DESCRIPTION OF TRIPLY-PERIODIC MINIMAL SURFACES. Andrew Fogden, Physical Chemistry 1, Center for Chemistry and Chemical Engineering, P.O. Box 124, S-221 00 Lund, Sweden

The special class of three-dimensionally periodic surfaces bearing uniform mean-curvature, and most notably the minimal (zero mean-curvature) surfaces, are reviewed with respect to their classification, mathematical description and their role in crystalline and liquid-crystalline structures, and illustrated with many examples and applications. The simplest and most familiar of these bicontinuous space partitions are the P, D and G minimal surfaces, possessing cubic symmetry and genus 3. The exact equation defining any minimal surface can be obtained via parametrisation in the complex plane, reducing the problem to the construction of its corresponding complex-analytic function. Using an elegant method of matching the distribution of functional singularities in the plane to the symmetry and topology in real space, a multitude of new minimal surfaces have been discovered, or verified, in recent years. This increased knowledge of minimal partitions has greatly helped in visualising and understanding the geometry underlying many complicated chemical and physical assemblies. These developments are illustrated by taking the 3 above-mentioned surfaces and considering the new families liberated by the different modes of distortion to lower symmetries. More elaborate topologies are also presented using the examples of the S'-S'', H'-T and T-WP minimal surfaces. These specific cases are of relevance to the description of zeolitic framework structure-types and also to the structure of novel bicontinuous liquid-crystalline phases formed by surfactants in water. Fitting of the minimal surfaces to the strucural information is assessed. The exact equations of the simple minimal surfaces can be approximated to great accuracy as level surfaces (equipotentials) of three-dimensional Fourier series, in which only the lowest frequency contributions are retained. Inclusion of higher frequencies yields good representations of more complicated examples, for which the exact parametrisations become unwieldly. For surfaces with uniform, but non-zero, mean-curvature the generalised parametrisation is difficult to apply and a Fourier description becomes even more beneficial. This is illustrated by the uniformly-curved companions of the H'-T minimal surface, which undergo necking transformations to layered "mesh" geometries of connectivity 3 or 6.