# Trigyrate rhombicosidodecahedron

Trigyrate rhombicosidodecahedron
Type Johnson
J74 - J75 - J76
Faces 2+2x3+2x6 triangles
4x3+3.6 squares
4x3 pentagons
Edges 120
Vertices 60
Vertex configuration 5x6(3.42.5)
4x3+3x6(3.4.5.4)
Symmetry group C3v
Dual polyhedron -
Properties convex
Net

In geometry, the trigyrate rhombicosidodecahedron is one of the Johnson solids (J75). It contains 20 triangles, 30 squares and 12 pentagons.

A Johnson solid is one of 92 strictly convex polyhedra that have regular faces but are not uniform (that is, they are not Platonic solids, Archimedean solids, prisms or antiprisms). They were named by Norman Johnson, who first listed these polyhedra in 1966.[1]

It can be constructed as a rhombicosidodecahedron with three pentagonal cupolae rotated through 36 degrees. Related Johnson solids are the gyrate rhombicosidodecahedron (J72) where one cupola is rotated, the parabigyrate rhombicosidodecahedron (J73) where two opposing cupolae are rotated and the metabigyrate rhombicosidodecahedron (J74) where two non-opposing cupolae are rotated.

## References

• Norman W. Johnson, "Convex Solids with Regular Faces", Canadian Journal of Mathematics, 18, 1966, pages 169–200. Contains the original enumeration of the 92 solids and the conjecture that there are no others.
• Victor A. Zalgaller (1969). Convex Polyhedra with Regular Faces. Consultants Bureau. No ISBN. The first proof that there are only 92 Johnson solids.