# Enneagonal-cubic duoprism

Enneagonal-cubic duoprism
Rank5
TypeUniform
Notation
Bowers style acronymEcube
Coxeter diagramx9o x4o3o ()
Elements
Tera9 tesseracts, 6 square-enneagonal duoprisms
Cells9+54 cubes, 12 enneagonal prisms
Faces54+108 squares, 8 enneagons
Edges72+108
Vertices72
Vertex figureTriangular scalene, edge lengths 2cos(π/9) (top), 2 (base triangle and sides)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {3+{\frac {1}{\sin ^{2}{\frac {\pi }{9}}}}}}{2}}\approx 1.69916}$
Hypervolume${\displaystyle {\frac {9}{4\tan {\frac {\pi }{9}}}}\approx 6.18182}$
Diteral anglesTes–cube–tes: 140°
Tes–cube–sendip: 90°
Sendip–ep–sendip: 90°
Height1
Central density1
Number of external pieces15
Level of complexity10
Related polytopes
ArmyEcube
RegimentEcube
DualEnneagonal-octahedral duotegum
ConjugatesEnneagrammic-cubic duoprism, Great enneagrammic-cubic duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryB3×I2(9), order 864
ConvexYes
NatureTame

The enneagonal-cubic duoprism or ecube, also known as a square-enneagonal duoprismatic prism, is a convex uniform duoprism that consists of 9 tesseracts and 6 square-enneagonal duoprisms. Each vertex joins 2 tesseracts and 3 square-enneagonal duoprisms. It is a duoprism based on a square and an enneagonal prism, which makes it a convex segmentoteron.

## Vertex coordinates

The vertices of an enneagonal-cubic duoprism of edge length 2sin(π/9) are given by:

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

where j = 2, 4, 8.

## Representations

A triangular-cubic duoprism has the following Coxeter diagrams:

• x9o x4o3o () (full symmetry)
• x x4o x9o () (square-enneagonal duoprismatic prism)
• x x x x9o () (enneagonal prismatic prismatic prism)