Great snub icosidodecahedron
Great snub icosidodecahedron | |
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Type | Uniform star polyhedron |
Elements | F = 92, E = 150 V = 60 (χ = 2) |
Faces by sides | (20+60){3}+12{5/2} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Wythoff symbol | | 2 5/2 3 |
Symmetry group | I, [5,3]+, 532 |
Index references | U57, C88, W113 |
Dual polyhedron | Great pentagonal hexecontahedron |
Vertex figure | ![]() 34.5/2 |
Bowers acronym | Gosid |
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.
In the book Polyhedron Models by Magnus Wenninger, the polyhedron is misnamed great inverted snub icosidodecahedron, and vice versa.
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).
Related polyhedra
Great pentagonal hexecontahedron
Great pentagonal hexecontahedron | |
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Type | Star polyhedron |
Face | ![]() |
Elements | F = 60, E = 150 V = 92 (χ = 2) |
Symmetry group | I, [5,3]+, 532 |
Index references | DU57 |
dual polyhedron | Great snub icosidodecahedron |
The great pentagonal hexecontahedron (or great petaloid ditriacontahedron) 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