The Fixed Point Theorem - Numberphile

[00:00] Brower's fixed point theorem, my bestie. So what we need, two pieces of paper of

[00:05] exactly the same thing, thanks James. And what you do, lay one flat. Did you

[00:09] just have them lying around at home? Don't out me like that. I do think

[00:14] he's amazing, but no, I printed them out last night and as you can see,

[00:17] I really need some more printering. So I didn't do him justice, I'm sorry. But

[00:22] anyway, so lay one flat, take the other, sorry James, and

[00:27] crumple and just lay it within the bounds of the other. Brower's

[00:32] fixed point theorem says there'll always be one point on the flat piece of paper

[00:37] that corresponds to exactly the same point on the crumpled piece of paper, like always.

[00:43] Brower's fixed point theorem, the idea is that you have one thing and a copy

[00:48] of the same thing that you've messed up. I think the technical word is transformed

[00:51] and it might be crazy transformation like screwing it up. But as long as it's,

[00:54] you've got a mapping between the two things, there will be a point on both

[00:59] that is the same in the sense that it's a fixed point and that when

[01:02] you do the transformation, everything else might change, but one point at least will be

[01:06] the same. So like if we were skewering this with like a pin or something,

[01:10] there would be one point on that part of James that's covered that would also...

[01:14] Yeah, you would skewer the same point as you go through the crumpled paper. Say

[01:18] it's like, imagine it's here, say it's his like pupil, that one pixel on his

[01:22] pupil. You would go through that pupil in the crumpled paper and then come out

[01:26] and go exactly into the same pixel on the pupil on the flat piece of

[01:31] paper. It's a fixed point, even though I've done all this. You can crumple, you

[01:36] can rotate, you can't tear, no tearing. It always holds, there's always at

[01:42] least one point. That seems weird. And it's the same with, if I can take

[01:47] another example, this is the best thing about Browers is that it has so many

[01:50] like real life visualisations. So imagine, well, we don't have to imagine, I'm here and

[01:56] I'm holding on to a map. There will be one point on this map that

[02:01] lies directly above the real worldly point like that we're studying. So for instance,

[02:07] there will be a point in the UK here, it'll go through the UK and

[02:11] it'll land exactly where it should be in the real world. So all the points

[02:15] covered by this rectangle right now, oh yeah, one of them is on the... One

[02:20] of them will be on the floor. If you put a map on top of

[02:23] another map, there will be a fixed point, even if it's twisted. If you rescale

[02:27] the map, make the second map smaller, there'll still be at least one fixed point,

[02:30] probably just one. If you screw the map up, there'll be a fixed point. If

[02:34] you are standing in a country and you're holding a map of that country and

[02:37] you drop that map on the floor, there's a fixed point on that map on

[02:40] the floor, which is in the same spot. At that point, the scales are so

[02:43] different, it's kind of hard to comprehend. But that's the sort of generality of this

[02:47] Brower's fixed point theorem. I don't know if you can tell, I'm from Yorkshire, I

[02:50] love a brew. So another example of Brower's is... A brew's tea, by the way.

[02:54] A brew, yeah, you want a cup of tea. Not beer. Maybe you like beer.

[02:58] No, I don't actually. Just a brew, just a cup of tea. Gin, if you're

[03:04] asking. You can take your cup and you'll stir your tea. You get your spoon,

[03:09] stir your tea, no violent motions. We don't want any of that. But basically, you

[03:15] stir it and then take your spoon out, let it come to rest. There'll be

[03:18] one point that goes back to the exact same point that it started, even though

[03:24] you stirred it. That is counterintuitive. I'd say I can't believe how much time I

[03:28] put into stirring tea and there's always at least one. There could be more that

[03:32] are just coming back to the same position. I'm just like, it doesn't make sense.

[03:37] But what I quite like is what happens when it breaks down. So we're going

[03:42] to go for a bond example, cocktails shaken or stirred. In the stirred

[03:48] example, Brower's holds. You stir that cocktail, there will be a point that goes back

[03:53] to its original point. But shaken, the criteria of the theorem break down, it doesn't

[03:59] hold. And I thought we could go into some of the bit more mathematical bits

[04:03] of the theorem and therefore show how it breaks down in the shaken scenario.

[04:09] For any continuous, we love a continuous function.

