Fractality in resistive circuits: The Fibonacci resistor networks

TL;DR

This paper introduces two novel infinite resistor networks based on Fibonacci sequences, demonstrating their convergence, self-similarity, and scale invariance, thus revealing fractal-like properties in electrical circuits.

Contribution

The paper presents new Fibonacci-based resistor networks with proven convergence and fractal characteristics, expanding the understanding of fractality in electrical systems.

Findings

Network equivalent resistance converges uniformly

Networks exhibit self-similarity and scale invariance

Generalizations include higher-order Fibonacci and recursive sequences

Abstract

We propose two new kinds of infinite resistor networks based on the Fibonacci sequence: a serial association of resistor sets connected in parallel (type 1) or a parallel association of resistor sets connected in series (type 2). We show that the sequence of the network's equivalent resistance converges uniformly in the parameter $\alpha=\frac{r_2}{r_1} \in [0,+\infty)$, where $r_1$ and $r_2$ are the first and second resistors in the network. We also show that these networks exhibit self-similarity and scale invariance, which mimics a self-similar fractal. We also provide some generalizations, including resistor networks based on high-order Fibonacci sequences and other recursive combinatorial sequences.