The chiral gyrating H’-T surface family: construction from the dual qtz – qzd nets and existence proof using a toroidal Weierstrass method

This paper provides an existence proof for a one-parameter family of chiral unbalanced triply periodic minimal surfaces of genus four. We name these gyrating H’-T surfaces because they are related to Schoen’s H’-T surfaces in a similar way as the gyroid is to the primitive surface. Their chirality is manifested in a screw symmetry of order six. The two labyrinthine domains on either side of the surface are not congruent, rather one representing the quartz net (qtz) and the other one the dual of the quartz net (qzd). The family tends to the Scherk saddle tower in one limit and to the doubly periodic Scherk surface in the other. The motivation for the construction was to construct a chiral tunable unbalanced surface family, originally as a template for photonic materials. The numeric construction is based on reverse engineering of the tubular surface of two suitably chosen dual nets, using the Surface Evolver to minimise area or curvature variations. The existence is proved using Weierstrass parametrisations defined on the branched torus. This article is part of the theme issue ‘Geometry, materials and the imagination’.