Kakeya's Needle Problem - Numberphile
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- Опубликовано: 7 сен 2024
- The famed Kakeya Needle Problem, discussed by Charles Fefferman from Princeton University.
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Edit and animation by Pete McPartlan
Film and interview by Brady Haran
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*draws a triangle*
"Let's say this is a triangle"
This man is an absolute genius
He won a fields medal
@@newkid9807 for drawing triangles?
when you are so smart you think no one knows what a triangle looks like
Maybe he knows that its not a perfect mathematical triangle and to avoid any confusion he therefore tells us to assume that it is?
@@pr1ckastley for assuming it to be a triangle
Usually I find numberphile videos pretty easy to follow, but this one was really taxing. I was particularly thrown off by all the numbers he was throwing around, while saying they were completely arbitrary and didn't matter. What was the point of saying the area was 1000 times theta or the length of the base was "3ish"? It didn't seem to contribute anything to the explanation.
Yes, exactly. I could follow because I knew the problem already but this guy is the worst kind of teacher. Hey kids, let me never explain the concept properly and at the same time boggle this interesting problem down with a million useless additional details that I'm making up.
@@LegionHimself The purpose of the video was to show that the shape they are drawing can be made as small as you want. That means that for any number x you might come up with, they can draw a version of the shape whose area is less than x. Saying that the area is "1000 times theta," and then saying that the "1000" doesn't matter, means that you can make the value of "1000 times theta" as small as you want by making theta really small. If you want to make it smaller than 0.001, for example, you just need to say that theta < 0.000001. Same thing with the "3-ish" and the other arbitrary numbers.
But I would agree that they didn't explain it very well. I was only able to follow it because I was a math major in college. But if I saw this video on my very first day of math classes, I would've been totally lost.
Tetraglot I understand perfectly what the problem is about. As I wrote, I was already familiar with it. The guy is atrocious as a teacher is what I'm saying.
Oh wow, so I googled this guy and it turns out he's a child prodigy. He's not an idiot who's unable to cobble together a coherent explanation, he's a bona fide genius who cannot be arsed to.
Interesting, I find the recent video uploaded on infinite series very well explained while this one was terrible.
It was a bit confusing along the way but in the end everything came together and I think I understood. The animation really helped, too, especially with the bit at the end removing the jumps. I don't think that would've made any sense without seeing how it worked... But this was really cool! Thanks for covering such an interesting, bizarre topic, and I hope you do more stuff like this in the future.
!2tone?! Didn't expect to see someone like you beneath one of these vids!
12TONE
Try 19-TET!
love you and your videos
The animation just got much more slicker! Kudos to the animator!
+Rick Chong (Homeworkace) that's Pete McPartlan - cheers Pete!!!!
+Rick Chong (Homeworkace) Thank you!
+Numberphile No, thank you for the mind scrambling projects.
thx.
I'll admit that I was completely lost until the end...
+FlesHBoX haha yeah me too!
I'm lost until 7:00 where I stopped watching :(
+FlesHBoX I was pretty confused too
+FlesHBoX I gave up with this one :/ The coconut problem, the bathroom problem, the marriage problem and the chess problem were much more interesting videos.
+FlesHBoX I think his way of explaining was somewhat confusing and difficult to follow.
this is a cool problem but i didnt like the explanation. it irks me when teachers say "say the number is 11 but it doesnt matter" then goes on to use variables in other places. make it all variables or all numbers. to jump between is unnecessarily confusing.
this is cool but randomly choosing numbers instead of using variables and stuff makes it more confusing
Its an asymptotic construction.
he should have explained why he was doing it and the outcome he wanted to make instead of just doing it and saying "This is indefinite but i dont care because its 1000 * Feta"
@@oerlikon20mm29 "Feta"
@@ion9084 that was my level of understanding
@@oerlikon20mm29 theta
7:16 Me solving a math problem.
Thank you Stranger! You made my day 😂 ~
loool
+Menno I Arachnid And that's also your teacher giving you grade in exam.
+Menno I Arachnid Haha It became even better if you included the sentence after as well
"It has... eleven, I don't care. It grows a pair of ears!"
+Menno I Arachnid That's ridiculous. It's not even funny.
4:10 "And if I can do that I will win" - the reason scientists do science
+Kostas1601 He's a mathematician, not a scientist, at least for this video.
