Enneagonal-truncated octahedral duoprism

Enneagonal-truncated octahedral duoprism
(OFF file)
Rank	5
Type	Uniform
Notation
Bowers style acronym	Etoe
Coxeter diagram	x9o o4x3x ()
Elements
Tera	6 square-enneagonal duoprisms, 8 truncated octahedral prisms, 8 hexagonal-enneagonal duoprisms
Cells	54 cubes, 72 hexagonal prisms, 12+24 enneagonal prisms, 9 truncated octahedra
Faces	54+108+216 squares, 72 hexagons, 24 enneagons
Edges	108+216+216
Vertices	216
Vertex figure	Digonal disphenoidal pyramid, edge lengths √2, √3, √3 (base triangle), cos(π/9) (top), √2 (side edges)
Measures (edge length 1)
Circumradius	${\frac {\sqrt {10+{\frac {1}{\sin ^{2}{\frac {\pi }{9}}}}}}{2}}\approx 2.15341$
Hypervolume	${\frac {18{\sqrt {2}}}{\tan {\frac {\pi }{9}}}}\approx 69.93936$
Diteral angles	Tope–toe–tope: 140°
	Sendip–ep–hendip: $\arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }$
	Hendip–ep–hendip: $\arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }$
	Sendip–cube–tope: 90°
	Hendip–hip–tope: 90°
Central density	1
Number of external pieces	23
Level of complexity	30
Related polytopes
Army	Etoe
Regiment	Etoe
Dual	Enneagonal-tetrakis hexahedral duotegum
Conjugates	Enneagrammic-truncated octahedral duoprism, Great enneagrammic-truncated octahedral duoprism
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	B3×I2(9), order 864
Convex	Yes
Nature	Tame

The enneagonal-truncated octahedral duoprism or etoe is a convex uniform duoprism that consists of 9 truncated octahedral prisms, 8 hexagonal-enneagonal duoprisms, and 6 square-enneagonal duoprisms. Each vertex joins 2 truncated octahedral prisms, 1 square-enneagonal duoprism, and 2 hexagonal-enneagonal duoprisms.

Vertex coordinates

The vertices of an enneagonal-truncated octahedral duoprism of edge length $2\sin(\pi /9)$ are given by all permutations of the last three coordinates of:

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

where j  = 2, 4, 8.