# Decagonal-small rhombicosidodecahedral duoprism

Decagonal-small rhombicosidodecahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymDasrid
Coxeter diagramx10o x5o3x ()
Elements
Tera20 triangular-decagonal duoprisms, 30 square-decagonal duoprisms, 12 pentagonal-decagonal duoprisms, 10 small rhombicosidodecahedral prisms
Cells200 triangular prisms, 300 cubes, 120 pentagonal prisms, 60+60 decagonal prisms, 10 small rhombicosidodecahedra
Faces200 triangles, 300+600+600 squares, 120 pentagons, 60 decagons
Edges600+600+600
Vertices600
Vertex figureIsosceles-trapezoidal scalene, edge lengths 1, 2, (1+5)/2, 2 (base trapezoid), (5+5)/2 (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {17+6{\sqrt {5}}}}{2}}\approx 2.75755}$
Hypervolume${\displaystyle 5{\frac {\sqrt {73825+33010{\sqrt {5}}}}{6}}\approx 320.19699}$
Diteral anglesTradedip–dip–squadedip: ${\displaystyle \arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 159.09484^{\circ }}$
Squadedip–dip–padedip: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }}$
Sriddip–srid–sriddip: 144°
Central density1
Number of external pieces72
Level of complexity40
Related polytopes
ArmyDasrid
RegimentDasrid
DualDecagonal-deltoidal hexecontahedral duotegum
ConjugateDecagrammic-quasirhombicosidodecahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH3×I2(10), order 2400
ConvexYes
NatureTame

The decagonal-small rhombicosidodecahedral duoprism or dasrid is a convex uniform duoprism that consists of 10 small rhombicosidodecahedral prisms, 12 pentagonal-decagonal duoprisms, 30 square-decagonal duoprisms, and 20 triangular-decagonal duoprisms. Each vertex joins 2 small rhombicosidodecahedral prisms, 1 triangular-decagonal duoprism, 2 square-decagonal duoprisms, and 1 pentagonal-decagonal duoprism.

## Vertex coordinates

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

• ${\displaystyle \left(0,\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}}\right),}$

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

• ${\displaystyle \left(0,\,\pm {\frac {1+{\sqrt {5}}}{2}},\,0,\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}}\right),}$
• ${\displaystyle \left(\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,0,\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,\pm {\frac {1}{2}},\,0,\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}}\right),}$
• ${\displaystyle \left(0,\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}}\right),}$
• ${\displaystyle \left(\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}}\right).}$

## Representations

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

• x10o x5o3x (full symmetry)
• x5x x5o3x (decagons as dipentagons)