I'm assuming everybody here also watches Numberphile Giuseppe, we're watching Numberphile. Oh yeah Numberphile is great. So just in case you don't watch numberphile You really should and we've got it was one of our recommended channels up there Now recently Numberphile posted a video just a couple of days ago about the very very famous Kakeya needle problem, it was great as usual, but it got a bit technical in parts and Quite a few people actually got lost and so we had a challenge from some of our viewers who also watch Numberphile to see whether we could do it a little bit simpler and Maybe even without any numbers without any formulas so here's our answer to this challenge, okay? So I'll take a slightly different approach one from what people usually do okay? I'm going to use a squeegee I'm not going to use a needle okay, the squeegee starts out in this screen position here and then it ends in the blue position and So you can go for the green position to the blue position many many different ways let's just do the most obvious one so we kind of start it out here, and then we could just swipe across like that, okay and So it swipes across area because it basically it's the area that gets cleaned, and we're interested in this area, okay? And we can swipe in many many different ways let's do it in a second way, so for example We could swipe up there, and then we could swipe down there That gives us a different area right, and now obviously, we can go really really wild but kind of swiping like this before we kind of get to blue and That will get like one more area, but now it's a slightly strange question What's the minimal area we can create like this so how can we get from left to right and create minimal wiping area? Okay, and it might seem that well I mean there was the square at the beginning, and there's this arrow here actually the same area. oh yeah, yeah Yeah, this one's a little bit bigger. Well. It isn't well anyway, maybe maybe the guys can sort it out the comments again Why are these two areas the same? Okay, now it turns out that you can actually make it a lot smaller than this And I'll just show you what the trick is So what you do is these two line segments are parallel and actually the squeegee is a mathematical as a mathematical line segment so it doesn't have any width for this sort of setup here and then what we do is we kind of just extend the Green Line and the Blue Line and the first move We do is we just kind of move it along this line and since it doesn't have any width It's actually not wiping out any area okay, so so far we've not wiped anything now we aim from this point up here to that point down there and now we wipe like that, okay, and now we just move along this line so no extra area created, and then we wipe again and then we just move Where we want to go? Finished right? Now, we will just compare the wiped area to what we had before to this one here You can see it's... smaller.
It's smaller and make sure we can make it even smaller. Can you imagine? How can we make it even smaller? And Giuseppe hasn't actually seen this before so just think about it. How can we make it even smaller? Yeah, so we just go way higher. The higher we go the smaller the angle will be. That's right. So the first move You just go higher up, okay? So you just go higher up and then these angles will get smaller actually it's pretty obvious that these two angles are always going to be the same much and So um you know they simultaneously get smaller okay, and I now if I push it really high up You know your basic can make this as small as you want So if you give me any positive number as small as you want I can make the wiping area that number basically Well because we really have to get from left to right and to get from left to right we have to Angle this thing at some point in time as as soon as you angle it It's going to be positive area doesn't matter how far we go everyone okay? So let's kind of just the warm-up exercise, okay, and So why did this work? Why does this kind of counter-intuitive thing come up? Well mainly because the initial position and the last position are parallel, right? So then you can kind of go up, and then with minimal swipe we can kind of bring this into the second position So what happens now if we make the second position here like not parallel to the first more like this right or like that Or whatever then we also can swipe from here to there What's the answer then? How small can we make things? And actually we're just going to go kind of for the for the worst possible scenario Where we kind of have? This as the initial position and as the final position we kind of have to the whole thing kind of turned 180 degrees So what we're requiring here. Is that during the motion We are going through 180 degrees before we can come back to what we started with up here And that's basically the Kakeya needle problem. It asks if we want to do this. What's the smallest area I can create?okay? The answer is also pretty surprising, but let's see how Anybody taken off the street like an expert in this would solve the problem of turning like from here to there Okay, so I mean I think what everybody would be doing is kind of try something like this all right? so we're just going to just go in a circle and With it swept out like this - can we do any better than this? Do you think Giuseppe can we