Enneagonal-truncated octahedral duoprism

Enneagonal-truncated octahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymEtoe
Coxeter diagramx9o o4x3x ()
Elements
Tera6 square-enneagonal duoprisms, 8 truncated octahedral prisms, 8 hexagonal-enneagonal duoprisms
Cells54 cubes, 72 hexagonal prisms, 12+24 enneagonal prisms, 9 truncated octahedra
Faces54+108+216 squares, 72 hexagons, 24 enneagons
Edges108+216+216
Vertices216
Vertex figureDigonal disphenoidal pyramid, edge lengths 2, 3, 3 (base triangle), cos(π/9) (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {10+{\frac {1}{\sin ^{2}{\frac {\pi }{9}}}}}}{2}}\approx 2.15341}$
Hypervolume${\displaystyle {\frac {18{\sqrt {2}}}{\tan {\frac {\pi }{9}}}}\approx 69.93936}$
Diteral anglesTope–toe–tope: 140°
Sendip–ep–hendip: ${\displaystyle \arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }}$
Hendip–ep–hendip: ${\displaystyle \arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }}$
Sendip–cube–tope: 90°
Hendip–hip–tope: 90°
Central density1
Number of external pieces23
Level of complexity30
Related polytopes
ArmyEtoe
RegimentEtoe
DualEnneagonal-tetrakis hexahedral duotegum
ConjugatesEnneagrammic-truncated octahedral duoprism, Great enneagrammic-truncated octahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryB3×I2(9), order 864
ConvexYes
NatureTame

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 ${\displaystyle 2\sin(\pi /9)}$ are given by all permutations of the last three coordinates of:

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

where j  = 2, 4, 8.