Two mathematicians just solved a century-old geometry problem

IN 1911, GERMAN MATHEMATICIAN Otto Toeplitz first posed the inscribed square problem, predicting that ‘any closed curve contains four points that can be connected to form a square.’

- CAROLINE DELBERT

A proof for Toeplitz’s theory still eludes experts, but according to Quanta Magazine, two mathematicians in quarantine have taken a huge leap towards a solution.

Imagine a serpentine belt in an engine. Like a closed curve, it can settle into almost any shape, as long as there are no ‘corners’. Toeplitz suggested that in any of these shapes, there must be four points along the curves that, when connected, create a perfect square (see fig. 1). Researchers proved this for ‘smooth, continuous’ closed curves in 1929, but throughout the COVID-19 quarantine, modern mathematicians Joshua Greene, of Boston University, and Andrew Lobb, of Durham University, sought to generalize the proof from squares to all kinds of rectangles, broadening Toeplitz’s ‘square peg problem’ into a ‘rectangular peg problem’.

For advanced mathematical concepts, proofs often begin with a special case within a generalisation so the learned qualities of that special case make the generalisation easier to prove. For Greene and Lobb, a critical change in perspective made the special case that ‘loosened the lid’ on decades of prior work.