The clarity of your videos are OUTSTANDING. The diagrams are simple and elegant and you take great care to explain and interpret subtle mathematical points without glossing them over. Thank you!
Just fool that cleaner with thick squeegee, he will end up cleaning infinite area of window just to skip the first centimeter of window/ first degree in the rotating problem😂😂
This is an example of truly excellent exposition. It's clear, concise, and close enough to precise to not make my teeth hurt. Very, very, very impressive. Congratulations on making a beautiful piece of pedagogy. Those who don't know the math behind this will learn it, and those who do will become better teachers for having watched it.
Wow. Numberphile also did a video on it (as you mention), but they lost me about halfway through. The animation is so incredibly helpful in understanding how this works. Good job.
Just got here from the Numberphile video. This version is so much clearer and makes me finally understand what the Numberphile video was saying. Excellent content!
He may never get another job as a window cleaner. But on his very first job he gets to charge for the infinite number of hours it takes him to clean 0% of a window. So I’d call him a great success as a window cleaner!
@@troyturner6498, I can see they're about the same, but it's not obvious to me how to prove they're the same. I'm sure I could do it but only after much fiddling with a pen, paper and Pythagorean theorem. I have master's degree in engineering, LOL.
You are amazing man, i loved everything you've posted and i literally spent watching some of your videos over and over again 4 hours i will never want to take back! Great work man!
Please keep these up, both you and numberphile are on every morning to pick up where my EE left off. More importantly, the lil one sees this stuff and she (although a bit oblivious to the technicality) appreciates it and it stirs Her Imagination (STEM). Thank you for all your hard work. I had a great math teacher as a young adult that would challenge us like you are, it makes all the difference in the world.
Amazing explanation. I did not get this intuition from the Numberphile video and really glad you make this video and provided those animations. Plus the equilateral triangle makes another appearance...math is Illuminati confirmed ;)
+MindYourDecisions This video seems to have worked very well for a lot of people trying to understand the Kakeya needle problem. Have to do more videos like this :)
+Mathologer Call me crazy, but wouldn't you have to take into account the are you are wiping as you swipe outwards to prepare for the turn? So unless the area of the needle was 0 the area of your drag would effectively add up to the entire area of the shape? I hope I explained that well enough to understand.
WrenAkula If the width of the needle is 0 then there is no point to the problem because you could spin it around in any which way and the area that the needle would have traversed over would still be 0.
Thank you for making this video, it simplified it so much, especially when he talked a lot about not using "magic jumps" but still used them anyways without explaining in a simple way how he justified it.
I read about Kakeya needle problem and its solutions in book '1089 and all that' but could not understand the figure formed by chopping the triangles into pieces and how would we move needle through them. Thanks to your video, I could thoroughly understand yet another concept without the slightest of confusion remaining. Your videos have the best animations of other similar channels on TH-cam. Nothing is left unexplained by you and nothing is left half-done by the animator.
You have an amazing gift for explaining this stuff. And that is no where near a strong enough compliment for the accomplishment this video is in the art of creating clarity.
I think it would be even more interesting if the line had width, because then we could figure out exactly how far we'd want to move it out before each rotation.
I just finished Highschool like and I just can think "Wish my teacher was as good explaining this stuff as you are". I was the third best student in my class and I don't feel like I have learned everything I should have. I really appreciate your videos, they're very clear and concise, congratulations.
The field medalist mathematicians are hunting infinitesimally small kakeya fish in the depths of an infinitely large ocean of mathematical forests they've planted
2:05 Well; you can just take the triangle-shaped top off, and place it to the bottom; and you’ve created the square, without increasing or decreasing the area.
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.
This is very interesting. One way I envisioned a physical example was to picture an enormous, dense field of wheat. In this particular field, there doesn't need to be any space for rows, every stalk is side by side with little space in between and stretches on for miles and miles in all directions. You have a long, straight, razor-thin wire that lies on the ground and you have to rotate it 360 degrees without cutting, breaking, or otherwise damaging any of the wheat. The stalks are flexible enough to bend a tiny amount, but any appreciable lateral movement from the wire will break the stalk. Or perhaps a different example is you have a razor blade pressed into a large sheet of carpet and need to rotate the blade without bending it or creating any noticeable cuts or indentations within the carpet. Let's say the carpet is such that you can't notice a straight cut, but once you start curving or sliding the blade in any other direction it breaks up the pattern or texture. You can't lift the blade off the surface, but you can go as far in any direction as you need. There is a problem with both physical examples having finite space to work within. The mathematical model requires you to be able to travel infinitely outwards to minimize the area. The logical extremes for both of these examples is to make the wheat field and carpet planet sized, but then you're not dealing with Euclidean geometry anymore. Instead of dealing with a line in a flat plane, you're messing about on the surface of a sphere, which might change things. I'm not knowledgeable enough on the subject to go down that particular rabbit hole.
But how come it doesn't get to zero when you go far enough up? 3:52 However far up you are, you can go further up and make the area smaller, so it would get infinitesimally close to zero, aka zero....not saying you're wrong, just asking you to explain it to me
Bosk Boskson if you STOP somewhere to perform the "move", you get a positive area because it will be more than if you went a bit further. To get to "actual" zero area, you'd have to go an infinite distance away, so you can never actually stop anywhere before you "reach" that infinite distance...
These graphical and visual examples with animations and sequences give a clarifying view about many of these math concepts that you would never get with just words and numbers alone in a boring blackboard. I bet that a lot of people are more interested in math now that they can see things in a very different way.
+pjmasterzz When mathematicians speak about the beauty of mathematics they mean the beauty, simplicity and elegance of abstract arguments that they have been trained to appreciate. The challenge for me is to capture this inherent beauty and make it visible. It always makes my day when people tell me that it worked :)
Hold on, at 10:00 you ignore the little wedge made by the sweep between the parallel lines saying is negligible. Why is this any different from the beginning? Surely given a long enough parallel line, you could create any sweep area of any small size right? I don't see how the final shape is any better than the first parallel method...
The beginning of the video was laying the foundation down when we use that trick here, so we don't have to explain again why that sweep area doesn't matter (since we can make it as small as we want by going far enough). He's not going that far only so we can see it happen, otherwise it would just look like the squeegee going up one line and coming down the other (since in reality it would be happening somewhere way over there close to infinity).
I think he divided the problem to two stages. 1 - how to move between two parallel lines swiping through minimal area, 2. how to move between non-parallel lines (demonstrated on full 180 degrees turn) swiping minimal area. He just used the solution for #1 (going through inifinitely long parallel lines) as one building block to the solution of #2
@@ehsantamrabadi The first part of the video explained how you can make them as small as you want, because of that they basically contribute no extra area (since you can make it as close to zero as you want, just not exactly zero). But you wouldn't be able to see it happening, so it would look like the squeegee jumps from one line to the other, so for illustrative purposes he shows it happening close to the origin, where the sweep angles are much larger. So just pretend he moves the squeegee to infinity, turns by infinitesimal angle, and then moves it back. The line segments themselves don't contribute any area even though they're really long, since mathematical line is a 2D construct, it has no width (infinite length * 0 width = 0).
It reminded me of a fractal in the end, like the Koch curve. Because the Koch curve, and other fractals, have these weird features of having infinite perimeter while occupying a finite area that you can push to 0 if you want. Very nice video!!
Hi Mathloger! Found your videos after being a subscriber of Numberphile and you guys are great! Your explainations are very well done and you have a lovely enthusiasm for teaching these ideas too! Love you vids! :)
Wasn't the point of this move, that every angular movement is covering the area? When You move the line up, it covers no area, but later u change the angle,which adds area and You don't colour this red. Why ?why when You move the line at the tree u cover the area,and when You do the same thing on those lines out of the tree the same rules does not apply
My thoughts exactly. Earlier in the video π shirt says that every angular motion counts towards a positive area as we are moving from one parallel line to the next. Then near the end he says that it is negligible or insignificant... but its still positive! If we multiply all the 'insignificant' areas by the number of such angular motions and add it to the area of the tree figure, we'll get the same area as the original equilateral triangle.
