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.’ 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 generalise 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.

The pair built upon a foundation laid by mathematician Herbert Vaughan. In the 1970s, Vaughan found that plotting all unordered pairs – or pairs of values that have no particular relationship to one another – on a closed curve in 2D space makes a Möbius strip, a one-sided ‘ribbon’ connected in such a way that you can trace the entire inside and outside without lifting your pen (see fig. 2). This is because when you plot unordered coordinate pairs on a plane, each pair (x, y) occupies the same place as its inverse (y, x).

For Greene and Lobb, Vaughan’s proof was like the first number in a sudoku puzzle. It was a helpful starting hint, sure, but still left an entire mystery to solve.

The next breakthrough arrived in 2019, when Cole Hugelmeyer, a PhD student studying mathematics at Princeton University, turned Vaughan’s Möbius strip into a 4D plot. Hugelmeyer extended (x, y) into (x, y, a, b), in which a represented the distance between x and y, and b represented the angle at which the chord connecting x and y met the x-axis. After Hugelmeyer plotted the Möbius strip in this fashion, he rotated it by adjusting only b.

Why? Because x, y, and a represent the properties of a rectangle: two equidistant pairs of points intersecting at a common midpoint. So, the point where the rotated Möbius strip intersected the original Möbius strip signified the vertices of a rectangle on the original closed curve.

Hugelmeyer’s proof applied to just a third of all possible rectangles, however. So in Greene and Lobb’s attempt to prove all rectangles could be found in a

closed curve, they used a different 4D concept called ‘symplectic space’ to plot their curves. Symplectic space applies a vector to an object in 3D space – for example, factoring a planet’s momentum into charting its orbit. Greene and Lobb sought to apply vectors to shapes like the Möbius strip to determine the conditions under which they would intersect with themselves (the indicator for plottable rectangles).

The mathematicians considered a shape called a Klein bottle (see fig. 3), which ‘overlaps itself’ in 3D space such that it looks like a pitcher pouring both through and into itself. The Klein bottle is made from two intersecting Möbius strips – the same signifier of a rectangle as in Hugelmeyer’s proof – so Greene and Lobb determined that wherever they had a Klein bottle, they also had a plottable rectangle.

And if two Möbius strips don’t intersect, it’s impossible to have a Klein bottle, so the two researchers deduced that any configuration of intersecting Möbius strips must produce a rectangle. That means infinite configurations, and infinite rectangles.

The solution combines decades of accumulated institutional memory – how generations of researchers expand on each other’s work – with sudden intuition. Nearly 110 years of mathematical progression made this solution possible. Well, that and a lot of time in quarantine.

Researchers have finally proved that all continuous, closed curves contain four points that create a rectangle (fig. 1).