100 thoughts on “Kakeya’s Needle Problem – Numberphile”

  1. In the jumping portion of problem's solution it is mentioned that the poles have a thickness T (because it felt weird to treat them as perfect line segments of 0 thickness). It turns out that this can be done if T is real, real small, but never mind we just pick T to be that small. How small of a T are we talking about? Less than the thickness of a hydrogen atom? What if we can't pick T as small as we want? Given some thickness, T, can we say what the arena size limit is before we can't turn the pole a full 360 degrees?

    Also, given the limitations can we construct an arena that allows you to turn a pole only through some specified angle? Say, I want to allow the pole to be turned, but not all the way around, say 180 degrees, or 190 degrees.

  2. Hop to the fourth dimension, if you can't do it then, hop to a higher dimension. It usually solves all your problems

  3. in reality, lets supose, you have a pole like 1×0,005 , when u going to an ear,(pretty far), u will make an surface with
    0,005 x pretty far , and thats just for one ear, so this will be more than a circle

  4. Can arena have a hole? If yes, probably it will be a circle of infinite radius with zero area.
    If not – probably you can use a spiral
    But I don't believe I'm smarter then all mathematicians. What I missing?

  5. I get the pole moving, but not his geometry…
    How do you add ears which must be smaller than the original triangle, but each has an area way larger than the original triangle?

  6. just make a track(movement space) in the shape of a stretched 8 with very minimal thickness. its surface area should also be calculable.

  7. Wouldn't a simpler solution just to be to move the needle across the path of a circle, with the direction of the needle being the tangent of the circle at that point? As the radius of the circle gets bigger the total area should go to 0 right?

  8. And now tell me what to do with that problem after understanding it any application? It took my a bit of effort tounderstansd it now im hooked

  9. So… if I give you a cm^3 box and I give you a meter stick and tell you to turn it inside. You need magic? Sounds about right

  10. Why could you not start with a 30 degree triangle and use 6 of the trees and slide the pole(needle) down to meet the other end of the first after a 180 rotation to rotate it another 180 therefore cutting the area in half

  11. What would be a useful real world application for this? And yes, this was way less complicated on Mathologer's channel.

  12. 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.

  13. Damn.The concept is hard enough to understand on its own, but it becomes enormously difficult if English isnt your mother language.

  14. Saying "never mind", "it doesn't matter" and not showing the math isn't how you explain something, just saying, mathologer's video is waaaaaaay better. And don't get me wrong I love your channel, just don't be that simplistic it make things uninteresting.

  15. The only reason why this works is because we assume that the line segment has a width of zero. Or in other words, we transforming a one dimensional across a 2 dimensional plane. Can this problem be abstracted to higher dimensions I wonder?

  16. If anyone here is confused by this, see Mathologer’s video on this puzzle; he gets less bogged down in all the numbers, not to mention his diagrams are way better. The butcher paper is certainly iconic, but there are some things that are just explained better with animations; imagine trying to cover some of 3Blue1Brown’s material (analytic continuation, Pythagorean triples from x^2, quaternion rotations…) without animations!

  17. I think a much simpler solution is an infinitely large but infinitely thin ring. As it gets larger, the ring can become thinner while still allowing the pole to turn slightly. This ring approaches a radius of infinity with a thickness approaching 0. This would theoretically have a infinitely small area.

  18. He failed miserably at simplifying or synopsizing the final explanation of purpose and significance (which should have been done at the beginning and the end of the video. The key I would have liked is that 'this geometric device allows you to build a (complex shape) tessellation of a much smaller total area than the obvious circle', in which you can gyre the 'log' a full 360deg. The scaling up and down, and the 1000theta, seemed unneeded distractions. The visual demonstration of the process of moving the log in the maze would have made more sense if it were animated by someone else!
    I've still no idea of how this solution might be useful to any kind of real world or even philosophical problem.

