Enneagonal-small rhombicuboctahedral duoprism Rank 5 Type Uniform Notation Bowers style acronym Esirco Coxeter diagram x9o x4o3x ( Elements Tera 8 triangular-enneagonal duoprisms , 6+12 square-enneagonal duoprisms , 9 small rhombicuboctahedral prisms Cells 72 triangular prisms , 54+108 cubes , 24+24 enneagonal prisms , 9 small rhombicuboctahedra Faces 72 triangles , 54+108+216+216 squares , 24 enneagons Edges 216+216+216 Vertices 216 Vertex figure Isosceles-trapezoidal scalene , edge lengths 1, √2 , √2 , √2 (base trapezoid), 2cos(π/9) (top), √2 (side edges)Measures (edge length 1) Circumradius
5
+
2
2
+
1
sin
2
π
9
2
≈
2.02343
{\displaystyle {\frac {\sqrt {5+2{\sqrt {2}}+{\frac {1}{\sin ^{2}{\frac {\pi }{9}}}}}}{2}}\approx 2.02343}
Hypervolume
3
18
+
15
2
2
tan
π
9
≈
53.86870
{\displaystyle 3{\frac {18+15{\sqrt {2}}}{2\tan {\frac {\pi }{9}}}}\approx 53.86870}
Diteral angles Tedip–ep–sendip:
arccos
(
−
6
3
)
≈
144.73561
∘
{\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 density 1 Number of external pieces 35 Level of complexity 40 Related polytopes Army Esirco Regiment Esirco Dual Enneagonal-deltoidal icositetrahedral duotegum Conjugates Enneagrammic-small rhombicuboctahedral duoprism , Great enneagrammic-small rhombicuboctahedral duoprism , Enneagonal-quasirhombicuboctahedral duoprism , Enneagrammic-quasirhombicuboctahedral duoprism , Great enneagrammic-quasirhombicuboctahedral duoprism Abstract & topological properties Euler characteristic 2 Orientable Yes Properties Symmetry B3 ×I2 (9) , order 864Convex Yes Nature Tame
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.
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:
(
1
,
0
,
±
sin
π
9
,
±
sin
π
9
,
±
(
1
+
2
)
sin
π
9
)
,
{\displaystyle \left(1,\,0,\,\pm \sin {\frac {\pi }{9}},\,\pm \sin {\frac {\pi }{9}},\,\pm (1+{\sqrt {2}})\sin {\frac {\pi }{9}}\right),}
(
cos
(
j
π
9
)
,
±
sin
(
j
π
9
)
,
±
sin
π
9
,
±
sin
π
9
,
±
(
1
+
2
)
sin
π
9
)
,
{\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),}
(
−
1
2
,
±
3
2
,
±
sin
π
9
,
±
sin
π
9
,
±
(
1
+
2
)
sin
π
9
)
,
{\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.
