# Paragyrate diminished rhombicosidodecahedron

Paragyrate diminished rhombicosidodecahedron
Rank3
TypeCRF
SpaceSpherical
Notation
Bowers style acronymPagydrid
Elements
Faces5+5+5 triangles, 5+5+5+10 squares, 1+5+5 pentagons, 1 decagon
Edges5+5+5+5+5+5+5+10+10+10+10+10+10+10
Vertices5+5+5+10+10+10+10
Vertex figures5+30 isosceles trapezoids, edge length 1, 2, (1+5)/2, 2
10 scalene triangles, edge lengths 2, (1+5)/2, (5+5)/2
10 irregular tetragons, edge lengths 1, 2, 2, (1+5)/2
Measures (edge length 1)
Circumradius${\displaystyle \frac{\sqrt{11+4\sqrt5}}{2} ≈ 2.23295}$
Volume${\displaystyle \frac{115+54\sqrt5}{6} ≈ 39.29128}$
Dihedral angles3–4: ${\displaystyle \arccos\left(-\frac{\sqrt3+\sqrt{15}}{6}\right) ≈ 159.09484°}$
3–5: ${\displaystyle \arccos\left(-\sqrt{\frac{65-2\sqrt5}{75}}\right) ≈ 153.94242°}$
4–4: ${\displaystyle \arccos\left(-\frac{2\sqrt5}{5}\right) ≈ 153.43495°}$
4–5: ${\displaystyle \arccos\left(-\sqrt{\frac{5+\sqrt5}{10}}\right) ≈ 148.28253°}$
4–10: ${\displaystyle \arccos\left(-\sqrt{\frac{5-\sqrt5}{10}}\right) ≈ 121.71747°}$
5–10: ${\displaystyle \arccos\left(-\frac{\sqrt5}{5}\right) ≈ 116.56505°}$
Central density1
Related polytopes
ArmyPagydrid
RegimentPagydrid
ConjugateParagyrate replenished quasirhombicosidodecahedron
Abstract & topological properties
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryH2×I, order 10
ConvexYes
NatureTame

The paragyrate diminished rhombicosidodecahedron is one of the 92 Johnson solids (J77). It consists of 5+5+5 triangles, 5+5+5+10 squares, 1+5+5 pentagons, and 1 decagon. It can be constructed by removing one of the pentagonal cupolaic caps of the small rhombicosidodecahedron, and rotating the opposite cap by 36°.

## Vertex coordinates

A paragyrate diminished rhombicosidodecahedron of edge length 1 has vertices given by:

• ${\displaystyle \left(±\frac{5+\sqrt5}{4},\,0,\,±\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle ±\left(0,\,-\frac{3+\sqrt5}{4},\,\frac{5+\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac{3+\sqrt5}{4},\,±\frac{5+\sqrt5}{4},\,0\right),}$
• ${\displaystyle \left(±\frac12,\,±\frac12,\,±\frac{2+\sqrt5}{2}\right),}$
• ${\displaystyle \left(±\frac{2+\sqrt5}{2},\,±\frac12,\,±\frac12\right),}$
• ${\displaystyle ±\left(±\frac12,\,-\frac{2+\sqrt5}{2},\,\frac12\right),}$
• ${\displaystyle \left(±\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{4},\,±\frac{1+\sqrt5}{2}\right),}$
• ${\displaystyle \left(±\frac{1+\sqrt5}{2},\,±\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{4}\right),}$
• ${\displaystyle ±\left(±\frac{1+\sqrt5}{4},\,-\frac{1+\sqrt5}{2},\,\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac12,\,-\frac{5+4\sqrt5}{10},\,-\frac{10+3\sqrt5}{10}\right),}$
• ${\displaystyle \left(±\frac{1+\sqrt5}{4},\,-\frac{5+2\sqrt5}{5},\,-\frac{15+\sqrt5}{20}\right),}$
• ${\displaystyle \left(0,\,-\frac{15+13\sqrt5}{20},\,-\frac{5+\sqrt5}{20}\right).}$