@@rewrose2838 If you make the angle you turn when moving from one parallel to another small you can make the area of the swipe small. We saw that in the video. We need two times as many of these swiped as there are splits of the original triangle. So let's say we want to make the area less than 1/1000. Let's say we need to cut the triangle into N smaller ones to get the area of the main piece to 1/1,000,000 (so 1000 times smaller than we want the area to be). Now we need 2*N swipes to move between these parallel lines. So we just go as far as we need to make the area of each swipe 1/,1000,000*N. So the swipes add a total area of 2*N/1000,000*N = 2/1,000,000. So the total area is 3/1,000,000. This is way less than 1/1000. And this is true for any small number you give me. The total angle we turn is still 180 degrees
I owe a lot of my mathematical knowledge to numberphile, but I have to say, you explain things with much greater clarity. More people will be able to appreciate maths if it is presented with this clarity, and such stylish T-shirts.
What I dont get is I thought that thole whole point of this was to be able to rotate a need within a boundary, but at around 10:00 you move the needle outside of the boundary to rotate it. How does that not totally nullify the enitre purpose of this process. Its almost like saying "we can rotate this needle in any boundary, as long as we first remove it from the boundary and then rotate it and then put it back in the boundary". Its really bothering me because it doesnt logically make sense at all
You mean those lines going out from the tips of the triangles? Those are a part of the boundary. You can make them very very thin rectangles and still be able to make the total area as small as you want. Also we can't do this with any boundary, this construction is very specific.
I get the interesting stuff at the end, but isn't there a much simpler solution to the original problem? Consider the needle to be a railway car. Biuld a „drop shape“ track so that the car turns around when it passes once around the track, consisting of a 60° left curve, then a 300° right curve, then a 60° left curve, all with same radius r). The (infinitely thin) car sweeps out some area around the track. If you scale the track, the length of the area increases linearly with r, but the width decreases quadratically with 1/r^2. It becomes as small as you like for sufficiently large r.
I believe you mean to say the x or y axis as rotating around the z would keep everything on the x y plane. Edit: The reply below is incorrect. You cannot rotate anything "in the z axis," only _around_ an axis (or within a plane for 2D objects) Although, saying "rotating it to the z axis" is technically a correct way of stating the OP's intended meaning.
@@HISEROD no, the problem is presented in the x and y axis, so rotating it to the z axis doesn't have any volume in the x and y axis. Rotating it 90 degrees in the z axis makes the line into a singular point, being an arbitrary small point means it doesn't have volume when moved around the x and y axis. Simply moving it to the finish line and rotating back 90 degrees in the z axis completes the problem and the shape was 0 volume. Get it?
I was wondering how the squeegee analogy would aid the demonstration and I think I’ve just learned something about what good teaching looks like today!
Amazing video as always.. one question.. Don't you think the solution to the Kakeya needle problem directly contradicts the Laws of Integration and Limits? Integration says if dx is infinitesimally small and we add all those chunks within an upper and lower limit then we can get the area bounded by those limits. On the other hand, the solution to the kakeya problem states that as the "small triangles" are infinitesimally small, the summation of the needle shifts will be zero, ie, infinity*0=0 Your thoughts?
+Inder Chopra I think it has to do with the fact that Kakeya seeks to minimize area of traversal between two bounds, where integration seeks to calculate a very finite, static, and extant area; but with a shape(a rectangle) that cannot remain loyal to a curve(at least one of the bounds),until its horizontal dimension approaches 0. If you make a cow pen out of 14ft of fence, in a 3x4 arrangement for example, a cow will fit in it undeniably, and a cow cannot fit in a space of no area. If the cow pen were circular, it would still have an area that could host a cow, but it would be more challenging to define that area in terms of smaller rectangular cow pens.
1 year late but... In both integration and the kakeya needle problem you shrink areas to 0 for each iteration of subdivision. In integration the area you shrink is the error area that makes your approximation for the area under a curve not exactly the actual area under the curve. So both kakeya needle problem and integration works by the same underlying principle. I recommend 3blue1brown "Essence of calculus" series on youtube, going through and refreshing the fundamentals helped me.
Inder Chopra he never said anything about zero area, he only said that you can make it arbitrary small. This refers to a thing called an episilon-delta approach. That is also formally how integration and differentiation is defined. So no they don't disprove each other, they fundamentally come from the same place. In integration you can't make the area under the curve arbitrary small, but you can bound it and that is what gives you the area
I got it from the Numberphile video before watching this, but I must say that even if your video don't explain it in a minute fashion, the general explanation is awfully good, I'm frankly a bit jealous on how good you are at simplifying.
The two first areas are the same because, on the second one, you can fit the "triangle" on the top on the "hole" on the bottom, and then you would get the square again!!!
Your explanation of the parallel lines trick made a lot more sense. In the NP video they somehow vanished after the very first diagram and I had no reference for how the 2•theta sweep came into it. Thanks!
With this logic of infinitesimal angular movements of the needle then why can't we move the needle from it's starting position some some small angle and have it travel a great distance only to torque it again some small angle until we create a giant semi circle? once 180 degrees but in a different position we make the original transform we did at the top of the video to move it back to it's original location.
I also thought of this solution, but I cannot calculate the area swiped this way... Intuitively it looks small - but gathered along a huge semi-circle, it may accumulate to some big area. calculating this area may incorporate some integration... I don't know.
My understanding (and if I understand you correctly) is that you wouldn't have the same amount of overlap that you have in this solution. With this solution, the amount of overlap is maximized, thus limiting the necessary area.
The process you’re describing doesn’t have any overlapping area that you’d turn the needle in and overlapping area is the only way to reduce the total area swept out in this problem. What you would find if you gathered together all the tiny sections traced out in the massive semi circle, would be a circle with a diameter that is the length of the needle at best
Thanks for making me think about this question, it has a delightful answer! The idea is you start by computing the region of the full circle sweep, then you can chop it in half and glue two half circle segments on each end. Such a full circle sweep is the region between the outer circle swept by the edges and the inner circle swept by the center of the squegee, so all you have to do is figure out a formula for the area two circles in terms of one of the radii, and then take the limit of the difference as the radii go to infinity. I was worried this would turn out to be quite scary, but... well I don't want to spoil it. Assume we have a line segment (our squegee) AB of length 2s, with midpoint M at distance s from both A and B. Pick a rotation origin O such that MO is of length r (the inner radius), and is at right-angles to AB. Call the distance between O and A (and B) R, as it is our outer radius. Thus, the triangles OMA and OMB are reflections around OM, right-angles at M, and have sides of length r, s and R. Now our outer circle has area C = pi R^2, and our inner circle has area c = pi r^2, and thus our swept area S = C-c = pi R^2 - pi r^2. But R^2 = r^2 + s^2 by pythagoras of OMA and OMB, so S = pi (r^2 + s^2) - pi r^2 = pi r^2 + pi s^2 - pi r^2 = pi s^2. That is, the area swept in a full circle does not depend on the radius at all, but only the line segment length! Clearly, the area of the half-circle case must be H = pi s^2 / 2 + E, where E is the (obviously positive) area of the end caps, so you can't get to 0 area this way. I tried to figure out the limit of E as r -> inf., but my brain melted around about taking the limit t -> 0 for t = s/r of s^2 ((t^2 + 1) arctan t - t) / t^2. I think it's just L'Hospital's at that point, but my brain can't do differentiation at 00:40 😵
Well, effectively, it proves that in the infinity, all parallel lines, in the limit, meet each other! Only technical or academic Assumption is that swiggy has ZERO width! Great way to explain a complicated maths concept! Love it.
Oftentimes when you know some complex subject really, really well, and you are very passionate about it, even trying to explain the "basics" of it to a layman can veer you into subjects that are too complex (and boring) for the average person. When something is very simple and clear to you, it can be hard to realize that it's not simple and clear to somebody who knows very little about the subject. Sometimes a friend of mine tries to explain some basics of electronics to me, but he often just starts meandering and rambling in a manner that makes me doze off, and even if I understood the beginning of it, I usually just lose track and don't understand anything further. It can be hard to explain a subject in a manner that's understandable and interesting to laypeople. And it doesn't even need to be something highly technical or theoretical either. I have experienced this with _board games_ of all things. I have been there when some passionate guy explains for 10 minutes the rules of a complex board game, the 9 latter minutes of it being completely useless rambling because it tells the listeners absolutely nothing because without the experience of having actually played the game several times they just can't understand the context. Those were 10 minutes that could have been used to actually play the game and learn in practice, rather than dozing off to a completely useless rambling that teaches absolutely nothing.
I assume it's because when you extend out far enough from the "fish", you only need the most minute of change in the angle to move to the other triangle, making it negligible. Like if the squeegee goes millions of kilometres away and rotates a nanometre, it's enough change in trajectory to have the squeegee change its position into the other triangle piece.
