# Hexagonal-small rhombicosidodecahedral duoprism

Hexagonal-small rhombicosidodecahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymHasrid
Coxeter diagramx6o x5o3x ()
Elements
Tera20 triangular-hexagonal duoprisms, 30 square-hexagonal duoprisms, 12 pentagonal-hexagonal duoprisms, 6 small rhombicosidodecahedral prisms
Cells120 triangular prisms, 180 cubes, 72 pentagonal prisms, 60+60 hexagonal prisms, 6 small rhombicosidodecahedra
Faces120 triangles, 180+360+360 squares, 72 pentagons, 60 hexagons
Edges360+360+360
Vertices360
Vertex figureIsosceles-trapezoidal scalene, edge lengths 1, 2, (1+5)/2, 2 (base trapezoid), 3 (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {15+4{\sqrt {5}}}}{2}}\approx 2.44664}$
Hypervolume${\displaystyle {\frac {60{\sqrt {3}}+29{\sqrt {15}}}{2}}\approx 108.11978}$
Diteral anglesThiddip–hip–shiddip: ${\displaystyle \arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 159.09484^{\circ }}$
Shiddip–hip–phiddip: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }}$
Sriddip–srid–sriddip: 120°
Thiddip–trip–sriddip: 90°
Shiddip–cube–sriddip: 90°
Phiddip–pip–sriddip: 90°
Central density1
Number of external pieces68
Level of complexity40
Related polytopes
ArmyHasrid
RegimentHasrid
DualHexagonal-deltoidal hexecontahedral duotegum
ConjugateHexagonal-quasirhombicosidodecahedral duoprism
Abstract & topological properties
Flag count57600
Euler characteristic2
OrientableYes
Properties
SymmetryH3×G2, order 1440
Flag orbits40
ConvexYes
NatureTame

The hexagonal-small rhombicosidodecahedral duoprism (OBSA: hasrid) is a convex uniform duoprism that consists of 6 small rhombicosidodecahedral prisms, 12 pentagonal-hexagonal duoprisms, 30 square-hexagonal duoprisms, and 20 triangular-hexagonal duoprisms. Each vertex joins 2 small rhombicosidodecahedral prisms, 1 triangular-hexagonal duoprism, 2 square-hexagonal duoprisms, and 1 pentagonal-hexagonal duoprism.

## Vertex coordinates

The vertices of a hexagonal-small rhombicosidodecahedral duoprism of edge length 1 are given by all permutations of the last three coordinates of:

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

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

• ${\displaystyle \left(0,\,\pm 1,\,0,\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}},\,0,\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(0,\,\pm 1,\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}}\right)}$.

## Representations

A hexagonal-small rhombicosidodecahedral duoprism has the following Coxeter diagrams:

• x6o x5o3x () (full symmetry)
• x3x x5o3x () (hexagons as ditrigons)