Hexagonal-enneagonal duoprismatic prism

Hexagonal-enneagonal duoprismatic prism
(OFF file)
Rank	5
Type	Uniform
Notation
Bowers style acronym	Haep
Coxeter diagram	x x6o x9o
Elements
Tera	9 square-hexagonal duoprisms, 6 square-enneagonal duoprisms, 2 hexagonal-enneagonal duoprisms
Cells	54 cubes, 6+12 enneagonal prisms, 9+18 hexagonal prisms
Faces	54+54+108 squares, 18 hexagons, 12 enneagons
Edges	54+108+108
Vertices	108
Vertex figure	Digonal disphenoidal pyramid, edge lengths √3 (disphenoid base 1), 2cos(π/9) (disphenoid base 2), √2 (remaining edges)
Measures (edge length 1)
Circumradius	${\frac {\sqrt {5+{\frac {1}{\sin ^{2}{\frac {\pi }{9}}}}}}{2}}\approx 1.84042$
Hypervolume	${\frac {27{\sqrt {3}}}{8\tan {\frac {\pi }{9}}}}\approx 16.06085$
Diteral angles	Shiddip–hip–shiddip: 140°
	Sendip–ep–sendip: 120°
	Sendip–cube–shiddip: 90°
	Hendip–hip–shiddip: 90°
	Sendip–ep–hendip: 90°
Height	1
Central density	1
Number of external pieces	17
Level of complexity	30
Related polytopes
Army	Haep
Regiment	Haep
Dual	Hexagonal-enneagonal duotegmatic tegum
Conjugates	Hexagonal-enneagrammic duoprismatic prism, Hexagonal-great enneagrammic duoprismatic prism
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	G2×I2(9)×A1, order 432
Convex	Yes
Nature	Tame

The hexagonal-enneagonal duoprismatic prism or haep, also known as the hexagonal-enneagonal prismatic duoprism, is a convex uniform duoprism that consists of 2 hexagonal-enneagonal duoprisms, 6 square-enneagonal duoprisms, and 9 square-hexagonal duoprisms. Each vertex joins 2 square-hexagonal duoprisms, 2 square-enneagonal duoprisms, and 1 hexagonal-enneagonal duoprism. Being a prism based on an orbiform polytope, it is also a convex segmentoteron.

Vertex coordinates

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

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

where j = 2, 4, 8.

Representations

A hexagonal-enneagonal duoprismatic prism has the following Coxeter diagrams:

• x x6o x9o (full symmetry)
• x x3x x9o (hexagons as ditrigons)
• xx6oo xx9oo&#x (hexagonal-enneagonal duoprism atop hexagonal-enneagonal duoprism)
• xx3xx xx9oo&#x