+NickRoman Mathematics: the abstract science of number, quantity, and space. Mathematics may be studied in its own right ( pure mathematics ), or as it is applied to other disciplines such as physics and engineering (applied mathematics ). (Google)
+NickRoman mathematicians are scientists
+Kostas1601 I would say science is mathematics, but science almost entirely fits within our real world. It would be an inaccuracy to relate all of maths to that.
+Kieran Kaempen You're right I suppose. Not all mathematics is science. Anyway my point was that scientists are mostly motivated by ambition and pushing the boundaries of knowledge. And I felt that this specific phrase embodies that.
is that official Numberphile brand brown paper he's using? it looks weird.
It seems like a different type of paper, the animated paper is also a different color
+j0nthegreat Hi, I was on the road (in Princeton) and that was the best we could get at short notice! :)
Numberphile if you need someone to travel with you to carry spare brown paper just in case, i can make adjustments to my life
+Numberphile In the US you'll find the paper that grocery store bags are made of, to be the perfect shade of brown. Their free here also.
I'm fairly certain that's just manila paper.
This video would benefit from a modern Numberphile revisit
"So I'm turning an N-long pole inside something whose area is going to be proportional to N, and if I can do that I will win."
- Prof. Fefferman 2015
him being so smart he can do that, but not able to find out why everyone is sleeping in class
??.
The whole "it's 1000" thing was kinda confusing. If it's just an arbitrary constant, just call it k or w/e :S
This one would've been a lot easier to understand if he'd given actual numbers and definitions rather than saying "it's a thing but it doesn't really matter what that thing is" over and over again. Even arbitrary numbers or variables would be easier to follow.
Yeh, but the arena has tiny little triangles an artbitrarily long distance away to which the pole has to travel through paths of zero width.
This particular video was not very accessible. Even though I have actually learned this concept before on another video I had a lot of trouble following the speaker.
PotatoMcWhiskey See Mathologer’s video on the problem; way more accessible and not as bogged down in unneeded numbers.
1st numberphile video that doesn't make sense. Mathologer did a much better job.
Thanks, will give it a look, this video is very unclearly explained.
the numbers are very distracting I'd say, but I understood it better here although I saw the other first so... I'd already had enough info to know where he was heading to.
Second actually. The first one would be sum of all natural numbers.
@@LakshaySura yep.
This was really confusing. Not at all up to numberphile's usual standard. Also why add even more complexity by introducing this scaling reality to a map? Maybe I missed it but that didn't seem to add anything to the explanation.
+Ulrich Petri As far as I can tell it's fairly fundamental, you make the pole longer with each iteration of ears as you move up to N. Which also makes the area marginally larger (by a smaller amount each step) when you renormalise everything back to 1 you then have a tiny area. I suppose you could start at some arbitrary tiny number (eg 1/a billion) and work towards 1 it wouldn't express the fact that N could be any value, the more steps the and smaller the area.
+Ulrich Petri I felt the same way. Glad some people were confused along with me. Watched it twice, which helped, but got lost on some of the explanation of scaling - I wondered why the need to discuss scaling, couldn't he have started with the unit length of 1, then explained the construction in terms of N where N was in the same units as 1, and create some constant which defined theta?
See Mathologer’s video on the Kakeya problem; much more accessible, and way better diagrams, although they did somewhat gloss over the issue of calculating the area that this video got so bogged down with.
Without the scaling argument you can get to arbitrary small area
I know a lot of people say that you guys are getting harder to understand or getting less accessible and I think I agree. Unfortunately, I enjoy this inaccessibility. I really enjoy when you guys discuss difficult subjects and some of the "deeper" mathematics. It's inspiring and helps motivate me to understand what you guys are talking about in my studies. Anyways, big fan of the channel and thanks for everything.
Am I the only one who thinks that this whole explanation was really weird and unnecessarily complicated?
+Andrew H I must admit that I dont even understand the drawing at the end. The illustration they do at 12.18 still doesn't turn all the way without jumping.
+FKlejs Every time there is a "jump" you are "actually" doing what the animation showed at 11:45. If you were to scale this actually process... well than the shape would be so small it would not even be possible to animate the steps.
