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

TL;DR

This paper introduces a new family of chiral, triply-periodic minimal surfaces called gyrating H'-T surfaces, constructed from dual quartz nets and proven to exist using a toroidal Weierstrass method, with potential applications in photonic materials.

Contribution

It provides a novel construction and existence proof for a family of chiral minimal surfaces related to quartz nets, expanding the understanding of triply-periodic minimal surfaces.

Findings

Family tends to Scherk saddle tower in one limit

Family tends to doubly periodic Scherk surface in another

Constructed surfaces exhibit screw symmetry of order six

Abstract

This paper provides a construction and existence proof for a 1-parameter family of chiral unbalanced triply-periodic minimal surfaces of genus 4. We name these {\textit{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 manifest 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 (\texttt{qtz}) and the other one the dual of the quartz net (\texttt{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 \textit{Surface Evolver}} to minimize area or curvature variations. The existence is proved using Weierstrass parametrizations defined on the branched torus.