Pentagrammic-heptagonal duoprism

Pentagrammic-heptagonal duoprism
	(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Stahedip
Coxeter diagram	x5/2o x7o ()
Elements
Cells	7 pentagrammic prisms, 5 heptagonal prisms
Faces	35 squares, 7 pentagrams, 5 heptagons
Edges	35+35
Vertices	35
Vertex figure	Digonal disphenoid, edge lengths (√5–1)/2 (base 1), 2cos(π/7) (base 2), √2 (sides)
Measures (edge length 1)
Circumradius
Hypervolume
Dichoral angles	Stip–5/2–stip: 5π/7 ≈ 128.57143°
	Stip–4–hep: 90°
	Hep–7–hep: 36°
Central density	2
Number of external pieces	17
Level of complexity	12
Related polytopes
Army	Semi-uniform pheddip
Regiment	Stahedip
Dual	Pentagrammic-heptagonal duotegum
Conjugates	Pentagonal-heptagonal duoprism, Pentagonal-heptagrammic duoprism, Pentagonal-great heptagrammic 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 pentagrammic-heptagonal duoprism, also known as stahedip or the 5/2-7 duoprism, is a uniform duoprism that consists of 7 pentagrammic prisms and 5 heptagonal prisms, with 2 of each at each vertex.

Vertex coordinates[edit | edit source]

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

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