In geometry, the snub icosidodecadodecahedron is a nonconvex uniform polyhedron, indexed as U46. It has 104 faces (80 triangles, 12 pentagons, and 12 pentagrams), 180 edges, and 60 vertices. As the name indicates, it belongs to the family of snub polyhedra.
The circumradius of the snub icosidodecadodecahedron with unit edge length is
where ρ is the plastic constant, or the unique real root of .
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 τ = (1+)/2 is the golden ratio; ρ is the plastic constant, or the unique real solution to ρ3=ρ+1; α = ρ+1 = ρ3; β = τ2ρ4+τ; and γ = ρ2+τρ. 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 polyhedra
Medial hexagonal hexecontahedron
The medial hexagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform snub icosidodecadodecahedron.