Trigyrate rhombicosidodecahedron

{{Short description|75th Johnson solid}}

{{Infobox polyhedron

|image=trigyrate_rhombicosidodecahedron.png

|type=Johnson
{{math|metabigyrate rhombicosidodecahedron – J{{sub|75}} – diminished rhombicosidodecahedron}}

|faces=2+2x3+2x6 triangles
4x3+3x6 squares
4x3 pentagons

|edges=120

|vertices=60

|symmetry={{math|C{{sub|3v}}}}

|vertex_config={{math|5x6(3.4{{sup|2}}.5)
4x3+3x6(3.4.5.4)}}

|dual=-

|properties=convex, canonical

|net=Johnson solid 75 net.png

}}

File:J75 trigyrate rhombicosidodecahedron.stl

In geometry, the trigyrate rhombicosidodecahedron is one of the Johnson solids ({{math|J{{sub|75}}}}). It contains 20 triangles, 30 squares and 12 pentagons. It is also a canonical polyhedron.

{{Johnson solid}}

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

• The gyrate rhombicosidodecahedron ({{math|J{{sub|72}}}}) where one cupola is rotated;
• The parabigyrate rhombicosidodecahedron ({{math|J{{sub|73}}}}) where two opposing cupolae are rotated;
• And the metabigyrate rhombicosidodecahedron ({{math|J{{sub|74}}}}) where two non-opposing cupolae are rotated.