Great icosidodecahedron

In geometry, the great icosidodecahedron is a nonconvex uniform polyhedron, indexed as U₅₄. It has 32 faces, 60 edges, and 30 vertices. It is given a Schläfli symbol r. It is the rectification of the great stellated dodecahedron and the great icosahedron. It was discovered independently by, and.

Related polyhedra

The figure is a rectification of the great icosahedron or the great stellated dodecahedron, much as the icosidodecahedron is related to the icosahedron and dodecahedron, and the cuboctahedron to the cube and octahedron.
It shares its vertex arrangement with the icosidodecahedron, which is its convex hull. Unlike the great icosahedron and great dodecahedron, the great icosidodecahedron is not a stellation of the icosidodecahedron, but a faceting of it instead.
It also shares its edge arrangement with the great icosihemidodecahedron, and with the great dodecahemidodecahedron.

Great icosidodecahedron	Great dodecahemidodecahedron	Great icosihemidodecahedron
Icosidodecahedron	-	-

The dual of the great icosidodecahedron is the great rhombic triacontahedron; it is nonconvex, isohedral and isotoxal. It has 30 intersecting rhombic faces. It can also be called the great stellated triacontahedron.
The great rhombic triacontahedron can be constructed by expanding the size of the faces of a rhombic triacontahedron by a factor of τ³ = 1+2τ = 2+√5, where τ is the golden ratio.