***** No matter how far out it goes you still have the cover the area to move over to change the angle, eventually the 2 lines meet and that's where there will be more area squeegeed that wasn't counted
+edstars101 No, only the unit line segment is swiped across that very small angle and therefore results in a negligible area swiped. The whole millions of kilometers line isn't swiped across that angle (that wouldn't be a negligible area).
The area swept by moving between parallels converges to zero by increasing number of triangle tips, but also then the number of triangle tips diverges to infinity. So in the calculation of the total area, the area per triangle and the number of triangles will chase each other to zero*infinity. My intuition says the swept area will always be equal to the area of the triangle, which is divided up an infinite number of times, but still, all the triangles are added back together to calculate the total area. The bottom edge of the tree structure would be the same length. The overlapping around the bottom of the tree structure does reduce the area somehow, but not by much.
Who finds the error 10:05 ? Weird but you're swiping outside of those triangles without marking the area of the swiping movement you're doing two times before re-entering the geometry. Don't you have to highlight those areas too?
I agree with this but what they are saying is that the line goes out very far until it converges with the other line beside it. My problem with this is that you have to travel a very long distance you shouldn't discount the lines path, I feel that is a cheat, the area is the length travelled plus the width of the needle.
@Jordan Cottle If it doesn’t have a width, then it is not a needle, and would not have a swiping motion. Also a zero width would have zero length. zero times anything is zero.
@@jcwalker3 It has a length, but not a width. It is a line, with only one dimension. That is why sliding it on the line it is in-line with without turning it doesn't generate a 'swipe' of area. Dragging it along a line perpendicular to it generates a 'swipe' that forms a rectangle. That rectangle has an area, but the individual lines that make it up (including the line we are talking about) do not. The same way a rectangle doesn't have a volume, a line cannot have an area. Length -> Area -> Volume
@Jordan Cottle , yes a line operates in a single dimension, with zero width, but you can’t drag a one dimension object in two dimensional space without giving it width. the problem originally used a needle.
Simple: the chevron you created has the same area as a square, because the concave angle on the chevron added to the angle at teh top of the triangle add up to 360, or 2 pi, and the other 2 angles add up to 90 degrees (pi/2) when cut off the top and pasted onto the bottom.
Now if you divide the total distance covered by the total area covered and vice versa, what would the smallest quotient for both of these scenarios be? So the smallest/most effective area possible with the least distance covered for the needle/squegee to travel?
Interesting to think about how we do "three point turns" in a car. With engines we are probably intuitively minimizing for time instead of distance, but the shape swept out approximates a Steiner curve.
the two areas are the same because the triangle subtracted from the bottom of the area is the same size as the one added on the top, so the total area does not change.
This guy said: Do you watch numberphile. Realised i am not subscribed to the channel i stopped the video; Searched for numberphile; Subscribed; Resumed the video.
Thank you sooooooooooo much. I read many articles or video about that problem and I was always a bit disappointed because I was always lost. First time I understand !!!
+Melinda Green I seem to recall owing you a reply to another comment you left somewhere else but I forgot what video it was. Do you remember ? (It's getting quite hard to keep up with all the comments.)
Mathologer I'm not sure but I am excited by your promise to explain why the main Mandelbrot shape repeats. Maybe that was it? And FTR, I am very impressed by how well you keep up with all the comments. I love your work!
Amazing content!! This is one channel that never disappoints!! Love his cube enthusiasm too!!❤️ As for the arrow being the same as the square, the triangle jutting out at the top fits at the bottom, so that's that, and as for the whole problem itself, I wonder if I could move the squeegee along the circumference of a circle big enough that the squeegee always fits while turning infinitesimally until it makes one complete turn thus turning 180° when it's covered approximately half the circumference..
OK, I have some questions/remarks. 1 - Several steps of this solution works with the concept of infinity, as in the perpendicular move one needle does. If I try to numerically implement this and for every time infinity is mentioned i replaced for x. Every time I increase x the closer my solution will be to 0? 2 - As a corollary of the video can you affirm that you can move any needle from any position in a 2-d space, to any position in a 2-d space? I wonder if that holds for any arbitrary n-d space. For my next question I'll assume 2 is true. (I'm not actually doing the math because I don't know the tools to work with abstract math and I have a job and stuff) 3 - I imagine that with the right math you can make a perfect analogy of the area this needle is covering as an techinique to optimize one existing polinomial of degree 2? Given that any polinomial of degree n can be mapped to any other polinomial of degree n by a simple coordinate transformation, does that mean you can optimize any polinomial of degree 2 to zero? If 2 and 3 are correct, is this a way to prove every polinomial has at least one maximum? If 1 holds true, does that mean this can be implemented an optimization method for any polinomial? That is absolutely mindfucking!!!!!
This is truly wonderful. It is such a great problem to discuss following a study of figures of Constant Width. It is so hard to explain to school students. You have made it so clear, and at an appropriate level both for me and them. Thankyou. That t-shirt looks a great idea too!
+Robyn Gregory That's great :) If you like shapes of constant width maybe you'll be interested in this article by me (make sure to check out the linked in movie clips). We'll also do a video about all this in the near future.
I love the shirt and the fish. I watched the Numberphile video a day or two ago and thought the concept was brilliant. But your explanation gave me a lot better grasp of it. Also, I too, was thinking that the chevron you swept early in the video was larger in area than the square. Once you pointed out that they are actually equal I realised what was going on. Great video, as always!
+Mathologer Nice. It looks like the boundaries define a perfect hexagon. Is there some way that we mortals are able to purchase some of your shirt designs? An online storefront, for instance?
+Steve French I just uploaded the "kakeya fish" t-shirt design to spreadshirt 1035536.spreadshirt.com Looks pretty good on the preview page but of course we'll only know how well it really looks when a hardcopy arrives. So, order at your own risk now or wait until I give the all-clear once I've seen the actual t-shirt. The same shop also has a few other math t-shirt designs by me :)
It seems like if you travel across the middle for each jump you'd really be able to shrink the angles down. I'm not sure how to explain it without a picture, but if you're on the xy plane, starting with the "needle" on (0,0) to (0,1) you could travel infinitely high (0,inf), cut a micro-sliver of area, travel down to like (0.01, -inf) and angle back across the origin. Seems like it'd be a tiny ring of slivers out at the extremes.
Mathologer the one that come up with those kind of solution to problems (or those kind of problems where the solution isn't something "real"). I get what you're implying, but I'm sure you get what I'm implying too.
This video is much easier to grasp than the one in Numberphile. However, I wish there was some kind of explanation of what is said at 8:40, that if you make the number of triangles high enough they can be moved into a shape with an area as small as desired. That is key, and left as unproved. My feeling is that much of the additional complexity in Numberphile's video is due to the fact that they do prove that statement.
thanks for this video - The Numberphile video was not at all.clear enough (from the start), never mind it being technical. This one helps to fill the gaps.
I just want to "second" what Joe Dinoto said below. I understand that it must be a lot of work to make these "simple explanations" but the result is well worth it. The only possible downside is that they might make us (the viewers) believe that we are a little bit smarter than we actually are ;) Thanks for all Your fantastic videos, Best regards.
+Mats Rdr Actually, in terms of presentations the advice I was given when I started out as an academic was to keep the first third of a talk understandable to everybody, the second third only to the experts and the last third to pretty much nobody. Sadly this turns out to be very good advice---if you actually succeed in presenting a difficult concept so that everybody can understand it a lot of people will automatically assume that what you are talking about is trivial. Anyway, as far as I am concerned, coming up with good ways of presenting tricky material is one of the things I really enjoy doing and having people like you confirm that I got an explanation right really makes my day :)
Great video, as always, and nice shirts. One questions that the tree will end up like fractals with infinite surface area but no volume?? Further, by using the first method 5 times will that not also end up with the needle facing the other way round??
No, the trees don't have infinite area, in fact the area gradually goes to 0 in this problem. That's the goal of the problem: to find the least area . Then about the volume, yes it has 0 volume cuz everything is in a plane and 2d. Lastly , I didn't understand what u meant by 5 times the original method. Could u elaborate on that?
great video. after watching Numberphile video I was looking for additional or clearer explanation and you really made it! congratulations! you've got a new follower!
It would be nice if you showed an animation of the number of branches increasing and the area of the whole thing decreasing. Just to help visualize the process. Great video! so easy to understand!
