Fractals and resistors run in parallel in the classroom

Fractals appear in everything from biology to quantum mechanics, and understanding and employing them for practical uses has resulted in signal processing tools, medical treatments, and countless other applications. However, fractals are mostly absent from undergraduate physics curricula, which means many physics graduates have little exposure to such a critically important topic.

Creffield described a lesson plan aimed at early undergraduate physics students that incorporates fractals and fractional dimensions into elementary resistor laws. The experiment builds upon the results of connecting resistors in series or parallel, asking students what would occur given a fractal resistor network in the shape of a Sierpinski’s Gasket.

“Calculating equivalent resistances is an essential part of the curriculum, but it tends not to be very interesting after a student has performed a few examples,” said author Charles Creffield. “To calculate the equivalent resistance of a fractal network is far more challenging.”

To perform the experiment, students are provided with small triangles formed from resistors, and then combine these triangles iteratively into increasingly larger versions while measuring the resistance at each step. At the same time, the undergraduates calculate the resistance theoretically, exploring the concept of fractional dimensions.

“My aim was to show how a seemingly well-understood concept like dimension can have surprisingly subtle consequences,” said Creffield. “Showing students that perhaps they don’t comprehend basic concepts (like that of dimension) as completely as they think they do is an excellent way of getting them to think more critically about their assumptions.”

Source: “Fractals on a benchtop: Observing fractal dimension in a resistor network,” by Charles Creffield, The Physics Teacher (2022). The article can be accessed at https://doi.org/10.1119/5.0054306 .