Hendecagonal-small rhombicosidodecahedral duoprism Rank 5 Type Uniform Notation Bowers style acronym Hensrid Coxeter diagram x11o x5o3x ( ) Elements Tera 20 triangular-hendecagonal duoprisms , 30 square-hendecagonal duoprisms , 12 pentagonal-hendecagonal duoprisms , 11 small rhombicosidodecahedral prisms Cells 220 triangular prisms , 330 cubes , 132 pentagonal prisms , 60+60 hendecagonal prisms , 11 small rhombicosidodecahedra Faces 220 triangles , 330+660+660 squares , 132 pentagons , 60 hendecagons Edges 660+660+660 Vertices 660 Vertex figure Isosceles-trapezoidal scalene , edge lengths 1, √2 , (1+√5 )/2, √2 (base trapezoid), 2cos(π/11) (top), √2 (side edges)Measures (edge length 1) Circumradius
11
+
4
5
+
1
sin
2
π
11
2
≈
2.85232
{\displaystyle {\frac {\sqrt {11+4{\sqrt {5}}+{\frac {1}{\sin ^{2}{\frac {\pi }{11}}}}}}{2}}\approx 2.85232}
Hypervolume
11
60
+
29
5
12
tan
π
11
≈
389.75414
{\displaystyle 11{\frac {60+29{\sqrt {5}}}{12\tan {\frac {\pi }{11}}}}\approx 389.75414}
Diteral angles Thendip–henp–shendip:
arccos
(
−
3
+
15
6
)
≈
159.09484
∘
{\displaystyle \arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 159.09484^{\circ }}
Shendip–henp–pahendip:
arccos
(
−
5
+
5
10
)
≈
148.28253
∘
{\displaystyle \arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }}
Sriddip–srid–sriddip:
9
π
11
≈
147.27273
∘
{\displaystyle {\frac {9\pi }{11}}\approx 147.27273^{\circ }}
Thendip–trip–sriddip: 90° Shendip–cube–sriddip: 90° Pahendip–pip–sriddip: 90° Central density 1 Number of external pieces 73 Level of complexity 40 Related polytopes Army Hensrid Regiment Hensrid Dual Hendecagonal-deltoidal hexecontahedral duotegum Conjugates Small hendecagrammic-small rhombicosidodecahedral duoprism , Hendecagrammic-small rhombicosidodecahedral duoprism , Great hendecagrammic-small rhombicosidodecahedral duoprism , Grand hendecagrammic-small rhombicosidodecahedral duoprism , Hendecagonal-quasirhombicosidodecahedral duoprism , Small hendecagrammic-quasirhombicosidodecahedral duoprism , Hendecagrammic-quasirhombicosidodecahedral duoprism , Great hendecagrammic-quasirhombicosidodecahedral duoprism , Grand hendecagrammic-quasirhombicosidodecahedral duoprism Abstract & topological properties Euler characteristic 2 Orientable Yes Properties Symmetry H3 ×I2 (11) , order 2640Convex Yes Nature Tame
The hendecagonal-small rhombicosidodecahedral duoprism or hensrid is a convex uniform duoprism that consists of 11 small rhombicosidodecahedral prisms , 12 pentagonal-hendecagonal duoprisms , 30 square-hendecagonal duoprisms , and 20 triangular-hendecagonal duoprisms . Each vertex joins 2 small rhombicosidodecahedral prisms, 1 triangular-hendecagonal duoprism, 2 square-hendecagonal duoprisms, and 1 pentagonal-hendecagonal duoprism.
The vertices of a hendecagonal-small rhombicosidodecahedral duoprism of edge length 2sin(π/11) are given by all permutations of the last three coordinates of:
(
1
,
0
,
±
sin
π
11
,
±
sin
π
11
,
±
(
2
+
5
)
sin
π
11
)
,
{\displaystyle \left(1,\,0,\,\pm \sin {\frac {\pi }{11}},\,\pm \sin {\frac {\pi }{11}},\,\pm (2+{\sqrt {5}})\sin {\frac {\pi }{11}}\right),}
(
cos
(
j
π
11
)
,
±
sin
(
j
π
11
)
,
±
sin
π
11
,
±
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 \sin {\frac {\pi }{11}},\,\pm \sin {\frac {\pi }{11}},\,\pm (2+{\sqrt {5}})\sin {\frac {\pi }{11}}\right),}
as well as all even permutations of the last three coordinates of:
(
1
,
0
,
0
,
±
(
3
+
5
)
sin
π
11
2
,
±
(
5
+
5
)
sin
π
11
2
)
,
{\displaystyle \left(1,\,0,\,0,\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}},\,\pm {\frac {(5+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}}\right),}
(
cos
(
j
π
11
)
,
±
sin
(
j
π
11
)
,
0
,
±
(
3
+
5
)
sin
π
11
2
,
±
(
5
+
5
)
sin
π
11
2
)
,
{\displaystyle \left(\cos \left({\frac {j\pi }{11}}\right),\,\pm \sin \left({\frac {j\pi }{11}}\right),\,0,\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}},\,\pm {\frac {(5+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}}\right),}
(
1
,
0
,
±
(
1
+
5
)
sin
π
11
2
,
±
(
1
+
5
)
sin
π
11
,
±
(
3
+
5
)
sin
π
11
2
)
,
{\displaystyle \left(1,\,0,\,\pm {\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}},\,\pm (1+{\sqrt {5}})\sin {\frac {\pi }{11}},\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}}\right),}
(
cos
(
j
π
11
)
,
±
sin
(
j
π
11
)
,
±
(
1
+
5
)
sin
π
11
2
,
±
(
1
+
5
)
sin
π
11
,
±
(
3
+
5
)
sin
π
11
2
)
,
{\displaystyle \left(\cos \left({\frac {j\pi }{11}}\right),\,\pm \sin \left({\frac {j\pi }{11}}\right),\,\pm {\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}},\,\pm (1+{\sqrt {5}})\sin {\frac {\pi }{11}},\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}}\right),}
where j = 2, 4, 6, 8, 10.
Klitzing, Richard. "n-srid-dip" .