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Trigyrate rhombicosidodecahedron

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Trigyrate rhombicosidodecahedron
TypeJohnson
J74J75J76
Faces2+2x3+2x6 triangles
4x3+3x6 squares
4x3 pentagons
Edges120
Vertices60
Vertex configuration5x6(3.42.5)
4x3+3x6(3.4.5.4)
Symmetry groupC3v
Dual polyhedron-
Propertiesconvex, canonical
Net
3D model of a trigyrate rhombicosidodecahedron

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

A Johnson solid is one of 92 strictly convex polyhedra that is composed of regular polygon faces but are not uniform polyhedra (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:

References

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  • 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.
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  1. ^ Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics, 18: 169–200, doi:10.4153/cjm-1966-021-8, MR 0185507, Zbl 0132.14603.