---
title: 'The Fixed Point Theorem - Numberphile'
source: 'https://youtube.com/watch?v=6PmuWFWVKDE'
video_id: '6PmuWFWVKDE'
date: 2026-06-30
duration_sec: 1093
---

# The Fixed Point Theorem - Numberphile

> Source: [The Fixed Point Theorem - Numberphile](https://youtube.com/watch?v=6PmuWFWVKDE)

## Summary

Brouwer's fixed point theorem is explained through intuitive visual demonstrations, including crumpling paper, stirring tea, and a treasure map puzzle. The video breaks down the mathematical conditions (continuous function on a non-empty, compact, convex set) that guarantee at least one point remains fixed during transformation.

### Key Points

- **Paper Crumpling Demo** [0:32] — Two identical sheets of paper: one flat, one crumpled and placed on top. Brouwer's theorem guarantees at least one point on the flat sheet corresponds to the same point on the crumpled sheet.
- **Map Analogy** [1:47] — Holding a map of a region, there will always be a point on the map that lies directly above the corresponding real-world location, even if the map is rotated or scaled.
- **Tea Stirring Example** [2:50] — After stirring a cup of tea and letting it settle, at least one point of liquid returns exactly to its original position, demonstrating a fixed point under continuous motion.
- **Mathematical Conditions** [4:03] — The theorem requires a continuous function f from a non-empty, compact, convex set to itself. Compact means closed and bounded; convex means any line between two points stays inside the set.
- **Treasure Map Puzzle Setup** [7:49] — Using two copies of a treasure map (one rotated/scaled), Brouwer's fixed point theorem ensures a single fixed point exists, which can be used to bury treasure.
- **Fixed Point on Interactive Map** [9:39] — When one map is placed on top of another with rotation or scaling, a fixed point appears. Moving the top map causes the fixed point to move in a non-intuitive, opposite direction.
- **Shaken vs Stirred Cocktail** [11:24] — Stirring preserves the convex set (liquid stays in glass), so Brouwer holds. Shaking creates splashes and separate blobs, breaking convexity – the theorem no longer applies.
- **Treasure Hunt Solution** [13:52] — By positioning a half-size map so that Cape of Ye Woods aligns with a small islet and Fourmast Hill points to the bulk of treasure, the fixed point lands on 'Graves' (dead man's chest).

### Conclusion

Brouwer's fixed point theorem is a powerful and counterintuitive result that guarantees at least one stationary point under continuous transformations of a compact convex set, offering insights that span from everyday phenomena to advanced mathematics.

