Triply periodic minimal surface

Monoclinic and triclinic examples are certain to exist, but have proven hard to parametrise.

TPMS have been observed as biological membranes,[2] as block copolymers,[3] equipotential surfaces in crystals[4] etc.

): from a mathematical standpoint they are the most interesting (since self-intersecting surfaces are trivially abundant).

[7] Embedded TPMS are orientable and divide space into two disjoint sub-volumes (labyrinths).

[9][10] In 1970 Alan Schoen came up with 12 new TPMS based on skeleton graphs spanning crystallographic cells.

[11] [12] 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.

[6] Members of this family can be continuously deformed into each other, remaining embedded in the process (although the lattice may change).

For surfaces containing lines the possible boundary polygons can be enumerated, providing a classification.