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In geometry, the truncated great icosahedron is a nonconvex uniform polyhedron, indexed as U55. It is given a Schläfli symbol t{3,5/2} or t0,1{3,5/2} as a truncated great icosahedron.
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Cartesian coordinates
Cartesian coordinates for the vertices of a truncated great icosahedron centered at the origin are all the even permutations of
(±1, 0, ±3/τ)(±2, ±1/τ, ±1/τ3)(±(1+1/τ2), ±1, ±2/τ)where τ = (1+√5)/2 is the golden ratio (sometimes written φ). Using 1/τ2 = 1 − 1/τ one verifies that all vertices are on a sphere, centered at the origin, with the radius squared equal to 10−9/τ. The edges have length 2.
Related polyhedra
This polyhedron is the truncation of the 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.
Great stellapentakis dodecahedron
The great stellapentakis dodecahedron is a nonconvex isohedral polyhedron. It is the dual of the truncated great icosahedron. It has 60 intersecting triangular faces.