So: Circle with diameter 1 squeegee-unit has an area of 0.7853 squeegee-units² Equalateral triangle with height 1squeegee-unit has an area of 0.5773 squeegee-units² Deltoid curve (as constructed) has an area of 0.3853 squeegee-units² ...what is the area of the needle set "fish" as constructed at 12:00 (not the proposed near-zero ideal)?
+SecularMentat Mission accomplished :) Maybe have a look at the survey that I linked in from the description part of the video to catch a glimpse of what the big shots see in this problem :)
It looks to me like it tends to 0/0, or all the possible slopes of a point as you shrink the shape. No idea if that makes any sense at all but I like the way you explain things, thanks!
For the "arrow", the height relative to the base remains the same as the rectangle, so the area remains the same. The base changes shape, but its still parallel to its top, similar to how a prism can have any base and the formula remains area of base, times height. You can imagine dividing it I to discrete chunks and pushing them up, which doesn't change the area. Then imagine the chunks getting smaller and smaller, like the proof of the area formula for a circle. Or, yes, you can just cut out the top and fit it on the bottom, since you know the top and bottom are parallel. But, to me, the above is actually more intuitive, even if takes longer to explain. And it connects directly with the "squeegee", which is the infinitely thin chunk from before.
The object that is characterised by this zero area move can only be held in transfinite 2D space (if you use infinitely many line segments irradiating from on point, you cover all infinite 2D space). Effectively, you need to use a 3rd dimension to move a line segment in a 2D plane without covering any area. You can also claim that as you approach such object, the area that you do end up using is infinite inasmuch as you are covering all space with your infinite line segments. Pretty interesting stuff.
Length of needle = 1 Needle alinged on the circle of radius = 10^3(very large as compared to needle's length) Needle moves along the circumference of circle(half way or along semi circle) Now needle has turned 180° with negligible area covered and is parallel to initial position (or at diagonally opposite side) Now apply that parallel trick as show in video which again cover negligible area I think this is simpler then the one in video or may be somewhere I m wrong , correct me if so
Explanation is very clear and easy to understand 😃 thanks. Same concept in Numberphile is explained in a very difficult way and I didn’t understand to be honest!
Very interesting. I completely understand the technique (and of course I’ll take Terry Tao’s word that this works), but it immediately seems counter-intuitive that this should work. I’ll have to check out the numberphile video next.
I know it can be a silly question but, why the areas do not add up? About 6'36'': The triangle's area is less than the circle, but there are some parts of this triangle wherein the 'needle' has been twice. If you cover the area the triangle's area is bigger. And another one. That can be a little arrogant but... At 10'20'' you actually justified why the tricky moves can be neglected. My second question is, why doesn't this justification do not also apply to any other moves we do (also the ones forming the orange area) and so we conclude that our area is indeed zero. That is more of a silly question but... Hope you can help me with it. Also congratulations. You do a really good job. Cannot put into words how much of a great job is this. Thanks man!
I've watched a couple of your videos now. They are amazing! Definitely equally as good as Brady Haran's channels (numberphile etc.). You have a new subscriber.
Everyone keeps talking about Banach-Tarski, but this video isn't related to it, since it doesn't use the axiom of choice. To me, what it seems most related to is the Cantor set. The Cantor set is equinumerous to the real numbers, but has a length of 0; the Kakeya set has a line segment of length 1 for every real number between 0 and π/2 (which is again equinumerous to the real numbers), but has an area of 0. They also both have the construction of dividing in half and reducing the area of each; if you imagine each of the intervals in each state of the Cantor set sliding partway open like a pair of curtains, it feels like the analogy is complete.
If i got it right , you suggest that while the segments 'N' tends to infinity, the areas 'A' of the triangles tends to zero and the product N*A tends to zero as well.....
As you up the number of little triangles at the start of the construnction, their overlap can be arranged to approach to zero. As well, we can make the total area of the little turning wedges go to zero by pushing things further and further out along those pairs of parallel lines.
maybe there is another elegant solution. imagine the shape of a huge rain drop. just the outside line. if you follow this shape starting from the tail along the line you would go around and come back on the starting point with an angle of 180 degree. the bigger the drop design the smaller will be the surface. isn't it more simple ? thanks for your video .
A question: If we count the area that is swept more than once by the needle, then do we get the same area as the circle that was initially swept out by the needle? If so, then the Kakeya problem basically utilises the fact that we do not count the overlapping areas. It’s a really clever trick, in that sense. The explanation was spot on. The Numberphile video was quite confusing mainly because the guy explaining the problem wasn’t consistent (with his formulae etc.) But this explanation really captures the essence. Thank you!
The clarity of your videos are OUTSTANDING. The diagrams are simple and elegant and you take great care to explain and interpret subtle mathematical points without glossing them over. Thank you!
+Joe DiNoto Comments like these make my day :)
+Joe DiNoto Agreed. Got my subscription.
+Brendan Redler may I also get your subscription?
I know. I was talking to numberphile in that comment lol
Simplicity is elegance.
You just explained the worst theoretical window cleaner.
+E N To make it the worst theoretical window cleaner is the point :)
Just fool that cleaner with thick squeegee, he will end up cleaning infinite area of window just to skip the first centimeter of window/ first degree in the rotating problem😂😂
The window cleaner will charge for the infinite number of hours it will take to clean 0% of a window!
😄💖😂
@@jimburrill8149 At the risk of being Mathematically guillotined, I’ll announce that the window cleaner owes me 1/12€. 🙃
This is an example of truly excellent exposition. It's clear, concise, and close enough to precise to not make my teeth hurt. Very, very, very impressive. Congratulations on making a beautiful piece of pedagogy. Those who don't know the math behind this will learn it, and those who do will become better teachers for having watched it.
+Harrison Smith Glad you like what we are doing and thank you very much for saying so :)
Wow. Numberphile also did a video on it (as you mention), but they lost me about halfway through. The animation is so incredibly helpful in understanding how this works. Good job.
Agreed, I finally understand it now.
+SlimThrull Great, mission accomplished :)
Mathologer I wish I had math teachers like you.
Just got here from the Numberphile video. This version is so much clearer and makes me finally understand what the Numberphile video was saying. Excellent content!
You are an amazing teacher. Such an calming effect while explaining the concept.
Failed as a window washer.
He may never get another job as a window cleaner. But on his very first job he gets to charge for the infinite number of hours it takes him to clean 0% of a window. So I’d call him a great success as a window cleaner!
@@jimburrill8149 Yes. He doesn’t need another job, anyways; since he’ll never get to finish his 1st job, either. 😅
Math profs HATE him:
Mathologer reveales this one weird trick to make the are needed by an needle to turn approach zero.
Find out how.
Nice
+TheLenmae Hahaha! Nice one :P
NEVER use these 5 numbers and u will always get a grade A+
Neil, you have 42 points so I can't decently upvote you, even if you deserve it!
TheLenmae He ignores the angles between parallel triangles, and that is a no-no for math, as theh draw a circle in totall themselves
The two areas are the same @2:10 because if you take the triangle at the top chunk, it fits exactly into the missing area below.
+SlimThrull Exactly :)
Alternatively, you can use Cavalieri's principle: Imagine it as a deck of cards on its side with the middle pushed out.
overlapping makes the difference
the two areas are the same from high school geometry. possibly middle school. if a viewer doesnt know this...are they watching this video?
@@troyturner6498, I can see they're about the same, but it's not obvious to me how to prove they're the same. I'm sure I could do it but only after much fiddling with a pen, paper and Pythagorean theorem. I have master's degree in engineering, LOL.
Was recommended to come here from a comment in the numberphile video. SO much clearer, thanks. Subbed.
You are amazing man, i loved everything you've posted and i literally spent watching some of your videos over and over again 4 hours i will never want to take back! Great work man!
+Fresh Good Glad you got so much out of our videos and thank you very much for saying so:)
Please keep these up, both you and numberphile are on every morning to pick up where my EE left off. More importantly, the lil one sees this stuff and she (although a bit oblivious to the technicality) appreciates it and it stirs Her Imagination (STEM). Thank you for all your hard work.
I had a great math teacher as a young adult that would challenge us like you are, it makes all the difference in the world.
