Two Mathematicians Crack Age-Old Geometry Problem in Quarantine

The inscribed square problem is over a hundred years old.

An age-old geometry problem, the inscribed square problem, has been cracked by two mathematicians during their quarantine time, adding to the list of fascinating discoveries made during quarantine.

The inscribed square problem was first posed by German mathematician Otto Toeplitz in 1911, in which he predicted that "any closed curve contains four points that can be connected to form a square," according to Quanta Magazine.

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A century-old problem

In order to be productive during their COVID-19 quarantine time, two friends and mathematiciansJoshua Greene and Andrew Lobb, decided to analyze a set of loopy shapes called smooth, continuous curves to prove that every one of these shapes contains four points that form a rectangle, and in doing so crack the inscribed square problem.

They posted the solution online for all to see.

“The problem is so easy to state and so easy to understand, but it’s really hard,” said Elizabeth Denne of Washington and Lee University told Quanta.

The inscribed square problem, also known as the "rectangular peg" problem has its basis in a closed loop — any curvy line that ends where it starts. The problem predicts that every closed loop contains sets of four points that form the vertices of rectangles of any desired proportion.

While the problem might seem simple on paper, it has actually stumped some of the world's best mathematicians for years.

As lockdown restrictions were eased, Greene and Lobb emerged with their final proof, after having collaborated over Zoom video calls. It showed for once and for all that Toeplitz's predicted rectangles do indeed exist.

Shifting the perspective

In order to reach their findings, they had to transport the problem into an entirely new geometric setting. Greene and Lobb’s proof is a great example of how a shift in perspective can help people find the correct answer to a problem.

Generations of mathematicians failed to solve the "rectangular peg" problem because they tried to solve it in more traditional geometric settings. The problem is so difficult because it deals with curves that are continuous, but not smooth — a type of curve can veer in all sorts of directions.

“These problems that were being thrown around in the 1910s and 1920s, they didn’t have the right framework to think about them,” Greene told Quanta. “What we’re realizing now is that they’re really hidden incarnations of symplectic phenomena.”