[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 : 96 points
Date : 2024-11-26 18:43 UTC (4 hours 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
| 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.
| MengerSponge wrote:
| This is relevant to my interests
| layer8 wrote:
| Teen mathematicians run circles inside you (if not around you).
| sakesun wrote:
| At my age, I really have to restrain myself of these interests
| to spare my time for some other stuffs. :(
| 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. :)
| 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.
| 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.
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