Heptagonal-small rhombicuboctahedral duoprism Rank 5 Type Uniform Notation Bowers style acronym Hesirco Coxeter diagram x7o x4o3x Elements Tera 8 triangular-heptagonal duoprisms , 6+12 square-heptagonal duoprisms , 7 small rhombicuboctahedral prisms Cells 56 triangular prisms , 42+84 cubes , 24+24 heptagonal prisms , 7 small rhombicuboctahedra Faces 56 triangles , 42+84+168+168 squares , 24 heptagons Edges 168+168+168 Vertices 168 Vertex figure Isosceles-trapezoidal scalene , edge lengths 1, √2 , √2 , √2 (base trapezoid), 2cos(π/7) (top), √2 (side edges)Measures (edge length 1) Circumradius
5
+
2
2
+
1
sin
2
π
7
2
≈
1.81248
{\displaystyle {\frac {\sqrt {5+2{\sqrt {2}}+{\frac {1}{\sin ^{2}{\frac {\pi }{7}}}}}}{2}}\approx 1.81248}
Hypervolume
7
6
+
5
2
6
tan
π
7
≈
31.66608
{\displaystyle 7{\frac {6+5{\sqrt {2}}}{6\tan {\frac {\pi }{7}}}}\approx 31.66608}
Diteral angles Theddip–hep–squahedip:
arccos
(
−
6
3
)
≈
144.73561
∘
{\displaystyle \arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }}
Squahedip–hep–squahedip: 135° Sircope–sirco–sircope:
5
π
7
≈
128.57143
∘
{\displaystyle {\frac {5\pi }{7}}\approx 128.57143^{\circ }}
Theddip–trip–sircope: 90° Squahedip–cube–sircope: 90° Central density 1 Number of external pieces 33 Level of complexity 40 Related polytopes Army Hesirco Regiment Hesirco Dual Heptagonal-deltoidal icositetrahedral duotegum Conjugates Heptagrammic-small rhombicuboctahedral duoprism , Great heptagrammic-small rhombicuboctahedral duoprism , Heptagonal-quasirhombicuboctahedral duoprism , Heptagrammic-quasirhombicuboctahedral duoprism , Great heptagrammic-quasirhombicuboctahedral duoprism Abstract & topological properties Euler characteristic 2 Orientable Yes Properties Symmetry B3 ×I2 (7) , order 672Convex Yes Nature Tame
The heptagonal-small rhombicuboctahedral duoprism or hesirco is a convex uniform duoprism that consists of 7 small rhombicuboctahedral prisms , 18 square-heptagonal duoprisms of two kinds, and 8 triangular-heptagonal duoprisms . Each vertex joins 2 small rhombicuboctahedral prisms, 1 triangular-heptagonal duoprism, and 3 square-heptagonal duoprisms.
The vertices of a heptagonal-small rhombicuboctahedral duoprism of edge length 2sin(π/7) are given by all permutations of the last three coordinates of:
(
1
,
0
,
±
sin
π
7
,
±
sin
π
7
,
±
(
1
+
2
)
sin
π
7
)
,
{\displaystyle \left(1,\,0,\,\pm \sin {\frac {\pi }{7}},\,\pm \sin {\frac {\pi }{7}},\,\pm (1+{\sqrt {2}})\sin {\frac {\pi }{7}}\right),}
(
cos
(
j
π
7
)
,
±
sin
(
j
π
7
)
,
±
sin
π
7
,
±
sin
π
7
,
±
(
1
+
2
)
sin
π
7
)
,
{\displaystyle \left(\cos \left({\frac {j\pi }{7}}\right),\,\pm \sin \left({\frac {j\pi }{7}}\right),\,\pm \sin {\frac {\pi }{7}},\,\pm \sin {\frac {\pi }{7}},\,\pm (1+{\sqrt {2}})\sin {\frac {\pi }{7}}\right),}
where j = 2, 4, 6.