Uniform tiling with electrical resistors

Abstract

Abstract.The electric resistance between two arbitrary nodes on any infinite lattice structure of resistors that is a periodic tiling of space is obtained. Our general approach is based on the lattice Green's function of the Laplacian matrix associated with the network. We present several non-trivial examples to show how efficient our method is. Deriving explicit resistance formulas it is shown that the Kagomé, the diced and the decorated lattice can be mapped to the triangular and square lattice of resistors. Our work can be extended to the random walk problem or to electron dynamics in condensed matter physics.

Modeling the electrical properties of three-dimensional printed meshes with the theory of resistor lattices

“…Analytical formulas are available for triangular, honeycomb, square, rectangular, diamond, simplecubic, body-centered cubic, face-centered cubic, and hypercubic (simple cubic in N dimensions) lattices [18][19][20][21][22]. Also, Green's function approach has been applied to uniform tiling of space with electrical resistors [23]. The lattice Green's function formalism can be developed to address several types of defects, including a broken resistor and an extra resistor between two initially nonconnected nodes [24,25].…”

Sum rules for effective resistances in infinite graphs

“…But by the way G is constructed, this voltage is the same if we replace all the pairs of edges between vertices originally in G with single edges with resistance 2. Once we have done that, we are back at the original grid G with all edges having resistance 2, with the same unit current flowing from i to j, and thus the voltage is double that of the original G, which was shown to be 2 k in (6). Now, if d(v, w) = 2 and v, w are subdividing vertices, then since G is 2-arc transitive s(G) will have exactly two different isomorphism classes of paths of length 2: those whose middle point is a subdividing vertex, and those whose middle point is a vertex in G. In a large ball in s(G) containing n vertices from G there will be nk 2 paths of length 2 whose middle point is a subdividing vertex, and the resistance across each of these paths is approximately 4 k , as well as n k(k−1) 2 paths of length 2 whose middle point is in G, and each of these has a resistance close to R vw .…”

On the perturbation of a uniform tiling with resistors

“…The formulation of calculating the resistance between two arbitrary sites in an infinite uniform lattice network of resistors is briefly reviewed (see, for more details Ref. 16). where k is the wave vector in the reciprocal lattice and is limited to the first Brillouin zone [18][19][20][21].…”