Snub dodecadodecahedron

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 has 84 faces (60 triangles, 12 pentagons, and 12 pentagrams), 150 edges, and 60 vertices.[1] It is given a Schläfli symbol sr{​52,5}, as a snub great dodecahedron.

3D model of a snub dodecadodecahedron

Cartesian coordinatesEdit

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 polyhedraEdit

Medial pentagonal hexecontahedronEdit

Medial pentagonal hexecontahedron
Type Star polyhedron
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 alsoEdit


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

External linksEdit