[04:15] Function f, that's like your action of stirring. From a, now this is the

[04:21] key bit. Non -empty, compact, convex, we're going to come on

[04:27] to those two. Let's come on to those two. Set to itself, there

[04:32] exists a point such that f of x, zero,

[04:38] is equal to x, zero, i .e. there's a fixed point. Can I just say,

[04:43] your handwriting is everything I dreamed. Let's break this down, shall we? So for any

[04:48] continuous function, f, so that function is the act of

[04:54] starting with exactly the same picture. And the function is the

[05:00] act of doing whatever you want, this, a rotation, a crumple. You

[05:06] can flip it that way, that's the function, that's the act. Stirring the tea. Stirring

[05:11] the tea, that's the function. From a non -empty, compact, convex set. So this

[05:16] is the picture of James is the set. Right, what do we mean by compact?

[05:22] What compact actually is asking for is for a set to be closed and bounded.

[05:29] So if we think of, let's come back to some of our set theory. So

[05:33] if we have the interval zero and one, this is bounded

[05:39] by zero and one. It's like got these, I guess, endpoints, but it's not closed

[05:45] because these endpoints aren't contained in the set. This set is closed

[05:51] and bounded because zero and one lie in the sets. So for anyone that doesn't

[05:56] maybe know set theory, these brackets mean that it's every value up to zero but

[06:01] not including zero. But when we have square brackets, we're including zero in that set.

[06:06] So that's what compact means. Convex is a bit different. So convex basically

[06:12] means that, let's take a circle, you've got a situation where you've got two points

[06:17] in your set, let's call them x and y. For a set to be convex,

[06:22] you can always draw a line between any two points in the set and the

[06:27] line is still contained within the set. So as a counter example, not

[06:33] convex because our straight line goes outside of our set. These

[06:39] are the criteria that we need for this theorem like to hold. So that's what

[06:44] we've got. our function, we've defined what we need from our set and then the

[06:48] last bit that we've written is there exists at least one x zero such that

[06:53] f of x zero is equal to x zero, i .e. there is a fixed

[06:57] point. So the x zero is like this pixel on James's

[07:03] pupil and we're basically saying after we've performed all these, this function, all these like

[07:09] crumples and rotations and we place it, there is that fixed point

[07:15] that dot on the pupil on the crumpled paper lies exactly above the dot on

[07:20] the flat piece of paper. There will always be that point. And Chelsea like I

[07:25] know for the example we've used James's pupil for fun but we don't have a

[07:29] control or choice over what that point will be. Not at all, yeah. I wouldn't

[07:34] even know where to start, you know. Actually his eye is quite quite close but

[07:39] yeah it's really hard to actually illustrate you know where it is but you know

[07:45] we'll put it there. Let's just say it's the pupil. So I found a treasure

[07:49] map. What you found the treasure map? The treasure map actually Robert Louis Stevenson's Treasure

[07:54] Island. He drew this map for his book and it's not the greatest treasure map

[07:59] ever but it was probably the first that used an x marks the spot. There's

[08:02] all sorts of nice history about it. Where was the treasure map? Well there's a

[08:04] little x here and it says bulk of treasure here. That was the phrase he

[08:08] used in the map and in the story to, spoilers, Ben Gunn who's like marooned

[08:14] on the island like moves like he's just marooned there and over the course of

[08:17] years and years he digs it all up and moves it somewhere else. So bulk

[08:19] of treasure was there and it's not anymore. Spoiler! Spoiler! I love it. Go and

[08:24] read the story. It's a fun old story. It has all the sort of tropes

[08:28] about pirates. A lot of them come from that story including the stuff about maps

[08:32] and parrots and... Oh let me get let me do something.

[08:40] It's a dot. It's the black spot. It's the black spot. You've read the book.

[08:44] Yeah. Don't give me the black spot. This video's gonna go bad. The point of

[08:50] getting all excited about maps is that once you've got a way of fixing a

[08:53] point on the map I think you've got a nice way to leave a treasure

[08:55] trail and so I want it occurred to me to like play with a simple

[08:59] version of Brouwer's fixed point theorem which is just two copies of a map not

[09:02] doing anything like scrumpling it up but if I had two copies of a map

[09:05] and put them on top of each other even if they're rotated, scaled, shifted it

[09:09] will define a point on the map. So I've got the map here on Jojibura

[09:13] and actually I've got another copy of the map here and the first thing to

[09:15] notice is this is an exact copy of the map. It's the same scale and

[09:19] the same orientation and there's no fixed point at the moment. So this is breaking

[09:25] one of our conditions and I think it's actually the condition that the set is

[09:29] like compact and convex. I don't want to get into the details but it's really

[09:32] obvious that if you have two exact copies of the map and you put them

[09:35] slightly shifted there is no going to be there's no point it will match up.

