Hendecagonal-dodecagonal duoprismatic prism

Hendecagonal-dodecagonal duoprismatic prism
Rank5
TypeUniform
Notation
Bowers style acronymHentwip
Coxeter diagramx x11o x12o
Elements
Tera12 square-hendecagonal duoprisms, 11 square-dodecagonal duoprisms, 2 hendecagonal-dodecagonal duoprisms
Cells132 cubes, 11+22 dodecagonal prisms, 12+24 hendecagonal prisms
Faces132+132+264 squares, 24 hendecagons, 22 dodecagons
Edges132+264+264
Vertices264
Vertex figureDigonal disphenoidal pyramid, edge lengths 2cos(π/11) (disphenoid base 1), 2+3 (disphenoid base 2), 2 (remaining edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {9+4{\sqrt {3}}+{\frac {1}{\sin ^{2}{\frac {\pi }{11}}}}}}{2}}\approx 2.67053}$
Hypervolume${\displaystyle 33{\frac {2+{\sqrt {3}}}{4\tan {\frac {\pi }{11}}}}\approx 104.85913}$
Diteral anglesShendip–henp–shendip: 150°
Sitwadip–twip–sitwadip: ${\displaystyle {\frac {9\pi }{11}}\approx 147.27273^{\circ }}$
Height1
Central density1
Number of external pieces25
Level of complexity30
Related polytopes
ArmyHentwip
RegimentHentwip
DualHendecagonal-dodecagonal duotegmatic tegum
ConjugatesHendecagonal-dodecagrammic duoprismatic prism, Small hendecagrammic-dodecagonal duoprismatic prism, Small hendecagrammic-dodecagrammic duoprismatic prism, Hendecagrammic-dodecagonal duoprismatic prism, Hendecagrammic-dodecagrammic duoprismatic prism, Great hendecagrammic-dodecagonal duoprismatic prism, Great hendecagrammic-dodecagrammic duoprismatic prism, Grand hendecagrammic-dodecagonal duoprismatic prism, Grand hendecagrammic-dodecagrammic duoprismatic prism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryI2(11)×I2(12)×A1, order 1056
ConvexYes
NatureTame

The hendecagonal-dodecagonal duoprismatic prism or hentwip, also known as the hendecagonal-dodecagonal prismatic duoprism, is a convex uniform duoprism that consists of 2 hendecagonal-dodecagonal duoprisms, 11 square-dodecagonal duoprisms, and 12 square-hendecagonal duoprisms. Each vertex joins 2 square-hendecagonal duoprisms, 2 square-dodecagonal duoprisms, and 1 hendecagonal-dodecagonal duoprism. Being a prism based on an orbiform polytope, it is also a convex segmentoteron.

Vertex coordinates

The vertices of a hendecagonal-dodecagonal duoprismatic prism of edge length 2sin(π/11) are given by all permutations of the third and fourth coordinates of:

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

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

Representations

A hendecagonal-dodecagonal duoprismatic prism has the following Coxeter diagrams:

• x x11o x12o (full symmetry)
• x x11o x6x (dodecagons as dihexagons)
• xx11oo xx12oo&#x (hendecagonal-dodecagonal duoprism atop hendecagonal-dodecagonal duoprism)
• xx11oo xx6xx&#x