Hendecagonal-great rhombicosidodecahedral duoprism Rank 5 Type Uniform Notation Bowers style acronym Hengrid Coxeter diagram x11o x5x3x Elements Tera 30 square-hendecagonal duoprisms , 20 hexagonal-hendecagonal duoprisms , 12 decagonal-hendecagonal duoprisms , 11 great rhombicosidodecahedral prisms Cells 330 cubes , 220 hexagonal prisms , 132 decagonal prisms , 60+60+60 hendecagonal prisms , 11 great rhombicosidodecahedra Faces 330+660+660+660 squares , 220 hexagons , 132 decagons , 120 hendecagons Edges 660+660+660+1320 Vertices 1320 Vertex figure Mirror-symmetric pentachoron , edge lengths √2 , √3 , √(5+√5 )/2 (base triangle), 2cos(π/11) (top edge), √2 (side edges) Measures (edge length 1) Circumradius
31
+
12
5
+
1
sin
2
π
11
2
≈
4.19617
{\displaystyle {\frac {\sqrt {31+12{\sqrt {5}}+{\frac {1}{\sin ^{2}{\frac {\pi }{11}}}}}}{2}}\approx 4.19617}
Hypervolume
55
19
+
10
5
4
tan
π
11
≈
1936.84617
{\displaystyle 55{\frac {19+10{\sqrt {5}}}{4\tan {\frac {\pi }{11}}}}\approx 1936.84617}
Diteral angles Shendip–henp–hahendip:
arccos
(
−
3
+
15
6
)
≈
159.09484
∘
{\displaystyle \arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 159.09484^{\circ }}
Shendip–henp–dahendip:
arccos
(
−
5
+
5
10
)
≈
148.28253
∘
{\displaystyle \arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }}
Griddip–grid–griddip:
9
π
11
≈
147.27273
∘
{\displaystyle {\frac {9\pi }{11}}\approx 147.27273^{\circ }}
Hahendip–henp–dahendip:
arccos
(
−
5
+
2
5
15
)
≈
142.62263
∘
{\displaystyle \arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }}
Shendip–cube–griddip: 90° Hahendip–hip–griddip: 90° Dahendip–dip–griddip: 90° Central density 1 Number of external pieces 73 Level of complexity 60 Related polytopes Army Hengrid Regiment Hengrid Dual Hendecagonal-disdyakis triacontahedral duotegum Conjugates Small hendecagrammic-great rhombicosidodecahedral duoprism , Hendecagrammic-great rhombicosidodecahedral duoprism , Great hendecagrammic-great rhombicosidodecahedral duoprism , Grand hendecagrammic-great rhombicosidodecahedral duoprism , Hendecagonal-great quasitruncated icosidodecahedral duoprism , Small hendecagrammic-great quasitruncated icosidodecahedral duoprism , Hendecagrammic-great quasitruncated icosidodecahedral duoprism , Great hendecagrammic-great quasitruncated icosidodecahedral duoprism , Grand hendecagrammic-great quasitruncated icosidodecahedral duoprism Abstract & topological properties Euler characteristic 2 Orientable Yes Properties Symmetry H3 ×I2(11) , order 2640Convex Yes Nature Tame
The hendecagonal-great rhombicosidodecahedral duoprism or hengrid is a convex uniform duoprism that consists of 11 great rhombicosidodecahedral prisms , 12 decagonal-hendecagonal duoprisms , 20 hexagonal-hendecagonal duoprisms , and 30 square-hendecagonal duoprisms . Each vertex joins 2 great rhombicosidodecahedral prisms, 1 square-hendecagonal duoprism, 1 hexagonal-hendecagonal duoprism, and 1 decagonal-hendecagonal duoprism.
The vertices of a hendecagonal-great 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
,
(
3
+
2
5
)
sin
π
11
)
,
{\displaystyle \left(1,\,0,\,\pm \sin {\frac {\pi }{11}},\,\pm \sin {\frac {\pi }{11}},\,(3+2{\sqrt {5}})\sin {\frac {\pi }{11}}\right),}
(
cos
(
j
π
11
)
,
±
sin
(
j
π
11
)
,
±
sin
π
11
,
±
sin
π
11
,
(
3
+
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}},\,(3+2{\sqrt {5}})\sin {\frac {\pi }{11}}\right),}
along with all even permutations of the last three coordinates of:
(
1
,
0
,
±
sin
π
11
,
±
(
2
+
5
)
sin
π
11
,
±
(
4
+
5
)
sin
π
11
)
,
{\displaystyle \left(1,\,0,\,\pm \sin {\frac {\pi }{11}},\,\pm (2+{\sqrt {5}})\sin {\frac {\pi }{11}},\,\pm (4+{\sqrt {5}})\sin {\frac {\pi }{11}}\right),}
(
cos
(
j
π
11
)
,
±
sin
(
j
π
11
)
,
±
sin
π
11
,
±
(
2
+
5
)
sin
π
11
,
±
(
4
+
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 (2+{\sqrt {5}})\sin {\frac {\pi }{11}},\,\pm (4+{\sqrt {5}})\sin {\frac {\pi }{11}}\right),}
(
1
,
0
,
±
2
sin
π
11
,
±
(
3
+
5
)
sin
π
11
2
,
±
(
7
+
3
5
)
sin
π
11
2
)
,
{\displaystyle \left(1,\,0,\,\pm 2\sin {\frac {\pi }{11}},\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}},\,\pm {\frac {(7+3{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}}\right),}
(
cos
(
j
π
11
)
,
±
sin
(
j
π
11
)
,
±
2
sin
π
11
,
±
(
3
+
5
)
sin
π
11
2
,
±
(
7
+
3
5
)
sin
π
11
2
)
,
{\displaystyle \left(\cos \left({\frac {j\pi }{11}}\right),\,\pm \sin \left({\frac {j\pi }{11}}\right),\,\pm 2\sin {\frac {\pi }{11}},\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}},\,\pm {\frac {(7+3{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}}\right),}
(
1
,
0
,
±
(
3
+
5
)
sin
π
11
2
,
±
3
(
1
+
5
)
sin
π
11
2
,
±
(
3
+
5
)
sin
π
11
)
,
{\displaystyle \left(1,\,0,\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}},\,\pm 3{\frac {(1+{\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
,
±
3
(
1
+
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 {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}},\,\pm 3{\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}},\,\pm (3+{\sqrt {5}})\sin {\frac {\pi }{11}}\right),}
(
1
,
0
,
±
(
1
+
5
)
sin
π
11
,
±
(
5
+
3
5
)
sin
π
11
2
,
±
(
5
+
5
)
sin
π
11
2
)
,
{\displaystyle \left(1,\,0,\,\pm (1+{\sqrt {5}})\sin {\frac {\pi }{11}},\,\pm {\frac {(5+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
)
,
±
(
1
+
5
)
sin
π
11
,
±
(
5
+
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),\,\pm (1+{\sqrt {5}})\sin {\frac {\pi }{11}},\,\pm {\frac {(5+3{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}},\,\pm {\frac {(5+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}}\right),}
where j = 2, 4, 6, 8, 10.