Transformations of 2-port networks and tiling by rectangles

TL;DR

This paper explores the properties of 2-port networks, introducing new concepts and demonstrating that planar networks can be simplified to at most five edges, with implications for tiling octagons with rectangles.

Contribution

It introduces the concepts of voltage drop and -equivalence for 2-port networks and proves a new bound on network simplification, extending tiling results from Kenyon's theorem.

Findings

Planar networks are -equivalent to networks with no more than 5 edges.

Octagons shaped as  can be tiled by at most 5 rectangles with rational aspect ratios.

Extension of Kenyon's theorem from 6 to 5 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.