Enneagonal-cuboctahedral duoprism

Enneagonal-cuboctahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymEco
Coxeter diagramx9o o4x3o ()
Elements
Tera9 cuboctahedral prisms, 8 triangular-enneagonal duoprisms, 6 square-enneagonal duoprisms
Cells72 triangular prisms, 54 cubes, 9 cuboctahedra, 24 enneagonal prisms
Faces72 triangles, 54+216 squares, 12 enneagons
Edges108+216
Vertices108
Vertex figureRectangular scalene, edge lengths 1, 2, 1, 2 (base rectangle), 2cos(π/9) (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {4+{\frac {1}{\sin ^{2}{\frac {\pi }{9}}}}}}{2}}\approx 1.77120}$
Hypervolume${\displaystyle {\frac {15{\sqrt {2}}}{4\tan {\frac {\pi }{9}}}}\approx 14.57070}$
Diteral anglesCope–co–cope: 140°
Tedip–ep–sendip: ${\displaystyle \arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }}$
Tedip–trip–cope: 90°
Sendip–cube–cope: 90°
Central density1
Number of external pieces23
Level of complexity20
Related polytopes
ArmyEco
RegimentEco
DualEnneagonal-rhombic dodecahedral duotegum
ConjugatesEnneagrammic-cuboctahedral duoprism, Great enneagrammic-cuboctahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryB3×I2(9), order 864
ConvexYes
NatureTame

The enneagonal-cuboctahedral duoprism or eco is a convex uniform duoprism that consists of 9 cuboctahedral prisms, 6 square-enneagonal duoprisms, and 8 triangular-enneagonal duoprisms. Each vertex joins 2 cuboctahedral prisms, 2 triangular-enneagonal duoprisms, and 2 square-enneagonal duoprisms.

Vertex coordinates

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

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

where j = 2, 4, 8.

Representations

An enneagonal-cuboctahedral duoprism has the following Coxeter diagrams:

• x9o o4x3o () (full symmetry)
• x9o x3o3x ()