Amazing explanation. I did not get this intuition from the Numberphile video and really glad you make this video and provided those animations. Plus the equilateral triangle makes another appearance...math is Illuminati confirmed ;)
+MindYourDecisions This video seems to have worked very well for a lot of people trying to understand the Kakeya needle problem. Have to do more videos like this :)
+Mathologer Call me crazy, but wouldn't you have to take into account the are you are wiping as you swipe outwards to prepare for the turn? So unless the area of the needle was 0 the area of your drag would effectively add up to the entire area of the shape?
I hope I explained that well enough to understand.
+MindYourDecisions Didn't expect to see you here!
+ShinyWaxBullet The needle has no width, so sliding it doesn't add any area.
WrenAkula
If the width of the needle is 0 then there is no point to the problem because you could spin it around in any which way and the area that the needle would have traversed over would still be 0.
Thank you for making this video, it simplified it so much, especially when he talked a lot about not using "magic jumps" but still used them anyways without explaining in a simple way how he justified it.
Glad this worked for you and thank you for saying so :)
Sans TheSkeleton I
Fantastic video; the numberphile video did in fact lose me and this was great in describing the solution. Great work!
+HJ Krypton Same here. Have to watch the Numberphile video again. Maybe it makes more sense now that I know what they were trying to say.
+HJ Krypton Mission accomplished :)
I read about Kakeya needle problem and its solutions in book '1089 and all that' but could not understand the figure formed by chopping the triangles into pieces and how would we move needle through them. Thanks to your video, I could thoroughly understand yet another concept without the slightest of confusion remaining. Your videos have the best animations of other similar channels on TH-cam. Nothing is left unexplained by you and nothing is left half-done by the animator.
You have an amazing gift for explaining this stuff. And that is no where near a strong enough compliment for the accomplishment this video is in the art of creating clarity.
+Trabber Shir Quite a few comments that make my day today :)
I think it would be even more interesting if the line had width, because then we could figure out exactly how far we'd want to move it out before each rotation.
We love your t-shirts! About 2/3rds of the way through the video my wife commented that the Kakeya tree would make a perfect Mathologer t-shirt ...
+Todd Cottle Well, I already designed and ordered the t-shirt. It will definitely feature in one of our future videos :)
Hit the jackpot with knowledge spreading knowledge for normal people. Good job you guys.
much better explanation
I just finished Highschool like and I just can think "Wish my teacher was as good explaining this stuff as you are". I was the third best student in my class and I don't feel like I have learned everything I should have. I really appreciate your videos, they're very clear and concise, congratulations.
The field medalist mathematicians are hunting infinitesimally small kakeya fish in the depths of an infinitely large ocean of mathematical forests they've planted
+marksmod Nice picture you conjure up there :)
These forests are acyclic, naturally :P
2:05 Well; you can just take the triangle-shaped top off, and place it to the bottom; and you’ve created the square, without increasing or decreasing the area.
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.
This is very interesting. One way I envisioned a physical example was to picture an enormous, dense field of wheat. In this particular field, there doesn't need to be any space for rows, every stalk is side by side with little space in between and stretches on for miles and miles in all directions. You have a long, straight, razor-thin wire that lies on the ground and you have to rotate it 360 degrees without cutting, breaking, or otherwise damaging any of the wheat. The stalks are flexible enough to bend a tiny amount, but any appreciable lateral movement from the wire will break the stalk.
Or perhaps a different example is you have a razor blade pressed into a large sheet of carpet and need to rotate the blade without bending it or creating any noticeable cuts or indentations within the carpet. Let's say the carpet is such that you can't notice a straight cut, but once you start curving or sliding the blade in any other direction it breaks up the pattern or texture. You can't lift the blade off the surface, but you can go as far in any direction as you need.
There is a problem with both physical examples having finite space to work within. The mathematical model requires you to be able to travel infinitely outwards to minimize the area. The logical extremes for both of these examples is to make the wheat field and carpet planet sized, but then you're not dealing with Euclidean geometry anymore. Instead of dealing with a line in a flat plane, you're messing about on the surface of a sphere, which might change things. I'm not knowledgeable enough on the subject to go down that particular rabbit hole.
if you chop off the top point in the arrow area and fit it into the bottom that makes a square.
That is correct :)
But how come it doesn't get to zero when you go far enough up? 3:52 However far up you are, you can go further up and make the area smaller, so it would get infinitesimally close to zero, aka zero....not saying you're wrong, just asking you to explain it to me
Bosk Boskson if you STOP somewhere to perform the "move", you get a positive area because it will be more than if you went a bit further. To get to "actual" zero area, you'd have to go an infinite distance away, so you can never actually stop anywhere before you "reach" that infinite distance...
beat me to it...
by 2 years XD
AyeJayCee You could've made your comment less useless if you mentioned it's about the dialogue at 1:55 - 2:10 ... :-\
These graphical and visual examples with animations and sequences give a clarifying view about many of these math concepts that you would never get with just words and numbers alone in a boring blackboard. I bet that a lot of people are more interested in math now that they can see things in a very different way.
Thank you for the video! All of you friends are super awesome!
Great video, as usual. I'm a great fan of the simplicity of how you explain things. Thank you for that :)
+pjmasterzz When mathematicians speak about the beauty of mathematics they mean the beauty, simplicity and elegance of abstract arguments that they have been trained to appreciate. The challenge for me is to capture this inherent beauty and make it visible. It always makes my day when people tell me that it worked :)
Hold on, at 10:00 you ignore the little wedge made by the sweep between the parallel lines saying is negligible. Why is this any different from the beginning? Surely given a long enough parallel line, you could create any sweep area of any small size right? I don't see how the final shape is any better than the first parallel method...
The requirement of the latter is that you rotate it 180 degrees.
The beginning of the video was laying the foundation down when we use
that trick here, so we don't have to explain again why that sweep area
doesn't matter (since we can make it as small as we want by going far enough).
He's not going that far only so we can see it happen, otherwise it would just
look like the squeegee going up one line and coming down the other
(since in reality it would be happening somewhere way over there close
to infinity).
I think he divided the problem to two stages. 1 - how to move between two parallel lines swiping through minimal area, 2. how to move between non-parallel lines (demonstrated on full 180 degrees turn) swiping minimal area. He just used the solution for #1 (going through inifinitely long parallel lines) as one building block to the solution of #2
I just noticed that and have question about that. I think they just ignored those tiny rotation areas.
@@ehsantamrabadi The first part of the video explained how you can make them as small as you want, because of that they basically contribute no extra area (since you can make it as close to zero as you want, just not exactly zero). But you wouldn't be able to see it happening, so it would look like the squeegee jumps from one line to the other, so for illustrative purposes he shows it happening close to the origin, where the sweep angles are much larger. So just pretend he moves the squeegee to infinity, turns by infinitesimal angle, and then moves it back. The line segments themselves don't contribute any area even though they're really long, since mathematical line is a 2D construct, it has no width (infinite length * 0 width = 0).
It reminded me of a fractal in the end, like the Koch curve. Because the Koch curve, and other fractals, have these weird features of having infinite perimeter while occupying a finite area that you can push to 0 if you want. Very nice video!!
I just realized that many parts of the "fishmouth" can be left out, some area is never reached by the linr
Hi Mathloger!
Found your videos after being a subscriber of Numberphile and you guys are great! Your explainations are very well done and you have a lovely enthusiasm for teaching these ideas too! Love you vids! :)
Wasn't the point of this move, that every angular movement is covering the area? When You move the line up, it covers no area, but later u change the angle,which adds area and You don't colour this red. Why ?why when You move the line at the tree u cover the area,and when You do the same thing on those lines out of the tree the same rules does not apply
My thoughts exactly.
Earlier in the video π shirt says that every angular motion counts towards a positive area as we are moving from one parallel line to the next.
Then near the end he says that it is negligible or insignificant... but its still positive! If we multiply all the 'insignificant' areas by the number of such angular motions and add it to the area of the tree figure, we'll get the same area as the original equilateral triangle.
@@rewrose2838 If you make the angle you turn when moving from one parallel to another small you can make the area of the swipe small. We saw that in the video. We need two times as many of these swiped as there are splits of the original triangle.
So let's say we want to make the area less than 1/1000. Let's say we need to cut the triangle into N smaller ones to get the area of the main piece to 1/1,000,000 (so 1000 times smaller than we want the area to be). Now we need 2*N swipes to move between these parallel lines. So we just go as far as we need to make the area of each swipe 1/,1000,000*N. So the swipes add a total area of 2*N/1000,000*N = 2/1,000,000. So the total area is 3/1,000,000. This is way less than 1/1000.
