Great snub icosidodecahedron

Great snub icosidodecahedron
TypeUniform star polyhedron
ElementsF = 92, E = 150
V = 60 (χ = 2)
Faces by sides(20+60){3}+12{5/2}
Wythoff symbol|2 5/2 3
Symmetry groupI, [5,3]+, 532
Index referencesU57, C88, W116
Dual polyhedronGreat pentagonal hexecontahedron
Vertex figure
34.5/2
Bowers acronymGosid

In geometry, the great snub icosidodecahedron is a nonconvex uniform polyhedron, indexed as U57. It can be represented by a Schläfli symbol sr{5/2,3}, and Coxeter-Dynkin diagram .

This polyhedron is the snub member of a family that includes the great icosahedron, the great stellated dodecahedron and the great icosidodecahedron.

Cartesian coordinates

Cartesian coordinates for the vertices of a great snub icosidodecahedron are all the even permutations of

(±2α, ±2, ±2β),
(±(α−βτ−1/τ), ±(α/τ+β−τ), ±(−ατ−β/τ−1)),
(±(ατ−β/τ+1), ±(−α−βτ+1/τ), ±(−α/τ+β+τ)),
(±(ατ−β/τ−1), ±(α+βτ+1/τ), ±(−α/τ+β−τ)) and
(±(α−βτ+1/τ), ±(−α/τ−β−τ), ±(−ατ−β/τ+1)),

with an even number of plus signs, where

α = ξ−1/ξ

and

β = −ξ/τ+1/τ2−1/(ξτ),

where τ = (1+√5)/2 is the golden mean and ξ is the negative real root of ξ3−2ξ=−1/τ, or approximately −1.5488772. Taking the odd permutations of the above coordinates with an odd number of plus signs gives another form, the enantiomorph of the other one.

The circumradius for unit edge length is

where is the appropriate root of . The four positive real roots of the sextic in

are the circumradii of the snub dodecahedron (U29), great snub icosidodecahedron (U57), great inverted snub icosidodecahedron (U69), and great retrosnub icosidodecahedron (U74).

Great pentagonal hexecontahedron

Great pentagonal hexecontahedron
TypeStar polyhedron
Face
ElementsF = 60, E = 150
V = 92 (χ = 2)
Symmetry groupI, [5,3]+, 532
Index referencesDU57
dual polyhedronGreat snub icosidodecahedron

The great pentagonal hexecontahedron is a nonconvex isohedral polyhedron and dual to the uniform great snub icosidodecahedron. It has 60 intersecting irregular pentagonal faces, 120 edges, and 92 vertices.

See also

References

  • Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 0730208
  • Weisstein, Eric W. "Great pentagonal hexecontahedron". MathWorld.
  • Weisstein, Eric W. "Great snub icosidodecahedron". MathWorld.
  • Uniform polyhedra and duals
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