# Trigyrate rhombicosidodecahedron

Trigyrate rhombicosidodecahedron
Rank3
TypeCRF
Notation
Bowers style acronymTagyrid
Elements
Faces1+1+3+3+6+6 triangles, 3+3+3+3+6+6+6 squares, 3+3+3+3 pentagons
Edges8×3+16×6
Vertices3+3+3+3+6+6+6+6+6+6+6+6
Vertex figures15+15 isosceles trapezoids, edge length 1, 2, (1+5)/2, 2
30 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${\displaystyle {\frac {60+29{\sqrt {5}}}{3}}\approx 41.61532}$
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: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }}$
Central density1
Number of external pieces62
Level of complexity80
Related polytopes
ArmyTagyrid
RegimentTagyrid
DualTrideltogyrate deltoidal hexecontahedron
ConjugateTrigyrate quasirhombicosidodecahedron
Abstract & topological properties
Flag count480
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryA2×I, order 6
ConvexYes
NatureTame

The trigyrate rhombicosidodecahedron is one of the 92 Johnson solids (J75). It consists of 1+1+3+3+6+6 triangles, 3+3+3+3+6+6+6 squares, and 3+3+3+3 pentagons. It can be constructed by rotating three mutually non-adjacent pentagonal cupolaic caps of the small rhombicosidodecahedron by 36°.

## Vertex coordinates

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

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