Heptagonal-small rhombicosidodecahedral duoprism Rank 5 Type Uniform Notation Bowers style acronym Hesrid Coxeter diagram x7o x5o3x Elements Tera 20 triangular-heptagonal duoprisms , 30 square-heptagonal duoprisms , 12 pentagonal-heptagonal duoprisms Cells 140 triangular prisms , 210 cubes , 84 pentagonal prisms , 60+60 heptagonal prisms , 7 small rhombicosidodecahedra Faces 140 triangles , 210+420+420 squares , 84 pentagons , 60 heptagons Edges 420+420+420 Vertices 420 Vertex figure Isosceles-trapezoidal scalene , edge lengths 1, √2 , (1+√5 )/2, √2 (base trapezoid), 2cos(π/7) (top), √2 (side edges)Measures (edge length 1) Circumradius ${\frac {\sqrt {11+4{\sqrt {5}}+{\frac {1}{\sin ^{2}{\frac {\pi }{7}}}}}}{2}}\approx 2.51278$ Hypervolume $7{\frac {60+29{\sqrt {5}}}{12\tan {\frac {\pi }{7}}}}\approx 151.22644$ Diteral angles Theddip–hep–squahedip: $\arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 159.09484^{\circ }$ Squahedip–hep–pheddip: $\arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }$ Sriddip–srid–sriddip: ${\frac {5\pi }{7}}\approx 128.57143^{\circ }$ Theddip–trip–sriddip: 90° Squahedip–cube–sriddip: 90° Pheddip–pip–sriddip: 90° Central density 1 Number of external pieces 69 Level of complexity 40 Related polytopes Army Hesrid Regiment Hesrid Dual Heptagonal-deltoidal hexecontahedral duotegum Conjugates Heptagrammic-small rhombicosidodecahedral duoprism , Great heptagrammic-small rhombicosidodecahedral duoprism , Heptagonal-quasirhombicosidodecahedral duoprism , Heptagrammic-quasirhombicosidodecahedral duoprism , Great heptagrammic-quasirhombicosidodecahedral duoprism Abstract & topological properties Euler characteristic 2 Orientable Yes Properties Symmetry H_{3} ×I_{2} (7) , order 1680Convex Yes Nature Tame

The heptagonal-small rhombicosidodecahedral duoprism or hesrid is a convex uniform duoprism that consists of 7 small rhombicosidodecahedral prisms , 12 pentagonal-heptagonal duoprisms , 30 square-heptagonal duoprisms , and 20 triangular-heptagonal duoprisms . Each vertex joins 2 small rhombicosidodecahedral prisms, 1 triangular-heptagonal duoprism, 2 square-heptagonal duoprisms, and 1 pentagonal-heptagonal duoprism.

The vertices of a heptagonal-small rhombicosidodecahedral duoprism of edge length 2sin(π/7) are given by all permutations of the last three coordinates of:

$\left(1,\,0,\,\pm \sin {\frac {\pi }{7}},\,\pm \sin {\frac {\pi }{7}},\,\pm (2+{\sqrt {5}})\sin {\frac {\pi }{7}}\right),$
$\left(\cos \left({\frac {j\pi }{7}}\right),\,\pm \sin \left({\frac {j\pi }{7}}\right),\,\pm \sin {\frac {\pi }{7}},\,\pm \sin {\frac {\pi }{7}},\,\pm (2+{\sqrt {5}})\sin {\frac {\pi }{7}}\right),$
as well as all even permutations of the last three coordinates of:

$\left(1,\,0,\,0,\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}},\,\pm {\frac {(5+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}}\right),$
$\left(\cos \left({\frac {j\pi }{7}}\right),\,\pm \sin \left({\frac {j\pi }{7}}\right),\,0,\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}},\,\pm {\frac {(5+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}}\right),$
$\left(1,\,0,\,\pm {\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}},\,\pm (1+{\sqrt {5}})\sin {\frac {\pi }{7}},\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}}\right),$
$\left(\cos \left({\frac {j\pi }{7}}\right),\,\pm \sin \left({\frac {j\pi }{7}}\right),\,\pm {\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}},\,\pm (1+{\sqrt {5}})\sin {\frac {\pi }{7}},\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}}\right),$
where j = 2, 4, 6.