Of the six cubic phases identified so far in lipid-containing systems, the structures of only two have been determined unambiguously. We tackle the structure determination of the other four. We use for that purpose a novel pattern recognition approach, which consists of generating all the sets of phase angles (phi-sets) compatible with the observed reflections, and of screening them in a search for the "best" one. Two criteria are used for screening: both involve the parameter [(delta rho)4] (delta rho is a dimensionless function proportional to the Fourier transform of the set of observed structure factors). One is a test of smoothness, based upon the postulate that the "best" phi-set is that whose [(delta rho)4] is minimum; this criterion, equivalent to maximum entropy, is fulfilled when the system is devoid of heavy atoms, and when the polar and the hydrocarbon moieties occupy almost equal volumes. The other criterion is based upon the notion that [(delta rho)4] takes the same (or similar) values in thermodynamic phases with the same (or similar) chemical composition, whatever the structure of the phases. The validity of the two criteria is verified using numerous examples. The six cubic phases are analysed using this approach. The structure of three of them (Q230, Q224, Q229) can be described in terms of two three-dimensional networks of connected rods, mutually intertwined and unconnected: in Q230 the rods are coplanarly joined 3 by 3; in Q224 the rods are tetrahedrally joined 4 by 4; in Q229 the rods are cubically joined 6 by 6. The structures of Q212 and Q227 are related to those of Q230 and Q224, respectively; one of the two networks of rods is preserved, the other is replaced by a lattice of closed micelles. The structure of Q223 appears to consist of a cage-like continuous three-dimensional network of connected globules, coplanarly joined 3 by 3 at one end and 4 by 4 at the other, enclosing a three-dimensional lattice of closed micelles. The analogies of the structures of Q230, Q224, Q229 with the three fundamental cubic infinite periodic minimal surfaces are discussed. More interestingly, the structures of, on the one hand Q230, Q224, Q229 and of Q212, Q227, Q223 on the other, are shown to provide topological generalizations of the two paradigms of lipid organization; namely, the bilayer and the monolayer.(ABSTRACT TRUNCATED AT 400 WORDS)