Enneagonal-octahedral duoprism

Enneagonal-octahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymEoct
Coxeter diagramx9o o4o3x ()
Elements
Tera9 octahedral prisms, 8 triangular-enneagonal duoprisms
Cells72 triangular prisms, 9 octahedra, 12 enneagonal prisms
Faces72 triangles, 108 squares, 6 enneagons
Edges54+108
Vertices54
Vertex figureSquare scalene, edge lengths 1 (base square), 2cos(π/9) (top), 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {2+{\frac {1}{\sin ^{2}{\frac {\pi }{9}}}}}}{2}}\approx 1.62393}$
Hypervolume${\displaystyle {\frac {3{\sqrt {2}}}{4\tan {\frac {\pi }{9}}}}\approx 2.91414}$
Diteral anglesOpe–oct–ope: 140°
Tedip–ep–tedip: ${\displaystyle \arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }}$
Tedip–trip–ope: 90°
Height${\displaystyle {\frac {\sqrt {6}}{3}}\approx 0.81650}$
Central density1
Number of external pieces17
Level of complexity10
Related polytopes
ArmyEoct
RegimentEoct
DualEnneagonal-cubic duotegum
ConjugatesEnneagrammic-octahedral duoprism, Great enneagrammic-octahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryB3×I2(9), order 864
ConvexYes
NatureTame

The enneagonal-octahedral duoprism or eoct is a convex uniform duoprism that consists of 9 octahedral prisms and 8 triangular-enneagonal duoprisms. Each vertex joins 2 octahedral prisms and 4 triangular-enneagonal duoprisms.

Vertex coordinates

The vertices of an enneagonal-octahedral duoprism of edge length 2sin(π/9) are given by all permutations and sign changes of the last three coordinates of:

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

where j = 2, 4, 8.