# Octagonal-small rhombicosidodecahedral duoprism

Octagonal-small rhombicosidodecahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymOsrid
Coxeter diagramx8o x5o3x
Elements
Tera20 triangular-octagonal duoprisms, 30 square-octagonal duoprisms, 12 pentagonal-octagonal duoprisms, 8 small rhombicosidodecahedral prisms
Cells160 triangular prisms, 240 cubes, 96 pentagonal prisms, 60+60 octagonal prisms, 8 small rhombicosidodecahedra
Faces160 triangles, 240+480+480 squares, 96 pentagons, 60 octagons
Edges480+480+480
Vertices480
Vertex figureIsosceles-trapezoidal scalene, edge lengths 1, 2, (1+5)/2, 2 (base trapezoid), 2+2 (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {15+2{\sqrt {2}}+4{\sqrt {5}}}}{2}}\approx 2.58712}$
Hypervolume${\displaystyle 2{\frac {60+60{\sqrt {2}}+29{\sqrt {5}}+29{\sqrt {10}}}{3}}\approx 200.93656}$
Diteral anglesTodip–op–sodip: ${\displaystyle \arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 159.09484^{\circ }}$
Sodip–op–podip: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }}$
Sriddip–srid–sriddip: 135°
Todip–trip–sriddip: 90°
Sodip–cube–sriddip: 90°
Podip–pip–sriddip: 90°
Central density1
Number of external pieces70
Level of complexity40
Related polytopes
ArmyOsrid
RegimentOsrid
DualOctagonal-deltoidal hexecontahedral duotegum
ConjugatesOctagrammic-small rhombicosidodecahedral duoprism, Octagonal-quasirhombicosidodecahedral duoprism, Octagrammic-quasirhombicosidodecahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH3×I2(8), order 1920
ConvexYes
NatureTame

The octagonal-small rhombicosidodecahedral duoprism or osrid is a convex uniform duoprism that consists of 8 small rhombicosidodecahedral prisms, 12 pentagonal-octagonal duoprisms, 30 square-octagonal duoprisms, and 20 triangular-octagonal duoprisms. Each vertex joins 2 small rhombicosidodecahedral prisms, 1 triangular-octagonal duoprism, 2 square-octagonal duoprisms, and 1 pentagonal-octagonal duoprism.

## Vertex coordinates

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

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

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

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

## Representations

An octagonal-small rhombicosidodecahedral duoprism has the following Coxeter diagrams:

• x8o x5o3x (full symmetry)
• x4x x5o3x (octagons as ditetragons)