On the symmetry properties and crystallographic generation of the Schwarz minimal P surface

The problem of how to split the 3D space into two identical triply periodic but disjointed objects, using a contiguous surface of minimal area, was solved by Hermann Schwarz. His minimal surface, later called P after the (two) underlying primitive cubic lattices used for the frameworks of the two labyrinths, has a formidable analytical definition based on elliptic functions and Weierstrass-Enneper integrals, although a simple trigonometric approximation also exists. Here, we derive the Schwarz minimal P surface from close tiling of the 3D space with regular polyhedra. This approach clearly exposes the Im3m symmetry of this surface and provides a starting point for its iterative generation through the minimization of the area of its crystallographic asymmetric unit.