# Gyrate rhombicosidodecahedron

Gyrate rhombicosidodecahedron
Rank3
TypeCRF
SpaceSpherical
Notation
Bowers style acronymGyrid
Elements
Faces5+5+5+5 triangles, 5+5+5+5+10 squares, 1+1+5+5 pentagons
Edges5+5+5+5+5+5+5+5+10+10+10+10+10+10+10+10
Vertices5+5+5+5+10+10+10+10
Vertex figures5+45 isosceles trapezoids, edge length 1, 2, (1+5)/2, 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${\displaystyle \frac{60+29\sqrt5}{3} ≈ 41.61532}$
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: ${\displaystyle \arccos\left(-\sqrt{\frac{5+\sqrt5}{10}}\right) ≈ 148.28253°}$
Central density1
Related polytopes
ArmyGyrid
RegimentGyrid
DualDeltogyrate deltoidal hexecontahedron
ConjugateGyrate quasirhombicosidodecahedron
Abstract & topological properties
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryH2×I, order 10
ConvexYes
NatureTame

The gyrate rhombicosidodecahedron is one of the 92 Johnson solids (J72). It consists of 5+5+5+5 triangles, 5+5+5+5+10 squares, and 1+1+5+5 pentagons. It can be constructed by rotating one of the pentagonal cupolaic caps of the small rhombicosidodecahedron by 36°.

## Vertex coordinates

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

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