# Hendecagonal-truncated icosahedral duoprism

Hendecagonal-truncated icosahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymHenti
Coxeter diagramx11o o5x3x
Elements
Tera12 pentagonal-hendecagonal duoprisms, 20 hexagonal-hendecagonal duoprisms
Cells132 pentagonal prisms, 220 hexagonal prisms, 30+60 hendecagonal prisms, 11 truncated icosahedra
Faces330+660 squares, 132 pentagons, 220 hexagons, 60 hendecagons
Edges330+660+660
Vertices660
Vertex figureDigonal disphenoidal pyramid, edge lengths (1+5)/2, 3, 3 (base triangle), 2cos(π/11) (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {29+9{\sqrt {5}}+{\frac {2}{\sin ^{2}{\frac {\pi }{11}}}}}{8}}}\approx 3.04799}$
Hypervolume${\displaystyle 11{\frac {125+43{\sqrt {5}}}{16\tan {\frac {\pi }{11}}}}\approx 517.80498}$
Diteral anglesTipe–ti–tipe: ${\displaystyle {\frac {9\pi }{11}}\approx 147.27273^{\circ }}$
Pahendip–henp–hahendip: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }}$
Hahendip–henp–hahendip: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18968^{\circ }}$
Pahendip–pip–tipe: 90°
Hahendip–hip–tipe: 90°
Central density1
Number of external pieces43
Level of complexity30
Related polytopes
ArmyHenti
RegimentHenti
DualHendecagonal-pentakis dodecahedral duotegum
ConjugatesSmall hendecagrammic-truncated icosahedral duoprism, Hendecagrammic-truncated icosahedral duoprism, Great hendecagrammic-truncated icosahedral duoprism, Grand hendecagrammic-truncated icosahedral duoprism, Hendecagonal-truncated great icosahedral duoprism, Small hendecagrammic-truncated great icosahedral duoprism, Hendecagrammic-truncated great icosahedral duoprism, Great hendecagrammic-truncated great icosahedral duoprism, Grand hendecagrammic-truncated great icosahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH3×I2(11), order 2640
ConvexYes
NatureTame

The hendecagonal-truncated icosahedral duoprism or henti is a convex uniform duoprism that consists of 11 truncated icosahedral prisms, 20 hexagonal-hendecagonal duoprisms, and 12 pentagonal-hendecagonal duoprisms. Each vertex joins 2 truncated icosahedral prisms, 1 pentagonal-hendecagonal duoprism, and 2 hexagonal-hendecagonal duoprisms.

## Vertex coordinates

The vertices of a hendecagonal-truncated icosahedral duoprism of edge length 2sin(π/11) are given by all even permutations of the last three coordinates of:

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

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