[09:39] As soon as I rotate it or scale it though we'll get one. With no

[09:42] rotation there is no fixed point but as soon as I rotate it there is

[09:46] a fixed point and as I move the top map around it moves around in

[09:49] a way that I can't really communicate how hard it is to control where this

[09:52] red dot goes. I move the map one way and the fixed point goes the

[09:54] other way. It's very non -intuitive. If I move the map up on the mouse

[09:59] the fixed point moves sideways and if I go sideways depending on the angle I

[10:02] spin the map at it's it's really non -intuitive how the fixed point moves around.

[10:06] So yeah there is a fixed point and now obviously if if the map is

[10:10] not on top of the other map the fixed point is not on the map

[10:13] actually if I zoom out there is a fixed point it's just kind of like

[10:16] way off there if the maps were bigger it would have it on it. So

[10:19] and this is where you have the conditions about you need to have one contained

[10:23] in the other one. So if I do scale the map like and I make

[10:26] sure that one map is entirely contained on top of the other one I guarantee

[10:30] you that somewhere on both maps is a point which is fixed and actually let's

[10:34] make it in a particular place and this is where moving it around is really

[10:36] hard to arrange where I want it to go. I'm trying to aim for the

[10:39] top of the mountain here. I can cheat by making the map a bit transparent

[10:44] and let's try and get the top of that mountain. There we go now I

[10:47] can there's a half scale map on top of the original map and it says

[10:52] the fixed point is on top of Spyglass Hill. I don't know if I remember

[10:55] that from the story and if I turn off the top map you can see

[10:58] that dot is indeed on Spyglass Hill on that map. You've got to do it

[11:01] on the X. Bulk of treasure here. You want it on the X, okay. So

[11:04] if I make my map semi -transparent and try and get the bulk of treasure

[11:08] here generally quite hard to see. I think it's about there isn't it?

[11:14] There we go. On the top map it's really hard to see the writing there

[11:17] the fixed point is that blob but if I go into the bigger map you

[11:19] can see bulk of treasure here. Red blob marks the spot. Now before

[11:24] I studied maths I did actually study fine art but maybe I shouldn't say this

[11:29] because this might not go so well. You have to

[11:35] draw a lot of cocktails when you're studying fine art. Well I've definitely studied them

[11:39] and it's in there and you're just

[11:43] stirring it round. Right really nice. You don't have any

[11:48] instances really like this is a nice example where things are breaking off you're

[11:54] not stirring violently so none of this cocktail is coming out of the glass or

[11:59] anything like that. So we have this situation where it's compact because we've

[12:05] got the boundary like you know inside the glass and it's it's bounded

[12:11] we're not letting it we're not letting the liquid come out of the glass or

[12:14] anything like that and it's also convex. The glass isn't a funky shape we can

[12:19] always pick two points and draw a line between them and that line be within

[12:23] this glass. But if we shake... That's a real sign of glass that

[12:29] you're like keep doing like best second mind shadow of the glass. You know I'm

[12:34] always like if you're going to do a job do it properly cocktail shaking it's

[12:39] a kind of act in it you you put some welly into it don't you

[12:43] and what you'll find is when we do that we kind of we might have

[12:47] a situation where we've kind of got this happening got some liquid up here

[12:53] oh there's this there's some you know this there's some up here we've got some

[12:57] maybe some blobs you know around here it's really oh and even this one's mid

[13:03] -flight. So we're going to have this situation

[13:09] we're still compact but we don't honor the convexivity rule

[13:15] because now any two points in our set we cannot draw a

[13:21] line between them that still lies within the set and so if you

[13:26] stir a cocktail Brower's fixed point theorem holds but if you shake a cocktail

[13:31] it doesn't. So I think we can take that James Bond potentially isn't a fan

[13:37] of fixed points but because he chose the shaken example the shaken cocktail whereas

[13:43] maybe it was maths folk would prefer the stirred one

[13:52] What I thought would be a nice demonstration is that assuming you know the rotation

[13:57] and the translation and the scaling of the second map you are

[14:02] defining one point on that map just by dumping the other one on top. So

[14:07] I thought I'd do a little treasure hunt. I found this document What have we

[14:13] got here? I have heard tell that a map of the much feared and much

[14:17] discussed skeleton island that's the one from the story may yet be found and if

