Why you can't comb a hairy ball, and why we care

0:00 [Submit subtitle corrections at criblate.com] These days,

0:01 whenever I look at the back of my beloved 7 month old baby's head,

0:04 this little swirl of tiny hairs reminds me

0:06 of one of the most ridiculously named facts in math, the hairy ball theorem.

0:11 I promise this is a genuinely serious bit of math,

0:14 where informally the statement is that if

0:16 you have a ball that's covered in hair, and you try to comb it down,

0:20 there is no way to do it without having the hair stick up at at least one point.

0:25 For example, let's say you try to comb it all counterclockwise around some axis.

0:29 Then at the top and the bottom, you end up with these little swirls,

0:33 and the hair at the centermost point of those swirls would have nowhere to go.

0:37 It's forced to stick up.

0:39 It's actually very fun to play around with this in your mind,

0:42 where no matter how you try to flatten out the hair,

0:44 it is a mathematical guarantee that you will

0:46 be left with at least one tuft like this.

0:49 In fact, even getting it down to just a single problem point,

0:52 as opposed to two, is a bit of a challenge.

0:55 It is possible, and if you like puzzles,

0:57 I encourage you to try thinking of how it could work.

1:00 Later on in this video,

1:01 I'm going to show you at least one way you can think about doing it.

1:04 For the moment, though, I imagine there's a more burning question,

1:07 which is that you might be wondering why

1:10 a mathematician would care about combing fluffy spheres like this.

1:13 And of course the answer is, they don't.

1:15 The name and the informal statement are a bit tongue-in-cheek.

1:18 I will of course share the more formal statement,

1:20 and in fact my real reason for making this video

1:22 is to share an unusually elegant way to prove it,

1:25 one that I think will delight any math lovers.

1:27 But before any of that, let's motivate things with an example

1:31 of the kind of situation where these fluffy spheres naturally arise in practice,

1:35 in a context that initially seems completely unrelated.

1:43 Okay, so imagine that you are a game developer,

1:46 and you're programming some game where you have a 3D model of an airplane,

1:50 and what you want is to be able to take

1:53 an arbitrary trajectory for this plane to fly along,

1:55 presumably something user-defined,

1:56 and your job is to write a function that orients

2:00 the plane correctly as it moves along that trajectory.

2:04 So, for example, let's say you're at a given point on some given trajectory.

2:08 You obviously want to move the center of the model to be on that point,

2:11 but you're left with ambiguity on how it should be rotated in 3D space.

2:15 The obvious constraint here is that you know the nose

2:18 of that plane should point along the tangent vector of the path,

2:22 but even that leaves some ambiguity.

2:24 How is the plane rotated about this nose-to-tail axis?

2:29 One way you could think about defining that last degree of freedom is

2:32 in terms of where this perpendicular

2:34 vector along the left wing direction points.

2:36 The task for you, as the programmer of this video game,

2:40 is to figure out what that perpendicular wing direction

2:43 should be at every single point along a given trajectory.

2:48 Now, there is a correct way to do this, which

2:50 would involve calculating the second derivative of the trajectory,

2:53 working out how to get this to match

2:55 the lift force from the wings together with gravity,

2:57 but maybe that seems a little complicated right now.

3:00 Resourceful and lazy programmer that you are, you might think, hey,

3:04 is there just some reasonable thing I can do to choose

3:07 some wing direction that's perpendicular to a given velocity vector,

3:10 the heading direction of the plane?

3:14 Here's one way you might think about it.

3:15 All of the possible ways this plane could point in space,

3:18 the various heading directions that I'm colouring in red,

3:21 make up the points of a unit sphere.

3:24 What you want is to write a function that takes in a given

3:27 vector on this sphere and returns some choice for a vector perpendicular to it,

3:31 the ones that I'm colouring in pink.

3:33 The only real constraint is that you want this association to be continuous,

3:37 otherwise it would mean the plane's orientation could sharply jump,

3:40 which would be a very clear glitch in the game.

3:43 And if you know nothing else,

3:45 it really feels like this should be a possible task.

3:48 After all, for a given heading direction, you are not starved for choices.

3:51 You have infinitely many wing directions to choose

3:54 from, an entire circle's worth of options.

3:57 So how hard could it be to make some reasonable

4:00 choice for every point on the sphere that varies continuously?

4:04 You might see where I'm going with this.

4:05 Choosing a perpendicular direction like this is equivalent to choosing

4:10 a unit tangent vector to that point of the sphere.

4:13 So if you're assigning a specific perpendicular

4:16 to every possible direction that plane could be pointed,

4:19 that's basically the same thing as defining

4:21 a tangent vector at every point on a sphere.

4:25 Now this is starting to look a little bit more like a hairy ball.

4:29 And in fact, now is as good a time as any to step

4:31 back and describe what the hairy ball theorem actually says more formally.

4:35 If you have a sphere and you choose some point

4:37 on that sphere and a plane tangent to the sphere at that point,

4:41 then any vector that you choose within that plane,

4:44 which is rooted at that point, is called a tangent vector of the sphere.

4:48 If you assign a tangent vector to every single point on the sphere,

4:52 one for each possible tangent plane, we call it a vector field on the sphere.

4:57 And whenever you're drawing vector fields like this, it's always standard

5:00 to scale the vectors down so that you can avoid clutter.

5:03 And the other thing to keep in mind is that even though

5:06 an illustration like this necessarily only shows a finite set of vectors,

5:09 rooted at a finite set of points on the sphere,

5:12 of course a vector field consists of infinitely many vectors,

5:15 one for every single point on the continuous surface.

5:18 So the theorem, our main character for today,

5:21 states that if your vector field is continuous,

5:23 meaning there are no sudden jumps in its direction,

5:26 then it must have at least one point with a null vector,

5:30 meaning a vector whose length is zero.

5:33 For example, look back at our 3D model case.

5:36 The function that I was using for many of the animations there was

5:39 essentially trying to keep the roof of the plane pointed as upward as possible.

5:44 And when you express this function as a vector field, where again,

5:47 each possible direction for the nose of the plane

5:49 is thought of as a point on the sphere,

5:52 and each corresponding wing direction is thought

5:54 of as a tangent vector at that point of the sphere,

5:57 then it turns out that function I was using

5:59 gives a vector field that spirals around the vertical axis.

6:03 This actually does give reasonable enough animations in most cases,

6:07 but the problem is that it has a discontinuity at the poles.

6:10 So if ever I let the plane point

6:12 straight up or straight down using this function,

6:14 you would get this glitching behavior as it passes through that direction.

6:19 Now, if you're just a programmer messing around with this, you might

6:22 think you can tweak things to avoid glitches like that, but actually,

6:25 the Harry Ball Theorem guarantees, no matter how clever you are,

6:28 you are doomed to have some direction producing this kind of glitch.

6:32 So for robust animations, you cannot simply use the direction of the nose

6:36 of the plane to determine its full orientation.

6:39 You have no choice but to step back and incorporate

6:42 more information from the trajectory than the velocity vector alone.

6:47 As another example, think about the wind velocity at every point on the Earth,

6:51 say, at some constant altitude.

6:53 A pretty reasonable assumption is that wind velocity varies continuously,

6:56 so the Harry Ball Theorem should apply.

6:59 The wind pattern I'm animating here

7:01 is completely unrealistic from a meteorological standpoint,

7:03 but the point is that whatever wind pattern you dream up, realistic or not,

7:08 the Harry Ball Theorem is going to guarantee that there is always one place

7:12 on the Earth for a given altitude where the wind velocity is exactly zero.

7:17 Now, if we're being pedantic, you could say atmosphere is three-dimensional,

7:20 so the more accurate statement would be that the component

7:23 of wind velocity parallel to the ground is zero.

7:26 You know, it could be going straight up or straight down,

7:28 but still, it is kind of counterintuitive.

7:31 A slightly more pragmatic example is if you want a radio

7:34 signal that is completely identical in every direction of 3D space,

7:38 in the sense that everyone a given distance

7:40 away from the source receives an identical radio wave,

7:43 same phase and amplitude at all points of time.

7:47 That might seem like a reasonable objective,

7:48 but if you know a little bit about electromagnetic waves,

7:51 you'll know that they are oscillations in two distinct vector fields,

7:55 the electric and the magnetic fields specifically.

7:58 Importantly, the direction of oscillation for each one

8:01 of these fields is always perpendicular to the direction of propagation,

8:05 at least far away from the source.

8:08 So, think about what that means.

8:10 At a given distance away from the source,

8:12 either one of these fields looks like a tangent vector field on the sphere,

8:16 and the hairy ball theorem states at least one

8:19 point of that vector field has to be zero,

8:22 so the only way to have a completely identical signal in every

8:25 direction of 3D space is for the signal itself to be zero,

8:29 which presumably defeats the point.

8:32 I bring up these examples just to say that this seemingly

8:35 playful fact about fluffy spheres really does pop up in unusual places,

8:38 but what I really want to do with this video, the fun that I want to have,

8:42 is to let you explore this idea the way that a pure mathematician might.

8:46 First, that puzzle that I mentioned at the start actually gives a really great

8:50 way to flex your mind and see how what feels obvious is not always true.

8:55 And then after that, I want to share a completely

8:57 beautiful proof that explains why this theorem is true.

9:01 So, to the puzzle.

9:03 I don't know about you, but when I was first playing around

9:06 with this idea in my mind to build some intuition,

9:08 it was really not at all obvious that reducing

9:10 to a single null point is even possible.

9:13 For most of the vector fields I could dream up,

9:15 you get at least one swirl going one way, and another swirl going the other way.

9:20 Or maybe a source at one point, and a sink at another.

9:24 This makes it really tempting to suggest

9:26 that there should be some universal law about needing

9:29 at least two different null points with something

9:32 opposite about them that has to cancel out.

9:35 Something like the north and south poles of a magnet.

9:38 Tempting as that is, with a little cleverness,

9:40 it is possible to get just one null point.

9:43 And a nice way to define this is

9:46 by using something known as a stereographic projection,

9:48 where every point on the sphere, except for the north pole,

9:52 gets mapped to a unique point on the xy-plane.

9:56 The way this works is very pretty.

9:57 You imagine a light shining from that north pole,

10:00 and every ray of light that passes some point on the sphere

10:04 also hits one and only one point of the xy-plane.

10:08 And it goes the other way around too.

10:09 Every point of the xy-plane corresponds to a unique point on that sphere,

10:13 meaning that plane can get mapped onto every point of the sphere,

10:17 except for the north pole.

10:19 This is a favorite mapping among mathematicians,

10:22 and the way we can use it here is to imagine

10:25 having some vector field on the xy-plane that's never zero.

10:28 That's simple enough to define.

10:30 You could just take a constant vector field,

10:32 always pointing one unit to the right.

10:34 If you project that vector field back onto the sphere,

10:37 this gives you something that's non-zero everywhere, except for the north pole.

10:43 Admittedly, the way I'm showing it right now makes it kind

10:45 of hard to parse what exactly is going on around that north pole.

10:49 The basic reason is that if you take a uniform sample of points on the plane,

10:52 they get infinitely dense around that north pole under this projection.

10:57 So let me show you a second way I could illustrate things,

11:00 which also, by the way,

11:01 lends itself to a more rigorous definition for what I even

11:04 mean by projecting a vector field onto a sphere like this.

11:06 Imagine a fluid flowing on the plane with a uniform

11:10 velocity one unit per second to the right,

11:12 and then consider what the projection of each particle

11:15 of that fluid would look like on the sphere during its motion.

11:19 If you take the velocity vectors for those projected particles on the sphere,

11:24 that defines the vector field that I'm talking about.

11:28 And illustrated this way, you can really nicely see how the flow

11:32 lines all form perfect circles on the sphere,

11:35 all of which are mutually tangent with the velocity of zero at that north pole.

11:40 It really is a lovely projection.

11:42 The point is, even if initial mental play and intuition might suggest

11:46 that vector fields on a sphere have to have at least two null points,

11:50 a little creativity can give you a field that just has one.

11:54 So, how do you know that it stops there?

11:57 How can you rigorously prove that no matter how clever and creative you are,

12:02 it is simply not possible to define a continuous vector field

12:05 without forcing at least one point to have a zero vector?

12:10 This is where the real cleverness kicks in.

12:12 The way that we're going to approach this is with a proof by contradiction,

12:16 meaning you will assume that such a non-zero

12:18 vector field on the sphere is possible,

12:20 and then deduce that something impossible would have to follow.

12:24 Like I said, the argument I want to show is just really beautiful,

12:27 and I think it's made all the more so if

12:29 you feel like it's something you could have discovered for yourself.

12:33 So, as always, please do pause and ponder

12:34 whenever you feel like you see the key idea.

12:37 This argument is not my own,

12:39 it came my way via the mathematician Senia Sheydvasser,

12:41 who also very kindly put together

12:43 the following animation to illustrate the core idea.

12:46 The basic outline is that if such

12:49 a non-zero continuous vector field really did exist,

12:52 you could use it to create a continuous deformation

12:55 of the sphere that turns that sphere inside out.

