Infinite periodic minimal surface

Luzzati V, Delacroix FI and Gulik A 1996 The micellar cubic phases of lipid-containing systems Analogies with foams, relations with the infinite periodic minimal surfaces, sharpness of the polar/apolar partition J. Physique. II 6 405-18... 

A. H. Schoen, Infinite Periodic Minimal Surfaces without Self-Intersections, NASA Technical Note No. D-5541 (1970). 

Schoen, A. H. 1970 Infinite periodic minimal surfaces without self-intersections. NASA Technical Note D-5541. 

The simplest three-periodic hyperbolic surfaces are "Infinite Periodic Minimal Surfaces" (EPMS, named by Alan Schoen [6]). For these surfaces, the mean curvature is constant on the surface, and everywhere identically zero. This is a defining characteristic of minimal surfaces. For these structures, the sub-volumes can be geometrically identical. This occurs if the IPMS contains straight lines. Such surfaces have been called "balance surfaces" by Koch and Fischer [7]. We focus primarily on IPMS in this book. Some further discussion of general properties of minimal surfaces is in order here, since a number of their geometrical and topological properties will be required for later chapters. 

S.T. Hyde, Infinite periodic minimal surfaces and crystal structures (1986), Ph. D. Thesis, Monash University. 

The structure of the PLB has been related to that of cubic phcises[7], discussed in Chapters 4 and 5. However, as we shall see, a description of these membrane morphologies as equilibrium phases seems to be applicable, if at all, in only a few cases that we have encountered. Independently of Larsson et al. [7], Linder and Staehelin [14] also suggested that a certain "membrane lattice" in a parasitic protozoa did indeed correspond to an infinite periodic minimal surface. However, no further structural details, such as the symmetry or form of IPMS, were deduced or discussed. Some ten additional examples of membrane assemblies displaying cubic symmetries have been pointed out [15,16] but no structural details were inferred. To the best of our knowledge, the above references ([7, 14-16]) are the only reports in which membrane assemblies have been related to the structure of IP. There are. 

DPMS, PMS, PCS, PNS All abbreviations for periodic hyperbolic surfaces infinite periodic minimal surfaces, periodic minimal surfaces, periodic cubic surfaces and periodic nodal surfaces respectively. (The abbreviations P-, D-and G- prepended to these indicate the topology and symmetry of the periodic surface, corresponding to the relative tuimel arangements and black-white sub-group of the P-surface, the D-surface and the gyroid respectively. 

The field of infinite periodic minimal surfaces (IPMS), that was introduced a few decades ago for the analysis of the topology of crystal structures [41], is a different approach to the analysis of nets many common nets are related to the known intersection-free IPMS [42], The IPMS studies have also produced a systematic enumeration of nets that has been recently proposed by the EPINET project (Euclidean Patterns in Non-Euclidean Tilings, see http //epinet.anu.edu.au) instead of working directly in three dimensions, the intrinsic hyperbolic geometry of IPMS is used to map 2D hyperbolic patterns into 3D Euclidean space [43],... 

R.B. King, Carbon Networks on Cubic Infinite Periodic Minimal Surfaces, in Discrete Mathematical Chemistry, DIMACS Series in Discrete Mathematical and Theoretical Computer Science, Vol. 51, eds. P. Hansen, P.W. Fowler and M. Zheng, American Mathematical Society, Providence, RI, 2000, pp. 235-248. 

The three dimensional (3D) cubic V2 phases are arranged as single continuous lipid curved bilayers forming a eomplex network containing two non-intersecting water channels [90]. Three different bicontinuous cubic nanostructures (a family of closely related phases) have been identified in the literature. They have a primitive (P), a gyroid (G), or a diamond (D) infinite periodic minimal surface (IPMS) [88, 89]. 

A special case is the much discussed infinite periodical minimal surface with a mean curvature H=Ho everywhere equal to zero, which separates equal volumes of water and oil. Evidently, interfacial structures of this principal nature can readily account for the water and oil bicontinuity observed for... 

Key words Cubic phase - infinite periodic minimal surface - phase transition - incubation time -... 

Besides the description of the stmcture on a molecular level, a discussion is given on the basis of an infinite periodic minimal surface (IPMS). In lyotropic systems it is well accepted that the interface between the hy-... 

The periodic network structures with known coordination number at a junctions, Nj, and distance between adjacent junctions. Dp e.g., the infinite periodic minimal surfaces (IPMS), were employed to confirm the accuracy of the 3D thinning algorithm. Various types of IPMS, such as the Schoen s... 

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