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Round Peg in a Square Hole - Numberphile
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⬐ wolfgkeTo me this is not very surprising to me, since in Germany there exists a well-known children's game called "Durch eine Postkarte steigen" ("step through a postcard"), where - as the naming suggests - you have to step through a postcard while you are only allowed to use scissors to cut the postcard in a clever way.
Here a German YouTube video that shows how it is done (you don't have to understand anything that is spoken, since I think the video is quite self-explanatory):⬐ jonshariat⬐ thegabezI think the reason the example in the video is so interesting is that its non-destructive. It only involves bending and the paper and coaster remain intact.⬐ wolfgke⬐ cortesoft> I think the reason the example in the video is so interesting is that its non-destructive. It only involves bending and the paper and coaster remain intact.
There is a quadratic hole cut into the quadratic paper (which I would clearly call "destructive"). It is just not shown in the video how the quadratic hole is cut into the quadratic paper.
If this argument does not convince you, here is another one: Also for stepping through a postcard after the hole is cut it is just bending the paper.⬐ TooEhh. They way you initially cut the paper constraints how you later can fold it. If the square was cut any smaller you wouldn't be able to fit the coaster through no matter how you try to fold the paper. The square and its size is a constant in this problem, using a scissor to make the square bigger after its given to you would be considered destructive. In the postcard problem there is no such constraint as you are allowed to cut in whichever form/size you want.Mr. Wizard had an episode where he did this same thing:⬐ dandareI think that is a different principle.⬐ wolfgke⬐ Dangeranger> I think that is a different principle.
I don't think so. In both cases we have some object with the property that the "perimeter of the convex hull of a 2D projection of it" is too large to fit the object through some hole in the paper. On the other hand, the length of the border of the hole in the paper is at least as large as this perimeter. So we twist the paper around such that the object passes through.Children in the USA play that game as well, but the moment when he shows that the gap in the 2D space is expanded to the two sides of the square by folding into 3D space is still amazing.
I screamed out loud it was so much fun to see.Goes to show how poorly adapted our intuition is in manipulating space across dimensions. Its scary to think about how much ground breaking science is still left uncovered because of this blindspot.⬐ joenathanone⬐ ghostwreckThe only thing I found surprising about this was that you were expected not to have known how it worked. It's the same sort of thing as trying to move a couch and having to angle it to fit it through a door, it's how 3D space works.⬐ 2bitencryption⬐ tw1010you're so smart!⬐ simoooooThank you for using your intelligence to give us such insight⬐ itgI think the equivalent would be a door that is too small now matter what angle the couch is in. The only way to get it through would be to "stretch" the door in 4-d space and collapse it back into 3-d to allow the couch through.While you're definitely onto something, I think it's also worth pointing out that a lot of the scientists that really revolutionized their field were those who excelled at either intuitively or explicitly tackling the areas where human intuition was especially likely of doing poorly on (which in many cases acts as a good proxy for assumptions that other scientists have, in an unknown unknown sense, failed to question).⬐ thegabez100% - This would make for an exciting opening lecture for any scientific educator.If you like the idea of solving problems using extra dimensions, Diaspora  takes the idea in some really interesting directions. Just be prepared to fight through a few sections of intense Riemannian geometry to bring you into that world.⬐ mirimir⬐ Starwatcher2001This is probably my favorite novel by Greg Egan. Most of its central characters are AIs.Love it. It's stuff like this that got me interested in science. I remember seeing the same thing (larger object through a smaller hole) demonstrated on "How" in the 1970s, which was a TV show with a quirky mixture of science and fun. It was aimed at children, but didn't talk down to them.
For example, I've been working on an essay for some time now, but I'm seriously considering making it into a companion video. I have no video experience whatsoever, so that's a tall order. But what are you to do? If you want to make an impact, are you sure it's still possible to do it by writing 173 essays over 15 years? It used to be, but the world seems to be changing.
