Mathematicians Wrote a Proof for a 100-Year-Old Problem—and May Have Just Changed Geometry
A 125-page proof posted to arXiv may represent a huge breakthrough in geometric measure theory.
This problem has implications in fields like encryption, computer science, and number theory.
How much area does it take up when you rotate a line segment 360 degrees?
Two mathematicians now say they've made progress on a very old unsolved math problem. The problem involves a subfield called geometric measure theory, in which sets of objects are generalized in an advanced way using properties like diameter and area. According to the duo's recent research (which is not yet peer reviewed), it turns out that examining things through the lens of geometry can shake loose other interesting qualities that objects may share, which has high value in the increasingly inter-subdisciplinary field of mathematics.
In a special question in geometry called a Kakeya set, mathematicians wonder how small the area can be where a line, or needle, is rotated completely through 360 degrees. You may picture something like a spinner in a board game or a baton twirler, so the spun needle must just form a circle. But the truth is much more complicated, because space can essentially be reused by different needles, and the needle positions don't need to have the same midpoint. '[I]f you slide it in clever ways, you can do much better,' Joseph Howlett explained for Quanta.
This creates shapes like the deltoid, which is a roughly triangular shape that may remind you of your old Spirograph drawing toy. The deltoid can have a much smaller area than the circle that would enclose the same needle spinning like a baton. Mathematicians who study this question are basically trying to find the smallest deltoid possible—whatever shape that ends up being—across a variety of types of spaces.
The Kakeya set—named for its discoverer Sōichi Kakeya—was complicated by a subsequent mathematician named Abram Samoilovitch Besicovitch. Besicovitch introduced the idea that a Kakeya set moved into a different number of dimensions could have an area of measure zero.
This is a specific definition that involves surrounding a particular item with points that can be made closer and closer together until they almost vanish, with an intuitive meaning that there is no area at all. Mathematicians can't write down and prove an intuitive meaning without an underpinning in the mathematics. As a result, this question—and others like it, all of which were tantalizing to those already immersed in similar concepts—tipped a line of dominoes that eventually helped to create the field of geometric measure theory. If you've ever seen an illustration of a Klein bottle (an iconic depiction of a four-dimensional shape crammed into a three-dimensional version that our human brains can parse), that's one example of a thought exercise from geometric measure theory.
Kakeya died in 1947 and Besicovitch died in 1970, so even the newest possible versions of these questions have been open and unproven for at least 55 years. But they really date back 100 years, to when both men were in their mathematical primes... so to speak. Since then, mathematicians have banged their heads against various types of Kakeya sets in different types of spaces and with different qualities. After all, there's no limit to how many dimensions something can have.
As is often the case in today's breakthroughs, the secret for these mathematicians—Hong Wang of New York University (NYU) and Joshua Zahl of the University of British Columbia (UBC)—was in reframing the thorny problem using lateral thinking. In an NYU statement, Zahl's UBC colleague Pablo Shmerkin explained that while it builds on 'recent advances in the area, this resolution combines many new insights together with remarkable technical mastery. For example, the authors were able to find a statement about tube intersections that is both more general than the Kakeya conjecture and easier to tackle with a powerful approach known as induction on scales.'
By making key substitutions and clarifications to the original problem, Wang and Zahl opened it to a type of proof called induction on scale. Classic proof by induction involves showing a relationship between, say, a value of 1 and a value of 2. If you can turn those concrete values into a generalization using mathematical notation instead, like n and (n + 1), then you can simplify and solve the math so that a solution applies to all possible values for n, not just 1 and 2.
Induction on scale is similar, but involves playing with... well... the scale of something. In their proof, Wang and Zahl consider tubes instead of simple lines or needle shapes. We all know what a tube is, but mathematically, it's a collection of points at a specific distance and position apart from a given line, curve, or shape outline—like a corkscrew, circle, or knot. That means it has a certain three-dimensionality unto itself when applied to a two-dimensional shape, turning a line segment into a straw. The size of those tubes can then be tuned in order to show properties about the needles they surround.
Fields Medal winner Terence Tao (himself a luminary in related mathematics) analyzed this 125-page (!!) proof in a detailed blog post, where he also calls the work 'spectacular progress.' Complicated proofs like this often emerge over decades as people iterate on small portions of the same problem—a process that's one part chiseling away and one part deciphering letter by letter. In his analysis, Tao already notes several places where the work can be iterated again, now that this portion is in place.
Zahl earned his bachelor's degree in 2008, meaning he was likely born in 1986. Wang's Wikipedia page says she was born in 1991. The next Fields Medal, which is limited to mathematicians under 40, will be given in 2026. The math, as the kids say, could be 'mathing' for these two mathematicians.
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