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

TL;DR

The paper addresses constructing and proving the existence of a 1-parameter family of chiral, unbalanced triply-periodic minimal surfaces of genus $g=4$, the gyrating H'-T surfaces, related to Schoen's H'-T and the Gyroid paradigm. The authors combine a numerical construction from dual nets (qtz and qzd) using Surface Evolver with an explicit toric Weierstrass parametrization on a branched torus, establishing a period problem that yields a continuous curve of embedded TPMS connecting a saddle-tower limit to a doubly periodic Scherk limit. An existence proof is provided for this curve, ensuring the surfaces are well-defined and embedded, with consistency to gluing constructions near the saddle-tower regime. The work highlights a tunable chiral surface template suitable for photonic materials and deepens the link between discrete interpenetrating nets and continuous minimal-surface geometry.

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.

Figures (11)

• Figure 1: The pair of interpenetrating dual nets that lead to the qtz-qzd surface family, the quartz (qtz) net (orange) and its proper dual qzd network (blue). (A) Perspective view; (B-D) top views along the $c$ axis of the qzd (A), qtz (B) qzd and qtz nets. The translational unit cell ($P6_222$) is the gray frame.
• Figure 2: The numerically evolved minimal surface that separates the qtz and the qzd net. (A-C) the networks shown are the qtz and the qzd net. (D-F) the networks shown are the medial axes as calculated from the minimal surface, as the curves tracing the center lines that are most distance from the minimal surface.
• Figure 3: Schoen's H'-T surface: (A) Photograph of a plasticine model created by Alan Schoen. (B) The H'-T surface can be constructed by copy-translations from a hexagonal catenoidal patch and two adjacent triangular catenoidal necks, all aligned in the vertical axis (that is, the crystallographic $c$ axis). (C) The hexagonal lattice (hex) representing the skeletal graphs on one side of the surface (D) The bnn net representing the other skeletal graph. (E) The dual pair of hex and bnn nets. (Image (A) reproduced from Figure 13 of Schoen_1970).
• Figure 4: Branched torus on which our surface is parametrized. Dashed segments are branched cuts. Solid circles are the zeros of $G^2$, empty circles are the poles, and $\times$ represent symmetry centers, corresponding to fixed points on the surface under order-$2$ rotations with horizontal axis.
• Figure 5: Image of Weierstrass representation assuming bonnet angle $\theta = \pi/2$, seen from below, showing two twisted triangular catenoids (corresponding to the lower third of the branched torus), and two twisted hexagonal catenoids (one of these corresponds to the upper two thirds of the branched torus, the other one is a lattice translation included for clarity).

Theorems & Definitions (13)

• Theorem 2.1
• Remark 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6: Compare weyhaupt2006
• Lemma 2.7
• Proposition 3.1
• Lemma B.1
• proof