(This is mostly a reaction to this post being one of the rare videos Deemed Worthy to be on the front page, when there are tens of thousands of others. It's not a good idea to mix up the content too much, but there really are a lot of quality videos and no central curation mechanism. Unlike articles. /r/videos is for mainstream content, like https://youtu.be/kJGGlVg5PpY. There really isn't any place that collects intellectually gratifying content like the current submission.)⬐ dizzystar⬐ will_dev_4_foodThe problem is that so many videos are full of fluff. This video is nice because it gets to the point and finishes in 3 minutes.
Article often have 3 paragraphs of preamble, but at least you can skip all that quickly.⬐ monochromaticSome topics lend themselves to videos. This is one of them.
Many (most?) topics can be explained just fine in text though. And I can read a hell of a lot faster than I can listen to someone talk.⬐ davidw⬐ notatoad> And I can read a hell of a lot faster than I can listen to someone talk.
Especially when you have to go through 1) an intro sequence 2) some guy asking how you are and telling you how he's doing 3) what upcoming awesome videos are going to happen 4) other chitchat - to even get started.the problem with videos is that they require a lot more of a time investment to determine if they're worth watching.
I can skim an essay or an article to determine if the content is worth a proper read, but i can't skim a video. So i'm a lot more likely to ignore a video, unless it's already been heavily upvoted or endorsed by somebody i trust. Video is a great medium to convey an idea to people who already trust that you aren't going to waste their time, and that's why youtube works so well - subscribers to a channel have made the decision to trust that channel up front. but it's a poor medium for a link aggregator site like HN. It's not that HN readers don't take video seriously, it's just video is not a good fit for the HN format, where the default assumption on clicking a link is that the content is probably not interesting, and it's up to the content to prove otherwise before losing the reader's interest.⬐ waynecochranThis is similar to the belief that visual computing will replace programming languages. New generations keep thinking this and it fails. Videos are great but they will not replace the written language.⬐ zer00eyz> I've been working on an essay for some time now, but I'm seriously considering making it into a companion video.
I think that this is the best route to go with. Your supporting multiple learning styles and providing an easily searchable chunk of reference material.
Others have pointed out that the ratio of videos like this one to junk is very high - and I do not disagree. The issue is that the quality videos are mixed in with the noise of "social media" as we have one (dominant) platform that supports both. Great videos are about as rare as great/informative social media content.
What IS nice is that there are a few channels that realize producing this sort of content is a good thing - numberphile tends to crank out great and quality stuff, but you can find the same sort of values on completely different topics - AVE, and big clive both spring to mind -⬐ 8bitsruleNobody ever said, "a picture is always worth a thousand words" !
In the case of the behavior of physical objects, though, if you tried to use words to describe complex motions, the reader would probably get lost. (And in this case, you'd lose the 'tension' of watching the race.)
For the same reasons, rock videos of people just playing instruments are rare.⬐ philfrastyPersonally I can learn MUCH faster (+ enjoying it) by looking at someone doing/explaining something in a video (e.g. coding). To take it even further: if one person (expert) explains something to another person (not an expert on this specific topic) and the non-expert frequently aks questions. This helps tremendously.⬐ jscharlack⬐ stochastic_monkAgreed though it depends on the topic. If it's very visual, like this is, video is perfect.
Video makes a lousy reference though.I typically don’t have the patience for most videos. I want information, and much of the time, the information bandwidth or payoff is too low. I think video can be great auxiliary, supporting material, but there should usually be a textual component when trying to inform.⬐ __sI get around this by watching videos in fast forward⬐ edanmI usually agree with you, but lately, I have seen amazing advances in using video as a medium to teach ideas.
I think the standout for me would be the channel 3Blue1Brown on Youtube, which teaches math concepts with animated videos, in a way that is just amazing. I highly recommend it if you're at all interested in math.
(And no, this doesn't replace the kind of textbook learning you need to actually understand math. It's more about concepts and ideas. But it's still amazingly valuable).⬐ sillysaurus3⬐ pdkl95To illustrate just how good 3Blue1Brown is: https://news.ycombinator.com/item?id=15003510
That was one of the other videos Deemed Worthy. There are so many more.