Yeah, see Mathologer’s video on the problem for a far clearer version with much better animations and fewer unnecessary numbers.
@@FKlejs That's because the animator was completely lost by this guy's bizarre explanation.
Yes, you are the only one.
There's something missing with this explanation: Dr. Fefferman says that the area of a pie segment is proportional to "something" times the angle, and calls that constant 1000 just for example. The problem is that "something" is not a fixed constant, but depends on the radius of the pie. For theta measured in radians, the area is (theta/2) * r^2.
Therefore, as he considers "ears" at stage 2, 3...N..., the needle's length at this stage is N, so the area it sweeps out increases in proportion to N^2. Now, a lot of that area is shared by overlapping pie segments: all that really matters is the increased area of the "ears". Dr. Fefferman didn't want to prove that the area of all the ears remains constant as N increases, but since other areas in the problem increase like N^2, and since the number of ears increases as 2^N, I think he'd probably better. I don't think it's wrong, but the proof is missing a vital step.
Literally did not understand a single thing this guy was talking about.
dombower
Mathologer explains it great.. ruclips.net/video/IM-n9c-ARHU/видео.html
dombower you’re not smart enough
you are not alone
dombower I take your understanding and subtract theta. I get that much.
@dombower was just about to comment the same thing.
He draws almost perfect lines.
This was a great video. The inaccuracy in the numbers and measurement was really getting to me, and I wondered where it was going, then it had that "oooooh!" moment, and the elegance of the solution was a great pay off!
Is it weird that math problems affect me more than most dramatic media these days?
+Wreqt
No, it's not weird. Different people have different interests. And logic problems are pretty fascinating even without knowing or understanding a solution.
+Wreqt maybe it is because there are interessant and creative approaches to solve problems in maths and in most cases somebody works out the solution. unlike in politics and media events
karottenkoenig If only politics would be as luxurious as pondering about the Riemann hypothesis...
+Zardo Dhieldor if only politics was focussed on solving problems...
+karottenkoenig
It is! Just not the sort of problems one might wish...
I always watch numberphile videos, rarely understand them, but always watch them. Can anyone solve this problem?
+anthony b be smarter
+anthony b can you pin point what topics you are having problems with?
This one is the worst yet.
But I could do with more Klein bottles. Anyone know if there's, say, a thousand lying around somewhere? :)
I find this a sometimes, but I like to research the problems and things they are using. It's always fun to learn Maths!
+anthony b cattle prods might work.
The shape at 9:13 may reach out farther in every direction than a circle and requires ten times the space, but think of all the infinitesimally small objects you could fit between those ears! Super efficient.
I feel like the solution should have come first and the explanation later, as it is I spent a large portion of the video rather lost, and only understood until the very end.
8:58 should be a poster. Or a T-shirt. Or a mug. Just saying!
definitely!
*LOTS*
OF
*EARS*
Many of the ideas about the problem came after the calculations, such as what needed to be done to stop the magic jumping; the solution should be given and then the maths demonstrated otherwise you get lost in "enth" this and "theta" that. Great maths problem though, even if my understanding of it only came at the very end, Brady!
totally confusing :-O let's say I'd have a car, or let's say I have 7 cars, the correct number is totally arbitrary, so we say, I have "n" cars with the size of 700, no matter if centimeter or miles and which size exactly. then I drive against a tree, so n cars with the size of 700 get damaged, now we see that the n and the 700 cancel when you multiply... whot?
Visiting this channel is always an experience of mindblown incomprehension. Its a rare and appreciated feeling for me. When asked what to call the shape of the arena, he probably could have just set "Fractal Ears"
10:45 - 11:32 Should not have been included. Perfect example of how you can cause confusion by including additional information.
You forgot the most important thing, you wouldn't be able to fit your hand inside there to turn it ;)
+Robert McGrath Of course a mathematician should always have a pair of infinitesimal tweezers at hand.
you use a magnet to move the needle :)
+Omar Z No need to go that far. I'm sure Planck's Pincette would do just fine for any real world application.
7:47: There's an (inconsequential) error here. It should be 2^N and not 2N.
+jotomicron That's my favourite type of error!
I normally enjoy all your videos Brady, but this seems like a prime example of the 'how many angels can fit on the head of a pin' exercise.
+Van Exel (Vanexel007) Well your question doesn't have a real answer, whereas this does.
