Great heptagrammic-enneagonal duoprism

Great heptagrammic-enneagonal duoprism
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Gishedip
Coxeter diagram	x7/3o x9o ()
Elements
Cells	9 great heptagrammic prisms, 7 enneagonal prisms
Faces	63 squares, 9 great heptagrams, 7 enneagons
Edges	63+63
Vertices	63
Vertex figure	Digonal disphenoid, edge lengths 2cos(3π/7) (base 1), 2cos(π/9) (base 2), √2 (sides)
Measures (edge length 1)
Circumradius	${\sqrt {{\frac {1}{4\sin ^{2}{\frac {3\pi }{7}}}}+{\frac {1}{4\sin ^{2}{\frac {\pi }{9}}}}}}\approx 1.54925$
Hypervolume	${\frac {63}{16\tan {\frac {3\pi }{7}}\tan {\frac {\pi }{9}}}}\approx 2.46918$
Dichoral angles	Giship–7/3–giship: 140°
	Giship–4–ep: 90°
	Ep–9–ep: ${\frac {\pi }{7}}\approx 25.71429^{\circ }$
Central density	3
Number of external pieces	23
Level of complexity	12
Related polytopes
Army	Semi-uniform heendip
Regiment	Gishedip
Dual	Great heptagrammic-enneagonal duotegum
Conjugates	Heptagonal-enneagonal duoprism, Heptagonal-enneagrammic duoprism, Heptagonal-great enneagrammic duoprism, Heptagrammic-enneagonal duoprism, Heptagrammic-enneagrammic duoprism, Heptagrammic-great enneagrammic duoprism, Great heptagrammic-enneagrammic duoprism, Great heptagrammic-great enneagrammic duoprism
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	I2(7)×I2(9), order 252
Convex	No
Nature	Tame

The great heptagrammic-enneagonal duoprism, also known as gishedip or the 7/3-9 duoprism, is a uniform duoprism that consists of 9 great heptagrammic prisms and 7 enneagonal prisms, with 2 of each at each vertex.

Vertex coordinates

The coordinates of a great heptagrammic-enneagonal duoprism, centered at the origin and with edge length 4sin(3π/7)sin(π/9), are given by:

• $\left(2\sin {\frac {\pi }{9}},\,0,\,2\sin {\frac {3\pi }{7}},\,0\right),$
• $\left(2\sin {\frac {\pi }{9}},\,0,\,2\sin {\frac {3\pi }{7}}\cos \left({\frac {k\pi }{9}}\right),\,\pm 2\sin {\frac {3\pi }{7}}\sin \left({\frac {k\pi }{9}}\right)\right),$
• $\left(2\sin {\frac {\pi }{9}},\,0,\,-\sin {\frac {3\pi }{7}},\,\pm {\sqrt {3}}\sin {\frac {3\pi }{7}}\right),$
• $\left(2\sin {\frac {\pi }{9}}\cos \left({\frac {j\pi }{7}}\right),\,\pm 2\sin {\frac {\pi }{9}}\sin \left({\frac {j\pi }{7}}\right),\,2\sin {\frac {3\pi }{7}},\,0\right),$
• $\left(2\sin {\frac {\pi }{9}}\cos \left({\frac {j\pi }{7}}\right),\,\pm 2\sin {\frac {\pi }{9}}\sin \left({\frac {j\pi }{7}}\right),\,2\sin {\frac {3\pi }{7}}\cos \left({\frac {k\pi }{9}}\right),\,\pm 2\sin {\frac {3\pi }{7}}\sin \left({\frac {k\pi }{9}}\right)\right),$
• $\left(2\sin {\frac {\pi }{9}}\cos \left({\frac {j\pi }{7}}\right),\,\pm 2\sin {\frac {\pi }{9}}\sin \left({\frac {j\pi }{7}}\right),\,-\sin {\frac {3\pi }{7}},\,\pm {\sqrt {3}}\sin {\frac {3\pi }{7}}\right),$

where j = 2, 4, 6 and k = 2, 4, 8.

External links

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

• Klitzing, Richard. "nd-mb-dip".