Hexagonal-hendecagonal duoprismatic prism

Hexagonal-hendecagonal duoprismatic prism
Rank5
TypeUniform
Notation
Bowers style acronymHahenip
Coxeter diagramx x6o x11o ()
Elements
Tera11 square-hexagonal duoprisms, 6 square-hendecagonal duoprisms, 2 hexagonal-hendecagonal duoprisms
Cells66 cubes, 6+12 hendecagonal prisms, 11+22 hexagonal prisms
Faces66+66+132 squares, 22 hexagons, 12 hendecagons
Edges66+132+132
Vertices132
Vertex figureDigonal disphenoidal pyramid, edge lengths 3 (disphenoid base 1), 2cos(π/11) (disphenoid base 2), 2 (remaining edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {5+{\frac {1}{\sin ^{2}{\frac {\pi }{11}}}}}}{2}}\approx 2.09754}$
Hypervolume${\displaystyle {\frac {33{\sqrt {3}}}{8\tan {\frac {\pi }{11}}}}\approx 24.33265}$
Diteral anglesShiddip–hip–shiddip: ${\displaystyle {\frac {9\pi }{11}}\approx 147.27273^{\circ }}$
Shendip–henp–shendip: 120°
Shendip–cube–shiddip: 90°
Hahendip–hip–shiddip: 90°
Shendip–henp–hahendip: 90°
Height1
Central density1
Number of external pieces19
Level of complexity30
Related polytopes
ArmyHahenip
RegimentHahenip
DualHexagonal-hendecagonal duotegmatic tegum
ConjugatesHexagonal-small hendecagrammic duoprismatic prism, Hexagonal-hendecagrammic duoprismatic prism, Hexagonal-great hendecagrammic duoprismatic prism, Hexagonal-grand hendecagrammic duoprismatic prism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryG2×I2(11)×A1, order 528
ConvexYes
NatureTame

The hexagonal-hendecagonal duoprismatic prism (OBSA: hahenip) also known as the hexagonal-hendecagonal prismatic duoprism, is a convex uniform duoprism that consists of 2 hexagonal-hendecagonal duoprisms, 6 square-hendecagonal duoprisms, and 11 square-hexagonal duoprisms. Each vertex joins 2 square-hexagonal duoprisms, 2 square-hendecagonal duoprisms, and 1 hexagonal-hendecagonal duoprism. Being a prism based on an orbiform polytope, it is also a convex segmentoteron.

Vertex coordinates

The vertices of a hexagonal-hendecagonal duoprismatic prism of edge length ${\displaystyle 2\sin(\pi /11)}$ are given by:

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

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

Representations

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

• x x6o x11o () (full symmetry)
• x x3x x11o () (hexagons as ditrigons)
• xx6oo xx11oo&#x (hexagonal-hendecagonal duoprism atop hexagonal-hendecagonal duoprism)
• xx3xx xx11oo&#x