Inverted snub dodecadodecahedron

Inverted snub dodecadodecahedron
Type	Uniform star polyhedron
Elements	F = 84, E = 150 V = 60 (χ = −6)
Faces by sides	60{3}+12{5}+12{5/2}
Wythoff symbol	|5/3 2 5
Symmetry group	I, [5,3]+, 532
Index references	U60, C76, W114
Dual polyhedron	Medial inverted pentagonal hexecontahedron
Vertex figure	3.3.5.3.5/3
Bowers acronym	Isdid

In geometry, the inverted snub dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U₆₀. It is given a Schläfli symbol sr{5/3,5}.

Cartesian coordinates

Cartesian coordinates for the vertices of an inverted snub dodecadodecahedron are all the even permutations of

(±2α, ±2, ±2β),

(±(α+β/τ+τ), ±(-ατ+β+1/τ), ±(α/τ+βτ-1)),

(±(-α/τ+βτ+1), ±(-α+β/τ-τ), ±(ατ+β-1/τ)),

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

(±(α+β/τ-τ), ±(ατ-β+1/τ), ±(α/τ+βτ+1)),

with an even number of plus signs, where

β = (α²/τ+τ)/(ατ−1/τ),

where τ = (1+√5)/2 is the golden mean and α is the negative real root of τα⁴−α³+2α²−α−1/τ, or approximately −0.3352090. 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

Medial inverted pentagonal hexecontahedron

Medial inverted pentagonal hexecontahedron
Type	Star polyhedron
Face
Elements	F = 60, E = 150 V = 84 (χ = −6)
Symmetry group	I, [5,3]+, 532
Index references	DU60
dual polyhedron	Inverted snub dodecadodecahedron

The medial inverted pentagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform inverted snub dodecadodecahedron.

See also

• List of uniform polyhedra
• Snub dodecadodecahedron

References

• Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 730208  p. 124

External links

• Weisstein, Eric W. "Medial inverted pentagonal hexecontahedron". MathWorld. 

• Weisstein, Eric W. "Inverted snub dodecadodecahedron". MathWorld.