The Kakeya needle problem (the squeegee approach)

by birtanpublished on September 11, 2020

I'm assuming everybody here also watches numberphile Giuseppe by watching number files numberphile is great so does 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 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 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 around here and then we could just swipe across like that okay and so it swipes across area because it basically area

Does it gets cleaned and we're interested in this area okay now 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 becomes 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 so

Yeah yeah yeah isn't well anyway maybe maybe the guys can sort it out in 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 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 and 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 it's smaller smaller and make sure we can make it even smaller can you get

Imagine how can we make it even smaller Josep it hasn't actually seen this before so this think about okay how can we make this even smaller that's right so the first move it just go higher up okay so you just go fire up and then

These angles will get smaller actually it's pretty obvious that these two angles are always going to be the same but and so um you know they simultaneously get get smaller okay and I now if I push it really high up you

Know you basically 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 encounter intuitive thing come up well mainly because the to the initial position the last position to parallel right so then we 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 one like this at all like that or whatever then we also can't 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 further 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 kekkai a needle

Problem it asks if we want to do this what's the smallest area you can create okay okay and the answer is also pretty surprising but let's see how anybody off 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 so we just go in a circle and with an air edit swept out like this can we do any better than this what do you think is that but

Can we do any better than this well obviously because I set it up like there's the feel better way so here's 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 matter whatever one unit of something okay they have 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 0.5 seven 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's doesn't have any kind of indentation I was like you know then then this 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 something better and I'll show you what

They came up with so what you start out with here's a big circle and inside the picture of the small circle it's the small circles 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 this 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 and that one so verse three points that are really on the boundary it's it's a very 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 in this guy yep so it's touching all three sides it's pretty good yeah okay now well the thing

Is you know it's a hundred years old it's important time to 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 I trying to sell the same same width right and 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 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 too 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 is the total area okay and now I'm going to show you how you can turn the needle around in 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 where 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 the parallel not the curl obviously well it's going to be

Important for later again okay now we're going to start turning our needle in here okay or else we squeegee so for me this is squeegee for you know the usual set up it's 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 we're 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'll 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 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 cannot just forget about it for forever what comes next

Okay becomes negative okay so now we just keep on going but putting in 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's about 6 degrees okay well with 60 degrees but we need 180 so what do we do next

Well we bring a second tree okay so a second tree like this just turn 60 degrees and then we make it 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 I'm just keep on going like this and we get a 120 and then 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 and it's arranged it like this and you get this nice fish-like and now I'm just going to show you how to 180 works so that'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 we 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 you give me

Any positive number it doesn't matter how small it is I can arrange this whole set up here so the total area swept our squeegee throughout the movement is smack on this number or less the number of branches

That's right yeah so that's placing one branch or well one tip little triangle that were using what so there's 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 past that you or

The 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 no we just kind of repeat the whole thing in here because we've done 108

Degrees and I would just kind of go for a second round and we get 360 degrees yeah so for this one here it's finite we could even do like a good start of the like a 90 degree triangle and I will

Just need to it basically that's 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 in the genius category right so he's like a Fields

Medalist like really really serious stuff basically equivalent of a Nobel Prize winner and Matz and he's actually interested in these sorts of things I mean it might seem like a pretty trivial problem you know it's it's cute but I

Mean what does it all to do with was real maths well it's kind of the starting point for some really really really deep stuff but actually quite a few of these fields medalists are interested in and I'm going to do a

Provide some links down here the bottom to some survey articles by Terry Tao in Australian Fields medalist who talks about this stuff and you can 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 the number files video complicated 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 trees really a very 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 a triangle sixteen triangles maybe doubling up every time and then we can make up these trees now the deep people they're interested in making up the

Trees 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 were just

Getting close to zero but 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

Cannot kind of see like the the individual moments that is kind of moves you can see like one line segment pointing this way one line segment pointing this way one that side 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 number files video you know just feel feel free post in the comments I'm happy to answer all these things and so now to finish off I

Mean this gives me a really really good idea I mean I just brought in the pumpkin pie because it's Halloween a couple of days time but of course I mean what I really am going to do now is go to a t-shirt of my kakeya fish

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