Snub dodecadodecahedron

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}
Coxeter diagram
Wythoff symbol	| 2 5/2 5
Symmetry group	I, [5,3]+, 532
Index references	U40, C49, W111
Dual polyhedron	Medial pentagonal hexecontahedron
Vertex figure	3.3.5/2.3.5
Bowers acronym	Siddid

In geometry, the snub dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U₄₀. It has 84 faces (60 triangles, 12 pentagons, and 12 pentagrams), 150 edges, and 60 vertices.^[1] It is given a Schläfli symbol sr{5⁄2,5}, as a snub great dodecahedron.

Cartesian coordinates

Cartesian coordinates for the vertices of a 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 positive real root of τα⁴−α³+2α²−α−1/τ, or approximately 0.7964421. 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 pentagonal hexecontahedron

Medial pentagonal hexecontahedron
Type	Star polyhedron
Face
Elements	F = 60, E = 150 V = 84 (χ = −6)
Symmetry group	I, [5,3]+, 532
Index references	DU40
dual polyhedron	Snub dodecadodecahedron

The medial pentagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the snub dodecadodecahedron. It has 60 intersecting irregular pentagonal faces.

See also

• List of uniform polyhedra
• Inverted snub dodecadodecahedron

References

• Wenninger, Magnus (1983), Dual Models, Cambridge University Press, doi:10.1017/CBO9780511569371, ISBN 978-0-521-54325-5, MR 0730208

External links

• Weisstein, Eric W. "Medial pentagonal hexecontahedron". MathWorld.
• Weisstein, Eric W. "Snub dodecadodecahedron". MathWorld.

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This page was last edited on 27 September 2023, at 17:19