# Gyrate bidiminished rhombicosidodecahedron

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Gyrate bidiminished rhombicosidodecahedron
Rank3
TypeCRF
Notation
Bowers style acronymGybadrid
Elements
Faces
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${\displaystyle {\frac {\sqrt {11+4{\sqrt {5}}}}{2}}\approx 2.23295}$
Volume${\displaystyle 5{\frac {11+5{\sqrt {5}}}{3}}\approx 36.96723}$
Dihedral angles3–4: ${\displaystyle \arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 159.09484^{\circ }}$
3–5: ${\displaystyle \arccos \left(-{\sqrt {\frac {65-2{\sqrt {5}}}{75}}}\right)\approx 153.94242^{\circ }}$
4–4: ${\displaystyle \arccos \left(-{\frac {2{\sqrt {5}}}{5}}\right)\approx 153.43495^{\circ }}$
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 complexity180
Related polytopes
ArmyGybadrid
RegimentGybadrid
DualGyrate bistellated deltoidal hexecontahedron
ConjugateGyrate bireplenished quasirhombicosidodecahedron
Abstract & topological properties
Flag count360
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryA1×I×I, order 2
Flag orbits180
ConvexYes
NatureTame

The gyrate bidiminished rhombicosidodecahedron (OBSA: gybadrid) 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:

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