ABSTRACT
Gauss Map and the Complex Plane ................................... 110 5.8 Homogeneity of Curvature on Surfaces.............................. 113 5.9 Molecular Shape and the Surfactant
Packing Parameter ............................................................... 114 5.10 Bending Energy and Relative Stabilities ........................... 115 5.11 TPMS Surface Symmetries and
Phase Transformations ........................................................ 117 5.12 Networks Embedded in TPMS ............................................ 118 5.13 A Brief Introduction to the Hyperbolic
Crystallography of Bicontinuous Mesomorphs and Their Use in Generating Surface Reticulations ................ 119
5.14 Polycontinuous Liquid Crystals .......................................... 121 5.15 Final Remarks ...................................................................... 125 Acknowledgments........................................................................... 126 References....................................................................................... 126
This chapter provides a semiformal, though largely qualitative, look at the mathematics of triply periodic minimal surfaces (TPMS) in relation to bicontinuous and polycontinuous liquid crystals. It is motivated largely by research of Stephen Hyde and others at the Applied Mathematics Department, Australian National University, Canberra, Australia and by the discovery of these TPMS liquid crystal partitions by Kåre Larsson and the late Krister Fontell at Lund University. The article is meant to complement the existing, more rigorous papers and provide an introduction to the more fundamental topics of differential geometry and topology of TPMS.
