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 name is constructed analogously as how a cube-octahedron creates a cuboctahedron, and how a dodecahedron-icosahedron creates a icosidodecahedron.
It shares the same vertex arrangement with the icosidodecahedron, 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 		--

This polyhedron can be considered a rectified great icosahedron:
The truncated great stellated dodecahedron is a degenerate polyhedron, with 20 triangular faces from the truncated vertices, and 12 pentagonal faces as truncations of the original pentagram faces, the latter forming a great dodecahedron inscribed within and sharing the edges of the icosahedron.
NameGreat
stellated
dodecahedron		Truncated great stellated dodecahedronGreat
icosidodecahedron		Truncated
great
icosahedron		Great
icosahedron
Coxeter
diagram
Picture

Great rhombic triacontahedron

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.