Higher Dimensional Electrical Circuits

Abstract

In this paper, we describe a physical problem, based on electromagnetic fields, whose topological constraints are two-dimensional versions of Kirchhoff’s laws, involving 2-simplicial complexes embedded in ${\mathfrak {R}}^3$ rather than graphs. The topological constraints are on flux and mmf vectors, which we show satisfy a ‘generalized Tellegen’s Theorem,’ i.e., that they constitute complementary orthogonal vector spaces analogous to voltage and current spaces for ordinary graph-based electrical circuits. We show that the problem of solving this two-dimensional electrical circuit reduces to that of solving an ordinary resistive electrical circuit that can be regarded as dual to it. We present a linear time algorithm for constructing, the above mentioned, dual electrical network. This algorithm is based on the construction of a triangle adjacency graph of the original embedded 2-simplicial complex. This graph contains the information of the order in which we encounter the triangles incident at an edge, when we rotate say clockwise with respect to the orientation of the edge. The triangle adjacency graph is processed through a ‘sliding’ algorithm which simulates sliding on the surfaces of the triangles, moving from one triangle to another which shares an edge with it but which also is adjacent with respect to the embedding of the complex in ${\mathfrak {R}}^3$. The connectedness information provided by this algorithm is used to construct, in linear time on the size of the 2-simplicial complex, its dual graph.