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Periodic minimal surfaces

Abstract

A minimal surface is one for which, like a soap film with the same pressure on each side, the mean curvature is zero and, thus, is one where the two principal curvatures are equal and opposite at every point. For every closed circuit in the surface, the area is a minimum. Schwarz1 and Neovius2 showed that elements of such surfaces could be put together to give surfaces periodic in three dimensions. These periodic minimal surfaces are geometrical invariants, as are the regular polyhedra, but the former are curved. Minimal surfaces are appropriate for the description of various structures where internal surfaces are prominent and seek to adopt a minimum area or a zero mean curvature subject to their topology; thus they merit more complete numerical characterization. There seem to be at least 18 such surfaces3, with various symmetries and topologies, related to the crystallographic space groups. Recently, glyceryl mono-oleate (GMO) was shown by Longley and McIntosh4 to take the shape of the F-surface. The structure postulated is shown here to be in good agreement with an analysis of the fundamental geometry of periodic minimal surfaces.

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Mackay, A. Periodic minimal surfaces. Nature 314, 604–606 (1985). https://doi.org/10.1038/314604a0

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