# Heptagonal-small rhombicosidodecahedral duoprism

Heptagonal-small rhombicosidodecahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymHesrid
Coxeter diagramx7o x5o3x
Elements
Tera20 triangular-heptagonal duoprisms, 30 square-heptagonal duoprisms, 12 pentagonal-heptagonal duoprisms
Cells140 triangular prisms, 210 cubes, 84 pentagonal prisms, 60+60 heptagonal prisms, 7 small rhombicosidodecahedra
Faces140 triangles, 210+420+420 squares, 84 pentagons, 60 heptagons
Edges420+420+420
Vertices420
Vertex figureIsosceles-trapezoidal scalene, edge lengths 1, 2, (1+5)/2, 2 (base trapezoid), 2cos(π/7) (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {11+4{\sqrt {5}}+{\frac {1}{\sin ^{2}{\frac {\pi }{7}}}}}}{2}}\approx 2.51278}$
Hypervolume${\displaystyle 7{\frac {60+29{\sqrt {5}}}{12\tan {\frac {\pi }{7}}}}\approx 151.22644}$
Diteral anglesTheddip–hep–squahedip: ${\displaystyle \arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 159.09484^{\circ }}$
Squahedip–hep–pheddip: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }}$
Sriddip–srid–sriddip: ${\displaystyle {\frac {5\pi }{7}}\approx 128.57143^{\circ }}$
Theddip–trip–sriddip: 90°
Squahedip–cube–sriddip: 90°
Pheddip–pip–sriddip: 90°
Central density1
Number of external pieces69
Level of complexity40
Related polytopes
ArmyHesrid
RegimentHesrid
DualHeptagonal-deltoidal hexecontahedral duotegum
ConjugatesHeptagrammic-small rhombicosidodecahedral duoprism, Great heptagrammic-small rhombicosidodecahedral duoprism, Heptagonal-quasirhombicosidodecahedral duoprism, Heptagrammic-quasirhombicosidodecahedral duoprism, Great heptagrammic-quasirhombicosidodecahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH3×I2(7), order 1680
ConvexYes
NatureTame

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.

## Vertex coordinates

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:

• ${\displaystyle \left(1,\,0,\,\pm \sin {\frac {\pi }{7}},\,\pm \sin {\frac {\pi }{7}},\,\pm (2+{\sqrt {5}})\sin {\frac {\pi }{7}}\right),}$
• ${\displaystyle \left(\cos \left({\frac {j\pi }{7}}\right),\,\pm \sin \left({\frac {j\pi }{7}}\right),\,\pm \sin {\frac {\pi }{7}},\,\pm \sin {\frac {\pi }{7}},\,\pm (2+{\sqrt {5}})\sin {\frac {\pi }{7}}\right),}$

as well as all even permutations of the last three coordinates of:

• ${\displaystyle \left(1,\,0,\,0,\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}},\,\pm {\frac {(5+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}}\right),}$
• ${\displaystyle \left(\cos \left({\frac {j\pi }{7}}\right),\,\pm \sin \left({\frac {j\pi }{7}}\right),\,0,\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}},\,\pm {\frac {(5+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}}\right),}$
• ${\displaystyle \left(1,\,0,\,\pm {\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}},\,\pm (1+{\sqrt {5}})\sin {\frac {\pi }{7}},\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}}\right),}$
• ${\displaystyle \left(\cos \left({\frac {j\pi }{7}}\right),\,\pm \sin \left({\frac {j\pi }{7}}\right),\,\pm {\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}},\,\pm (1+{\sqrt {5}})\sin {\frac {\pi }{7}},\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}}\right),}$

where j = 2, 4, 6.