And this is true for any small number you give me. The total angle we turn is still 180 degrees
I owe a lot of my mathematical knowledge to numberphile, but I have to say, you explain things with much greater clarity. More people will be able to appreciate maths if it is presented with this clarity, and such stylish T-shirts.
+Jonathan Fowler Great, thank you very much for the compliment :)
What I dont get is I thought that thole whole point of this was to be able to rotate a need within a boundary, but at around 10:00 you move the needle outside of the boundary to rotate it. How does that not totally nullify the enitre purpose of this process. Its almost like saying "we can rotate this needle in any boundary, as long as we first remove it from the boundary and then rotate it and then put it back in the boundary". Its really bothering me because it doesnt logically make sense at all
The point is to minimize the area. To create the solution, you create the boundary which defines the area.
you go so far out you have to turn essentially zero degrees to make the change
You mean those lines going out from the tips of the triangles? Those are a part of the boundary. You can make them very very thin rectangles and still be able to make the total area as small as you want.
Also we can't do this with any boundary, this construction is very specific.
I get the interesting stuff at the end, but isn't there a much simpler solution to the original problem?
Consider the needle to be a railway car.
Biuld a „drop shape“ track so that the car turns around when it passes once around the track, consisting of a 60° left curve, then a 300° right curve, then a 60° left curve, all with same radius r).
The (infinitely thin) car sweeps out some area around the track.
If you scale the track, the length of the area increases linearly with r, but the width decreases quadratically with 1/r^2.
It becomes as small as you like for sufficiently large r.
Just rotate the whole thing 90 degrees along the z axis and the area is zero.
This guy solved a problem with this little trick, mathematicians hate him!
I believe you mean to say the x or y axis as rotating around the z would keep everything on the x y plane.
Edit: The reply below is incorrect. You cannot rotate anything "in the z axis," only _around_ an axis (or within a plane for 2D objects) Although, saying "rotating it to the z axis" is technically a correct way of stating the OP's intended meaning.
@@HISEROD no, the problem is presented in the x and y axis, so rotating it to the z axis doesn't have any volume in the x and y axis.
Rotating it 90 degrees in the z axis makes the line into a singular point, being an arbitrary small point means it doesn't have volume when moved around the x and y axis. Simply moving it to the finish line and rotating back 90 degrees in the z axis completes the problem and the shape was 0 volume. Get it?
wAiT
Area, not volume
I was wondering how the squeegee analogy would aid the demonstration and I think I’ve just learned something about what good teaching looks like today!
Amazing video as always..
one question.. Don't you think the solution to the Kakeya needle problem directly contradicts the Laws of Integration and Limits? Integration says if dx is infinitesimally small and we add all those chunks within an upper and lower limit then we can get the area bounded by those limits. On the other hand, the solution to the kakeya problem states that as the "small triangles" are infinitesimally small, the summation of the needle shifts will be zero, ie, infinity*0=0
Your thoughts?
+Inder Chopra I think it has to do with the fact that Kakeya seeks to minimize area of traversal between two bounds, where integration seeks to calculate a very finite, static, and extant area; but with a shape(a rectangle) that cannot remain loyal to a curve(at least one of the bounds),until its horizontal dimension approaches 0.
If you make a cow pen out of 14ft of fence, in a 3x4 arrangement for example, a cow will fit in it undeniably, and a cow cannot fit in a space of no area. If the cow pen were circular, it would still have an area that could host a cow, but it would be more challenging to define that area in terms of smaller rectangular cow pens.
1 year late but... In both integration and the kakeya needle problem you shrink areas to 0 for each iteration of subdivision. In integration the area you shrink is the error area that makes your approximation for the area under a curve not exactly the actual area under the curve. So both kakeya needle problem and integration works by the same underlying principle. I recommend 3blue1brown "Essence of calculus" series on youtube, going through and refreshing the fundamentals helped me.
Inder Chopra he never said anything about zero area, he only said that you can make it arbitrary small. This refers to a thing called an episilon-delta approach. That is also formally how integration and differentiation is defined. So no they don't disprove each other, they fundamentally come from the same place. In integration you can't make the area under the curve arbitrary small, but you can bound it and that is what gives you the area
Numberphile's explanation was clearer for me.
Like always, Mathologer is amazing :-)
I love this guy ❤️.....I especially like his laugh/giggle. It makes me happy.
Can anyone make a gif of 1:34?
Italego6 definitely
Italego6 lol
I got it from the Numberphile video before watching this, but I must say that even if your video don't explain it in a minute fashion, the general explanation is awfully good, I'm frankly a bit jealous on how good you are at simplifying.
+The Major Cool, thank you very much for the compliment :)
The two first areas are the same because, on the second one, you can fit the "triangle" on the top on the "hole" on the bottom, and then you would get the square again!!!
+Marcos Leitão Chamis Yep :)
I dont mean to be mean, but well done genius.
+25greengoblin He's literally following directions, so...
25greengoblin you just contradicted your other statement.
Your explanation of the parallel lines trick made a lot more sense. In the NP video they somehow vanished after the very first diagram and I had no reference for how the 2•theta sweep came into it. Thanks!
+Anolaana Seranaar Great, glad my explanation worked for you and thank you very much for saying so :)
With this logic of infinitesimal angular movements of the needle then why can't we move the needle from it's starting position some some small angle and have it travel a great distance only to torque it again some small angle until we create a giant semi circle? once 180 degrees but in a different position we make the original transform we did at the top of the video to move it back to it's original location.
I also thought of this solution, but I cannot calculate the area swiped this way... Intuitively it looks small - but gathered along a huge semi-circle, it may accumulate to some big area. calculating this area may incorporate some integration... I don't know.
My understanding (and if I understand you correctly) is that you wouldn't have the same amount of overlap that you have in this solution. With this solution, the amount of overlap is maximized, thus limiting the necessary area.
The process you’re describing doesn’t have any overlapping area that you’d turn the needle in and overlapping area is the only way to reduce the total area swept out in this problem. What you would find if you gathered together all the tiny sections traced out in the massive semi circle, would be a circle with a diameter that is the length of the needle at best
Thanks for making me think about this question, it has a delightful answer!
The idea is you start by computing the region of the full circle sweep, then you can chop it in half and glue two half circle segments on each end. Such a full circle sweep is the region between the outer circle swept by the edges and the inner circle swept by the center of the squegee, so all you have to do is figure out a formula for the area two circles in terms of one of the radii, and then take the limit of the difference as the radii go to infinity. I was worried this would turn out to be quite scary, but... well I don't want to spoil it.
Assume we have a line segment (our squegee) AB of length 2s, with midpoint M at distance s from both A and B. Pick a rotation origin O such that MO is of length r (the inner radius), and is at right-angles to AB. Call the distance between O and A (and B) R, as it is our outer radius. Thus, the triangles OMA and OMB are reflections around OM, right-angles at M, and have sides of length r, s and R.
Now our outer circle has area C = pi R^2, and our inner circle has area c = pi r^2, and thus our swept area S = C-c = pi R^2 - pi r^2.
But R^2 = r^2 + s^2 by pythagoras of OMA and OMB, so S = pi (r^2 + s^2) - pi r^2 = pi r^2 + pi s^2 - pi r^2 = pi s^2.
That is, the area swept in a full circle does not depend on the radius at all, but only the line segment length!
Clearly, the area of the half-circle case must be H = pi s^2 / 2 + E, where E is the (obviously positive) area of the end caps, so you can't get to 0 area this way.
I tried to figure out the limit of E as r -> inf., but my brain melted around about taking the limit t -> 0 for t = s/r of s^2 ((t^2 + 1) arctan t - t) / t^2. I think it's just L'Hospital's at that point, but my brain can't do differentiation at 00:40 😵
Well, effectively, it proves that in the infinity, all parallel lines, in the limit, meet each other! Only technical or academic Assumption is that swiggy has ZERO width! Great way to explain a complicated maths concept! Love it.
Why with the fish he doesn’t show the area produced by the rotation of the segment in the lines outside te figure? 😧
Oftentimes when you know some complex subject really, really well, and you are very passionate about it, even trying to explain the "basics" of it to a layman can veer you into subjects that are too complex (and boring) for the average person. When something is very simple and clear to you, it can be hard to realize that it's not simple and clear to somebody who knows very little about the subject.
