Fractals on a benchtop: Observing fractal dimension in a resistor network

TL;DR

This paper presents a simple, cost-effective experimental method for observing fractal dimensions by measuring the resistance scaling in a resistor network, enhancing understanding of fractional dimensions and complex scaling laws.

Contribution

It introduces a novel, hands-on approach for students to experimentally observe and analyze fractal dimensions using resistor networks, bridging theory and practical visualization.

Findings

Resistor network resistance scales with network size following a power law.

Students can construct and analyze fractal resistor networks easily.

The method effectively demonstrates fractional dimension concepts.

Abstract

Our first experience of dimension typically comes in the intuitive Euclidean sense: a line is one dimensional, a plane is two-dimensional, and a volume is three-dimensional. However, following the work of Mandelbrot \cite{mandelbrot}, systems with a fractional dimension, ``fractals'', now play an important role in science. The novelty of encountering fractional dimension, and the intrinsic beauty of many fractals, have a strong appeal to students and provide a powerful teaching tool. I describe here a low-cost and convenient experimental method for observing fractal dimension, by measuring the power-law scaling of the resistance of a fractal network of resistors. The experiments are quick to perform, and the students enjoy both the construction of the network and the collaboration required to create the largest networks. Learning outcomes include analysis of resistor networks beyond the elementary series and parallel combinations, scaling laws, and an introduction to fractional dimension.