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One word you don’t often associate with math is creativity. Sure, mathematicians are logical, but they are rarely seen as having the same flair that artists do. However, many mathematical questions require an out-of-the-box style of thinking, and the “Rectangular Peg Problem” is the perfect example of this.
Way back in 1911, German mathematician Otto Toeplitz proposed that any closed curve (i.e. that starts and ends at the same place) will contain four points that when connected form a square. Whilst Toeplitz’s square peg conjecture was quickly proven for a continuous (no breaks) and smooth (no corners) closed curve in 1929, the puzzle has yet to be solved for continuous non-smooth curves.
As a square is a particular type of rectangle (one with sides of an equal length), a variation on Toeplitz’s problem was also borne – the rectangular peg problem. In this version, you had to prove that rectangles of every aspect ratio (i.e. the ratio of its sides – for a square it is 1:1 and for HDTV’s it is often 16:9) could be formed from four points on the same closed curve.
“The problem is so easy to state and so easy to understand, but it’s really hard,” Associate Professor Elizabeth Denne of Washington and Lee University, told Quanta Magazine.
But a collaboration between mathematicians Joshua Greene of Boston College, US, and Andrew Lobb from Durham University, UK, formed to cope with the Covid-19 lockdown, has ended the century-long wait for a solution to the rectangular peg problem. Focused on the smooth, continuous closed curve conditions the duo set their creative juices flowing, and combined old thoughts with new perspectives to land upon the proof. Buckle up, everyone…
Greene and Lobb’s first piece of the jigsaw came from mathematician Herbert Vaughan, who in the late 1970s found that when pairs of points from the curve were plotted without worrying about the co-ordinates' order, they produced a Möbius strip (a twisted loop that you may well remember making at school). Vaughan used this tool to prove that the curves in question contained at least four points that form a rectangle – but that fell short of the plethora of rectangles asked for in the problem.
Forty years later, Vaughan’s Möbius strip got a revamp by Princeton graduate Cole Hugelmeyer. Hugelmeyer took the strip and embedded it in a four dimensional space. “Essentially, you’ve got your Möbius strip, and for each point on it you’re going to give it four coordinates,” Lobb explained to Quanta Magazine.
In this 4D space, Hugelmeyer could “rotate” the strip. The point at which the “rotated” strips intersected with the “unrotated” strip, exactly corresponded to the four corners of a sought-after rectangle on the curve. However in only one-third of rotations did this overlap exist, hence proving the existence of only one-third of all possible aspect ratios of rectangles – still short of a full solution.
To find the missing two-thirds, Greene and Lobb introduced yet another shape – the Klein Bottle. “The Klein bottle is supposed to be a surface, but the handle, to get from the outside to the inside, has to crash through the bottle,” Richard Schwartz, of Brown University, told Quanta Magazine.
In a particular type of 4D space, the symplectic space, Klein bottles, the equivalent of two intersecting Möbius strips, can be embedded. Also, by mathematical law, in the symplectic space, a Klein Bottle can never not intersect itself. Therefore, as Greene and Lobb proved that in every “orientation” the surface intersected, they consequently proved that every possible ratio of rectangle could be formed. The rectangular peg problem finally has a full solution.
If you want to read their full proof, Lemmas and all, you can do so on the pre-print server arXiv. One thing is for sure, Greene and Lobb didn't reach this endgame without some creative moves.
A deeper look into the rectangular peg problem. 3Blue1Brown/YouTube
[H/T: Quanta Magazine]
