[HN Gopher] 'Once in a Century' Proof Settles Math's Kakeya Conj...
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'Once in a Century' Proof Settles Math's Kakeya Conjecture
Author : pseudolus
Score : 89 points
Date : 2025-03-14 23:21 UTC (23 hours ago)
(HTM) web link (www.quantamagazine.org)
(TXT) w3m dump (www.quantamagazine.org)
| hsshhshshjk wrote:
| The article stops half way through?
| asplake wrote:
| Reader mode?
| shakna wrote:
| It waits until some packages from unpkg.com load, and then
| loads in the rest of the article.
| kbmr wrote:
| need to wait another century for the rest
| jan_Inkepa wrote:
| I had to reload a few times - eventually it loaded everything.
| nextts wrote:
| Animation looks like how a skilled fighter might used a double
| barbed weapon. Maybe there is a connection? Minimise movement
| (time, energy use) while maximising agility?
| larodi wrote:
| https://archive.is/YzZUA
| keming wrote:
| Calling all Blue Blaze Irregulars...
|
| We're one step closer to the Oscillation Overthruster!
| gcanyon wrote:
| Big-boo-tay!! Tay, Tay, tay!
|
| (hello, fellow old!)
| vrnvu wrote:
| Time to add a leetcode medium problem for this.
| kevinventullo wrote:
| Joking aside, I feel like a variation of this could make a good
| Project Euler problem...
| jan_Inkepa wrote:
| "Consider a pencil lying on your desk. Try to spin it around so
| that it points once in every direction, but make sure it sweeps
| over as little of the desk's surface as possible."
|
| I'm really stuck at the start here - moving a pen so that it
| pointing in all directions is basically impossible - the space of
| directions is two-dimensional and you can only trace out a one-
| dimensional curve (or pair of curves).
|
| Ok, wikipedia makes it clearer:
|
| "In mathematics, a Kakeya set, or Besicovitch set, is a set of
| points in Euclidean space which contains a unit line segment in
| every direction."
|
| Quanta writers are generally very good at explaining things, but
| wikipedia wins hands down in this case...
| tromp wrote:
| The Wikipedia entry [1] has some good illustrations for the
| 2-dimensional case.
|
| [1] https://en.wikipedia.org/wiki/Kakeya_set
| seanhunter wrote:
| It's not pointing in all directions at once, it's pointing once
| in each direction. So if you spin the pencil so it does exactly
| one complete rotation that works, doesn't it?
| OJFord wrote:
| Yes, and spinning the pencil on its centre like that shows
| that the circle (of pencil length diameter) is such a set. (I
| think you're thinking about it the wrong way around: it's
| which containing shapes allow this, not how can it be done at
| all.)
| rsaarelm wrote:
| I think OP is thinking about covering the sphere of
| directions in 3D space, not just directions in a 2D plane. No
| matter how hard you spin the pencil, you're drawing a one-
| dimensional curve that has no area, so any finite amount you
| draw will cover zero percent of the area of the two-
| dimensional sphere surface.
| flufluflufluffy wrote:
| > No matter how hard you spin the pencil, you're drawing a
| one-dimensional curve that has no area
|
| The object doesn't matter, using pencil as the example was
| what threw you off - it's not about what the pencil
| "draws". Consider a thin cylinder, or rectangular prism, or
| just a stick - if you spin it around, its endpoints trace
| out a circle whose diameter is the length of the stick. You
| can move and spin such an object in another way where the
| shape traced out by its endpoints has smaller area than
| that circle.
| jan_Inkepa wrote:
| in 2d, but not in 3d though (like in the video on top of the
| article)!
| Someone wrote:
| > Ok, wikipedia makes it clearer:
|
| > "In mathematics, a Kakeya set, or Besicovitch set, is a set
| of points in Euclidean space which contains a unit line segment
| in every direction."
|
| Is that definition correct/complete? It leaves open the option
| that such a set isn't connected. I think there's an additional
| requirement that, for any two directions D and E, you can move
| a line segment oriented in direction D so that it's oriented in
| direction E without any point on it ever leaving the set.
| shiandow wrote:
| The minimal set is trivially connected. Though showing any
| two unit line segments have a homology between them is
| trickier.
| gsf_emergency_2 wrote:
| https://youtu.be/hySXxdlKdNI?t=17m28s
| blincoln wrote:
| The animation in the article looks like a simplified diagram of a
| rotary engine or a drill that will cut polygon-shaped holes
| instead of round holes.
|
| The article mentions a connection to the Fourier transform, which
| makes sense because nested rotations are essentially summed sine
| waves in a different coordinate space.[1]
|
| Is there more of a connection than that between the Kakeya
| conjecture, physical machines like rotary engines, and additive
| wave synthesis, or are they all fairly different branches of
| "interesting things one can do with nested rotations / summed
| sine waves"?
|
| I'm curious if proving the conjecture opens up new possibilities
| in mechanical engineering or sound/EM wave synthesis/analysis, in
| other words.
|
| [1] Apologies in advance if I mauled this description.
| arutar wrote:
| This is a very phenomenal result and everyone in the field is
| excited about this! Josh and Hong both gave talks about this at a
| conference 2 weeks ago in Berkeley, and the videos are online:
| [1] [2]. Josh's talk (at least from my perspective of someone who
| is adjacent to this field) is quite approachable, whereas Hong
| talks more about the induction scheme on Guth's grains
| decomposition which is quite a bit more technical.
|
| I visited Josh at UBC last year around this time and I recall
| asking him if he thought Kakeya in dimension 3 would be solved
| soon. I remember that he believed it would be (though perhaps he
| was worried by someone other than Hong and himself). In the end
| they were able to complete the proof themselves.
|
| Hong is probably quite a serious candidate for the fields medal
| because of this. She has already made impressive progress on
| problems in harmonic analysis and geometric measure theory and
| she is one of few people has a firm foothold in both fields at
| the same time.
|
| The quanta article talks about a 'tower of conjectures' in
| harmonic analysis; at the top of the tower is the so-called
| "local smoothing conjecture" (a conjecture about how much waves,
| such as 'idealized' sound waves, can amplify from some initial
| configuration when averaged over time). A Kakeya set is a certain
| type of geometric obstruction to local smoothing; resolving the
| full conjecture also requires handling so-called 'oscillatory'
| obstructions. In dimension 2 + 1 (2 spatial and 1 time dimension)
| the local smoothing was only recently resolved (also by Hong and
| co-authors [3]); even though the corresponding result for Kakeya
| sets in dimension 2 has been known for over 40 years.
|
| [1] https://player.vimeo.com/video/1062254156
|
| [2] https://player.vimeo.com/video/1063428579
|
| [3] https://annals.math.princeton.edu/2020/192-2/p06
| Avshalom wrote:
| Wait, "once in a century"? How many times has this been proven
| such that they feel comfortable talking about the statistical
| rate of proofs?
| xenocratus wrote:
| I can tell a HN article is from Quanta from the obnoxious
| clickbaity title. I'm shocked that they have to do this given
| their target audience (because I'd rather not believe they just
| choose to)
| aoki wrote:
| Each area has its own "most important" conjecture(s), so
| Quanta is in no danger of running out of proofs to write
| about that somebody legitimately thinks are super important.
|
| (Alas, while "most important" conjectures are a renewable
| resource, lay reader tolerance for such headlines may not
| be.)
| aoki wrote:
| The study of Kakeya sets is over 100 years old -- so if you
| think this generalized conjecture is the most important one in
| their subfield, you can see why somebody would describe it this
| way.
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