# Pentagonal-small rhombicosidodecahedral duoprism

Pentagonal-small rhombicosidodecahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymPesrid
Coxeter diagramx5o x5o3x
Elements
Tera20 triangular-pentagonal duoprisms, 30 square-pentagonal duoprisms, 12 pentagonal duoprisms
Cells100 triangular prisms, 150 cubes, 60+60+60 pentagonal prisms, 5 small rhombicosidodecahedra
Faces150 triangles, 150+300+300 squares, 60+60 pentagons
Edges300+300+300
Vertices300
Vertex figureIsosceles-trapezoidal scalene, edge lengths 1, 2, (1+5)/2, 2 (base trapezoid), (1+5)/2 (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {65+22{\sqrt {5}}}{20}}}\approx 2.38949}$
Hypervolume${\displaystyle 5{\frac {\sqrt {14765+6602{\sqrt {5}}}}{12}}\approx 71.59822}$
Diteral anglesTrapedip–pip–squipdip: ${\displaystyle \arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 159.09484^{\circ }}$
Squipdip–pip–pedip: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }}$
Sriddip–srid–sriddip: 108°
Trapedip–trip–sriddip: 90°
Squipdip–cube–sriddip: 90°
Pedip–pip–sriddip: 90°
Central density1
Number of external pieces67
Level of complexity40
Related polytopes
ArmyPesrid
RegimentPesrid
DualPentagonal-deltoidal hexecontahedral duotegum
ConjugatePentagrammic-quasirhombicosidodecahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH3×H2, order 1200
ConvexYes
NatureTame

The pentagonal-small rhombicosidodecahedral duoprism or pesrid is a convex uniform duoprism that consists of 5 small rhombicosidodecahedral prisms, 12 pentagonal duoprisms, 30 square-pentagonal duoprisms, and 20 triangular-pentagonal duoprisms. Each vertex joins 2 small rhombicosidodecahedral prisms, 1 triangular-pentagonal duoprism, 2 square-pentagonal duoprisms, and 1 pentagonal duoprism.

## Vertex coordinates

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

• ${\displaystyle \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}}\right),}$

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

• ${\displaystyle \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,0,\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,0,\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,0,\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}}\right),}$
• ${\displaystyle \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}}\right).}$