Pentagonal-great heptagrammic duoprism

Pentagonal-great heptagrammic duoprism
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Pagishdip
Coxeter diagram	x5o x7/3o ()
Elements
Cells	7 pentagonal prisms, 5 great heptagrammic prisms
Faces	35 squares, 7 pentagons, 5 great heptagrams
Edges	35+35
Vertices	35
Vertex figure	Digonal disphenoid, edge lengths (1+√5)/2 (base 1), 2cos(3π/7) (base 2), √2 (sides)
Measures (edge length 1)
Circumradius	${\sqrt {{\frac {5+{\sqrt {5}}}{10}}+{\frac {1}{4\sin ^{2}{\frac {3\pi }{7}}}}}}\approx 0.99329$
Hypervolume	${\frac {7{\sqrt {25+10{\sqrt {5}}}}}{16\tan {\frac {3\pi }{7}}}}\approx 0.68720$
Dichoral angles	Giship–7/3–giship: 108°
	Pip–4–giship: 90°
	Pip–5–pip: ${\frac {\pi }{7}}\approx 25.71429^{\circ }$
Central density	3
Number of external pieces	19
Level of complexity	12
Related polytopes
Army	Semi-uniform pheddip
Regiment	Pagishdip
Dual	Pentagonal-great heptagrammic duotegum
Conjugates	Pentagonal-heptagonal duoprism, Pentagonal-heptagrammic duoprism, Pentagrammic-heptagonal duoprism, Pentagrammic-heptagrammic duoprism, Pentagrammic-great heptagrammic duoprism
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	H2×I2(7), order 140
Convex	No
Nature	Tame

The pentagonal-great heptagrammic duoprism, also known as pagishdip or the 5-7/3 duoprism, is a uniform duoprism that consists of 7 pentagonal prisms and 5 great heptagrammic prisms, with two of each at each vertex.

Vertex coordinates

The coordinates of a pentagonal-great heptagrammic duoprism, centered at the origin and with edge length 2sin(3π/7), are given by:

• $\left(\pm \sin {\frac {3\pi }{7}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{5}}}\sin {\frac {3\pi }{7}},\,1,\,0\right),$
• $\left(\pm \sin {\frac {3\pi }{7}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{5}}}\sin {\frac {3\pi }{7}},\,\cos \left({\frac {j\pi }{7}}\right),\,\pm \sin \left({\frac {j\pi }{7}}\right)\right),$
• $\left(\pm {\frac {1+{\sqrt {5}}}{2}}\sin {\frac {3\pi }{7}},\,{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\sin {\frac {3\pi }{7}},\,1,\,0\right),$
• $\left(\pm {\frac {1+{\sqrt {5}}}{2}}\sin {\frac {3\pi }{7}},\,{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\sin {\frac {3\pi }{7}},\,\cos \left({\frac {j\pi }{7}}\right),\,\pm \sin \left({\frac {j\pi }{7}}\right)\right),$
• $\left(0,\,2{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\sin {\frac {3\pi }{7}},\,1,\,0\right),$
• $\left(0,\,2{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\sin {\frac {3\pi }{7}},\,\cos \left({\frac {j\pi }{7}}\right),\,\pm \sin \left({\frac {j\pi }{7}}\right)\right),$

where j = 2, 4, 6.

External links

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

• Klitzing, Richard. "nd-mb-dip".