- Cartesian coordinates
- Related polyhedra Great pentagonal hexecontahedron
- See also
- References
- External links
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 {{CDD|node_h|5|rat|d2|node_h|3|node_h}}.
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+{{radic|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
{{Uniform polyhedra db|Uniform dual polyhedron stat table|Gosid}}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
- List of uniform polyhedra
- Great inverted snub icosidodecahedron
- Great retrosnub icosidodecahedron
References
- {{Citation | last1=Wenninger | first1=Magnus | author1-link=Magnus Wenninger | title=Dual Models | publisher=Cambridge University Press | isbn=978-0-521-54325-5 |mr=730208 | year=1983}}
External links
- {{mathworld | urlname = GreatPentagonalHexecontahedron| title =Great pentagonal hexecontahedron}}
- {{mathworld | urlname = GreatSnubIcosidodecahedron| title = Great snub icosidodecahedron}}
- Uniform polyhedra and duals
- White Soft Paraffin
- White Somalians
- White Somalis
- Whiteson Changwe
- Whites only
- White Soul
- White Souls in Black Suits
- White South African people
- White South African policy
- White Southerners
- White Sox 2015
- White Sox (New Zealand)
- White Sox-Tigers Rivalry
- White Sox-Twins rivalry
- White Sox–Cubs rivalry
- White Sox – Cubs rivalry
- White Space (film)
- Whitespace (language)
- Whitespace (programming language
- White spaces device
- White space technology
- White spanishbroom
- White Spanish broom
- White-speckled Laughingthrush
- White Speck Ringlet