Snub dodecadodecahedron

From Wikipedia, the free encyclopedia

Snub dodecadodecahedron

Type	Uniform 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
Index references	U₄₀, C₄₉, W₁₁₁
3.3.⁵/₂.3.5 (Vertex figure)	Medial pentagonal hexecontahedron (dual polyhedron)

In geometry, the snub dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U₄₀.

This polyhedron can be considered a snub great dodecahedron.

[edit] 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 solution to 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.