Great snub icosidodecahedron

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Great snub icosidodecahedron
Great snub icosidodecahedron.png
Type Uniform star polyhedron
Elements F = 92, E = 150
V = 60 (χ = 2)
Faces by sides (20+60){3}+12{5/2}
Wythoff symbol |2 5/2 3
Symmetry group I, [5,3]+, 532
Index references U57, C88, W116
Dual polyhedron Great pentagonal hexecontahedron
Vertex figure Great snub icosidodecahedron vertfig.png
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 CDel node h.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node h.pngCDel 3.pngCDel node h.png.

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

Cartesian coordinates[edit]

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

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

Related polyhedra[edit]

Great pentagonal hexecontahedron[edit]

Great pentagonal hexecontahedron
DU57 great pentagonal hexecontahedron (2).png
Type Star polyhedron
Face DU57 facets.png
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 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[edit]

References[edit]

External links[edit]