Hendecagonal-small rhombicuboctahedral duoprism Rank 5 Type Uniform Notation Bowers style acronym Hensirco Coxeter diagram x11o x4o3x Elements Tera 8 triangular-hendecagonal duoprisms , 6+12 square-hendecagonal duoprisms Cells 88 triangular prisms , 66+132 cubes , 24+24 hendecagonal prisms , 11 small rhombicuboctahedra Faces 88 triangles , 66+132+264+264 squares , 24 hendecagon Edges 264+264+264 Vertices 264 Vertex figure Isosceles-trapezoidal scalene , edge lengths 1, √2 , √2 , √2 (base trapezoid), 2cos(π/11) (top), √2 (side edges)Measures (edge length 1) Circumradius
5
+
2
2
+
1
sin
2
π
11
2
≈
2.25982
{\displaystyle {\frac {\sqrt {5+2{\sqrt {2}}+{\frac {1}{\sin ^{2}{\frac {\pi }{11}}}}}}{2}}\approx 2.25982}
Hypervolume
11
6
+
5
2
6
tan
π
11
≈
81.61261
{\displaystyle 11{\frac {6+5{\sqrt {2}}}{6\tan {\frac {\pi }{11}}}}\approx 81.61261}
Diteral angles Sircope–sirco–sircope:
9
π
11
≈
147.27273
∘
{\displaystyle {\frac {9\pi }{11}}\approx 147.27273^{\circ }}
Thendip–henp–shendip:
arccos
(
−
6
3
)
≈
144.73561
∘
{\displaystyle \arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }}
Shendip–henp–shendip: 135° Thendip–trip–sircope: 90° Shendip–cube–sircope: 90° Central density 1 Number of external pieces 37 Level of complexity 40 Related polytopes Army Hensirco Regiment Hensirco Dual Hendecagonal-deltoidal icositetrahedral duotegum Conjugates Small hendecagrammic-small rhombicuboctahedral duoprism , Hendecagrammic-small rhombicuboctahedral duoprism , Great hendecagrammic-small rhombicuboctahedral duoprism , Grand hendecagrammic-small rhombicuboctahedral duoprism , Hendecagonal-quasirhombicuboctahedral duoprism , Small hendecagrammic-quasirhombicuboctahedral duoprism , Hendecagrammic-quasirhombicuboctahedral duoprism , Great hendecagrammic-quasirhombicuboctahedral duoprism , Grand hendecagrammic-quasirhombicuboctahedral duoprism Abstract & topological properties Euler characteristic 2 Orientable Yes Properties Symmetry B3 ×I2 (11) , order 1056Convex Yes Nature Tame
The hendecagonal-small rhombicuboctahedral duoprism or hensirco is a convex uniform duoprism that consists of 11 small rhombicuboctahedral prisms , 18 square-hendecagonal duoprisms of two kinds, and 8 triangular-hendecagonal duoprisms . Each vertex joins 2 small rhombicuboctahedral prisms, 1 triangular-hendecagonal duoprism, and 3 square-hendecagonal duoprisms.
The vertices of a hendecagonal-small rhombicuboctahedral 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
,
±
(
1
+
2
)
sin
π
11
)
,
{\displaystyle \left(1,\,0,\,\pm \sin {\frac {\pi }{11}},\,\pm \sin {\frac {\pi }{11}},\,\pm (1+{\sqrt {2}})\sin {\frac {\pi }{11}}\right),}
(
cos
(
j
π
11
)
,
±
sin
(
j
π
11
)
,
±
sin
π
11
,
±
sin
π
11
,
±
(
1
+
2
)
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 (1+{\sqrt {2}})\sin {\frac {\pi }{11}}\right),}
where j = 2, 4, 6, 8, 10.