Infinite resistive lattices

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Abstract
Was introduced by Venezian, 1 and works by superposing the potentials and currents corresponding to two configurations in each of which only one finite node carries a current from outside the lattice. The currents in such a uniterminal circuit enjoy the full symmetry of the lattice, whereas the currents of the diterminal circuit that is actually of interest have lower symmetry. Use of a similar higher symmetry was analo- gously made in a recent treatment of the resistance between the vertices of regular polyhedra constructed from identical resistors. 2 Despite the emphasis on the high symmetry of a unitermi- nal lattice, the method proceeds, oddly enough, through the imposition of an ansatz that does not possess the full sym- metry in a manifest manner. By using complex Fourier trans- forms we illuminate the method of Venezian and generalize it from the original square lattice in two dimensions to cubic and hypercubic lattices in higher dimensions, as well as to triangular and hexagonal lattices in two dimensions. The final result for the resistance between two nodes is expressed as an integral transform, and it does not seem pos- sible to express this as a known higher transcendental func- tion, like a generalized hypergeometric function, for ex- ample. Nevertheless, for specific choices of the lattice, and of the nodes, it turns out to be possible to evaluate the integrals algebraically, in terms of p and of the square roots of inte- gers. It was within the capacity of Mathematica 3.0 to evalu- ate sample results for two-dimensional lattices, yielding ex- act answers in place of the numerical approximations given in Venezian's paper. Special cases of the results obtained in this paper have been published before. The resistance between adjacent mesh points in the square lattice was derived by Aitchison, 3 while Bartis 4 treated adjacent mesh points in more general two- dimensional lattices. Purcell 5 gives the resistance between diagonally opposedmesh points of the square lattice without proof, while a proof of this result is provided by Lavatelli, 6 who emphasizes how the resistive net can be thought of as a discrete approximation to a continuous resistive medium. Trier 7 obtains a double integral representation for the resis- tances in the case of the square lattice, together with a table of exact results; this work was further evaluated by Cameron, 8 who considered carefully the conditions for the existence and uniqueness of Trier's solution. In the present work we tacitly obtain uniqueness by requiring that the cur- rents at infinity vanish, and this condition accords with one of Cameron's criteria. Finally, the work of Zemanian 9 should be noted: he also considers the uniqueness problem, as well as the double integral representation for the square lattice, and analogous expressions for uniform and nonuniform lat- tices in more than two dimensions.

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