
±
sin
π
11
,
±
(
3
+
5
)
sin
π
11
2
,
±
(
3
+
5
)
sin
π
11
)
,
{\displaystyle \left(1,\,0,\,\pm \sin {\frac {\pi }{11}},\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}},\,\pm (3+{\sqrt {5}})\sin {\frac {\pi }{11}}\right),}
(
1
,
0
,
±
(
3
+
5
)
sin
π
11
2
,
±
(
1
+
5
)
sin
π
11
,
±
(
2
+
5
)
sin
π
11
)
,
{\displaystyle \left(1,\,0,\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}},\,\pm (1+{\sqrt {5}})\sin {\frac {\pi }{11}},\,\pm (2+{\sqrt {5}})\sin {\frac {\pi }{11}}\right),}
(
cos
(
j
π
11
)
,
±
sin
(
j
π
11
)
,
0
,
±
sin
π
11
,
±
(
5
+
3
5
)
sin
π
11
2
)
,
{\displaystyle \left(\cos \left({\frac {j\pi }{11}}\right),\,\pm \sin \left({\frac {j\pi }{11}}\right),\,0,\,\pm \sin {\frac {\pi }{11}},\,\pm {\frac {(5+3{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}}\right),}
(
cos
(
j
π
11
)
,
±
sin
(
j
π
11
)
,
±
sin
π
11
,
±
(
3
+
5
)
sin
π
11
2
,
±
(
3
+
5
)
sin
π
11
)
,
{\displaystyle \left(\cos \left({\frac {j\pi }{11}}\right),\,\pm \sin \left({\frac {j\pi }{11}}\right),\,\pm \sin {\frac {\pi }{11}},\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}},\,\pm (3+{\sqrt {5}})\sin {\frac {\pi }{11}}\right),}
(
cos
(
j
π
11
)
,
±
sin
(
j
π
11
)
,
±
(
3
+
5
)
sin
π
11
2
,
±
(
1
+
5
)
sin
π
11
,
±
(
2
+
5
)
sin
π
11
)
,
{\displaystyle \left(\cos \left({\frac {j\pi }{11}}\right),\,\pm \sin \left({\frac {j\pi }{11}}\right),\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}},\,\pm (1+{\sqrt {5}})\sin {\frac {\pi }{11}},\,\pm (2+{\sqrt {5}})\sin {\frac {\pi }{11}}\right),}
where j = 2, 4, 6, 8, 10.
