Great icosidodecahedron

In geometry, the great icosidodecahedron is a nonconvex uniform polyhedron, indexed as U54. It has 32 faces (20 triangles and 12 pentagrams), 60 edges, and 30 vertices.[1] It is given a Schläfli symbol r{3,​5⁄2}. It is the rectification of the great stellated dodecahedron and the great icosahedron. It was discovered independently by Hess (1878), Badoureau (1881) and Pitsch (1882).

Great icosidodecahedron
TypeUniform star polyhedron
ElementsF = 32, E = 60
V = 30 (χ = 2)
Faces by sides20{3}+12{5/2}
Wythoff symbol3 5/2
2 | 3 5/3
2 | 3/2 5/2
2 | 3/2 5/3
Symmetry groupIh, [5,3], *532
Index referencesU54, C70, W94
Dual polyhedronGreat rhombic triacontahedron
Vertex figure
3.5/2.3.5/2
Bowers acronymGid
3D model of a great icosidodecahedron

The name is constructed analogously as how a cube-octahedron creates a cuboctahedron, and how a dodecahedron-icosahedron creates a (small) 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 (having the triangular faces in common), and with the great dodecahemidodecahedron (having the pentagrammic faces in common).


Great icosidodecahedron
Great dodecahemidodecahedron
Great icosihemidodecahedron

Icosidodecahedron (convex hull)
Animated truncation sequence from {​5⁄2, 3} to {3, ​5⁄2}

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 (hidden) pentagonal faces as truncations of the original pentagram faces, the latter forming a great dodecahedron inscribed within and sharing the edges of the icosahedron.

Name Great
stellated
dodecahedron		 Truncated great stellated dodecahedron Great
icosidodecahedron		 Truncated
great
icosahedron		 Great
icosahedron
Coxeter
diagram
Picture

Great rhombic triacontahedron
Great rhombic triacontahedron
TypeStar polyhedron
Face
ElementsF = 30, E = 60
V = 32 (χ = 2)
Symmetry groupIh, [5,3], *532
Index referencesDU54
dual polyhedronGreat icosidodecahedron
3D model of a 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.

See also

Notes
  1. Maeder, Roman. "54: great icosidodecahedron". MathConsult.

References


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