- Researchers have solved a century-old math problem using cutting-edge geometry.
- Cracking age-old math puzzles can require decades of thinking and new perspectives.
- In this case, a curve becomes a Mobius strip which in turn becomes ... well, you'll see.

In 1911, German mathematician Otto Toeplitz first posed the inscribed square problem, in which he predicted that "any closed curve contains four points that can be connected to form a square," according to *Quanta*. For more than a century, it's remained unsolved.

But now, two mathematician friends have used their quarantine time to crack a variation of the age-old geometry problem. They analyzed a set of loopy shapes called smooth, continuous curves to prove why every one of these shapes contains four points that form a rectangle.

The secret to beginning to solve this problem came from a simple change of perspective. Let’s start with the string-like closed curve itself. For the rectangle problem in question, the closed curve must be smooth and continuous, meaning it forms a loop and does not have corners.

First, in the late 1970s, mathematician Herbert Vaughan found that plotting all the pairs of points that make up an example closed curve, when plotted indiscriminately with the x or y coordinates taken in any order, makes the shape of a Mobius strip. Somehow, removing the x-y context lets the pairs of coordinates arrange in exactly the right way.

But that was 40 years ago, and even that milestone came after decades of wondering. While the problem was proven for squares in 1929, that proof was much simpler. A square is a special case of rectangle, but generalizing to *all* rectangles took away the elegant and neat way the case for squares is proven.

Mathematical proofs of tough ideas often work this way, starting with a special case with qualities that make it easier to prove. This is like the first number you place in the sudoku or the pop culture clue that lets you start to fill in the *New York Times* crossword. But from there, mathematicians spent more than 40 years trying to determine the next step.

Like many other cutting-edge proofs, mathematicians Joshua Greene and Andrew Lobb used newer developments in mathematics to shift their perspective and resituate the problem they were trying to solve. In this case, the same way a decades-prior colleague turned the curve into a Mobius strip, in 2019, another peer turned that strip into a 4D plot. To do that, Cole Hugelmeyer considered the X and Y coordinates as before and added the length of the “chord” the two points formed and the angle at which that chord would meet the X axis. (Check out this *Quanta *article for those images.)

As in calculus, where calculating a line into a curve and into an accumulation of area beneath that curve reveals information about the system that generated it, plotting a Mobius strip into four-dimensional space revealed data that helped Hugelmeyer prove the rectangle problem. In the 4D plot, he “rotated” the Mobius strip by adjusting one of the four coordinates. Like a spinning top, he found that a core remained static and overlapped as the strip rotated. Where the rotated Mobius strips overlap, that forms the rectangle back in regular two-dimensional space.

That additional special proof worked for just a third of rectangles. But, again, it was a hook into the next step. With the knowledge of some portion of overlaps, Greene and Lobb extended Hugelmeyer’s important finding into one more critical step. They decided to consider a special shape called a Klein bottle, below, which is a charismatic example of a shape that overlaps itself in the right dimensionality of space.

The two mathematicians found that while just one-third of curves conformed to Hugelmeyer’s specific condition, it worked when they took the shape into a slightly different four-dimensional space: symplectic space, where the rules are slightly different. All the smooth, continuous curves can be represented as Klein bottles in this symplectic space, and only Klein bottles with overlaps can exist in this space by nature of the rules. From there, they concluded that every curve has an overlap that represents the coordinates of a rectangle.

Something great about this solution is how it combines both decades of accumulated institutional memory within this kind of math—where researchers expand on and study each other’s work over literal generations of scholars—and the way individuals with an inkling of an idea can let it roll around for months or even years before the final piece falls into place.

In this case, the decades of latency corresponded with the new developments in mathematics that made this solution possible at all. That, and an unplanned quarantine.