Enneagonal-decagonal duoprismatic prism

Enneagonal-decagonal duoprismatic prism
(OFF file)
Rank	5
Type	Uniform
Notation
Bowers style acronym	Eddip
Coxeter diagram	x x9o x10o ()
Elements
Tera	10 square-enneagonal duoprisms, 9 square-decagonal duoprisms, 2 enneagonal-decagonal duoprisms
Cells	90 cubes, 9+18 decagonal prisms, 10+20 enneagonal prisms
Faces	90+90+180 squares, 20 enneagons, 18 decagons
Edges	90+180+180
Vertices	180
Vertex figure	Digonal disphenoidal pyramid, edge lengths 2cos(π/9) (disphenoid base 1), √(5+√5)/2 (disphenoid base 2), √2 (remaining edges)
Measures (edge length 1)
Circumradius	${\frac {\sqrt {7+2{\sqrt {5}}+{\frac {1}{\sin ^{2}{\frac {\pi }{9}}}}}}{2}}\approx 2.23723$
Hypervolume	${\frac {45{\sqrt {5+2{\sqrt {5}}}}}{8\tan {\frac {\pi }{9}}}}\approx 47.56425$
Diteral angles	Sendip–ep–sendip: 144°
	Squadedip–dip–squadedip: 140°
	Squadedip–cube–sendip: 90°
	Edidip–ep–sendip: 90°
	Squadedip–dip–edidip: 90°
Height	1
Central density	1
Number of external pieces	21
Level of complexity	30
Related polytopes
Army	Eddip
Regiment	Eddip
Dual	Enneagonal-decagonal duotegmatic tegum
Conjugates	Enneagonal-decagrammic duoprismatic prism, Enneagrammic-decagonal duoprismatic prism, Enneagrammic-decagrammic duoprismatic prism, Great enneagrammic-decagonal duoprismatic prism, Great enneagrammic-decagrammic duoprismatic prism
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	I2(9)×I2(10)×A1, order 720
Convex	Yes
Nature	Tame

The enneagonal-decagonal duoprismatic prism or eddip, also known as the enneagonal-decagonal prismatic duoprism, is a convex uniform duoprism that consists of 2 enneagonal-decagonal duoprisms, 9 square-decagonal duoprisms, and 10 square-enneagonal duoprisms. Each vertex joins 2 square-enneagonal duoprisms, 2 square-decagonal duoprisms, and 1 enneagonal-decagonal duoprism. Being a prism based on an orbiform polytope, it is also a convex segmentoteron.

Vertex coordinates

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

• $\left(0,\,1,\,0,\,\pm (1+{\sqrt {5}})\sin {\frac {\pi }{9}},\,\pm \sin {\frac {\pi }{9}}\right),$
• $\left(0,\,1,\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{2}}}\sin {\frac {\pi }{9}},\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{9}}}{2}},\,\pm \sin {\frac {\pi }{9}}\right),$
• $\left(0,\,1,\,\pm {\sqrt {5+2{\sqrt {5}}}}\sin {\frac {\pi }{9}},\,\pm \sin {\frac {\pi }{9}},\,\pm \sin {\frac {\pi }{9}}\right),$
• $\left(\cos \left({\frac {j\pi }{9}}\right),\,\pm \sin \left({\frac {j\pi }{9}}\right),\,0,\,\pm (1+{\sqrt {5}})\sin {\frac {\pi }{9}},\,\pm \sin {\frac {\pi }{9}}\right),$
• $\left(\cos \left({\frac {j\pi }{9}}\right),\,\pm \sin \left({\frac {j\pi }{9}}\right),\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{2}}}\sin {\frac {\pi }{9}},\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{9}}}{2}},\,\pm \sin {\frac {\pi }{9}}\right),$
• $\left(\cos \left({\frac {j\pi }{9}}\right),\,\pm \sin \left({\frac {j\pi }{9}}\right),\,\pm {\sqrt {5+2{\sqrt {5}}}}\sin {\frac {\pi }{9}},\,\pm \sin {\frac {\pi }{9}},\,\pm \sin {\frac {\pi }{9}}\right),$
• $\left(-{\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,0,\,\pm (1+{\sqrt {5}})\sin {\frac {\pi }{9}},\,\pm \sin {\frac {\pi }{9}}\right),$
• $\left(-{\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{2}}}\sin {\frac {\pi }{9}},\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{9}}}{2}},\,\pm \sin {\frac {\pi }{9}}\right),$
• $\left(-{\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm {\sqrt {5+2{\sqrt {5}}}}\sin {\frac {\pi }{9}},\,\pm \sin {\frac {\pi }{9}},\,\pm \sin {\frac {\pi }{9}}\right),$

where j = 2, 4, 8.

Representations

An enneagonal-decagonal duoprismatic prism has the following Coxeter diagrams:

• x x9o x10o () (full symmetry)
• x x9o x5x () (decagons as dipentagons)
• xx9oo xx10oo&#x (enneagonal-decagonal duoprism atop enneagonal-decagonal duoprism)
• xx9oo xx5xx&#x