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.

Snub dodecadodecahedron
TypeUniform star polyhedron
ElementsF = 84, E = 150
V = 60 (χ = −6)
Faces by sides60{3}+12{5}+12{5/2}
Wythoff symbol| 2 5/2 5
Symmetry groupI, [5,3]+, 532
Index referencesU40, C49, W111
Dual polyhedronMedial pentagonal hexecontahedron
Vertex figure
3.3.5/2.3.5
Bowers acronymSiddid
3D model of a snub dodecadodecahedron

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.

Medial pentagonal hexecontahedron
Medial pentagonal hexecontahedron
TypeStar polyhedron
Face
ElementsF = 60, E = 150
V = 84 (χ = −6)
Symmetry groupI, [5,3]+, 532
Index referencesDU40
dual polyhedronSnub 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

References
  1. Maeder, Roman. "40: snub dodecadodecahedron". MathConsult.


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