# Enneagonal-small rhombicuboctahedral duoprism

Enneagonal-small rhombicuboctahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymEsirco
Coxeter diagramx9o x4o3x ()
Elements
Tera8 triangular-enneagonal duoprisms, 6+12 square-enneagonal duoprisms, 9 small rhombicuboctahedral prisms
Cells72 triangular prisms, 54+108 cubes, 24+24 enneagonal prisms, 9 small rhombicuboctahedra
Faces72 triangles, 54+108+216+216 squares, 24 enneagons
Edges216+216+216
Vertices216
Vertex figureIsosceles-trapezoidal scalene, edge lengths 1, 2, 2, 2 (base trapezoid), 2cos(π/9) (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {5+2{\sqrt {2}}+{\frac {1}{\sin ^{2}{\frac {\pi }{9}}}}}}{2}}\approx 2.02343}$
Hypervolume${\displaystyle 3{\frac {18+15{\sqrt {2}}}{2\tan {\frac {\pi }{9}}}}\approx 53.86870}$
Diteral anglesTedip–ep–sendip: ${\displaystyle \arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }}$
Sircope–sirco–sircope: 140°
Sendip–ep–sendip: 135°
Tedip–trip–sircope: 90°
Sendip–cube–sircope: 90°
Central density1
Number of external pieces35
Level of complexity40
Related polytopes
ArmyEsirco
RegimentEsirco
DualEnneagonal-deltoidal icositetrahedral duotegum
ConjugatesEnneagrammic-small rhombicuboctahedral duoprism, Great enneagrammic-small rhombicuboctahedral duoprism, Enneagonal-quasirhombicuboctahedral duoprism, Enneagrammic-quasirhombicuboctahedral duoprism, Great enneagrammic-quasirhombicuboctahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryB3×I2(9), order 864
ConvexYes
NatureTame

The enneagonal-small rhombicuboctahedral duoprism or esirco is a convex uniform duoprism that consists of 9 small rhombicuboctahedral prisms, 18 square-enneagonal duoprisms of two kinds, and 8 triangular-enneagonal duoprisms. Each vertex joins 2 small rhombicuboctahedral prisms, 1 triangular-enneagonal duoprism, and 3 square-enneagonal duoprisms.

## Vertex coordinates

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

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

where j = 2, 4, 8.