[HN Gopher] Teen mathematicians tie knots through a mind-blowing...
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Teen mathematicians tie knots through a mind-blowing fractal
Author : GavCo
Score : 311 points
Date : 2024-11-26 18:43 UTC (1 days ago)
(HTM) web link (www.quantamagazine.org)
(TXT) w3m dump (www.quantamagazine.org)
| lovegrenoble wrote:
| A browser puzzle, based on "Knot Theory". Not sure I learned
| anything from playing this, but that was fun:
|
| https://brainteaser.top/knot/index.html
| awesome_dude wrote:
| Kind of reminds me of https://www.jasondavies.com/planarity/
| zvr wrote:
| Or the "Untangle" puzzle from Simon Tatham's Portable Puzzle
| Collection
| https://www.chiark.greenend.org.uk/~sgtatham/puzzles/
|
| Constantly one of the first additions to any new device I
| acquire (Android, Linux, Windows).
| nolroz wrote:
| How do you install this?
| Koshkin wrote:
| > _Every knot is "homeomorphic" to the circle_
|
| Here's an explanation:
|
| https://math.stackexchange.com/questions/3791238/introductio...
| bmitc wrote:
| Intuitively, 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 wrote:
| Somewhat _counter_ intuitively, all knots are homeomorphic to
| each other.
| xanderlewis wrote:
| If 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 wrote:
| Only 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 wrote:
| Homeomorphic is also interesting because it's a very
| fancy sounding word for a concept that has very few
| requirements to be met. It's usually a basic requirement
| for concepts in topology, but it does very little to
| distinguish topological spaces. It's essentially a
| highfalutin word for "these things are basically the same
| thing in a very basic way".
| noqc wrote:
| This isn't true. Knots are _not_ topological spaces, they are
| imbeddings, K:S^1 - > S^3 (more generally, S^n-> S^n+2).
| Therefore there isn't an obvious notion of homeomorphism. What
| the poster points out is that restriction to the image is a
| homeomorphism from the circle (because it is an imbedding of
| the circle)
|
| As maps _between_ topological spaces (and almost always we pick
| these to be from the smooth or piecewise linear categories,
| which further restricts them), the closest "natural"
| interpretation of isomorphism is pairs of homeomorphisms, f:S^1
| -> S^1, g:S^3 -> S^3 satisfying Kf=gK. Ie natural
| transformations or commuting squares.
|
| This gives us almost what we want, except that I can flip the
| orientation of space, or the orientation of the knot with
| homeomorphisms, neither of which correspond to the physical
| phenomenon of knots, so we give our spaces an orientation,
| which requires us to move to the PL or smooth category, or use
| homotopy/isotopy.
| gowld wrote:
| https://en.wikipedia.org/wiki/Embedding#General_topology
|
| "In general topology, an embedding is a homeomorphism onto
| its image."
| MengerSponge wrote:
| This is relevant to my interests
| sakesun wrote:
| At my age, I really have to restrain myself of these interests
| to spare my time for some other stuffs. :(
| MengerSponge wrote:
| Don't feel bad: it's mostly a Menger Sponge joke
| emptiestplace wrote:
| > 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 wrote:
| > 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 wrote:
| I 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 wrote:
| I 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
| Koshkin wrote:
| Think of a 3-dimensional object (unlike a surface, which is
| 2-dimensional, regardless of the shape), with the volume zero.
| That's not easy to wrap your head around.
| Nevermark wrote:
| > That's not easy to wrap your head around.
|
| I am trying to figure out the formal version of this
| topological conjecture. Even that isn't easy.
| wruza wrote:
| I don't see why it would be hard. Surprising, maybe, if one
| never thought about a limit of iterative processes.
| glial wrote:
| I love quanta so much. I wish there were a print version.
| Koshkin wrote:
| I, on the other hand, prefer the modern media for the ability
| to include animations etc.
| jll29 wrote:
| I 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 wrote:
| You'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 wrote:
| And 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.
| julianeon wrote:
| I'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 wrote:
| Yes that's the education system. But I suspect you mean in some
| automated turk fashion.
| julianeon wrote:
| Sorry, I meant while they were still in high school; I've
| edited my original comment to make that clear.
| afry1 wrote:
| It 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
| thaumasiotes wrote:
| > a feat considered impossible until they did it
|
| Hm? 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 wrote:
| Unfortunately 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.
| lupire wrote:
| That's not true at all.
|
| To understand why, read Euclid.
| SJC_Hacker wrote:
| Geometry has made a bit of progress since Euclid's time.
| Its become a bit more rigorous.
|
| Euclidean geometry is based on five axioms, and some
| other terms left undefined.
|
| The fifth postulate - the parallel postulate - was
| considered so irksome that for hundreds of years, many
| attempted to prove it using the other four, but failed to
| do so, and almost drove some crazy. In the late 19th
| century it was shown you can generate perfectly valid
| geometries if you assume it to be false somehow - either
| no-parallel (spherical geometry) or infinite parallel
| (hyperbolic)
|
| Euclid's third postulate - "a circle can be drawn with
| any center and radius - doesn't define how to do it. Like
| I could draw a "circle with a radius of 1" using taxicab
| distance, and it would look like a diamond shape.
|
| Conversely, if you take the "conventional" definition,
| than the Pythagorean theorem falls out almost
| immediately.
| gowld wrote:
| The (non-generalized) Pythagorean theorem is part of
| Euclidean geometry, so non-Euclidean geometry is
| irrelevant to this discusion.
|
| > Euclid's third postulate - "a circle can be drawn with
| any center and radius - doesn't define how to do it.
|
| You do it using an axiomatic compass, a device that
| _copies_ length in a circular pattern but does not
| _measure_ it. Lengths are measured using constructable
| line segments.
|
| Are you implying that nearly all the hundreds of proofs
| of Pythagorean theorem, which do not use modern rigorous
| definitions, are not valid proofs?
|
| > Conversely, if you take the "conventional" definition,
| than the Pythagorean theorem falls out almost
| immediately.
|
| So? The Pythagorean theorem is very easy to prove. There
| are hundreds of proofs created by amateurs. That doesn't
| make them "not proofs" simply because other proofs exist.
| znyboy wrote:
| Monkeys with typewriters, or teenagers with MacBooks?
| cevi wrote:
| MIT's PRIMES program does exactly this - they give advanced
| high school students a mentor who picks out a problem, gets
| them up to speed on what is known, and then they work on the
| problem for a year and publish their results. It tends to work
| best with problems which have a computational aspect, so that
| the students can get some calculations done on the computer to
| get the ball rolling.
| adrianN wrote:
| Formulating a problem well is half the solution already.
| RoboTeddy wrote:
| Quanta 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 wrote:
| Unfortunately 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-...
|
| [1] https://www.youtube.com/watch?v=uOJCS1W1uzg
|
| [2] https://www.quantamagazine.org/wormhole-experiment-called-
| in...
|
| [3] https://www.nature.com/articles/s41586-022-05424-3
|
| [4]
| https://web.archive.org/web/20230329191417/https://www.quant...
|
| [5] https://arstechnica.com/science/2022/12/no-physicists-
| didnt-...
|
| [6] https://arxiv.org/abs/2302.07897
|
| [7] https://scottaaronson.blog/?p=6871
|
| [8] https://www.math.columbia.edu/~woit/wordpress/?p=13181
| msephton wrote:
| I'm sure the article author would love to know this!
| keiferski wrote:
| It's a project funded by the recently passed Jim Simons:
|
| https://en.wikipedia.org/wiki/Quanta_Magazine
|
| https://en.wikipedia.org/wiki/Simons_Foundation
| nsoonhui wrote:
| Sorry to ask this, but is the result itself significant enough to
| the community, if it's not discovered by teens?
| moomin wrote:
| I 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.
| lupire wrote:
| You could if you had a good mentor.
| moomin wrote:
| Fair, but I think the article itself is very clear on the
| importance of the mentor. It's an fun question that hasn't
| previously been posed, which itself is quite impressive.
| rizs12 wrote:
| Quanta looks like a magnificent magazine. Thank you for bringing
| it into my life! This is the first time I've come across it
| 2wrist wrote:
| they have a couple of cool podcasts too, I quite like the Joy
| of Wh(y)
| dpig_ wrote:
| Super cool. I would have liked to have seen a similar
| visualisation for how they solved it on the Sierpinski gasket.
| itronitron wrote:
| Interesting, I'm tempted to apply this towards routing minecart
| rails in Minecraft.
| l3x4ur1n wrote:
| Can you explain?
| teeray wrote:
| Not 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 wrote:
| Like 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 wrote:
| Can 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 wrote:
| The 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 Sierpinski tetrahedron?
| calibas wrote:
| > a tetrahedral version of the Menger sponge
|
| Better known as a Sierpinski tetrahedron, AKA the 3d version of a
| Sierpinski triangle.
| singularity2001 wrote:
| I love that the proof is so elementary and understandable (
| almost reminiscent of the Pythagorean theorem proofs) yet it
| might have some significance
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