In both cases any answer would be arbitrary and furthermore useless.
+Van Exel (Vanexel007) The answer to this problem is nowhere near arbitrary :/
It's a small area but it covers a huge space...
_ Bender the space it covers is its area...thats literally the definition of area. The space it covers.
If however you mean its overall length in 2 dimensions, those are large, yes.
If anyone still finds this confusing, as it was a rather technical explanation, look for the Mathologer's video on this problem, as he explains it to us laymen pretty well. Though I personally had no problem with this explanation, as I think Professor Fefferman did a great job simplifying it so I could understand it, I can see in some of the comments here that a lot of us did not quite get it, and since this is a complicated thing, it is totally understandable that it is eluding some of us. Of course Pete's animation was an important lynch pin in my ability to understand this as well. I thought this video was spectacular, and I think I understand why this weird little trick is important, as the formula involved let's you reduce an area problem to almost zero, and that is amazing to me. I just hope Brady has not been dissuaded from giving us more complex and technical videos like this, as I thought it was one of the best ones yet. I really like that it was a bit deeper than the usual fare, and I hope Brady sticks with it and keeps mixing it up by giving us both kinds of videos in both kinds of formats.
Man, I just Looked up Professor Fefferman's credentials...holy smoke this guy is something else. Possibly the most accomplished mathematician you've had on this channel Brady...and that is no small feat, as you've had a lot of insanely smart and talented and accredited people on this channel. We need a higher echelon of intellectual study to describe some people, like Einstein and Gell-mann and Hawking and Goedel. I think Professor Fefferman could easily be included int hat group, of mega-geniuses.
4:18
*draws a triangle*
Let's say this is a triangle!
Ahh those mathematicians
Platon: "But a true triangle does not exist in the physical world and thus cannot be drawn."
Charles Fefferman: "That's damn right. That's why I only suppose it is one."
:)
This is a ridiculously inventive solution for this problem!
For everyone complaining that this solution is too complicated (it is as complicated as it needs to be) or there is cheater magic jumping (there is not), watch the video as many times as you need until you realize why you are wrong.
Dear reader , we have presented this problem by another point of view which shows that needle can not be rotated in zero area. Plz watch our video and give your feedback.
The "magic jumping" trick is really clever. He should have explained that first. Then the rest of it would have made (more) sense. I had to watch this 3 times to get it.
The intuition behind the fractal triangle isn't explained really well either. The important part to understand is that 2 triangles overlapping have less area than one big triangle, but still allow the same degree of rotation (with magic jumping). And 4 triangles have even less area, and so on. The animations help a lot though.
13:18 - "someone trying to turn a long poll in a dense forest" - You know what's on his mind.
My First math graduation thesis (3 years degree - Italy) Fantastic! I studied also the n-dimensional version and algebra connections in finite fields ;)
Wow! Can you share anything neat about those
I found some of the explanation confusing. It was like too much info. n, k, 1000*theta...just not the way I am used to understanding a problem I guess. I sort of understood the answer, but I think I will have to think about this more for it to be intuitive
this guy is brilliant at drawing straight lines
ok, I am officially confused.
There is nothing quite like watching a genius work
In the real world the bar would need thickness, and this would necessitate thickness in the other constructions. Also, the lengths of the ears... Well, maybe not. I suppose the bar could be a photon bouncing between two mirrors. Positioning the mirrors would be a problem then, though.
+Nehmo Sergheyev Yeah, there can't surely be a real world application for this.
+MagikGimp But it does make you think that anything can be achieved if you go at it at increasingly high detail.
+Nehmo Sergheyev It's obvious you never needed to move a pole in a dense forest
+MagikGimp It could easily have some applications.
Even e^i has tons of use and it uses imaginary numbers etc.
Hats off to the animator! I would not have followed at all without the animations.
Finally some substance
This is one of those numberphile videos where I have no idea what hes talking about when he's explaining, but when the conclusion comes, I actually understand. o.O
Time to make a videogame about turning long poles in dense forests.
The number 9 models all digits in the universe while simultaneously remaining void.
Degrees of a circle:
360°(3+6+0=9)
180°(1+8+0=9)
90°(9+0=9)
45°(4+5=9)
22.5°(2+2+5=9)
11.25°(1+1+2+5=9)
The resulting angle always reduces to 9.
