The other day, I finally managed to simplify Alfred Gray’s parametric equations for Costa’s minimal surface. I might edit this post later, with details on how to manipulate the Weierstrass elliptic functions that show up in the equations, but enjoy these for now:

Here, and are respectively the Weierstrass elliptic function, its derivative, and the Weierstrass zeta function, with invariants and , and and are the usual beta and gamma functions. The invariants are the ones corresponding to the semi-periods and . The parameter ranges are and .

I was very delighted at my result, that I decided to make a stylized plot of the surface in Mathematica, using Perlin’s simplex noise for the coloring: