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Teen mathematicians tie knots through a mind-blowing fractal
- deleted 2y ago[deleted]
- lovegrenoble 2y agoA browser puzzle, based on "Knot Theory". Not sure I learned anything from playing this, but that was fun: https://brainteaser.top/knot/index.html https://brainteaser.top/knot/index.html
- awesome_dude 2y agoKind of reminds me of https://www.jasondavies.com/planarity/ https://www.jasondavies.com/planarity/
- zvr 2y agoOr the "Untangle" puzzle from Simon Tatham's Portable Puzzle Collection https://www.chiark.greenend.org.uk/~sgtatham/puzzles/ https://www.chiark.greenend.org.uk/~sgtatham/puzzles/ Constantly one of the first additions to any new device I acquire (Android, Linux, Windows).
- nolroz 2y agoHow do you install this?
- zvr 2y agoI am not sure I understand the question. Doesn't the "Download" section of the page provide links to the binaries for different platforms? https://www.chiark.greenend.org.uk/~sgtatham/puzzles/#download https://www.chiark.greenend.org.uk/~sgtatham/puzzles/#downlo...
- fsckboy 2y agoyou can play in a browser on the web. you can download the web versions and play locally in your browser. you can download the source and build. you can download builds for many different platforms.
- fsckboy 2y agoI've played that to completion with 2500 dots. I tried to go bigger but on my old machine I don't have the memory (or maybe nobody can with that architecture, i dunno, it got really slow)
- Koshkin 2y ago> Every knot is “homeomorphic” to the circle Here's an explanation: https://math.stackexchange.com/questions/3791238/introduction-to-topology-knots-and-circles-are-homeomorphic https://math.stackexchange.com/questions/3791238/introductio...
- bmitc 2y agoIntuitively, just imagine picking a starting point on each of the circle and the knot. Now walk at different speeds such that you get back to the starting point at the same time. In fact, that's what the knot is: a continuous, bijective mapping from the circle to the image of the mapping, i.e., the knot. (As the linked answer says.) Edit: I see now that the article already has this intuitive explanation but with ants.
- Koshkin 2y agoSomewhat counterintuitively, all knots are homeomorphic to each other.
- xanderlewis 2y agoIf you regard two spaces being homeomorphic as meaning — roughly — that if you lived in either space you’d not notice a difference, it makes sense. To a one-dimensional being (that has no concept of curvature or length, since we’re talking about topology here), they’d all feel like living in a circle.
- lupire 2y agoOnly because "homeomorphic" is a highly technical term that most people don't even know the definition of, let alone have an intuition for, and because "knot" in math is a closed loop, unlikely "knot" in common language. Once you know the definitions and have learned to tie your shoes, it's quite intuitive. Even a small child can easily constructively prove that knots are homeomorphic.
- bmitc 2y ago
- MengerSponge 2y agoThis is relevant to my interests
- sakesun 2y agoAt my age, I really have to restrain myself of these interests to spare my time for some other stuffs. :(
- MengerSponge 2y agoDon't feel bad: it's mostly a Menger Sponge joke
- deleted 2y ago[deleted]
- emptiestplace 2y ago> But most important, the fractal possesses various counterintuitive mathematical properties. Continue to pluck out ever smaller pieces, and what started off as a cube becomes something else entirely. After infinitely many iterations, the shape’s volume dwindles to zero, while its surface area grows infinitely large. I'm struggling to understand what is counterintuitive here. Am I missing something? Also, it's still (always) going to be in the shape of a cube. And if we are going to argue otherwise, we can do that without invoking infinity—technically it's not a cube after even a single iteration. This feels incredibly sloppy to me.
- betenoire 2y ago> shape’s volume dwindles to zero, while its surface area grows infinitely large I think it's easy to grok when you get it, but that's certainly counter-intuitive on the surface, no?
- emptiestplace 2y agoI won't say it isn't possible that someone might struggle with this—it's quite subjective, obviously—but I do think it's unlikely that anyone with a general understanding of both volume and surface area would struggle here. Even just comparing two consecutive iterations, I feel confident that any child who has learned the basic concepts would be able to reliably tell you which has more enclosed volume or surface area. I will happily concede that the part you quoted could be quite unintuitive without the context of the article or the animation included in it. :)
- benbayard 2y agoI think Gabriel's Horn is a great explanation of how this is counter-intuitive[1]. This is a shape which you could fill with a finite amount of water, say a gallon. Yet it would take an infinite amount of paint to paint the surface. Of course, part of the reason it's counter-intuitive is that there is no 0-thickness paint that exists. [1] https://en.wikipedia.org/wiki/Gabriel%27s_horn https://en.wikipedia.org/wiki/Gabriel%27s_horn
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- glial 2y agoI love quanta so much. I wish there were a print version.
- Koshkin 2y agoI, on the other hand, prefer the modern media for the ability to include animations etc.
- jll29 2y agoI agree - I would happily pay for a print subscription and donate a second one to the nearest high school (although teens may no longer visit the library, so you'd probably have to spread the magazines in toilets, corridors and cafes).
- kasey_junk 2y agoYou’d probably have a harder time finding a school library or even harder a librarian in the school. Between funding prioritization and challenging public policy requirements many schools have removed their libraries.
- marxisttemp 2y agoAnd soon we’ll have an administration with a major goal of deliberately defunding and crippling what remains of our school system to make sure nobody reads enough to realize what’s happening to their country.
- ompogUe 2y agoReminds me of one of my sophomore crushes: Leonardo @ MIT: https://direct.mit.edu/leon https://direct.mit.edu/leon
- julianeon 2y agoI've always wondered if it's possible to harness teen minds to solve significant math problems in high school, if you formulated them well and found the right scope. I think it's possible.
- colordrops 2y agoYes that's the education system. But I suspect you mean in some automated turk fashion.
- julianeon 2y agoSorry, I meant while they were still in high school; I've edited my original comment to make that clear.
- afry1 2y agoIt is very possible! Just this year these girls discovered a proof for the Pythagorean theorem using nothing but trigonometry, a feat considered impossible until they did it: https://youtu.be/VHeWndnHuQs https://youtu.be/VHeWndnHuQs
- thaumasiotes 2y ago> a feat considered impossible until they did it Hm? https://www.cut-the-knot.org/pythagoras/TrigProof.shtml https://www.cut-the-knot.org/pythagoras/TrigProof.shtml > J. Zimba, On the Possibility of Trigonometric Proofs of the Pythagorean Theorem, Forum Geometricorum, Volume 9 (2009) And Zimba's proof terminates in a finite number of steps.
- SJC_Hacker 2y agoUnfortunately it seems their proof already had the Pythagorean Theorem embedded within its implicit assumptions - they define measure of an angle through rotation of a circle. They don't explicitly define circle, but from their diagram they hint at the "understood" definition, namely a set of points equidistance from a central point, while using Euclidean distance as the metric.
- RoboTeddy 2y agoQuanta Magazine consistently explains mathematics/physics for an advanced lay audience in ways that don't terribly oversimplify / still expose you to the true ideas. It's really nice! I don't know of any other sources like this.
- godelski 2y agoUnfortunately I have not found this to be true (though it is in this case). There are quite a number of articles that are misleading or flat out false. The best example is the quantum wormhole article and video[0,1], because it is egregious and doesn't take much nuance or expertise see the issues. I'm glad they made a note and wrote a follow-up[2], but all this illustrates what is wrong with the picture. For one, the article and video were published the same day as it was published in Nature[3]. Sure, they are getting wind of the preprints, but in this case there was none! They're often acting as a PR firm for many of the big universities and companies, unfortunately so is Nature. The article was published Nov 30th, but the note didn't come till March 29th![4] You might think, oh it took that much time to figure out that there were problems, but no, only a few days after the publication (Dec 2nd) even Ars Technica was posting about the misinformation. They even waited over a month after Kobrin, Schuster, and Yao placed their comment on ArXiv[6]. Scott Aaronson had already written about it[7]. There was so much dissent in that time frame that it is