I wasn't able to visualize higher dimensions until watching that. And now it seems totally obvious how to do it.> there should
Or, we could accept that different people learn better in different ways. It's wonderful that a variety of learning resources are being made, so more people have the opportunity to learn complex and interesting topics.
> the information bandwidth or payoff is too low
Perhaps consider that you might not be the target audience?⬐ laretluval> Or, we could accept that different people learn better in different ways.
No evidence for interpersonal difference in learning styles: https://www.theguardian.com/education/2017/mar/12/no-evidenc...⬐ mjw1007That's saying that some recently fashionable, and fairly elaborate, theories about teaching children are basically junk science.
That includes the ideas that there are left/right brain learners, children who are holistic as opposed to serialists, and children who are inherently "verbalisers" or "visualisers".
I don't doubt they're right. But I think it would be a mistake to read that article and conclude that adults who simply say they do much better with videos than books, or vice versa, are mistaken about themselves.⬐ jehlakjUneducated individual here. What is the proper term for arguing against a claim that was never made or incorrectly assumed, but ambiguous enough to become a target? I see this a lot in political debates. Something more technical than a misunderstanding.
Anyway, I think that’s what’s happening here. It’s not a simple matter of different learning styles, but rather the strength of stimuli that are involved when learning something new. Some people may find text sufficient, but a lot of people prefer videos because it’s far more engaging. Breaking down learning styles into visual, audio, etc is meaningless imo.⬐ hexane360⬐ PhasmaFelisI think this falls under a strawman argument. I don't think there's any requirement that strawmans have to be deliberate.It's trivially obvious that some people get more from reading than from watching, and vice versa. If you think a study disproves something trivially obvious, odds are that either the study is flawed or you've misunderstood it.
I'm guessing the latter, as mjw1007 points out.⬐ posterboy> If you think a study disproves something ...
That wasn't the comment. Absence of evidence is not evidence of absence. So, I don't really know why the parent posted that. Or why you try to imply counter evidence, but don't name any.
> trivially obvious
It's something you see on a street intersection? I'm kidding, but i don't agree. I don't even agree that the problem is framed correctly. Somebody mentioned "isomorphism" in this thread, which is a very important notion. Different methods of learning are used to learn different things. The things are in principle in different domains. The things might be isomorphic, ie. readily transferred from one domain to another, but they are not equal. A painting of a square is not the same as a verbal description of it. On the one hand, nobody is equally skilled in all domains. On the other hand, there should be some basic skills that are necessary to solve certain problems. So, either everyone who isn't comfortable with a given "learning style" will inherently fail to solve corresponding fundamental problems, or there are multiple ways to solve a problem effectively -- which ideally would mean the student can choose the problems of interest and capability.
In conclusion, both hypothesis are true at the same time, but you are assuming a misguided context. Of course there might exist different ideas of what's elementary or fundamental.Could you be so kind as to provide a non YouTube link for this? Sites that don’t honor the 1st Amendment are blocked on my devices.
Thanks.⬐ andr⬐ neogodlessYou probably know that already, but the First Amendment does not apply to private companies.⬐ will_dev_4_food⬐ hexane360On the other hand, I suppose having big tech act as the ministry of truth is great, as long as you’re on the top of big tech.⬐ will_dev_4_foodUnless they represent what is classified as a utility. How to classify a utility? Ask what would be nigh impossible without it.Do you own a website? I have some files I'd like to publish on it.
You do honor the 1st Amendment, right?⬐ will_dev_4_foodI’m not a functioning utility, but when I get there, yes.⬐ will_dev_4_foodI do. And I will honor all requests of the CIA to your metadata. Still ok?Ahhh Easter Egg! Firmware version! Love it!⬐ IshKebabThought this was going to get clever but... it didn't.⬐ OskarSI love Tadashi Tokieda so much, he's just one of the most delightful human beings I've ever seen. I highly recommend all of his Numberphile videos, and you should also check out his Toy Models lecture . It's so much fun to see him show some weird phenomenon with marbles or magnets or whatever (that should be seemingly simple to explain with mathematics and/or classical physics) and then say "There is no theory for this. We have no idea why this happens. I'm working on it though".⬐ WaterluvianHe's also got the wonderful pairing of both a smooth ear butter voice and the talent of communicating well.⬐ sethbannonIncredible. Is there still no solution to the Newton's cradle like problem with magnets?⬐ imglorp⬐ luisbHe said there is no solution for the chiral tippy-top as well.⬐ Gravityloss⬐ OskarSIsnt it related to that rotating widget demo on the space station? Terry Tao had a good explanation of it.⬐ imglorpWould you have a link? If you're thinking of the Tennis Racket theorem, I don't think it's the same thing as chirality, is it? The special tippy-top would erect itself only when spun in one direction but not the other.
> The tennis racket theorem asserts that when rotating a rigid body with three distinct moments of inertia, the rotation around the axes with the largest or smallest moments of inertia is stable, but the rotation around the axis with the intermediate moment of inertia is unstable.⬐ GravitylossYes, https://news.ycombinator.com/item?id=14125097
Does not explain chirality... I didn't watch the original video so closely, sorry.I'm really curious about this as well. The lecture is from 2008, so it's entirely possible that he or someone else has solved it by now, and I'd really be curious to know what the answer is. It's such a simple system of marbles and magnets, so it's pretty incredible that such an unintuitive thing would happen. You'd also expect that, given the simplicity of the system, any weird behavior would have some elementary explanation (elementary to trained physicists and mathematicians, anyway), but apparently not.
Stick around for the end of the video where he demonstrates an equally incredible and (at the time of the lecture) unexplained phenomenon with "chiral tippe tops".⬐ hi41Is it just me or does someone else also thinks that the comments section is incredibly enlightening to read! :-)Tadashi is awesome! I love his video explaining how someone found a prime that looks like the Arms of Trinity Hall . Highly recommend all videos from Numberphile  and Fermat's Library .⬐ mistermannTadashi playlist: Tadashi Tokieda: https://www.youtube.com/playlist?list=PLt5AfwLFPxWI9eDSJREzp...⬐ NoneNone⬐ imglorpIf you aren't convinced rolling downhill is complicated, physicist Walter Lewin has a demonstration involving cylinders, no fluids this time.⬐ arbieThat was incredible. Every combination hypothesized, then experimented with, ending with an intuitive explanation.⬐ lisperI'm afraid I have to dissent here.
Compare the footage at 2:10-2:15 with that at 3:20-3:25. He wants the result to be that the box starts sliding at the same angle in both cases, and that's what should happen if what they teach in physics 101 is correct, but that is not actually what happens in the video. In fact, the box starts to slide noticeably earlier in the second case (which is exactly what the off-screen crew members predicted). But he simply glosses over this and says "within one degree the same". What is what should have happened, but is not what actually happened. This is a very unscientific attitude, and I think this does a tremendous amount of damage to the popularization of science, particularly in the face of accusations that science is just another religion with its own dogmas, high-priesthood, etc.
The correct reaction here would have been, "Hm, that's interesting, it started sliding earlier. That's not what I expected, and is not what theory predicts. Let's see if we can figure out why this happened."
It's also kind of lame that we have to rely on his callouts to know what angle the board is at. It would have been trivial to set up a pointer that would have been visible to the camera.
P.S. Yes, I know that he "explains" the discrepancy at 3:40, but his explanation is bogus. First, the discrepancy is much bigger than the "half a degree" that he explains away. And second, his explanation is actually a hypothesis that is easily tested: just run the experiment multiple times. He could easily have done this and again demonstrated how science is actually supposed to work, but he didn't.
