Minimal Surfaces à la Weierstrass

A small example in differential geometry.

Exact parametrizations of minimal surfaces can be found by integration.

∫((E^ξ) (Sqrt[ξ]) +  (ξ^(1/9))) 
        {(1 - (ξ^2)), I (1 + (ξ^2)), 2 ξ} 
        \[DifferentialD]ξ

Here is a small piece of the surface that is generated.

ParametricPlot3D[
      Evaluate[Re[
          % /. ξ -> 
                r (E^(I φ)) /. 
            (((r (E^(I φ))))^n)_ 
              -> (r^n) 
                (E^(I n φ))]], 
       {r,  0.6,  0.65},  {φ,  0,  18  π}, 
      PlotPoints  ->  {3, 800}];