# Gyrate bidiminished rhombicosidodecahedron

Gyrate bidiminished rhombicosidodecahedron
Rank3
TypeCRF
SpaceSpherical
Notation
Elements
Faces4×1+3×2 triangles, 4×1+8×2 squares, 4×1+3×2 pentagons, 2 decagons
Edges8×1+41×2
Vertices4×1+23×2
Vertex figures5+15 isosceles trapezoids, edge length 1, 2, (1+5)/2, 2
20 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${\displaystyle 5\frac{11+5\sqrt5}{3} ≈ 36.96723}$
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
DualGyrate bistellated deltoidal hexecontahedron
ConjugateGyrate bireplenished quasirhombicosidodecahedron
Abstract & topological properties
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryA1×I×I, order 2
ConvexYes
NatureTame

The gyrate bidiminished rhombicosidodecahedron is one of the 92 Johnson solids (J82). It consists of 4×1+3×2 triangles, 4×1+8×2 squares, 4×1+3×2 pentagons, and 2 decagons. It can be constructed by removing two non-opposite pentagonal cupolaic caps of the small rhombicosidodecahedron, and rotating a third cap by 36°.

## Vertex coordinates

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

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