A chiral minimal surface from space group symmetries

Triply periodic minimal surfaces form the interface between bicontinuous structures in diblock copolymers and lipid bilayers in the membranes of certain organelles. Here we study a one parameter family of chiral triply periodic minimal surfaces which partition space into two disjoint regions, enclosing a quartz network on one side and its dual qzd network on the other. The three dimensional orbifold of the underlying space group, $P6_{2}22$, is used to analyze the symmetries of the surface. The symmetry elements combined with the flat points of the surface are used to obtain a regular parametrization known as the Weierstrass-Ennepper representation.