Transformations of 2-port networks and tiling by rectangles

Abstract

This paper presents a novel investigation into the properties of 2-port networks, introducing the concepts of voltage drop and Π-equivalence. The primary contribution is the demonstration that any planar network is Π-equivalent to a network with a maximum of five edges. This finding has significant implications for tiling problems, specifically in relation to octagons shaped like the letter Π. We establish that if such an octagon can be tiled by squares, it can also be tiled by no more than five rectangles with rational aspect ratios. The theorem by Y. C. de Verdière, I. Gitler, and D. Vertigan from 1996 proves this only for 6 rectangles. In our approach, we use Π-equivalent transformations to simplify the network's structure. A novel transformation, which we have named Box-H, plays a crucial role in this process. By applying these transformations, we are able to significantly reduce the complexity of 2-port networks.