Big deal math alert: The Kakeya Conjecture has been solved for 3D space!
Sōichi Kakeya in 1917 when he was 31
Quanta Magazine discusses the importance of solving the Kakeya Conjecture, a conjecture that has bedeviled mathematicians for 50 years. Quanta calls it a "once in a century" mathematical proof. Quanta writes:
In 1917, Sōichi Kakeya posed the problem, but with an infinitely thin pencil [or needle or line segment]. He found a way of sliding the pencil that covered less area than the instinctual circular motion.
Kakeya's original 1917 2D space solution
Kakeya wondered how small an area the pencil could possibly sweep. Two years later, the Russian mathematician Abram Besicovitch found the answer: a complicated set of narrow turns that, counterintuitively, covers no space at all.**** Unfortunately, there is no image of the Besicovitch solution. Besicovitch's construction can be mathematically described, but creating a visual image of the full shebang ["infinite iteration process"] is fundamentally impossible due to its fractal-like nature and reliance on an uncountable set of directions.[1] 🤔That more or less settled the question until 1971, when Charles Fefferman was studying something apparently unrelated to twirling lines: the Fourier transform, a foundational mathematical tool that lets you reimagine any mathematical function as a combination of waves. In Fefferman’s work, a tweaked version of Kakeya’s problem kept coming up. In this case, the pencil has a thickness and twirls in three dimensions. Here, Kakeya’s question becomes the following: As you change the width of the pencil, how does it affect the volume of space that it traces out?
The spinning needle in 3D
The conjecture’s resolution is a seismic shift for the field of harmonic analysis, which studies the details of the Fourier transform.
A tower of three monumental conjectures in harmonic analysis rests atop the Kakeya conjecture. Each story in the tower needs to be sturdy for the stories above it to stand a chance themselves. If the Kakeya conjecture had been proved false — if Wang and Zahl had found a counterexample — the entire tower would have come tumbling down.
But now that they’ve proved it, mathematicians might be able to work their way up the tower, using Kakeya to build up proofs of these successively more ambitious conjectures. “All these problems that [mathematicians] dreamed about someday solving, they all look approachable now,” [Larry Guth, a mathematician at MIT] said.
The four-dimensional Kakeya conjecture remains open [unresolved], with a tower of four-dimensional conjectures above it as well. New difficulties will arise, Guth said, but he thinks that the jump from two dimensions to three was the hardest, and that Wang and Zahl’s proof can likely be adapted to that tower, and beyond.
From what I can tell, this mathematical proof is a really big deal. The benefits are likely to be significant, but buried deep in various consumer and commercial products that will generally be hard to see. And, it will probably translate into some "improved" kinetic and cyber weapons.
Footnote:
1. The Kakeya conjecture's resolution (and this) by Besicovitch involves a counterintuitive geometric construction known as a Besicovitch set — a configuration of overlapping line segments pointing in every direction while occupying arbitrarily small area. The structure can be visualized through its mathematical description. The solution relies on iteratively splitting a shape (like a triangle) into thinner pieces, sliding them to overlap extensively, and repeating this process infinitely. Each iteration reduces the total area covered, approaching zero in the limit while maintaining all directional orientations.
Ha -- thin, sliding shapes! Bet you didn't think of that way out! Hm, neither did I.
By Germaine: Math boob
See, it's easy peasy!
(or not)
