Snub icosidodecadodecahedron

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Snub icosidodecadodecahedron

Type	Uniform polyhedron
Elements	F=104, E=180, V=60 (χ=-16)
Faces by sides	(20+60){3}+12{5}+12{⁵/₂}
Wythoff symbol	\|⁵/₃ 3 5
Symmetry group	I
Index references	U₄₆, C₅₈, W₁₁₂
3.3.3.5.3.⁵/₃ (Vertex figure)	Medial hexagonal hexecontahedron (dual polyhedron)

In geometry, the snub icosidodecadodecahedron is a nonconvex uniform polyhedron, indexed as U₄₆.

[edit] Cartesian coordinates

Cartesian coordinates for the vertices of a snub icosidodecadodecahedron are all the even permutations of

(±2α, ±2γ, ±2β),

(±(α+β/τ+γτ), ±(-ατ+β+γ/τ), ±(α/τ+βτ-γ)),

(±(-α/τ+βτ+γ), ±(-α+β/τ-γτ), ±(ατ+β-γ/τ)),

(±(-α/τ+βτ-γ), ±(α-β/τ-γτ), ±(ατ+β+γ/τ)) and

(±(α+β/τ-γτ), ±(ατ-β+γ/τ), ±(α/τ+βτ+γ)),

with an even number of plus signs, where

α = ρ+1,

β = ρ²+τρ,

γ = τ²ρ²+τ²ρ+τ,

and where τ = (1+√5)/2 is the golden mean and ρ is the real solution to ρ³=ρ+1, or approximately 1.3247180. ρ is called the plastic constant. Taking the odd permutations of the above coordinates with an odd number of plus signs gives another form, the enantiomorph of the other one.