Transformations of 2-port networks and tiling by rectangles

Abstract

In this paper, we study 2-port networks and introduce new concepts of voltage drop and $\Pi$-equivalence. The main result is that each planar network is $\Pi$-equivalent to a network with no more than 5 edges. This implies that if an octagon in the shape of the letter $\Pi$ can be tiled by squares then it can be tiled by no more than 5 rectangles with rational aspect ratios. Kenyon's theorem from 1998 proves this only for 6 rectangles.

In English and in Russian. English translation added and minor improvement of exposition performed