  19. The problem is an interesting one but this guy is terrible at explaining things. Do not let him near any students.

  20. 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.

  21. sorry but Numberphile is supposed to bring interesting Maths to people that didn't spend their life studying it. This one failed IMHO.

  22. This was the most convoluted, difficult to follow numberphile video i have ever seen… there were so many extra unnecessary details, they made it tough to pick out the details that mattered and the point he was working towards. For example, why did he keep saying area and the constant did not matter and then continue to assign them arbitrary values throughout the video?

  23. this is poorly explained. The original problem is: you have a needle on the table, what is the path the needle has to take to sweep the less area possible? if there is sand on the table, you want to flip the needle without removing sand from the table.

  24. Is there an important postulate missing that perhaps parallel lines converge at infinite? Or tend towards convergence at infinite?

  25. Do these arenas point at every direction no matter the arbitrarily small chosen area? or they point at every direction as area tends to zero?

  26. 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.

  27. Why don't the ears push outside the arena? I feel as though that they would. If the ears push outside the arena, then you're not using the area of the arena to turn 360 degrees in…

  28. The scaling was totally irrelevant and unnecessary. If were going to scale the map and the pole up by the same factor to the "real world" or then scale both down by the same factor down to our piece of paper. That has no bearing on anything. you can take any equation and multiply it by a billion and then later divide it by a billion again. Totally irrelevant.

  29. Why not draw a star of n points and radious 2 -1/2πn. Then your needle of unit 1 could like follow around the edges in a star pattern.

  30. 4:18
    draws a triangle
    Let's say this is a triangle!

    Ahh those mathematicians

  31. If there is enough space in each ear and each jumping mechanism to facilitate a full turn, wouldn't that area be the same as a circle anyway? Isn't this basically just a circle sliced into thousands of segments and arrange in a large array?

  32. This is just turning a large car around in a narrow street. The joys of London and the infinite point turn.

  33. This is a perfect case of someone who is suppose to be a "Teacher" who is in reality as a confuser adding all sorts of dialogue that is not germane to the subject at hand.

  34. This still scales with N-squared though. The area might be minimal but it's still just a shape.
    Double N and the area of this arena quadruples.

  35. Aren't mathematic professionals supposed to be doing something useful to reality instead of solving puzzles?

  36. They forgot to tell us WHY would we do that. Instead of some other solution. Is this the smallest area to turn a needle? Can we make it smaller by adding thinner triangles? Is it really smaller than that triangular shape from the beginning? Cause this shape looks like it takes an awful lot of additional space for turning.

  37. 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?

  38. The explanation could've been much simpler. It's pretty clear that what he cared was the big/small O notation in a sense than any function from O(N) will eventually become smaller than N^2. But there is literally no need to assign arbitrary confusing constants. Call it kN or O(N) or just say that its linear with the increase of N or w/e – the 1000N part is just so arbitrary that is simply confuses more than it clarifies things.
    Other than that, that's a really interesing trick.

  39. I just can't seem to tie it all together.

    You're spinning a needle – or a pole, in this case – around an area so that, after however many moves are necessary, it has rotated through 360 degrees.

    We start with the example of a circle and how, while it works, it's grossly inefficient in the sense that the result can be achieved with a much smaller area, with the next example being a triangle with indented sides – I know there's a name for this but I can't remember.

    But then we move into triangles and making triangles with ears and then we make the triangles bigger and we throw arbitrary numbers into the pile with Greek symbols and algebra and none of it really matters because the goal of what we are attempting to do is, what? Find the smallest possible area necessary for a given length of pole, okay. So the area can be 1,000 times theta. What? What was the deal with squaring the size to scale it up? What does it matter? But now we're going to ignore rotation too and just say the pole can "jump" from one region to the next. Why? What is the problem even trying to solve in the first place?

    Perhaps a more accessible way to present this would have been to start with the above example of the indented triangle of a given area, and explain the maximum length of pole it could accommodate and why.

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