Enneagonal duoprismatic prism

Enneagonal duoprismatic prism
	(OFF file)
Rank	5
Type	Uniform
Notation
Bowers style acronym	Eep
Coxeter diagram	x x9o x9o ()
Elements
Tera	18 square-enneagonal duoprisms, 2 enneagonal duoprisms
Cells	81 cubes, 18+36 enneagonal prisms
Faces	162+162 squares, 36 enneagons
Edges	81+324
Vertices	162
Vertex figure	Tetragonal disphenoidal pyramid, edge lengths 2cos(π/9) (disphenoid bases) and √2 (remaining edges)
Measures (edge length 1)
Circumradius
Hypervolume
Diteral angles	Sendip–ep–sendip: 140°
	Sendip–cube–sendip: 90°
	Edip–ep–sendip: 90°
Height	1
Central density	1
Number of external pieces	20
Level of complexity	15
Related polytopes
Army	Eep
Regiment	Eep
Dual	Enneagonal duotegmatic tegum
Conjugates	Enneagrammic duoprismatic prism, Great enneagrammic duoprismatic prism
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	I2(9)≀S2×A1, order 1296
Convex	Yes
Nature	Tame

The enneagonal duoprismatic prism or eep, also known as the enneagonal-enneagonal prismatic duoprism, is a convex uniform duoprism that consists of 2 enneagonal duoprisms and 18 square-enneagonal duoprisms. Each vertex joins 4 square-enneagonal duoprisms and 1 enneagonal duoprism. Being a prism based on an orbiform polytope, it is also a convex segmentoteron.

Vertex coordinates[edit | edit source]

The vertices of an enneagonal duoprismatic prism of edge length 2sin(π/9) are given by:

$\left(1,\,0,\,1,\,0,\,\pm \sin {\frac {\pi }{9}}\right),$
$\left(1,\,0,\,\cos {\frac {k\pi }{9}},\,\pm \sin {\frac {k\pi }{9}},\,\pm \sin {\frac {\pi }{9}}\right),$
$\left(1,\,0,\,-{\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm \sin {\frac {\pi }{9}}\right),$
$\left(\cos {\frac {j\pi }{9}},\,\pm \sin {\frac {j\pi }{9}},\,1,\,0,\,\pm \sin {\frac {\pi }{9}}\right),$
$\left(\cos {\frac {j\pi }{9}},\,\pm \sin {\frac {j\pi }{9}},\,\cos {\frac {k\pi }{9}},\,\pm \sin {\frac {k\pi }{9}},\,\pm \sin {\frac {\pi }{9}}\right),$
$\left(\cos {\frac {j\pi }{9}},\,\pm \sin {\frac {j\pi }{9}},\,-{\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm \sin {\frac {\pi }{9}}\right),$
$\left(-{\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,1,\,0,\,\pm \sin {\frac {\pi }{9}}\right),$
$\left(-{\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,\cos {\frac {k\pi }{9}},\,\pm \sin {\frac {k\pi }{9}},\,\pm \sin {\frac {\pi }{9}}\right),$
$\left(-{\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,-{\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm \sin {\frac {\pi }{9}}\right),$