## Transcript

Brower's fixed point theorem, my bestie. So what we need, two pieces of paper of
exactly the same thing, thanks James. And what you do, lay one flat. Did you
just have them lying around at home? Don't out me like that. I do think
he's amazing, but no, I printed them out last night and as you can see,
I really need some more printering. So I didn't do him justice, I'm sorry. But
anyway, so lay one flat, take the other, sorry James, and
crumple and just lay it within the bounds of the other. Brower's
fixed point theorem says there'll always be one point on the flat piece of paper
that corresponds to exactly the same point on the crumpled piece of paper, like always.
Brower's fixed point theorem, the idea is that you have one thing and a copy
of the same thing that you've messed up. I think the technical word is transformed
and it might be crazy transformation like screwing it up. But as long as it's,
you've got a mapping between the two things, there will be a point on both
that is the same in the sense that it's a fixed point and that when
you do the transformation, everything else might change, but one point at least will be
the same. So like if we were skewering this with like a pin or something,
there would be one point on that part of James that's covered that would also...
Yeah, you would skewer the same point as you go through the crumpled paper. Say
it's like, imagine it's here, say it's his like pupil, that one pixel on his
pupil. You would go through that pupil in the crumpled paper and then come out
and go exactly into the same pixel on the pupil on the flat piece of
paper. It's a fixed point, even though I've done all this. You can crumple, you
can rotate, you can't tear, no tearing. It always holds, there's always at
least one point. That seems weird. And it's the same with, if I can take
another example, this is the best thing about Browers is that it has so many
like real life visualisations. So imagine, well, we don't have to imagine, I'm here and
I'm holding on to a map. There will be one point on this map that
lies directly above the real worldly point like that we're studying. So for instance,
there will be a point in the UK here, it'll go through the UK and
it'll land exactly where it should be in the real world. So all the points
covered by this rectangle right now, oh yeah, one of them is on the... One
of them will be on the floor. If you put a map on top of
another map, there will be a fixed point, even if it's twisted. If you rescale
the map, make the second map smaller, there'll still be at least one fixed point,
probably just one. If you screw the map up, there'll be a fixed point. If
you are standing in a country and you're holding a map of that country and
you drop that map on the floor, there's a fixed point on that map on
the floor, which is in the same spot. At that point, the scales are so
different, it's kind of hard to comprehend. But that's the sort of generality of this
Brower's fixed point theorem. I don't know if you can tell, I'm from Yorkshire, I
love a brew. So another example of Brower's is... A brew's tea, by the way.
A brew, yeah, you want a cup of tea. Not beer. Maybe you like beer.
No, I don't actually. Just a brew, just a cup of tea. Gin, if you're
asking. You can take your cup and you'll stir your tea. You get your spoon,
stir your tea, no violent motions. We don't want any of that. But basically, you
stir it and then take your spoon out, let it come to rest. There'll be
one point that goes back to the exact same point that it started, even though
you stirred it. That is counterintuitive. I'd say I can't believe how much time I
put into stirring tea and there's always at least one. There could be more that
are just coming back to the same position. I'm just like, it doesn't make sense.
But what I quite like is what happens when it breaks down. So we're going
to go for a bond example, cocktails shaken or stirred. In the stirred
example, Brower's holds. You stir that cocktail, there will be a point that goes back
to its original point. But shaken, the criteria of the theorem break down, it doesn't
hold. And I thought we could go into some of the bit more mathematical bits
of the theorem and therefore show how it breaks down in the shaken scenario.
For any continuous, we love a continuous function.
Function f, that's like your action of stirring. From a, now this is the
key bit. Non -empty, compact, convex, we're going to come on
to those two. Let's come on to those two. Set to itself, there
exists a point such that f of x, zero,
is equal to x, zero, i .e. there's a fixed point. Can I just say,
your handwriting is everything I dreamed. Let's break this down, shall we? So for any
continuous function, f, so that function is the act of
starting with exactly the same picture. And the function is the
act of doing whatever you want, this, a rotation, a crumple. You
can flip it that way, that's the function, that's the act. Stirring the tea. Stirring
the tea, that's the function. From a non -empty, compact, convex set. So this
is the picture of James is the set. Right, what do we mean by compact?
What compact actually is asking for is for a set to be closed and bounded.
So if we think of, let's come back to some of our set theory. So
if we have the interval zero and one, this is bounded
by zero and one. It's like got these, I guess, endpoints, but it's not closed
because these endpoints aren't contained in the set. This set is closed
and bounded because zero and one lie in the sets. So for anyone that doesn't
maybe know set theory, these brackets mean that it's every value up to zero but
not including zero. But when we have square brackets, we're including zero in that set.
So that's what compact means. Convex is a bit different. So convex basically
means that, let's take a circle, you've got a situation where you've got two points
in your set, let's call them x and y. For a set to be convex,
you can always draw a line between any two points in the set and the
