Great snub icosidodecahedron

From Wikipedia, the free encyclopedia
Jump to: navigation, search
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.

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]