Snub dodecadodecahedron

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Snub dodecadodecahedron
Snub dodecadodecahedron.png
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 Snub dodecadodecahedron vertfig.png
Bowers acronym Siddid

In geometry, the snub dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U40. It is given a Schläfli symbol sr{5/2,5}, as a snub great dodecahedron.

Cartesian coordinates[edit]

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[edit]

Medial pentagonal hexecontahedron[edit]

Medial pentagonal hexecontahedron
DU40 medial pentagonal hexecontahedron.png
Type Star polyhedron
Face DU40 facets.png
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[edit]


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

External links[edit]