Sometimes a friend of mine tries to explain some basics of electronics to me, but he often just starts meandering and rambling in a manner that makes me doze off, and even if I understood the beginning of it, I usually just lose track and don't understand anything further.
It can be hard to explain a subject in a manner that's understandable and interesting to laypeople.
And it doesn't even need to be something highly technical or theoretical either. I have experienced this with _board games_ of all things. I have been there when some passionate guy explains for 10 minutes the rules of a complex board game, the 9 latter minutes of it being completely useless rambling because it tells the listeners absolutely nothing because without the experience of having actually played the game several times they just can't understand the context. Those were 10 minutes that could have been used to actually play the game and learn in practice, rather than dozing off to a completely useless rambling that teaches absolutely nothing.
Why is the area he is squeegeeing outside of the fish not counted? to do the "magic move" he is creating a triangle are that has been wiped?
I assume it's because when you extend out far enough from the "fish", you only need the most minute of change in the angle to move to the other triangle, making it negligible.
Like if the squeegee goes millions of kilometres away and rotates a nanometre, it's enough change in trajectory to have the squeegee change its position into the other triangle piece.
***** No matter how far out it goes you still have the cover the area to move over to change the angle, eventually the 2 lines meet and that's where there will be more area squeegeed that wasn't counted
Yes, you're right. In fact, that's shown and explained in the numberphile video he mentioned (link in the description) ;-)
+edstars101 No, only the unit line segment is swiped across that very small angle and therefore results in a negligible area swiped. The whole millions of kilometers line isn't swiped across that angle (that wouldn't be a negligible area).
Assume you can extend the tip out to infinity. Then you add no area by rotating.
The area swept by moving between parallels converges to zero by increasing number of triangle tips, but also then the number of triangle tips diverges to infinity. So in the calculation of the total area, the area per triangle and the number of triangles will chase each other to zero*infinity. My intuition says the swept area will always be equal to the area of the triangle, which is divided up an infinite number of times, but still, all the triangles are added back together to calculate the total area. The bottom edge of the tree structure would be the same length. The overlapping around the bottom of the tree structure does reduce the area somehow, but not by much.
Who finds the error 10:05 ? Weird but you're swiping outside of those triangles without marking the area of the swiping movement you're doing two times before re-entering the geometry. Don't you have to highlight those areas too?
I agree with this but what they are saying is that the line goes out very far until it converges with the other line beside it. My problem with this is that you have to travel a very long distance you shouldn't discount the lines path, I feel that is a cheat, the area is the length travelled plus the width of the needle.
@@jcwalker3 The width of the needle is 0 though. It doesn't have a width.
@Jordan Cottle If it doesn’t have a width, then it is not a needle, and would not have a swiping motion. Also a zero width would have zero length. zero times anything is zero.
@@jcwalker3 It has a length, but not a width. It is a line, with only one dimension. That is why sliding it on the line it is in-line with without turning it doesn't generate a 'swipe' of area. Dragging it along a line perpendicular to it generates a 'swipe' that forms a rectangle. That rectangle has an area, but the individual lines that make it up (including the line we are talking about) do not.
The same way a rectangle doesn't have a volume, a line cannot have an area. Length -> Area -> Volume
@Jordan Cottle , yes a line operates in a single dimension, with zero width, but you can’t drag a one dimension object in two dimensional space without giving it width.
the problem originally used a needle.
Simple: the chevron you created has the same area as a square, because the concave angle on the chevron added to the angle at teh top of the triangle add up to 360, or 2 pi, and the other 2 angles add up to 90 degrees (pi/2) when cut off the top and pasted onto the bottom.
Now if you divide the total distance covered by the total area covered and vice versa, what would the smallest quotient for both of these scenarios be? So the smallest/most effective area possible with the least distance covered for the needle/squegee to travel?
calculate it for yourself
no
yes
the most efficient in area AND distance is probably the shape in 7:12
Interesting to think about how we do "three point turns" in a car. With engines we are probably intuitively minimizing for time instead of distance, but the shape swept out approximates a Steiner curve.
the two areas are the same because the triangle subtracted from the bottom of the area is the same size as the one added on the top, so the total area does not change.
This guy said: Do you watch numberphile. Realised i am not subscribed to the channel i stopped the video; Searched for numberphile; Subscribed; Resumed the video.
A completely reasonable course of action :)
Did you not know numberphile before? I think they're a lot more approachable than Mathologer
Thank you sooooooooooo much. I read many articles or video about that problem and I was always a bit disappointed because I was always lost. First time I understand !!!
PUMPKIN PIE!!!!
+Daan Roelofs I kept thinking "Why does he have a T-shirt with tomato pie?? I guess that would be a pizza.
+Melinda Green I was thinking tomatoes as well, but probably because I have a shirt featuring suicidal cartoon tomatoes jumping off a ladder.
+Melinda Green I seem to recall owing you a reply to another comment you left somewhere else but I forgot what video it was. Do you remember ? (It's getting quite hard to keep up with all the comments.)
Mathologer
I'm not sure but I am excited by your promise to explain why the main Mandelbrot shape repeats. Maybe that was it? And FTR, I am very impressed by how well you keep up with all the comments. I love your work!
Great, all under control then. Mandelbrot set is definitely high up on the to-do list :)
Amazing content!! This is one channel that never disappoints!! Love his cube enthusiasm too!!❤️ As for the arrow being the same as the square, the triangle jutting out at the top fits at the bottom, so that's that, and as for the whole problem itself, I wonder if I could move the squeegee along the circumference of a circle big enough that the squeegee always fits while turning infinitesimally until it makes one complete turn thus turning 180° when it's covered approximately half the circumference..
My last name is Kakeya! I feel honoured :)
Bruh why were you such a problem back then? Like leave my mathematicians alone, smh :\
OK, I have some questions/remarks.
1 - Several steps of this solution works with the concept of infinity, as in the perpendicular move one needle does. If I try to numerically implement this and for every time infinity is mentioned i replaced for x. Every time I increase x the closer my solution will be to 0?
2 - As a corollary of the video can you affirm that you can move any needle from any position in a 2-d space, to any position in a 2-d space? I wonder if that holds for any arbitrary n-d space.
For my next question I'll assume 2 is true. (I'm not actually doing the math because I don't know the tools to work with abstract math and I have a job and stuff)
3 - I imagine that with the right math you can make a perfect analogy of the area this needle is covering as an techinique to optimize one existing polinomial of degree 2?
Given that any polinomial of degree n can be mapped to any other polinomial of degree n by a simple coordinate transformation, does that mean you can optimize any polinomial of degree 2 to zero?
If 2 and 3 are correct, is this a way to prove every polinomial has at least one maximum?
If 1 holds true, does that mean this can be implemented an optimization method for any polinomial?
That is absolutely mindfucking!!!!!
"Hot to turn needles into squeegees" ;)
If you cut the triangle from the top, flip it and place in the bottom and you will end up with the same square.
skveegee
This is truly wonderful. It is such a great problem to discuss following a study of figures of Constant Width. It is so hard to explain to school students. You have made it so clear, and at an appropriate level both for me and them. Thankyou.
That t-shirt looks a great idea too!
+Robyn Gregory That's great :) If you like shapes of constant width maybe you'll be interested in this article by me (make sure to check out the linked in movie clips). We'll also do a video about all this in the near future.
O ja
I love the shirt and the fish. I watched the Numberphile video a day or two ago and thought the concept was brilliant. But your explanation gave me a lot better grasp of it.
Also, I too, was thinking that the chevron you swept early in the video was larger in area than the square. Once you pointed out that they are actually equal I realised what was going on. Great video, as always!
+Steve French The fish materialized all by itself. Looks great doesn't it? I already made and ordered the t-shirt.
+Steve French Oh, and have a look at this. If you use six of the trees you also get a nice snowflakelike picture www.qedcat.com/Kakeya_snowflake.jpg
+Mathologer Nice. It looks like the boundaries define a perfect hexagon. Is there some way that we mortals are able to purchase some of your shirt designs? An online storefront, for instance?
+Steve French I just uploaded the "kakeya fish" t-shirt design to spreadshirt 1035536.spreadshirt.com Looks pretty good on the preview page but of course we'll only know how well it really looks when a hardcopy arrives. So, order at your own risk now or wait until I give the all-clear once I've seen the actual t-shirt. The same shop also has a few other math t-shirt designs by me :)
+Mathologer Great! I'll check them out.
