Parameterization of the Schwarz P surface

Question

Is there a closed form parameterization of the Schwarz P minimal surface?

Answer 1

The following description is adapted and corrected from Gandy and Klinowski (2000), Exact computation of the triply periodic Schwarz P minimal surface, Chemical Physics Letters 322, 579–586. The expression for $z$ below is a major correction, also talked about here.

Let $w$ be a complex number with $0\le\arg(w)\le\frac\pi4$ and $|w+e^{i\pi/4}|\le\sqrt2$. The following coordinates, viewed as functions of $w$, give a fundamental patch of the P surface: $$x=-\kappa\operatorname{Im}\left(\frac1{\sqrt8}F\left(\sin^{-1}\frac{\sqrt8w}{\sqrt{w^4+4w^2+1}},\frac14\right)\right)$$ $$y=\kappa\operatorname{Re}\left(\frac1{\sqrt8}F\left(\sin^{-1}\frac{-\sqrt8w}{\sqrt{w^4+4w^2+1}},\frac34\right)\right)$$ $$z=-\kappa\operatorname{Im}\left(\frac1{2+\sqrt3}F\left(\sin^{-1}\frac{w^2}{2-\sqrt3},97-56\sqrt3\right)\right)$$ $\kappa=\frac2{K(3/4)}$ is a normalising factor and arguments to elliptic integrals $F,K$ follow the conventions of Mathematica/mpmath. Now take this patch and transform it according to the following $12$ transformation matrices: $$\left[\begin{array}{ccc|c} -q&-q&0&0\\-q&q&0&1\\0&0&1&\frac12 \end{array}\right]\text{and all permutations of rows}$$ $$\left[\begin{array}{ccc|c} q&q&0&1\\q&-q&0&0\\0&0&-1&\frac12 \end{array}\right]\text{and all permutations of rows}$$ Here $q=\frac{\sqrt2}2$ and applying the transformation $[\mathbf A\mid\mathbf b]$ to a point $\mathbf x$ gives $\mathbf{Ax}+\mathbf b$. The union of the $12$ transformed patches is a hexagonal plate within the cube $[0,1]^3$:

Mirroring this plate in the coordinate planes gives a unit cell of the Schwarz P surface with side length $2$.