Enneagonal-great rhombicuboctahedral duoprism Rank 5 Type Uniform Notation Bowers style acronym Egirco Coxeter diagram x9o x4x3x ( Elements Tera 12 square-enneagonal duoprisms , 8 hexagonal-enneagonal duoprisms , 6 octagonal-enneagonal duoprisms Cells 108 cubes , 72 hexagonal prisms , 54 octagonal prisms , 24+24+24 enneagonal prisms , 9 great rhombicuboctahedra Faces 108+216+216+216 squares , 72 hexagons , 54 octagons Edges 216+216+216+432 Vertices 432 Vertex figure Mirror-symmetric pentachoron , edge lengths √2 , √3 , √2+√2 (base triangle), 2cos(π/9) (top edge), √2 (side edges) Measures (edge length 1) Circumradius
13
+
6
2
+
1
sin
2
π
9
2
≈
2.74016
{\displaystyle {\frac {\sqrt {13+6{\sqrt {2}}+{\frac {1}{\sin ^{2}{\frac {\pi }{9}}}}}}{2}}\approx 2.74016}
Hypervolume
9
11
+
7
2
2
tan
π
9
≈
258.39401
{\displaystyle 9{\frac {11+7{\sqrt {2}}}{2\tan {\frac {\pi }{9}}}}\approx 258.39401}
Diteral angles Sendip–ep–hendip:
arccos
(
−
6
3
)
≈
144.73561
∘
{\displaystyle \arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }}
Gircope–girco–gircope: 140° Sendip–ep–oedip: 135° Hendip–ep–oedip:
arccos
(
−
3
3
)
≈
125.26439
∘
{\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 density 1 Number of external pieces 35 Level of complexity 60 Related polytopes Army Egirco Regiment Egirco Dual Enneagonal-disdyakis dodecahedral duotegum Conjugates Enneagrammic-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 characteristic 2 Orientable Yes Properties Symmetry B3 ×I2(9) , order 864Convex Yes Nature Tame
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.
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:
(
1
,
0
,
±
(
1
+
2
2
)
sin
π
9
,
±
(
1
+
2
)
sin
π
9
,
±
sin
π
9
)
,
{\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),}
(
cos
(
j
π
9
)
,
±
sin
(
j
π
9
)
,
±
(
1
+
2
2
)
sin
π
9
,
±
(
1
+
2
)
sin
π
9
,
±
sin
π
9
)
,
{\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),}
(
−
1
2
,
±
3
2
,
±
(
1
+
2
2
)
sin
π
9
,
±
(
1
+
2
)
sin
π
9
,
±
sin
π
9
)
,
{\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.
