Infinity, self-similarity, and continued fractions in physics: Applications to resistor network puzzles

Abstract

Puzzles involving infinite networks of resistors are an engaging way for students to explore the idea of infinity and self-similarity in physics. Recently K Atkin has described one such puzzle, alongside a solution based on an equivalent finite network (2022 Phys. Educ. 57 025015). Here we present a generalisation of this problem which showcases an important perspective on infinity as the limit of a process, and an alternative method of solution using continued fractions. We illustrate how our method can be used to devise new network puzzles suitable for class challenge problems.