Sum of angles in regular polygons:
Triangle: 60x3=180 (1+8+0=9)
Square: 90x4=360 (3+6+0=9)
Pentagon: 108x5=540 (5+4+0=9)
Hexagon: 120x6=720 (7+2+0=9)
Heptagon: 135x8=1080 (1+0+8+0=9)
Octagon: 140x9=1260 (1+2+6+0=9)
Nonagon: 144x10=1440 (1+4+4+0=9)
When bisecting a circle the resulting angle always reduces to 9. Converging into a singularity.
The polygons however, revealed the complete opposite. Their vectors communicate an outward divergence.
The 9 reveals the linear duality. It is both the singularity and the vacuum. Nine models everything and nothing simultaneously. The sum of all numbers excluding 9 is 36:
0+1+2+3+4+5+6+7+8=36(3+6=9)
Paradoxically, nine plus any digit returns the same digit.
i.e: 9+5=14(1+4=5)
Therefore 9 equals all digits (36) and nothing (0).
thanks for posting this one!
+Mr Autistic our pleasure
I do not know what common people are not able to understand. Simply, the problem was about mapping a line segment (he used a small t for its thickness) around a 360° trajectory. The rest was only explaining a great way to create a geometric construction where the numbers associated with it are just to give a reference for the proportion of sizes.
The idea of jumps is just stupid. If you ignore that it is an interesting problem
Dr. Fefferman is a natural story teller, I could listen to him all day. More seamless transitions from the Euclidian plane to the Bolshevik revolution to the magic jumps solution found by a Hungarian mathematician named Pal, bravo!
Before the prof gave his solution, I paused the video to think up my own solution: all I could think of was an infinite circumference of zero thickness (technically zero area, right?). The circumference has to be infinite so that it has zero curvature and you can place the needle "inside" of it. The kicker is that a full rotation takes infinite time!
After watching the prof's solution, I have a question? Does there need to be infinite ears in the final structure? I didn't quite figure this out. Or is it a number of ears that's proportional to the starting tip angle?
+Jose Villegas There doesn't need to be infinite ears but the more ears you have, the smaller the area is going to be
+Jose Villegas your solution is a lot more elegant but, being infinite, can only exist in theory.
The prof's shape could be physically constructed at a given size N but his problem (if I'm understanding the geometry correctly) is that a shape of N+1 would have less area.
Therefore a perfect solution would have an infinite number of infinitesimally small ears. I think that as N approached infinity, the limit would converge to a finite value but not knowing the area formula for this shape I'm not sure.
+Jose Villegas I also thought of an infinite circle (or any closed path that has an infinitesimally small rate of change in direction), but I didn't accept it as a solution because it has a hole, has to be infinite in size and has no area.
+Jose Villegas Well, our friend 'Phraser' here gave an explanation to prove why the circumference type solution is incorrect. Essentially, you will end up with infinite area in the end. You had made a convenient mistake of assuming the thickness as absolute zero.
One could say that if you make the circle large enough, you would approach zero thickness. At the same time the radius would approach infinity. So, we need to look at the Calculation of area of such a figure. As it turns out, if you increase radius the area of your figure increases. So we are not getting anywhere with that solution, are we?
Charudatt Awaghate What I will admit is that the figure requires infinite area (which is definitely a problem and irredemable), but the figure itself is of zero area.
I'm not a mathematician but I did actually find his explenation quite intuitive.
Buy something from china and you will see how to fit something into the smallest area possible.
while this solution works mathematically, if I was told this in any other setting I would not accept it as an actual solution.
The legendary Charlie Fefferman! Youngest full time professor in the history of US and oh, a Fields Medalist at the age of 29. How about that?
7:50 should be 2^N, not 2N (maybe you just meant it as a cute notation thing but it's just wrong now).
+tgwnn He really doesn't care because it goes awat, that's why he put 11 area, 1000 as the constant of the area of the slice, etc.
+pipemoreno94 of course, those (11 and 1000) are constants, but this was actually 2 to the power of N, not just some number. This was just an error. And it wasn't made by Charles Fefferman but the makers of the video.
+pipemoreno94 at 7:40 you can clearly see that Fefferman wrote 2^N and not 2N.