P.P.S. He does it again at 7:02 where the longer cylinder is clearly beating the shorter one by nearly a full cylinder radius right before he proclaims "no difference".⬐ wruzaWe had post-soviet physics class in school and it had ‘laboratory work’ — hours when you experiment yourself and report results to your teacher to discuss them later. What we learned from these hours is that in low-precision conditions errors can go as high as 70% and more. No single formula worked as intended, because everything in experiment was homemade, heterogeneous and roughly measured.
I have no clear conclusion for how this relates to your point though. Maybe modern education should include this as abc basics, not as high school revelations. Uneducated people are hard to convince, since they don’t know that reality is hard, while beliefs are easy.⬐ lisper⬐ baddoxMy point was about pedagogy, not about physics per se. If you're going to do a basic experiment (on a video no less, where you can prepare and edit) then you should design the thing so it will work. Either that, or make experimental error, statistics, etc. one of the presentation topics. Under no circumstances is it permissible in an educational video to do a single experiment that didn't produce the expected results and then try to sweep the failure under the rug.I understand your complaint, but this isn’t a formal proof or a research paper seeking peer review. This is an educational demonstration of physics concepts that have already been very well-established through the much more rigorous methodology and peer review you are asking for. Of course the educational demonstration won’t have the same rigor, and it shouldn’t be expected to.⬐ sagoWith respect, I don't think that was his complaint.
The problem is pretending to demonstrate by experiment something that involves abstractions to the real world.
It's the same thing with 'demonstrations' that a heavy object will fall at the same speed as a light object where inevitably its 'close enough,' but not actually true (they do in perfect vacuum, but not in a simple experiment).
This happens a lot in demonstrating science, because science is really messy. And even relatively simple physical phenomena we don't understand fully (a recent topical example is we don't properly understand why curling stones curl the way they do).
The problem is pretending the answer is simpler than it is. Which inevitably leads to motivated 'pseudo-experiments'.⬐ lisper⬐ tripzilch> The problem is pretending to demonstrate by experiment something that involves abstractions to the real world.
Not quite. My complaint is pretending that the result of an experiment was something other than what it clearly was and saying, implicitly, "despite the fact that you could plainly see that the result of this experiment was NOT what I said it would be, you should nonetheless believe what I tell you about the laws of physics because I'm the professor and you are the student and I know what's what and you don't. Pay no attention to the actual outcome of the experiment that you saw with your own eyes and instead trust my authoritah."
> This happens a lot in demonstrating science, because science is really messy.
That's no excuse here IMO. In this case the demo was simply very poorly designed. It's just not that hard to construct a friction and moment-of-inertia demo that actually shows what it is intended to show.⬐ exodustObviously he is gearing up to demonstrate the significant difference between hollow and solid cylinders. It's the only race out of all that anyone could place a confident bet on the outcome, and this is what matters. Kids learn that weight and size make no difference. Kids would probably call out the differences on the day and then that could be explained and repeated if needed.⬐ JudgmentalityI think you're glossing over something very important though - which is that he visibly demonstrated that he was wrong. I understand his intention and how he was building up towards the end, but this video makes me question everything he says rather than believe him. He is very much telling us to ignore what we saw with our eyes because it disagrees with his prediction, with very little explanation as to why. He made a prediction, he demonstrated for us that his prediction was wrong, and then he told us he was still right. If that's not an appeal to authority then I don't know what is.⬐ lisperExactly right.⬐ baddoxI just don’t think it’s anywhere near as egregious as you make it sound. If he had literally told us to ignore our eyes, yes that would be bad. But he explicitly said that things were close, and that variations in the plank can cause slight differences, and that averaging a few trials would show extremely close results. Sure, you have to “take it on faith,” but it’s a very reasonable explanation, and by the end of the demonstration (when you see the much larger difference with the hollow cylinders) it becomes clear (or at least extremely easy to believe) that the tiny variations in earlier experiments were indeed the result of “impurities” in the physical medium rather than attempts to conceal inaccuracies in the physics claims being demonstrated.⬐ wsyIf this was a scientific experiment, then what you consider 'very reasonable explanation' I would call a lot of hand-waiving to explain away the difference between the theory and the experimental outcome.
