A chiral family of triply-periodic minimal surfaces derived from the quartz network

TL;DR

This paper introduces a new family of triply-periodic minimal surfaces with hexagonal symmetry, derived from the quartz network, providing a parametrisation and solution to the period problem using the Weierstrass-Enneper formalism.

Contribution

It presents a novel analytical description of these minimal surfaces based on dual graphs and the generalized Voronoi construction, expanding the methods for generating such surfaces.

Findings

Successfully parametrized the new minimal surfaces.

Demonstrated the applicability of the method to other dual graph topologies.

Provided a solution to the period problem for this family.

Abstract

We describe a new family of triply-periodic minimal surfaces with hexagonal symmetry, related to the quartz (qtz) and its dual (the qzd net). We provide a solution to the period problem and provide a parametrisation of these surfaces, that are not in the regular class, by the Weierstrass-Enneper formalism. We identified this analytical description of the surface by generating an area-minimising mesh interface from a pair of dual graphs (qtz & qzd) using the generalised Voronoi construction of De Campo, Hyde and colleagues, followed by numerical identification of the flat point structure. This mechanism is not restricted to the specific pair of dual graphs, and should be applicable to a broader set of possible dual graph topologies and their corresponding minimal surfaces, if existent.