Snub Dodecahedron - Cartesian Coordinates

Cartesian Coordinates

Cartesian coordinates for the vertices of a snub dodecahedron 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

α = ξ − 1 / ξ

and

β = ξτ + τ2 + τ /ξ,

where τ = (1 + √5) / 2 is the golden ratio and ξ is the real solution to ξ3 − 2ξ = τ, which is the number:

or approximately 1.7155615.

This snub dodecahedron has an edge length of approximately 6.0437380841.

Taking the even permutations of the above coordinates with an odd number of plus signs gives another form, the enantiomorph of the other one.