@7:05, My thoughts go immediately to Wankel. Then my old girlfriend who drove an rx7. Then 80085. 😂
It seems like if you travel across the middle for each jump you'd really be able to shrink the angles down.
I'm not sure how to explain it without a picture, but if you're on the xy plane, starting with the "needle" on (0,0) to (0,1) you could travel infinitely high (0,inf), cut a micro-sliver of area, travel down to like (0.01, -inf) and angle back across the origin. Seems like it'd be a tiny ring of slivers out at the extremes.
How to measure the area of a singularity:
"Imagine that we can"
-Theoretical Mathematicians
What is a "theoretical mathematician"?
Mathologer
the one that come up with those kind of solution to problems (or those kind of problems where the solution isn't something "real").
I get what you're implying, but I'm sure you get what I'm implying too.
@@svampebob007 you mean a "mathematician"
Thank you for this video, I will try to apply this to my brush strokes when I'm painting patterns.
9:24 that SUPER WEIRD moment, when he says BLUE
but it's PURPLE...
I thought I was color blind for a second... Anyone else??
This video is much easier to grasp than the one in Numberphile. However, I wish there was some kind of explanation of what is said at 8:40, that if you make the number of triangles high enough they can be moved into a shape with an area as small as desired. That is key, and left as unproved. My feeling is that much of the additional complexity in Numberphile's video is due to the fact that they do prove that statement.
I like the Kakeya Fish! Nice symbol for the representation of the Kakeya problem/solution!
thanks for this video - The Numberphile video was not at all.clear enough (from the start), never mind it being technical. This one helps to fill the gaps.
I just want to "second" what Joe Dinoto said below.
I understand that it must be a lot of work to make these "simple explanations" but the result is well worth it.
The only possible downside is that they might make us (the viewers) believe that we are a little bit smarter than we actually are ;)
Thanks for all Your fantastic videos,
Best regards.
+Mats Rdr Actually, in terms of presentations the advice I was given when I started out as an academic was to keep the first third of a talk understandable to everybody, the second third only to the experts and the last third to pretty much nobody. Sadly this turns out to be very good advice---if you actually succeed in presenting a difficult concept so that everybody can understand it a lot of people will automatically assume that what you are talking about is trivial.
Anyway, as far as I am concerned, coming up with good ways of presenting tricky material is one of the things I really enjoy doing and having people like you confirm that I got an explanation right really makes my day :)
Great video, as always, and nice shirts. One questions that the tree will end up like fractals with infinite surface area but no volume?? Further, by using the first method 5 times will that not also end up with the needle facing the other way round??
No, the trees don't have infinite area, in fact the area gradually goes to 0 in this problem. That's the goal of the problem: to find the least area . Then about the volume, yes it has 0 volume cuz everything is in a plane and 2d.
Lastly , I didn't understand what u meant by 5 times the original method. Could u elaborate on that?
Great explanation for a complex problem. Thanks.
I have no idea what You're talking about, but I love it!
great video. after watching Numberphile video I was looking for additional or clearer explanation and you really made it! congratulations! you've got a new follower!
+Xavier Climent Glad you like the video and thank you very much for saying so :)
It would be nice if you showed an animation of the number of branches increasing and the area of the whole thing decreasing. Just to help visualize the process. Great video! so easy to understand!
They're the same because the second one is just the first one, but there is a triangle cut out of the bottom and put on top
So:
Circle with diameter 1 squeegee-unit has an area of 0.7853 squeegee-units²
Equalateral triangle with height 1squeegee-unit has an area of 0.5773 squeegee-units²
Deltoid curve (as constructed) has an area of 0.3853 squeegee-units²
...what is the area of the needle set "fish" as constructed at 12:00 (not the proposed near-zero ideal)?
Indeed you clarified that problem a WHOLE lot. Thank you. I'm sorta curious as to what lies beneath all this.
+SecularMentat Mission accomplished :) Maybe have a look at the survey that I linked in from the description part of the video to catch a glimpse of what the big shots see in this problem :)
Ah wave equations. I'm guessing there's a lot of really deep physics going on here.
It looks to me like it tends to 0/0, or all the possible slopes of a point as you shrink the shape. No idea if that makes any sense at all but I like the way you explain things, thanks!
For the "arrow", the height relative to the base remains the same as the rectangle, so the area remains the same. The base changes shape, but its still parallel to its top, similar to how a prism can have any base and the formula remains area of base, times height.
You can imagine dividing it I to discrete chunks and pushing them up, which doesn't change the area. Then imagine the chunks getting smaller and smaller, like the proof of the area formula for a circle.
Or, yes, you can just cut out the top and fit it on the bottom, since you know the top and bottom are parallel. But, to me, the above is actually more intuitive, even if takes longer to explain. And it connects directly with the "squeegee", which is the infinitely thin chunk from before.
The object that is characterised by this zero area move can only be held in transfinite 2D space (if you use infinitely many line segments irradiating from on point, you cover all infinite 2D space). Effectively, you need to use a 3rd dimension to move a line segment in a 2D plane without covering any area. You can also claim that as you approach such object, the area that you do end up using is infinite inasmuch as you are covering all space with your infinite line segments. Pretty interesting stuff.
Length of needle = 1
Needle alinged on the circle of radius = 10^3(very large as compared to needle's length)
Needle moves along the circumference of circle(half way or along semi circle)
Now needle has turned 180° with negligible area covered and is parallel to initial position (or at diagonally opposite side)
Now apply that parallel trick as show in video which again cover negligible area
I think this is simpler then the one in video or may be somewhere I m wrong , correct me if so
Explanation is very clear and easy to understand 😃 thanks.
Same concept in Numberphile is explained in a very difficult way and I didn’t understand to be honest!
Very interesting. I completely understand the technique (and of course I’ll take Terry Tao’s word that this works), but it immediately seems counter-intuitive that this should work. I’ll have to check out the numberphile video next.
I know it can be a silly question but, why the areas do not add up? About 6'36'': The triangle's area is less than the circle, but there are some parts of this triangle wherein the 'needle' has been twice. If you cover the area the triangle's area is bigger. And another one. That can be a little arrogant but... At 10'20'' you actually justified why the tricky moves can be neglected. My second question is, why doesn't this justification do not also apply to any other moves we do (also the ones forming the orange area) and so we conclude that our area is indeed zero. That is more of a silly question but... Hope you can help me with it.
Also congratulations. You do a really good job. Cannot put into words how much of a great job is this. Thanks man!
Thank You sooo much for making this! I was really confused by the numberphile video and this cleared up very well! Overall another great video!
+danlmd1 Great, glad it worked for you and thank you very much for saying so :)
I've watched a couple of your videos now. They are amazing!
Definitely equally as good as Brady Haran's channels (numberphile etc.).
You have a new subscriber.
+jnxmaster Great, thanks for subscribing and for confirming that we are on the right way :)
Everyone keeps talking about Banach-Tarski, but this video isn't related to it, since it doesn't use the axiom of choice. To me, what it seems most related to is the Cantor set. The Cantor set is equinumerous to the real numbers, but has a length of 0; the Kakeya set has a line segment of length 1 for every real number between 0 and π/2 (which is again equinumerous to the real numbers), but has an area of 0. They also both have the construction of dividing in half and reducing the area of each; if you imagine each of the intervals in each state of the Cantor set sliding partway open like a pair of curtains, it feels like the analogy is complete.
If i got it right , you suggest that while the segments 'N' tends to infinity, the areas 'A' of the triangles tends to zero and the product N*A tends to zero as well.....
As you up the number of little triangles at the start of the construnction, their overlap can be arranged to approach to zero. As well, we can make the total area of the little turning wedges go to zero by pushing things further and further out along those pairs of parallel lines.
maybe there is another elegant solution. imagine the shape of a huge rain drop. just the outside line. if you follow this shape starting from the tail along the line you would go around and come back on the starting point with an angle of 180 degree.
the bigger the drop design the smaller will be the surface.
isn't it more simple ?
thanks for your video .
A question:
If we count the area that is swept more than once by the needle, then do we get the same area as the circle that was initially swept out by the needle?
If so, then the Kakeya problem basically utilises the fact that we do not count the overlapping areas. It’s a really clever trick, in that sense.
The explanation was spot on. The Numberphile video was quite confusing mainly because the guy explaining the problem wasn’t consistent (with his formulae etc.)
But this explanation really captures the essence. Thank you!
yes, if you count the aera gone over more than once, than the trick he's usung has no purpose.