# Enneagonal-great rhombicuboctahedral duoprism

Enneagonal-great rhombicuboctahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymEgirco
Coxeter diagramx9o x4x3x ()
Elements
Tera12 square-enneagonal duoprisms, 8 hexagonal-enneagonal duoprisms, 6 octagonal-enneagonal duoprisms
Cells108 cubes, 72 hexagonal prisms, 54 octagonal prisms, 24+24+24 enneagonal prisms, 9 great rhombicuboctahedra
Faces108+216+216+216 squares, 72 hexagons, 54 octagons
Edges216+216+216+432
Vertices432
Vertex figureMirror-symmetric pentachoron, edge lengths 2, 3, 2+2 (base triangle), 2cos(π/9) (top edge), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {13+6{\sqrt {2}}+{\frac {1}{\sin ^{2}{\frac {\pi }{9}}}}}}{2}}\approx 2.74016}$
Hypervolume${\displaystyle 9{\frac {11+7{\sqrt {2}}}{2\tan {\frac {\pi }{9}}}}\approx 258.39401}$
Diteral anglesSendip–ep–hendip: ${\displaystyle \arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }}$
Gircope–girco–gircope: 140°
Sendip–ep–oedip: 135°
Hendip–ep–oedip: ${\displaystyle \arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }}$
Sendip–cube–gircope: 90°
Hendip–hip–gircope: 90°
Oedip–op–gircope: 90°
Central density1
Number of external pieces35
Level of complexity60
Related polytopes
ArmyEgirco
RegimentEgirco
DualEnneagonal-disdyakis dodecahedral duotegum
ConjugatesEnneagrammic-great rhombicuboctahedral duoprism, Great enneagrammic-great rhombicuboctahedral duoprism, Enneagonal-quasitruncated cuboctahedral duoprism, Enneagrammic-quasitruncated cuboctahedral duoprism, Great enneagrammic-quasitruncated cuboctahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryB3×I2(9), order 864
ConvexYes
NatureTame

The enneagonal-great rhombicuboctahedral duoprism or egirco is a convex uniform duoprism that consists of 9 great rhombicuboctahedral prisms, 6 octagonal-enneagonal duoprisms, 8 hexagonal-enneagonal duoprisms, and 12 square-enneagonal duoprisms. Each vertex joins 2 great rhombicuboctahedral prisms, 1 square-enneagonal duoprism, 1 hexagonal-enneagonal duoprism, and 1 octagonal-enneagonal duoprism.

## Vertex coordinates

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

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

where j = 2, 4, 8.