Triply periodic minimal surface

In differential geometry, a triply periodic minimal surface (TPMS) is a minimal surface in ℝ³ that is invariant under a rank-3 lattice of translations.

These surfaces have the symmetries of a crystallographic group. Numerous examples are known with cubic, tetragonal, rhombohedral, and orthorhombic symmetries. Monoclinic and triclinic examples are certain to exist, but have proven hard to parametrise.

TPMS are of relevance in natural science. TPMS have been observed as biological membranes, as block copolymers, equipotential surfaces in crystals etc. They have also been of interest in architecture, design and art.

Nearly all studied TPMS are free of self-intersections (i.e. embedded in ℝ³): from a mathematical standpoint they are the most interesting (since self-intersecting surfaces are trivially abundant).

All connected TPMS have genus ≥ 3, and in every lattice there exist orientable embedded TPMS of every genus ≥3.

Embedded TPMS are orientable and divide space into two disjoint sub-volumes (labyrinths). If they are congruent the surface is said to be a balance surface.

The first examples of TPMS were the surfaces described by Schwarz in 1865, followed by a surface described by his student E. R. Neovius in 1883.

In 1970 Alan Schoen came up with 12 new TPMS based on skeleton graphs spanning crystallographic cells. While Schoen's surfaces became popular in natural science the construction did not lend itself to a mathematical existence proof and remained largely unknown in mathematics, until H. Karcher proved their existence in 1989.

Using conjugate surfaces many more surfaces were found. While Weierstrass representations are known for the simpler examples, they are not known for many surfaces. Instead methods from Discrete differential geometry are often used.

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