Pentagrammic-hendecagrammic duoprism

Pentagrammic-hendecagrammic duoprism
(OFF file)
Rank	4
Type	Uniform
Notation
Coxeter diagram	x5/2o x11/3o ()
Elements
Cells	11 pentagrammic prisms, 5 hendecagrammic prisms
Faces	55 squares, 11 pentagrams, 5 hendecagrams
Edges	55+55
Vertices	55
Vertex figure	Digonal disphenoid, edge lengths (√5–1)/2 (base 1), 2cos(3π/11) (base 2), √2 (sides)
Measures (edge length 1)
Circumradius	${\sqrt {{\frac {5-{\sqrt {5}}}{10}}+{\frac {1}{4\sin ^{2}{\frac {3\pi }{11}}}}}}\approx 0.84505$
Hypervolume	$11{\frac {\sqrt {25-10{\sqrt {5}}}}{16\tan {\frac {3\pi }{11}}}}\approx 0.96781$
Dichoral angles	Stip–4–shenp: 90°
	Stip–5/2–stip: ${\frac {5\pi }{11}}\approx 81.81818^{\circ }$
	Shenp–11/3–shenp: 36°
Central density	6
Number of external pieces	32
Level of complexity	24
Related polytopes
Army	Semi-uniform pahendip
Dual	Pentagrammic-hendecagrammic duotegum
Conjugates	Pentagonal-hendecagonal duoprism, Pentagonal-small hendecagrammic duoprism, Pentagonal-hendecagrammic duoprism, Pentagonal-great hendecagrammic duoprism, Pentagonal-grand hendecagrammic duoprism, Pentagrammic-hendecagonal duoprism, Pentagrammic-small hendecagrammic duoprism, Pentagrammic-great hendecagrammic duoprism, Pentagrammic-grand hendecagrammic duoprism
Abstract & topological properties
Flag count	1320
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	H2×I2(11), order 220
Convex	No
Nature	Tame

The pentagrammic-hendecagrammic duoprism, also known as the 5/2-11/3 duoprism, is a uniform duoprism that consists of 11 pentagrammic prisms and 5 hendecagrammic prisms, with 2 of each at each vertex.

The name can also refer to the pentagrammic-small hendecagrammic duoprism, the pentagrammic-great hendecagrammic duoprism, or the pentagrammic-grand hendecagrammic duoprism.

Vertex coordinates

The coordinates of a pentagrammic-hendecagrammic duoprism, centered at the origin and with edge length 2sin(3π/11), are given by:

• $\left(\pm \sin {\frac {3\pi }{11}},\,-{\sqrt {\frac {5-2{\sqrt {5}}}{5}}}\sin {\frac {3\pi }{11}},\,1,\,0\right)$,
• $\left(\pm \sin {\frac {3\pi }{11}},\,-{\sqrt {\frac {5-2{\sqrt {5}}}{5}}}\sin {\frac {3\pi }{11}},\,\cos \left({\frac {j\pi }{11}}\right),\,\pm \sin \left({\frac {j\pi }{11}}\right)\right)$,
• $\left(\pm {\frac {{\sqrt {5}}-1}{2}}\sin {\frac {3\pi }{11}},\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\sin {\frac {3\pi }{11}},\,1,\,0\right)$,
• $\left(\pm {\frac {{\sqrt {5}}-1}{2}}\sin {\frac {3\pi }{11}},\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\sin {\frac {3\pi }{11}},\,\cos \left({\frac {j\pi }{11}}\right),\,\pm \sin \left({\frac {j\pi }{11}}\right)\right)$,
• $\left(0,\,-2{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\sin {\frac {3\pi }{11}},\,1,\,0\right)$,
• $\left(0,\,-2{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\sin {\frac {3\pi }{11}},\,\cos \left({\frac {j\pi }{11}}\right),\,\pm \sin \left({\frac {j\pi }{11}}\right)\right)$,

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

External links

• Bowers, Jonathan. "Category A: Duoprisms".

• Klitzing, Richard. "nd-mb-dip".