Barth Sextic is a algebraic surface of degree 6. Its formula is:

b = 0; ϕ=GoldenRatio; ContourPlot3D[ 4*(ϕ^2*x^2 - y^2)*(ϕ^2*y^2 - z^2)* (ϕ^2*z^2 - x^2) - (1 + 2*ϕ)*(x^2 + y^2 + z^2 - b^2)^2*b^2, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}, PlotPoints -> {6, 6}]

where ϕ is the golden ratio (ϕ = (1 + Sqrt[5])/2 ≈ 1.61803) and b is the parameter. In the above plot, b=1.

When b is 0, the surface is 6 intersecting planes, arranged in a way like 3 sets of x-crossed planes intersecting from mutually orthogonal directions. In the above image, the cleavages is an artifact of plotting software.

barth_sextic_para_change.mov barth_sextic.gcf (b goes from 0 to 1)