Schwarz surfaces generalized

By decoration of these various infinite two-dimensional manifolds (just as the sphere has been decorated with closed networks) several related structures have been proposed for graphite nets. These are mostly based on the P, D and G surfaces (the first two due to Schwarz (1890) and the last, the gyroid, discovered by Schoen (1970). However, many other surfaces (perhaps 50) are available for consideration. Some fit naturally with hexagonal sheets and others with sheets of square or lower symmetry. In general, the P, D and G surfaces are the least curved from planarity. Surfaces parallel to the surfaces of zero mean curvature have lower symmetry than those with H = 0. When decorated with graphite nets the symmetry may be further lowered to that of a sub-group of the symmetry group of the surface itself. 

The last equation demonstrates that the starting point for the solution of the problem is the calculation of ci(double layer (this makes low-frequency dielectric dispersion [LFDD] measurements a most valuable electrokinetic technique). Probably, the first theoretical treatment is the one due to Schwarz [61], who considered only surface diffusion of counterions (it is the so-called surface diffusion model). In fact, the model is inconsistent with any explanation of dielectric dispersion based on double-layer polarization. The generalization of the theory of diffuse atmosphere polarization to the case of alternating external fields and its application to the explanation of LFDD were first achieved by Dukhin and Shilov [20]. A full numerical approach to the LFDD in suspensions is due to DeLacey and White [60], and comparison with this numerical model allowed to show that the thin double-layer approximations [20,62,63] worked reasonably well in a wider than expected range of values of both and ku [64]. Figure 3.12 is an example of the calculation of As. From this it will be clear that (i) at low frequencies As can be very high and (ii) the relaxation of the dielectric constant takes place in the few-kHz frequency range, in accordance with Equations (3.56) and (3.57). 

This shape is, as one would expect, in the form of a saddle in addition to the sides of the skew quadrilateral, it admits two other straight hues, which would respectively join the midpoint of each of the sides to the midpoint on the opposite side. Finally Mr. Schwarz continues the surface beyond the contour indicated, and to deteimine the general shape of it the total surface is made of portions identical to that above, juxtaposed in a certain way, and it is very curious ... 

Mr. Schwarz determination of surface supported on the sides of a skew quadrilateral, in a case more general than that of 141 the same of another surface deriving from the preceding one by inflection. - Symmetry in surfaces of minimum area on which one can trace a straight line, and in those which can be cut by a plane in such a way that, all along the section, their elements are perpendicular to this plane. Determination, by elliptic functions, of two particular surfaces. 143... 

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