Trigyrate rhombicosidodecahedron explained

Type:	Johnson
Faces:	2+2x3+2x6 triangles 4x3+3x6 squares 4x3 pentagons
Edges:	120
Vertices:	60
Dual:	-
Properties:	convex, canonical
Net:	Johnson solid 75 net.png

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

It can be constructed as a rhombicosidodecahedron with three pentagonal cupolae rotated through 36 degrees. Related Johnson solids are:

• The gyrate rhombicosidodecahedron where one cupola is rotated;
• The parabigyrate rhombicosidodecahedron where two opposing cupolae are rotated;
• And the metabigyrate rhombicosidodecahedron 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.
• Book: Victor A. Zalgaller. Victor Zalgaller. Convex Polyhedra with Regular Faces. Consultants Bureau. 1969. No ISBN. The first proof that there are only 92 Johnson solids.