+pipemoreno94 he meaning who? I never said it was the main thing or anything, just that it is a mistake. 1000*theta or 11 are not mistakes, but just choosing an arbitrary constant, because in the end he (Fefferman) cares about the scaling. 2N instead of 2^N was a mistake. But I think we basically agree.
One of the more confusing / less clear videos on this channel. I think the first minute is the problem.
I think there is a much easier way to do it. Imagine an annulus whose radius is arbitrarily big and the thickness is arbitrarily small. If the ring is big enough you can turn a pole of an arbitrarily length in it. Obviously the longer the pole the bigger the radius has to be. But if you take the limit R -> ∞ it should work. The bigger the radius is the thinner the annulus can be to move the pole in it.
Ah damn i think i have found a problem^^ The circumference scales with R but the Area of the annulus scales with R^2 so the area will be infinite at the end xD
+Phraser also when the pole goes all the way around to the same point it will not have actually rotated it will be facing the same direction.
correct ^^
it will have been rotated, around the centre of the annulus surley? the problem wasnt to have the pole come back to the same point at 180 phase or anything like that...
if the thickness of the pole was infinitesimal then wouldnt the area between the two circles of the annulus be infinitly small, regardless of the infinte radius of both circles? I really dont know!
Christopher Taylor nope because if you increase the thickness by a factor of x the area will increase with x^2.
Not only did nobody watching this video understand what this guy is talking about, it's obvious the animator didn't either.
I mean I understand the core concept, that a "spiky circleoid" shape is optimal for being able to rotate an infinitely thin line within that shape while taking up the least possible area. I even paused at around 3 minutes and thought about it myself, and also arrived at the "spiky" conclusion. Then after watching the rest of the video, all I can say is I'm glad this guy isn't my teacher.
If you have to do these magical jumps then I do not really see why this is either important or impressive. It seems that if you are doing these jumps then what is even the point of this whole arrangement? If you can just jump from one to another couldn't you just use any sort of extended shape and make the jumps further. I am sure this would then reduce the overall area. Simply make 2 jumps and you would have rotated the object. I may be completely lost and that is why I do not understand the significance of this object. If anyone could explain why it is not simply cheating when you are doing the jumps I would very much appreciate it?
+Mark G the magical jumps don't turn the rod at all, only by a very small angle theta. if you did two of the magical jumps, you'd need to do two half-circle turns, defeating the whole purpose.
The jumps were only magical until he explained how to do them. Did you watch the last quarter?
+Stephen Halliwell Yeah, "magical" was the wrong word. He should have simply said "We'll deal with this later".
+Nehmo Sergheyev he does say that at 7:02
zach burke I am saying that the _term_ "magic" is inappropriate and even misleading. What's magic about it if you can accomplish it within the rules? It even turned off one poster to the point of not viewing the rest of the video, where, proving the point, the magic was removed.
You could also just slide the pole along the circumference of an extremely large circle, where of course the only area actually being taken into account would be an almost infinitely narrow ring.
Not sure if that might call into action some sort of napkin ring type issue though
I was thinking of something similar too. Kind of a teardrop shape, where the point would be infinitely long to let the needle turn before going back down (like the "ears" mentioned in the video). If it were just a full circle, the needle would not have flipped once it got back to to the start (it would have it's original orientation).
I was pretty sure that the teardrop solution was a simpler way to approach zero area , but after running the numbers on it, I found that it does run into a sort of napkin ring type issue. A circle like you suggested would have a minimum area of π/4, while the teardrop would approach π/8.
I can share my math with you if you would like.
Mathematically correct but practically impossible answer. As an engineer, I'm not satisfied with this solution.
I normally perfectly understand numberfile videos but this one I was completely lost the whole time
I am lost.
I was completely lost and then he brought it all together within about a minute and all of a sudden it made sense. I think it was the only video in the series where I was ever completely dumbfounded for a majority of the video.
this video is very bad at explaining. and even has missing information.
for example the construction at 8:50 how many degrees could you turn the needle in it? how many such construction do you need to get 360 degrees? (at 9:00 you just stitched 8 of them - where is the proof for that?)
how could you connect them?
is the 45 degree head angle of the triangle we started with at 6:10 makes any difference?
+elraviv "is the 45 degree head angle of the triangle we started with at 6:10 makes any difference?"
