Minimal Twin Surfaces

TL;DR

This paper explores minimal twin surfaces as symmetric copies of classical triply periodic minimal surfaces, using numerical methods to provide evidence for their existence and proposing new models and deformations.

Contribution

It introduces the concept of minimal twin surfaces related by reflections, constructs examples of D and G twins, and develops new models and deformation ideas for these surfaces.

Findings

Numerical evidence supports the existence of D and G twins.

New cubic polyhedral models for D and G surfaces are proposed.

Insights into rPD twins are enhanced by previous constructions.

Abstract

We report some minimal surfaces that can be seen as copies of a triply periodic minimal surface (TPMS) related by reflections in parallel mirrors. We call them minimal twin surfaces for the resemblance with twin crystal. Brakke's Surface Evolver is employed to construct twinnings of various classical TPMS, including Schwarz' Primitive (P) and Diamond (D) surfaces, their rhombohedral deformations (rPD), and Schoen's Gyroid (G) surface. Our numerical results provide strong evidences for the mathematical existence of D twins and G twins, which are recently observed in experiment by material scientists. For rPD twins, we develop a good understanding, by noticing examples previously constructed by Traizet (2008) and Fujimori and Weber (2009). Our knowledge on G twins is, by contrast, very limited. Nevertheless, our experiments lead to new cubic polyhedral models for the D and G surfaces, inspired by which we speculate new TPMS deformations in the framework of Traizet.