do any better than this? Well obviously because I set it up like Yeah There's definitely a better way, so here's a nice one so we just start with our line segment like that, then we swipe like 30 degrees and We swipe like that And we swipe like that, and we swipe like that and basically what we've done. Now is we've swept out an equilateral triangle whose height is the length of the squeegee and actually we're going to make the lengths of the squeegee one? One centimetre, one meter or whatever one unit of something, okay there one And now we can actually compare the areas of those two guys and so the area of that one here is 0.7 something and the area of this one here is 0.57 so it's actually a lot better and for some Setup, this is actually the best so if you've got like aiming for an area That's swept, that doesn't have any kind of indentation or like you know then this is one's actually the best But if you are a little bit more relaxed about what you can do There's actually better ways of doing this and the problem is like a hundred years old and even then they came up with Think better and I'll show you what they came up with So what you start out with here is a big circle and inside the big circle is a small circle The small circle is one-third of the diameter of the big one and I will just draw this guy here and so a Point on the boundary trace out is a curve and if you just scale things right we get this figure so the our One segment, just fits in like that and now we can turn it around in here and It's actually quite an amazing Movement. You can see like at any any point throughout the movement We've got that end bit here touching The boundary of the curve this one here that runs all those three points that are really on the boundary it's a very tight fit that's it's something really really quite miraculous by itself, so that shape is called the deltoid and So how small is [that] one now? Well quite a bit smaller than this guy. So any part of the moving segment is always touching all three sides. Yep, always! Okay, now The thing is, you know it's a hundred years old, at some point of time they actually figure out we can do even better You can do it significantly better and here's what we do So we'll start out with the same triangle that we'll just use to kind of swipe things out, okay and now we'll cut this into little triangles, so the Basis of all those little triangles are the same width We'll cut it apart like this and overlap those little triangles and overlap more and overlap more okay? So now we've gone from a equilateral triangle to a shape that's got an area that's a lot less all right, and actually if you use more triangles and are really smart about this pushing together You can actually make the area of this shape as small as we want. That's pretty surprising It's actually pretty tricky to prove - but it's true So you know chop it up into maybe a gazillion little triangles And you know push them together in just the right way and actually Getting pretty close to zero with the total area okay? And now I'm going to show you how you can turn needle around, it is something like this so we've got this little tree here, and we've got the initial triangle, okay, and Now what we're going to do is let's just focus on the two little triangles on the left, okay And I'm going to highlight them here in purple and then brown and I'm also highlighting the common You know edge okay? So in yellow, and now [we're] these two guys here in in the tree well the Brown Triangle is here and The blue triangle is there and obviously those two yellow lines are they parallel They are parallel obviously, it's going to be important for later again, okay now We're going to start turning our needle in here, or our squeegee So for me the squeegee for you know the usual celibates It's a needle that's being turned in the plane okay, so there is the squeegee and we're turning it no problem What's on the left and on the right now obviously here is stuck, right? There's actually no way you can turn this thing further in just inside the figure But now we remember the trick the magic move that we designed at the very beginning right and we've got these two parallel lines So we do is exactly there right remember that I? Didn't come back and now we can turn Alright, so that works and well. I mean there's these little wedges that also get swept out here But what you really have to imagine is that we've going a lot further out right so basically I said we can push this too Close to zero so we kind of just forget about it for forever. What comes next okay? So that's why I mean becomes negative, okay So now we just keep on going But putting in a lot more of those connections and just kind of keep repeating this trick over and over and you can see That we're succeeding in slowly turning the squeegee through 60 degrees smart So almost there and just the last What and with wipe sixty degrees if it sort of [wycliffe] degrees? Okay, well with [sixty] degrees, but we need 180 so what do we do next well? We bring a second tree in okay, so a second tree like this [sixty] degrees and Then we make a transfer using the same kind of Magic move and again Imagine that this is really pushed out forever right and coming back and we're starting our second 60 degrees and just keep on going like this and we get a 120 and we need a third tree And I'm just to