[14:21] you be able I should have gone pirate on this shouldn't I if you be

[14:24] able to set a half -sized copy of such a map atop an original I'm

[14:28] sorry about the voice in such a way that the cape of ye woods lies

[14:32] above the small islet in the extreme southeast and the summit of four mast hill

[14:35] points to where the bulk of the treasure used to be you'll be able to

[14:39] find fix the point at which lies buried the remainder of the treasure in a

[14:42] dead man's chest Yo -ho. I don't know why they're putting yo -ho. Yo -ho

[14:47] -ho and a bottle of rum. Oh you have read the story. Yeah. How many

[14:51] men is on a dead man's chest? Fifteen I believe. Fifteen men on a dead

[14:53] man's chest. Yo -ho -ho and a bottle of rum. Insert your pirate drink there.

[14:57] Rum not grog apparently. Grog is watered down rum anyway isn't it? I'm off I'm

[15:01] off on a tangent. The point is is that I mean this is obviously ridiculous

[15:04] and silly I had a bit of fun making a treasure story but I think

[15:07] that document is as good as putting an x on a map but you don't

[15:11] have to put the x on the map so you reward the treasure to anyone

[15:13] who can sort of decode it. So I don't want to be said that I'm

[15:16] spoiling the experience for Numberphile viewers but I will show you the solution but if

[15:20] you want to find it yourself, you know, time to pause.

[15:30] So what we need is a half -size copy of the map. Oh look I

[15:35] found one. And according to this I need to set the Cape of Ye Woods

[15:40] which is this this Cape down here on top of a small islet in the

[15:45] extreme southeast. There's an islet down here in the southeast and the summit of Fourmast

[15:50] Hill which is this one at the top of the map. It needs to point

[15:53] to where the bulk of the treasure is down here so we should be able

[15:55] to line this up. So there's the Cape of Ye Woods and this this is

[15:58] obviously really hard to do with non -opaque non -transparent paper but it's kind of

[16:02] it's got to be on top of there and then the Fourmast Hill's got to

[16:05] point to where the x is. Yeah that's roughly pointing to where the x is

[16:11] there. Now that's the arrangement. Now somewhere on this diagram is a fixed point by

[16:16] Brouwer's Fixed Point Theorem. But this piece of paper doesn't find it for me and

[16:21] actually it's really non -trivial to know where their fixed point is and unless you've

[16:24] got a neat bit of software to show you the fixed point, it's really quite

[16:27] irritating to find. Do you want to see where it is? Yeah! This is where

[16:31] Ben Gunn moved it to. Well in my version of the story yeah so what

[16:35] I'm going to do is make the map sort of transparent which makes it much

[16:38] easier to work with than my version. Like there's the Cape down here and I'm

[16:42] going to move it on top of the little island so that bit's got to

[16:44] be there but the Fourmast Hill is at the top of this island, the small

[16:48] one, and it's not yet pointing to where the treasure is over there. That's it.

[16:53] So the fixed point is there and if I make the map bigger we can

[16:56] see there is a little thing there. Can you read that word? Graves. The graves.

[17:01] It's where the graves are. Ah, in a dead man's chest. Ah, look at that.

[17:06] I'm trying to tell you why my pun was really worth, worth constructing. Okay, well

[17:10] done. But the point I like about Brouwer's Fixed Point Theorem is that as soon

[17:13] as you've got two copies of a thing, one of which is lying contained in

[17:16] the other one, true when you're in a landscape with a map or you put

[17:19] a map on top of the other one, it does define a single fixed point

[17:22] and for me that was enough to set a treasure trail, X marks the spot

[17:24] based on how you position two maps. Do you go stir it or what's your,

[17:29] or you don't just order it? Yeah, I just order it and drink. I never

[17:31] tell them. That's really bad isn't it? I'm like, maybe with a, I'm potentially more

[17:36] serious about tea. I do like a well -made tea, in fact. But no one

[17:41] shakes tea. Oh, no. Yeah, but there's crazy people who put milk in first. So

[17:46] we connect, we never know what they're going to do. So my mom genuinely sends

[17:51] tea back if there's the, not the correct amount of milk in. She's very serious

[17:56] about it. I'm turning into her and I do, I do like, I can't be

[18:00] doing with a substandard cup of tea. Did you make that, that thing? Is that

[18:03] yours? This is mine. That's your puzzle. It's my puzzle. Oh, well done. Thank you.

[18:08] don't have to read it in a pyro voice, but you can if you want

[18:10] to have the full experience.