In geometry, the gyrate rhombicosidodecahedron is one of the Johnson solids (J72).

They were named by Norman Johnson, who first listed these polyhedra in 1966.

As a result, these pentagonal cupolas cover its dodecagonal faces, so the resulting polyhedron has 20 equilateral triangles, 30 squares, and 10 regular pentagons as its faces.

The difference between those two polyhedrons is that one of two pentagonal cupolas from the gyrate rhombicosidodecahedron is rotated through 36°.

[2] A convex polyhedron in which all faces are regular polygons is called the Johnson solid, and the gyrate rhombicosidodecahedron is among them, enumerated as the 72th Johnson solid

[3] Because the two aforementioned polyhedrons have similar construction, they have the same surface area and volume.

The gyrate rhombicosidodecahedron is one of the five Johnson solids that do not have Rupert property, meaning a polyhedron of the same or larger size and the same shape as it cannot pass through a hole in it.

[4] Alternative Johnson solids, constructed by rotating different cupolae of a rhombicosidodecahedron, are: