geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, the parabigyrate rhombicosidodecahedron is one of the

Johnson solid In geometry, a Johnson solid, sometimes also known as a Johnson–Zalgaller solid, is a convex polyhedron whose faces are regular polygons. They are sometimes defined to exclude the uniform polyhedrons. There are ninety-two Solid geometry, s ...

s (). It can be constructed as a

rhombicosidodecahedron In geometry, the rhombicosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces. It has a total of 62 faces: 20 regular triangular faces, 30 square f ...

with two opposing

pentagonal cupola Properties The pentagonal cupola (geometry), cupola's faces are five equilateral triangles, five squares, one regular pentagon, and one regular decagon. It has the property of Convex set, convexity and regular polygonal faces, from which it is ...

e rotated through 36 degrees. It is also a

canonical polyhedron In geometry, the midsphere or intersphere of a convex polyhedron is a sphere which is tangent to every edge of the polyhedron. Not every polyhedron has a midsphere, but the uniform polyhedra, including the regular, quasiregular and semiregul ...

. Alternative Johnson solids, constructed by rotating different cupolae of a rhombicosidodecahedron, are: * The

gyrate rhombicosidodecahedron In geometry, the gyrate rhombicosidodecahedron is one of the Johnson solids (). It is also a Midsphere#Canonical polyhedron, canonical polyhedron. Construction The gyrate rhombicosidodecahedron can be constructed similarly as rhombicosidodec ...

() where only one cupola is rotated; * The metabigyrate rhombicosidodecahedron () where two non-opposing cupolae are rotated; * And the trigyrate rhombicosidodecahedron () where three cupolae are rotated.

External links

* Johnson solids {{Polyhedron-stub