see what the whole figure here looks like So that's the whole thing you know let's arrange it like this and get this nice [fish-like] thing And now I'm just going to show you how the whole hundred eighty works, so [it's] a starting position We want to get back to that at the end, but you know we've want to have turned 180 degrees So now we go all this complicated movement It gets us to there then we make the transfer, alright And we get another complicated movement now the transfer and another complicated movement gets us here And then we can just push up and we're back to where we started from so pretty pretty cool And now what does this say? Well what I said is that if we kind of cut up the triangle at the beginning Into lots lots of pieces you can make this overlapping tree as small as we want that means we can make the three trees as small as we want and by pushing out like these transfers you know really really far. We can make everything Inside here as small as we want so give me any positive number Doesn't matter how small it is, I can arrange this whole set up here so the total area swept by the squeegee throughout the movement is Smack on this number or less So the number of tree the number of trees will still be three But then what increases as we're getting smaller and smaller is the number of branches. The number of branches. That's right yeah So that's basically one branch or well one tip Little triangle that we're using, so there's going to be a lot more tips a lot more of these Like kind of Lines starting out and also kind of the pass that you or the distance that you're covering Throughout that your journey here is going to be Ridiculous and So that's that's basically all there is right.
And if I wanted to do a full circle would I need more trees? No
We just kind of repeat the whole thing in here because we've done 108 degrees never just kind of go for a second round We get 360 degrees So that's the maximum amount of trees? Yeah, so for this one here it's fine We could even do like - we could start off at a 90 degree triangle And we would just need two basically, that's also doable. Now why did the Numberphile video turn out a little bit technical? Well the guy who was explaining it I mean He's actually the genius category, right? so he's like a Fields medallist like really really serious stuff basically equivalent of a Nobel prize winner in maths and He's actually interested in these sorts of things. I mean it might seem like a pretty trivial problem You know it's cute, but I mean, what does that all to do with real maths well? It's kind of the starting point for some really really deep stuff, that actually quite a few of these Fields medallists are are interested in. And I'm going to do a provide some links down here at the bottom to some survey articles by Terry Tao. An Australian Fields medallist who talks about this stuff and he just have a look at it and kind of get a feel for what's going on and so there were two things that made Numberphile's video complicated and What the guy was trying to do was actually get you a little bit closer to this To the deep end right, by really giving you a good idea of why we can make these things arbitrarily small I was skipping over that, right? So he was giving quite a bit of detail there why he can make the the trees really arbitrarily small, but also He talked a lot more about how exactly you're going to push them together And that's very important for later on so basically we can we can subdivide into you know Four triangles, eight triangles, sixteen triangles. Maybe doubling up every time and then we can make up these trees Now the deep people they are interested in making up the tree So they're kind of connected to this making it a sequence of trees that kind of converges to something and in the limit This something has very interesting properties, so what it tears is it has area zero So we are just getting close to zero this thing. They're really interested in has area zero and also, what it still has is basically like a line segment of unit lengths in every single direction right so I mean As we turn around we can kind of see like the individual moments that it kind of moves You can see like one line segment pointing this way, one line segment pointing this way, one outside, so in all possible directions We've got line segments here in our constructions, but the total area is still positive. So what they're getting there is something that has Zero area it still has all lines in all possible directions and with that sort of stuff you can then do some really amazing things and Just in general you want to ask anything about this or anything about The Numberphile's video? You know just feel free post in the comments. I'm happy to answer all these things and so now to finish off I mean, it just gave me a really really good idea. I mean, I just brought in the pumpkin Pi because it's Halloween in a couple days time but of course I mean what I really am going to do now is do a t-shirt of my kakeya fish And that's it for today
Man, I really love Mathologer
Can this be analogized to the Moving Sofa problem, or are they fundamentally different?
I recognize most of the background but what is 1729?
http://imgur.com/NlqRel7
He "swiped" smaller area in that "fish" than in triangle, but needed much more area to actually do that movement. Doesn't it bother anybody?