In geometry, the truncated great icosahedron (or great truncated icosahedron) is a nonconvex uniform polyhedron, indexed as U55. It has 32 faces (12 pentagrams and 20 hexagons), 90 edges, and 60 vertices. It is given a Schläfli symbol t{3,5⁄2} or t0,1{3,5⁄2} as a truncated great icosahedron.
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 φ ) ( ± 2 , ± 1 φ , ± 1 φ 3 ) ( ± [ 1 + 1 φ 2 ] , ± 1 , ± 2 φ ) {\displaystyle {\begin{array}{crccc}{\Bigl (}&\pm \,1,&0,&\pm \,{\frac {3}{\varphi }}&{\Bigr )}\\{\Bigl (}&\pm \,2,&\pm \,{\frac {1}{\varphi }},&\pm \,{\frac {1}{\varphi ^{3}}}&{\Bigr )}\\{\Bigl (}&\pm {\bigl [}1+{\frac {1}{\varphi ^{2}}}{\bigr ]},&\pm \,1,&\pm \,{\frac {2}{\varphi }}&{\Bigr )}\end{array}}}
where φ = 1 + 5 2 {\displaystyle \varphi ={\tfrac {1+{\sqrt {5}}}{2}}} is the golden ratio. Using 1 φ 2 = 1 − 1 φ {\displaystyle {\tfrac {1}{\varphi ^{2}}}=1-{\tfrac {1}{\varphi }}} one verifies that all vertices are on a sphere, centered at the origin, with the radius squared equal to 10 − 9 φ . {\displaystyle 10-{\tfrac {9}{\varphi }}.} 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.
See also List of uniform polyhedra
References
Wenninger, Magnus (1983), Dual Models, Cambridge University Press, doi:10.1017/CBO9780511569371, ISBN 978-0-521-54325-5, MR 0730208
External links Weisstein, Eric W. "Truncated great icosahedron". MathWorld. Weisstein, Eric W. "Great stellapentakis dodecahedron". MathWorld. Uniform polyhedra and duals
