# Parabidiminished rhombicosidodecahedron

Parabidiminished rhombicosidodecahedron
Rank3
TypeCRF
Notation
Bowers style acronymPabidrid
Elements
Faces
Edges3×10+3×20
Vertices10+20+20
Vertex figures30 isosceles trapezoids, edge length 1, 2, (1+5)/2, 2
20 scalene triangles, edge lengths 2, (1+5)/2, (5+5)/2
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {11+4{\sqrt {5}}}}{2}}\approx 2.23295}$
Volume${\displaystyle 5{\frac {11+5{\sqrt {5}}}{3}}\approx 36.96723}$
Dihedral angles3–4: ${\displaystyle \arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 159.09484^{\circ }}$
4–5: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }}$
4–10: ${\displaystyle \arccos \left(-{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\right)\approx 121.71747^{\circ }}$
5–10: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }}$
Central density1
Number of external pieces42
Level of complexity18
Related polytopes
ArmyPabidrid
RegimentPabidrid
DualParabistellated deltoidal hexecontahedron
ConjugateParabireplenished quasirhombicosidodecahedron
Abstract & topological properties
Flag count360
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
Symmetry(I2(10)×A1)/2, order 20
Flag orbits18
ConvexYes
NatureTame

The parabidiminished rhombicosidodecahedron (OBSA: pabidrid) is one of the 92 Johnson solids (J80). It consists of 10 triangles, 10+10 squares, 10 pentagons, and 2 decagons. It can be constructed by removing two opposite pentagonal cupolaic caps of the small rhombicosidodecahedron.

## Vertex coordinates

A parabidiminished rhombicosidodecahedron of edge length 1 has vertices given by:

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