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}
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
	Siddid

3D model of a snub dodecadodecahedron

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

3D model of a medial pentagonal hexecontahedron

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.
• Uniform polyhedra and duals

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