Amphiphiles are molecules which have both hydrophilic and hydrophobic parts. In water- and/or oil-like solvent, they self-assemble into extended sheet-like structures due to the hydrophobic effect. The free energy of an amphiphilic system can be written as a functional of its interfacial geometry, and phase diagrams can be calculated by comparing the free energies following from different geometries. Here we focus on bicontinuous structures, where one highly convoluted interface spans the whole sample and thereby divides it into two separate labyrinths. The main models for surfaces of this class are triply periodic minimal surfaces, their constant mean curvature and parallel surface companions, and random surfaces. We discuss the geometrical properties of each of these types of surfaces and how they translate into the experimentally observed phase behavior of amphiphilic systems.
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