# Hendecagonal-truncated dodecahedral duoprism

Hendecagonal-truncated dodecahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymHentid
Coxeter diagramx11o x5x3o
Elements
Tera20 triangular-hendecagonal duoprisms, 12 decagonal-hendecagonal duoprisms
Cells220 triangular prisms, 132 decagonal prisms, 30+60 hendecagonal prisms, 11 truncated dodecahedra
Faces220 triangles, 330+660 squares, 132 decagons, 60 hendecagons
Edges330+660+660
Vertices660
Vertex figureDigonal 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${\displaystyle {\sqrt {\frac {37+15{\sqrt {5}}+{\frac {2}{\sin ^{2}{\frac {\pi }{11}}}}}{8}}}\approx 3.45938}$
Hypervolume${\displaystyle 55{\frac {99+47{\sqrt {5}}}{48\tan {\frac {\pi }{11}}}}\approx 796.45088}$
Diteral anglesTiddip–tid–tiddip: ${\displaystyle {\frac {9\pi }{11}}\approx 147.27273^{\circ }}$
Thendip–henp–dahendip: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }}$
Dahendip–henp–dahendip: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }}$
Thendip–trip–tiddip: 90°
Dahendip–dip–tiddip: 90°
Central density1
Number of external pieces43
Level of complexity30
Related polytopes
ArmyHentid
RegimentHentid
DualHendecagonal-triakis icosahedral duotegum
ConjugatesSmall 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 characteristic2
OrientableYes
Properties
SymmetryH3×I2(11), order 2640
ConvexYes
NatureTame

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.

## Vertex coordinates

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:

• ${\displaystyle \left(1,\,0,\,0,\,\pm \sin {\frac {\pi }{11}},\,\pm {\frac {(5+3{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}}\right),}$
• ${\displaystyle \left(1,\,0,\,\pm \sin {\frac {\pi }{11}},\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}},\,\pm (3+{\sqrt {5}})\sin {\frac {\pi }{11}}\right),}$
• ${\displaystyle \left(1,\,0,\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}},\,\pm (1+{\sqrt {5}})\sin {\frac {\pi }{11}},\,\pm (2+{\sqrt {5}})\sin {\frac {\pi }{11}}\right),}$
• ${\displaystyle \left(\cos \left({\frac {j\pi }{11}}\right),\,\pm \sin \left({\frac {j\pi }{11}}\right),\,0,\,\pm \sin {\frac {\pi }{11}},\,\pm {\frac {(5+3{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}}\right),}$
• ${\displaystyle \left(\cos \left({\frac {j\pi }{11}}\right),\,\pm \sin \left({\frac {j\pi }{11}}\right),\,\pm \sin {\frac {\pi }{11}},\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}},\,\pm (3+{\sqrt {5}})\sin {\frac {\pi }{11}}\right),}$
• ${\displaystyle \left(\cos \left({\frac {j\pi }{11}}\right),\,\pm \sin \left({\frac {j\pi }{11}}\right),\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{11}}}{2}},\,\pm (1+{\sqrt {5}})\sin {\frac {\pi }{11}},\,\pm (2+{\sqrt {5}})\sin {\frac {\pi }{11}}\right),}$

where j = 2, 4, 6, 8, 10.