He gets away with it because he doesn't need to convince his fellow scientists, only a bunch of pupils. But it is nevertheless quite unscientific.⬐ JudgmentalityI admittedly could be wrong, but I feel you're approaching this video from the assumption that he is a physics professor and what he says must make sense. I suspect you've already taken physics at some point and this reinforces what you've already learned.
Consider this from the perspective of someone who knows nothing of physics and is skeptical of science and academia. Here is a person that shows us he is wrong, does almost nothing to rectify this (there is a very brief explanation, but why not more demonstrations? Or really, why not just prepare a better demonstration that doesn't have these problems?), and then continues to pontificate as an authority figure.
If his goal is to offer a refresher to people that already know the physics, okay I guess this makes sense, but I sure as hell would've liked a refresher on the equations personally (so I am not the target demographic). If his goal is to teach people that know nothing of physics, he really could've done a lot more to actually teach and explain things.
Getting back to my first paragraph though, I feel like this comment thread might be a great example of humans being humans - everybody is susceptible to confirmation bias (again I could be wrong, but that hypothesis makes the most sense to me). If you didn't already understand the physics and you didn't take him at his word because he is a college professor, honestly his tutorial is unconvincing. He completely ignores the scientific method!Yeah, no. I actually paused the video at that point and said "what the hell?? that was 12 degrees (as someone in the crew predicted), not 16-17 degrees!". It wasn't even 15 degrees, which was clearly when the object actually started sliding in the first experiment.
I wanted to continue watching the video because I should watch the whole thing if I'm gonna criticise it.
But then he lied about the stapler. Just because he skipped counting "15, 16, 18, 19.." doesn't mean it didn't start sliding at 17 degrees, which is in fact within one degree of the unweighted yellow box (if you take his word for it, or within two degrees if you go by what you saw), either way definitely closer than when the weighted yellow box started sliding.
If there's one thing this video tells me it's that the angles that he measures are accurate to no more than +/- 5 degrees. Therefore drawing conclusions about differences that are less than that is FLAWED.
What he is demonstrating is how NOT to do a physics experiment.
If he just wanted the audience to take away from this video that friction is independent of weight or whatever, it would have been better if he had just told the story in front of a blackboard or something.
I stopped watching when he claimed the two cylinders of different length made "no difference". https://i.imgur.com/SXg4FRk.png FFS this is just embarassing.
It's pretty obvious that this guy is just going through the motions telling a rehearsed story that has not much to do with the actual outcomes of the "experiment" he is trying to make us believe he is performing.
If you think that's okay, then imagine replacing this "demonstration" with an animated cartoon of the very same. How convincing would that be to you? Because he does say "you just gotta take my word for it", a lot.
> this isn’t a formal proof or a research paper seeking peer review
No but it is a youtube video of a physics experiment. They don't get peer reviewed. What is it exactly that you are trying to say?
Am I not allowed to expect some amount of rigour and honesty unless it's a research paper seeking peer review?
If it intends to be educational, at least don't show that apparently it's okay to fudge results in a physics experiment as long as you're a MIT professor and say "just take my word for it". Because that's my takeaway from this video: "this guy probably shouldn't do physics experiments".
Because most Youtube videos on scientific topics I've seen, they either perform the experiment correctly, but if they don't get the results that they "should", they own up to it. Sometimes they record the experiment again, and they TELL you it failed at first and what they changed. Sometimes I even hear them lament that they filmed a great experiment, got the results they wanted, but had to trash the video because it just looked fake and unconvincing (because of the angle, timing, just didn't work on camera, etc).
It sucks for their time and effort but they know they can't use it because it would completely undermine their point when the comments point out how fake it looks.
Guess who disabled commenting on their video? You're absolutely right about not seeking peer review, or in fact any review.
Quick check of some other vids on that Physics World youtube-channel, in particular a few videos with a professor or PHD talking or doing an experiment: NONE (of the ones I clicked so far) had their comments disabled!