line is still contained within the set. So as a counter example, not
convex because our straight line goes outside of our set. These
are the criteria that we need for this theorem like to hold. So that's what
we've got. our function, we've defined what we need from our set and then the
last bit that we've written is there exists at least one x zero such that
f of x zero is equal to x zero, i .e. there is a fixed
point. So the x zero is like this pixel on James's
pupil and we're basically saying after we've performed all these, this function, all these like
crumples and rotations and we place it, there is that fixed point
that dot on the pupil on the crumpled paper lies exactly above the dot on
the flat piece of paper. There will always be that point. And Chelsea like I
know for the example we've used James's pupil for fun but we don't have a
control or choice over what that point will be. Not at all, yeah. I wouldn't
even know where to start, you know. Actually his eye is quite quite close but
yeah it's really hard to actually illustrate you know where it is but you know
we'll put it there. Let's just say it's the pupil. So I found a treasure
map. What you found the treasure map? The treasure map actually Robert Louis Stevenson's Treasure
Island. He drew this map for his book and it's not the greatest treasure map
ever but it was probably the first that used an x marks the spot. There's
all sorts of nice history about it. Where was the treasure map? Well there's a
little x here and it says bulk of treasure here. That was the phrase he
used in the map and in the story to, spoilers, Ben Gunn who's like marooned
on the island like moves like he's just marooned there and over the course of
years and years he digs it all up and moves it somewhere else. So bulk
of treasure was there and it's not anymore. Spoiler! Spoiler! I love it. Go and
read the story. It's a fun old story. It has all the sort of tropes
about pirates. A lot of them come from that story including the stuff about maps
and parrots and... Oh let me get let me do something.
It's a dot. It's the black spot. It's the black spot. You've read the book.
Yeah. Don't give me the black spot. This video's gonna go bad. The point of
getting all excited about maps is that once you've got a way of fixing a
point on the map I think you've got a nice way to leave a treasure
trail and so I want it occurred to me to like play with a simple
version of Brouwer's fixed point theorem which is just two copies of a map not
doing anything like scrumpling it up but if I had two copies of a map
and put them on top of each other even if they're rotated, scaled, shifted it
will define a point on the map. So I've got the map here on Jojibura
and actually I've got another copy of the map here and the first thing to
notice is this is an exact copy of the map. It's the same scale and
the same orientation and there's no fixed point at the moment. So this is breaking
one of our conditions and I think it's actually the condition that the set is
like compact and convex. I don't want to get into the details but it's really
obvious that if you have two exact copies of the map and you put them
slightly shifted there is no going to be there's no point it will match up.
As soon as I rotate it or scale it though we'll get one. With no
rotation there is no fixed point but as soon as I rotate it there is
a fixed point and as I move the top map around it moves around in
a way that I can't really communicate how hard it is to control where this
red dot goes. I move the map one way and the fixed point goes the
other way. It's very non -intuitive. If I move the map up on the mouse
the fixed point moves sideways and if I go sideways depending on the angle I
spin the map at it's it's really non -intuitive how the fixed point moves around.
So yeah there is a fixed point and now obviously if if the map is
not on top of the other map the fixed point is not on the map
actually if I zoom out there is a fixed point it's just kind of like
way off there if the maps were bigger it would have it on it. So
and this is where you have the conditions about you need to have one contained
in the other one. So if I do scale the map like and I make
sure that one map is entirely contained on top of the other one I guarantee
you that somewhere on both maps is a point which is fixed and actually let's
make it in a particular place and this is where moving it around is really
hard to arrange where I want it to go. I'm trying to aim for the
top of the mountain here. I can cheat by making the map a bit transparent
and let's try and get the top of that mountain. There we go now I
can there's a half scale map on top of the original map and it says
the fixed point is on top of Spyglass Hill. I don't know if I remember
that from the story and if I turn off the top map you can see
that dot is indeed on Spyglass Hill on that map. You've got to do it
on the X. Bulk of treasure here. You want it on the X, okay. So
if I make my map semi -transparent and try and get the bulk of treasure
here generally quite hard to see. I think it's about there isn't it?
There we go. On the top map it's really hard to see the writing there
the fixed point is that blob but if I go into the bigger map you
can see bulk of treasure here. Red blob marks the spot. Now before
I studied maths I did actually study fine art but maybe I shouldn't say this
because this might not go so well. You have to
draw a lot of cocktails when you're studying fine art. Well I've definitely studied them
and it's in there and you're just
stirring it round. Right really nice. You don't have any
instances really like this is a nice example where things are breaking off you're
not stirring violently so none of this cocktail is coming out of the glass or
anything like that. So we have this situation where it's compact because we've
got the boundary like you know inside the glass and it's it's bounded
we're not letting it we're not letting the liquid come out of the glass or
