# Pentagonal-icosidodecahedral duoprism

Pentagonal-icosidodecahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymPid
Coxeter diagramx5o o5x3o ()
Elements
Tera20 triangular-pentagonal duoprisms, 12 pentagonal duoprisms, 5 icosidodecahedral prisms
Cells100 triangular prisms, 60+60 pentagonal prisms, 5 icosidodecahedra
Faces100 triangles, 300 squares, 30+60 pentagons
Edges150+300
Vertices150
Vertex figureRectangular scalene, edge lengths 1, (1+5)/2, 1, (1+5)/2 (base rectangle), (1+5)/2 (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {10+3{\sqrt {5}}}{5}}}\approx 1.82802}$
Hypervolume${\displaystyle 5{\frac {\sqrt {6530+2918{\sqrt {5}}}}{24}}\approx 23.80371}$
Diteral anglesTrapedip–pip–pedip: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }}$
Iddip–id–iddip: 108°
Trapedip–trip–iddip: 90°
Pedip–pip–iddip: 90°
Central density1
Number of external pieces37
Level of complexity20
Related polytopes
ArmyPid
RegimentPid
DualPentagonal-rhombic triacontahedral duotegum
ConjugatePentagrammic-great icosidodecahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH3×H2, order 1200
ConvexYes
NatureTame

The pentagonal-icosidodecahedral duoprism or pid is a convex uniform duoprism that consists of 5 icosidodecahedral prisms, 12 pentagonal duoprisms, and 20 triangular-pentagonal duoprisms. Each vertex joins 2 icosidodecahedral prisms, 2 triangular-pentagonal duoprisms, and 2 pentagonal duoprisms.

## Vertex coordinates

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

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

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

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