Short answer: no, if it adds up to 360 it's all good.
Longer answer, Prof Fefferman accidentally said/drew a 45degree triangle instead of 90. So I had to cut out a lot where he talked about dividing the arena into 4 quadrants and sweeping 90degrees etc. I'm not sure if one is a more efficient solution than the other but they both end up with the area being proportional to N and not N^2. (The even longer story involves me having to tidy my bedroom to skype with a Fields Medallist to ask if maybe he got his maths wrong. However all smugness was dropped when he told me I'd spelled Kakeya wrong in the titles.)
"at 9:00 you just stitched 8 of them - where is the proof for that?" Because you can sweep through these angles in the first sector and end up at the start of the next one you can continue around the rest of the circle. (They are identical, just rotated, and you are performing the same motion.)
Is that any help?
+Pete McPartlan You're not the numberphile guy, are you?
+Pete McPartlan You're not the numberphile guy, are you?
For anyone who is confused, watching it multiple times helps A LOT in terms of beginning to understand it all
What paper is that??? I want the official numberphile brown paper!
There were a lot of annoying "let's call it " moments, but I can appreciate that he was probably using that to replace wordy formalization that would turn out to be inconsequential anyway.
Am i the only one that doesnt know any thing about math and is still watching this
now you know more. haha. but yeah, I get the idea but most of these videos are more than I know
+Caljkusic1 Here is your first lesson: 1 + 1 = 2
Caljkusic1
Lesson 1:
Addition
Great video. I did not mind that it was complicated.
I don't understand this at all. How can N spin in something so much smaller than itself? Or is this purely mathematical and wouldn't work in real life? Is that why I don't understand it, because this is made up geometry?
+cortster12 It isn't made up -_- please read "Apology of a Mathermatitian" before you start asking about the "real world"
+cortster12 it *would* work out in the real world. The point is that you can make N as large as possible and thereby make the area arbitrarily small. In a real application you would choose N judiciously to suit our purpose. As a simple example: Taylor's series. In principle, the use of Taylor's series only really "converges" to some nonlinear function when you include an infinite number of terms. However, computers usually only store numbers up to 8 (sometimes 16) decimal places, so if you need a function for e^x or sin(x), you can use the Taylor's series (if you set it up carefully enough) and stop it roughly when you see that none of the future terms are bigger than 10^-8. So in real life computers, Taylor's series are exact even in finite cases. In a real-life turning example, we would agree as to what an acceptably small turning area is (I struggle to think of any real-life applications but maybe there are), and iterate as many times as it takes.
+cortster12 A distant relative of this problem is exploiting surface tension by insects. Surface tension is the force between a liquid and a physical object touching its surface, and its magnitude is proportional to the *circumference* of the contact area. So insects needed to come up with a contact surface with a finite area (because they don't want to be as big as an elephant) but a practically infinite circumference. Of course, mathematically you can make the circumference arbitrarily large by making a bunch of triangles or rectangles that are arbitrarily narrow and tying them together. Insects can't grow arbitrarily narrow triangles but they do grow very thin hairs that do make a lot of contact circumference in relatively small areas. So yea, that's kind of the case of the video here too, you have a lot of circumference but very little total area. Well, like I said, it's a distant relative, but the insects also want to maximize the circumference (i.e., something that is proportional to N) and not the area (something that is proportional to N^2).
+cortster12 In the real world everything is discrete and finite. In this problem we assume we have a zero-thickness line with infinitely small areas. So this will not work in the real world.
+tgwnn Wow, I am stupendously impressed you managed to find a real world example to answer the question. I didn't subscribe to this channel thinking I would learn anything about the real world. Thanks!
Mathematician: "...and that's how you solve Kakeya's needle problem."
Me: "The only thing I asked you to do was calculate the tip on this check."
I now hate the letter N
just like Kakeya's triangle, I can make n-pointed stars where n is odd and n tends to infinity, and the stick would simply be riding on the edges of the star, following the same path taken by a pen used to draw an odd-pointed star without lifting the pen from paper.
In the limiting case, the area of the star would be tending to zero. And our stick makes a full 360 degree rotation inside of it.
Each arm of the star does get very skinny, but there are many, many arms! How do you know that the "density" of your arms is sparse enough that the area really does tend to zero?