anything like that and it's also convex. The glass isn't a funky shape we can
always pick two points and draw a line between them and that line be within
this glass. But if we shake... That's a real sign of glass that
you're like keep doing like best second mind shadow of the glass. You know I'm
always like if you're going to do a job do it properly cocktail shaking it's
a kind of act in it you you put some welly into it don't you
and what you'll find is when we do that we kind of we might have
a situation where we've kind of got this happening got some liquid up here
oh there's this there's some you know this there's some up here we've got some
maybe some blobs you know around here it's really oh and even this one's mid
-flight. So we're going to have this situation
we're still compact but we don't honor the convexivity rule
because now any two points in our set we cannot draw a
line between them that still lies within the set and so if you
stir a cocktail Brower's fixed point theorem holds but if you shake a cocktail
it doesn't. So I think we can take that James Bond potentially isn't a fan
of fixed points but because he chose the shaken example the shaken cocktail whereas
maybe it was maths folk would prefer the stirred one
What I thought would be a nice demonstration is that assuming you know the rotation
and the translation and the scaling of the second map you are
defining one point on that map just by dumping the other one on top. So
I thought I'd do a little treasure hunt. I found this document What have we
got here? I have heard tell that a map of the much feared and much
discussed skeleton island that's the one from the story may yet be found and if
you be able I should have gone pirate on this shouldn't I if you be
able to set a half -sized copy of such a map atop an original I'm
sorry about the voice in such a way that the cape of ye woods lies
above the small islet in the extreme southeast and the summit of four mast hill
points to where the bulk of the treasure used to be you'll be able to
find fix the point at which lies buried the remainder of the treasure in a
dead man's chest Yo -ho. I don't know why they're putting yo -ho. Yo -ho
-ho and a bottle of rum. Oh you have read the story. Yeah. How many
men is on a dead man's chest? Fifteen I believe. Fifteen men on a dead
man's chest. Yo -ho -ho and a bottle of rum. Insert your pirate drink there.
Rum not grog apparently. Grog is watered down rum anyway isn't it? I'm off I'm
off on a tangent. The point is is that I mean this is obviously ridiculous
and silly I had a bit of fun making a treasure story but I think
that document is as good as putting an x on a map but you don't
have to put the x on the map so you reward the treasure to anyone
who can sort of decode it. So I don't want to be said that I'm
spoiling the experience for Numberphile viewers but I will show you the solution but if
you want to find it yourself, you know, time to pause.
So what we need is a half -size copy of the map. Oh look I
found one. And according to this I need to set the Cape of Ye Woods
which is this this Cape down here on top of a small islet in the
extreme southeast. There's an islet down here in the southeast and the summit of Fourmast
Hill which is this one at the top of the map. It needs to point
to where the bulk of the treasure is down here so we should be able
to line this up. So there's the Cape of Ye Woods and this this is
obviously really hard to do with non -opaque non -transparent paper but it's kind of
it's got to be on top of there and then the Fourmast Hill's got to
point to where the x is. Yeah that's roughly pointing to where the x is
there. Now that's the arrangement. Now somewhere on this diagram is a fixed point by
Brouwer's Fixed Point Theorem. But this piece of paper doesn't find it for me and
actually it's really non -trivial to know where their fixed point is and unless you've
got a neat bit of software to show you the fixed point, it's really quite
irritating to find. Do you want to see where it is? Yeah! This is where
Ben Gunn moved it to. Well in my version of the story yeah so what
I'm going to do is make the map sort of transparent which makes it much
easier to work with than my version. Like there's the Cape down here and I'm
going to move it on top of the little island so that bit's got to
be there but the Fourmast Hill is at the top of this island, the small
one, and it's not yet pointing to where the treasure is over there. That's it.
So the fixed point is there and if I make the map bigger we can
see there is a little thing there. Can you read that word? Graves. The graves.
It's where the graves are. Ah, in a dead man's chest. Ah, look at that.
I'm trying to tell you why my pun was really worth, worth constructing. Okay, well
done. But the point I like about Brouwer's Fixed Point Theorem is that as soon
as you've got two copies of a thing, one of which is lying contained in
the other one, true when you're in a landscape with a map or you put
a map on top of the other one, it does define a single fixed point
and for me that was enough to set a treasure trail, X marks the spot
based on how you position two maps. Do you go stir it or what's your,
or you don't just order it? Yeah, I just order it and drink. I never
tell them. That's really bad isn't it? I'm like, maybe with a, I'm potentially more
serious about tea. I do like a well -made tea, in fact. But no one
shakes tea. Oh, no. Yeah, but there's crazy people who put milk in first. So
we connect, we never know what they're going to do. So my mom genuinely sends
tea back if there's the, not the correct amount of milk in. She's very serious
about it. I'm turning into her and I do, I do like, I can't be
doing with a substandard cup of tea. Did you make that, that thing? Is that
yours? This is mine. That's your puzzle. It's my puzzle. Oh, well done. Thank you.
don't have to read it in a pyro voice, but you can if you want
to have the full experience.