In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, the inverted snub dodecadodecahedron is a nonconvex uniform polyhedron

Nonconvex uniform polyhedron

In geometry, a uniform star polyhedron is a self-intersecting uniform polyhedron. They are also sometimes called nonconvex polyhedra to imply self-intersecting...

, indexed as U₆₀. It is given a Schläfli symbol s{5/3,5}.

Cartesian coordinates

Cartesian coordinates for the vertices of an inverted 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

Golden mean

Golden mean may refer to:*Doctrine of the Golden Mean , a chapter in Li Ji, one of the Four Books of Confucianism*Golden mean , the felicitous middle between the extremes of excess and deficiency...

 and
α is the negative real root of τα⁴−α³+2α²−α−1/τ, or approximately −0.3352090.
Taking the odd permutations of the above coordinates with an odd number of plus signs gives another form, the enantiomorph

Chirality (mathematics)

In geometry, a figure is chiral if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone. For example, a right shoe is different from a left shoe, and clockwise is different from counterclockwise.A chiral object...

of the other one.