hard to explain it as an accident. A week or two and it wouldn't be an issue. But I think Peter Woit explains it best[8] (published, yes, Nov 30th). This work is getting the full-press promotional package: no preprint on the arXiv, embargoed info to journalists, with reveal at a press conference, a cover story in Nature, accompanied by a barrage of press releases. This is the kind of PR effort for a physics result I’ve only seen before for things like the Higgs and LIGO gravitational wave discoveries. It would be appropriate I suppose if someone actually had built a wormhole in a lab and teleported information through it, as advertised. I hate to say it, but you need to be careful with Quanta and others that __should__ be respectable. And I don't think we should let these things go. They are unhealthy for science and fundamentally create more social distrust for science. Now science skeptics can point to these same things as if there isn't more nuance all because they were more willing to take money from Google and CIT than wait a day and get some comments from other third party sources. (The whole peer review thing is another problem, but that's a different rabbit hole). [0] https://www.quantamagazine.org/physicists-create-a-wormhole-using-a-quantum-computer-20221130/ https://www.quantamagazine.org/physicists-create-a-wormhole-... [1] https://www.youtube.com/watch?v=uOJCS1W1uzg https://www.youtube.com/watch?v=uOJCS1W1uzg [2] https://www.quantamagazine.org/wormhole-experiment-called-into-question-20230323/ https://www.quantamagazine.org/wormhole-experiment-called-in... [3] https://www.nature.com/articles/s41586-022-05424-3 https://www.nature.com/articles/s41586-022-05424-3 [4] https://web.archive.org/web/20230329191417/https://www.quantamagazine.org/physicists-create-a-wormhole-using-a-quantum-computer-20221130/ https://web.archive.org/web/20230329191417/https://www.quant... [5] https://arstechnica.com/science/2022/12/no-physicists-didnt-make-a-real-wormhole-what-they-did-was-still-pretty-cool/ https://arstechnica.com/science/2022/12/no-physicists-didnt-... [6] https://arxiv.org/abs/2302.07897 https://arxiv.org/abs/2302.07897 [7] https://scottaaronson.blog/?p=6871 https://scottaaronson.blog/?p=6871 [8] https://www.math.columbia.edu/~woit/wordpress/?p=13181 https://www.math.columbia.edu/~woit/wordpress/?p=13181
- nsoonhui 2y agoSorry to ask this, but is the result itself significant enough to the community, if it's not discovered by teens?
- moomin 2y agoI don’t think the question is an active research area, or the problem would probably already be solved. However, it’s nonetheless extremely impressive. I couldn’t have done this at 21 with a lot more experience under my belt.
- rizs12 2y agoQuanta looks like a magnificent magazine. Thank you for bringing it into my life! This is the first time I've come across it
- 2wrist 2y agothey have a couple of cool podcasts too, I quite like the Joy of Wh(y)
- dpig_ 2y agoSuper cool. I would have liked to have seen a similar visualisation for how they solved it on the Sierpinski gasket.
- itronitron 2y agoInteresting, I'm tempted to apply this towards routing minecart rails in Minecraft.
- l3x4ur1n 2y agoCan you explain?
- teeray 2y agoNot parent, but I assume the idea is that you could form this fractal by hand (all or in part) in any 3D Minecraft terrain. Then you could route rails according to the theorem.
- itronitron 2y agoLike teeray mentions, if we can define any knot as a path through a Menger Sponge then that path could be realized in a Minecraft world (since it is cube/block/voxel based.) If you placed minecart rails along the knot plot then you could ride it like a roller coaster. Maybe this was the initial motivation for the teenagers.
- err4nt 2y agoCan anyone explain why they bothered with the fractal at all, instead of using a 3 dimensional grid? Doesn't a grid of the appropriate resolution provide the exact same? Or is it to show that they can do everything within even a subset of a 3D grid limited in this way?
- boothby 2y agoThe 3D grid result is well known, perhaps it would be fair to call it trivial (you can even throw away all but 2 layers of one of your dimensions). As you say, the Menger Sponge is a subset of the 3D grid, so the students had to find a construction which dodges the holes. To me, the result isn't surprising, but it is pleasing. But the really cool part of the article in my mind is the open problem at the end: can you embed every knot in the Sierpiński tetrahedron?
- err4nt 2y agoThanks, I think that helps me understand the 'why' a lot more! That's kind of cool, and I do hope they can keep making progress in that effort!
- calibas 2y ago> a tetrahedral version of the Menger sponge Better known as a Sierpiński tetrahedron, AKA the 3d version of a Sierpiński triangle.
- singularity2001 2y agoI love that the proof is so elementary and understandable ( almost reminiscent of the Pythagorean theorem proofs) yet it might have some significance