I get it!
(I don't get it)
+유형준 just the opposite for me :-)
Of all the Numberphile videos, I found this one the most difficulty to follow.
I lost you like 2 minutes in.
Not gonna lie, I can usually follow most of these videos but this went over my head completely.
Much much TL:DW - Draw the arena so it looks like a sea urchin.
Skip about 10 minutes in to avoid him rambling on about how scaling objects up and down works and being vague.
+Jonathan Chappell He doesn't do a great job at some of the explanations but the whole N^2...N...1/N part is quite obviously important in his geometric scaling but also comes in handy when dealing with big-O and algorithm growth rates.
I'm pretty sure there is some comparable functions between the two and could be very useful (hence why he and other mathematicians are studying it)
it wast obvious to me, Kai Hulud
I had some problems understanding what was the problem we were trying to solve. But after I saw it a couple times I saw how beautiful this problem is. Thanks for the video!
wat
Really liked this one. He is not only giving the solution, but the thought process that led to it as well.
For a mathematician, this guy cares very little about numbers.
+Lazhward Kirmist He's just demonstrating that the numbers aren't specific. The important thing he's showing off is the construction, not a particular example of the construction.
+Lazhward Kirmist One of the really nice things about maths is that once you've done enough of it, you kind of slowly stop using numbers.
+Lazhward Kirmist That's because they aren't important for this problem. The only important thing is to make sure that one number is bigger than another so that the shape is possible.
Suppose, assume, whatever, we don't care etc. Most common words in this video.
"Never mind what the arena is but the arena includes a pole of length 1"... Yeah I'm just gonna stop watching this, because he doesn't know how to explain things.
Please don't use this guy again. He's nice and all, but Grimes et al. are just so much better at making complex things understandable.
+Firstname Lastname
He's much better than my maths lecturer who tried to explain this to me last winter. Now I actually understood the problem.
+Firstname Lastname I second this.
+Firstname Lastname To be honest, that's kind of because this is inherently more complex than most mathematics on this channel. I agree Grimes, Parker, etc might be able to explain something like this slightly better but I don't think this guy was too bad.
+Firstname Lastname I thought he did a great job..perhaps the problem, dear Brutus...
Since I didn't understand this video, I came up with a simpler solution. Basically it's just a star with N points, and diameter D, but having a circular unbroken area in the middle of radius R. The length of the pole is D/2. You just move it up and down in a clockwise direction until it has rotated. But it's not clear to me if the limit of the size of the arena in this case approaches 0 or some other positive value. My guess is that if you push R to zero, while increasing N to infinity, then the size of the arena would be zero. I came up with this solution instantly while looking at the image at 3:23 in this video. For that region, N=3 and R appears to be about 0.4.
The limit does not go to zero. Each arm of your star has an area of around (π/N)(D/2)^2, but there are N arms.
The circular unbroken area gets vanishingly small the more you increase N.
Here's how i try to explain the solution to myself. warning- you might need to re-read this several times before you get it.
Firstly, assume that the needle that you are moving around is a painting needle. So whatever area the needle crosses over during its full journey, gets painted. Think about it.
Now, you have to turn the needle throughout your journey by 360 degrees. So it might seem that whenever you turn a small angle theta, you would paint and area of k*theta.(k is a constant- its equal to 'half of r squared' if you take theta in radians
This is just turning a large car around in a narrow street. The joys of London and the infinite point turn.
Typo at 7:46 : the video says "2N" in the numerator and denominator of the expression. It should say "2^N" in both places.
I was entirely lost until the end of the video. I'm glad I hung on because it did tie together for me a little at the end... but yeah, the first bit there was super confusing.
If I was 17 again and deciding on a major, after watching this there would be no question.
I think I understand... It was confusing as hell to follow along, the animation really helped.
PLEASE go over to the wikipedia page and FIX it...the explanation there is terrible, yours is great ! They never show the triangles getting longer (all the same height had me confused step 1) and they never intimated that MANY triangles would be needed to add up to 360...they started with an equilateral triangle that can do the whole 360 and without explaining they were NOT just transforming that solution started the segmentation idea. Its really misleading ! This is the better explanation by far.
I was lost at a moment in this video, but I had that EUREKA moment and did in fact understand the whole idea that was conveyed. Great video!