Minimal Twin Surfaces

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 08] and [Fujimori and Weber 09]. 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.

KEYWORDS:

• Triply periodic minimal surfaces
• twin lattice
• Surface Evolver

Mathematics Subject Classification:

• Primary 53A10
• 49Q05

Acknowledgments

This manuscript, while self-contained, presents mathematical context and technical details as a complement and extension to another paper in preparation on material science. I would like to thank my collaborators in that project, especially Chenyu Jin, Lu Han, and Nobuhisa Fujita, for their helps on this note. I am very grateful to Karsten Große-Brauckmann for valuable suggestions, to Matthias Weber for his comments and Mathematica programs, and to Han Yu and Martin Traizet for helpful discussions. Most of the work was done while the author is visiting St Andrews University, and the author thanks Louis Theran for his kind invitation.

Notes

1 Paper in preparation.

2 The crystallographic term is “primitive unit cell.”

3 Confirmed by Traizet through personal communication.

4 The author thanks Mathias Weber and the anonymous referee for pointing this out.

Additional information

Funding

National Science Foundation (DMS-1440140)

A revision of the manuscript was done while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2017 semester, supported by the National Science Foundation under Grant No. DMS-1440140.