So what I'm guessing happened here is, a LOT of people had issues with the fudging of the experimental results and dis-ingenuity of what they supposedly demonstrate. They commented and pointed out that what the man says is happening isn't actually what they see is happening, and that makes it really hard to "just take his word for it".
I doubt it even got really ugly before someone's ego got a dent in it and instead of deleting the crummy video, they took the much easier road of deleting the criticism.
If he's doing this all the time in Skype demonstrations all over the world, why put up a video where the first couple of experiments clearly don't show what he's saying they show?
 That example btw, was Tom Scott a couple of times, huge respect. Not only owning up to failed experiments, but clearly talking about the experiments he DID do, that DID NOT work out as he hoped, so much that he didn't want to publish them (on YouTube), but he still said he did the experiment. That's a HIGHER STANDARD actually than a proper scientific formal proof or peer-reviewed research paper is held to!!!! Peer review doesn't check for publication bias.⬐ mncharityContent for science education commonly mixes aspects done with great care, aspects done with less, and aspects that are... artistic license and trippy psychotic bogosity. Regrettably, students are often unsure which aspects are which. Unsurprisingly so - they're new to the area. This feeds and diversifies their rich ecologies of misconceptions.
So some lecturer might think of themselves as demonstrating only friction. But they're also demonstrating how to present material, how to do lab safety, and perhaps also how to silently subordinate observation and data to expectation and narrative.
When doing a demo, one can easily acknowledge and flag a divergence with a "<expectation>... more or less".
And it's not just a problem for students. In the context of science education research, a first-rate stone-cold-empircial experimental physicist, when thinking about their teaching, might be "like, I have gut feel for what works - I trust my gut". Sigh. I know a biologist, who does respected work, but get them started on <other thing that's not their research focus>, and it's a bizarre sidestep into earthy-crunchy crystal-crazy land. Pity the poor student trying to figure out which of all these are the behaviors to emulate, and which not.
Just today on numberphile channel they showed how to pass circle through smaller square hole by bending it in higher (3rd) dimension: https://m.youtube.com/watch?v=AvFNCNOyZeE
⬐ empath75The did a numberphile video on this exact topic a few months ago: https://www.youtube.com/watch?v=mceaM2_zQd8⬐ ghusbands⬐ NoneThe video you linked to has nothing at all to do with the parent one. You probably meant to reply to the topic, rather than the comment you actually replied to.⬐ NoneNone⬐ itpI suspect you're wrong, and that the connection between video and parent was that they were both Numberphile videos. An expanded comment might have read something like:
"Not only is today's Numberphile video potentially relevant, Numberphile actually directly addressed the article's exact topic several months ago."None⬐ dhimesThat short little video gave me a lot of pleasure!⬐ ilitiritThere's a (sort of) intuitive way of understanding why things like these are possible without even seeing it visually.
How can you make two solid 2D squares occupy the space? They can't travel through each other, so...? Easy. Lift one of the squares into the 3rd Dimension, move it to the same X,Y coords, then drop it on the other (think of putting stacking 2 pieces of paper).
Similarly, you can have two 3D objects occupy the same space by "lifting" one into the 4th dimension, moving it to the same X,Y,Z coords as the other, and then "dropping it". You can "cheat" by thinking of the 4th dimension as time - one object travels through time, places itself at the same position of where the other object is, then returns. But as I said, that's "cheating".⬐ saiya-jinI do similar mental exercise with higher dimensions, but for me with easier situation - escaping from certain area without crossing boundaries.
1 dimensional - an endless line with 2 spots marking boundaries to some subpart X. If you're inside, you need to enter 2nd dimension to escape that subpart X without crossing boundaries (and ie come back to original dimension).
2 dimensional - circle, you're inside, you need to enter 3rd dimension to escape without touching the circle.
3 dimensional - sphere, entering 4th dimension to escape without touching the sphere.
I can't go with my simpleton mind much further but it should scale indefinitely :)