12:58 And then we're going to prove why it's

13:00 actually impossible to turn a sphere inside out,

13:02 at least in a certain manner of speaking.

13:05 It's at this point that viewers of classic

13:07 math YouTube will be yelling at their screens,

13:09 but bear with me, I promise I will get to that.

13:15 Okay, so this continuous deformation is a little weird to define,

13:17 but here's how it works.

13:19 Imagine that your sphere is centered at the origin

13:22 for some coordinate system in 3D space.

13:24 For a given point on that sphere,

13:26 consider the vector attached to that point, the one from our vector field.

13:31 If you slice the sphere along a plane,

13:33 which is defined by that vector and the radial line to the origin,

13:37 the plane intersects the sphere at a certain great circle,

13:40 meaning a circle that's also centered at the origin.

13:44 What you're going to do is let that point of the sphere move along

13:48 this circle in the direction of that initial

13:51 vector until it gets precisely halfway around.

13:54 Just to be clear, I'm not saying that it

13:56 flows along the general vector field of the sphere.

13:58 Its motion is entirely determined just by the one vector that it started out on.

14:04 The two important facts to highlight are that it

14:06 ends up on the negative of where it started,

14:09 and then also because its motion is entirely defined by what vector

14:13 it started on, and because we're assuming the whole vector field is continuous,

14:17 nearby points are going to have nearby trajectories.

14:20 Right now, I'm just showing you one

14:22 point moving along its prescribed half-circle path,

14:24 but we could just as well highlight a handful of other points on the sphere,

14:28 each of which has its own vector associated with it,

14:31 each of which defines a great circle to walk along,

14:34 and you could watch all of those points wander along the assigned paths.

14:38 Now remember, we're assuming that the vector field is non-zero everywhere.

14:42 We hope to contradict that, but that's the assumption.

14:44 And what that means is that every single point

14:47 on the infinite continuous sphere has a similarly well-defined trajectory.

14:52 So naturally, you want to see what it looks

14:53 like for the entire sphere to undergo that motion.

14:56 But it's at this point that animations become a little tricky,

14:59 because of something intrinsically paradoxical

15:01 about illustrating a proof by contradiction.

15:04 Think about it.

15:04 We want to show what this motion looks like,

15:06 as defined by some hypothetical vector field that is non-zero everywhere.

15:11 But of course, the whole point is that no such vector field exists.

15:14 As the next best thing,

15:15 we're going to use that special vector field that we just defined,

15:18 only a single null point at the North Pole.

15:21 That way, if we chop away the North Pole, we can at least see the kind of thing

15:25 that this motion would do to most of the sphere,

15:27 even if there's something impossible about

15:29 this being applied to all of the sphere.

15:32 I'll go ahead and remove the vectors themselves to avoid clutter.

15:35 And this right here is what it looks like

15:38 for every point on the surface to undergo that bizarre,

15:41 specially defined motion, each one marching along its own half-circle path,

15:46 defined by whatever vector it started on.

15:50 And actually, that's kind of confusing to follow.

15:53 So let me roll back the clock a bit here,

15:55 where you see that the whole sphere ends up awkwardly crossing through itself.

15:58 To clarify things, we might perturb the motion a little

16:01 by letting the radius of the points vary during the motion.

16:05 And it also makes things clear if we widen out that hole on the top.

16:09 This makes it much, much easier to follow,

16:11 at least for this subset of the sphere,

16:14 you can clearly see two important features of the motion.

16:18 Number one, the sphere gets turned inside out.

16:22 Number two, at no point in time does any part of the sphere cross the origin.

16:26 And that should make sense.

16:27 Each individual point is just following a half-circle centered at the origin,

16:31 so of course it never passes through the origin.

16:34 Those are the two key ingredients.

16:36 For our proof, what we want to say

16:39 is that these two facts are somehow incompatible.

16:42 Before we can do that though, we need to linger on this first point.

16:45 Why exactly does this motion turn the sphere inside out?

16:49 And actually, what do we even mean by the phrase inside out here?

16:52 And in fact, let me start with an even more basic question.

16:55 If you're standing at some point on the sphere,

16:57 how do you know which way is outside and which way is inside?

17:01 I realize that might sound like a very dumb question.

17:04 You might say just look at whichever way is pointed away from the origin.

17:08 What's wrong with you?

17:09 The real conundrum here though comes from the fact that we intend to let

17:13 the surface warp and deform and get all manipulated in some crazy way.

17:17 So really what you want is a clear notion of what we mean by inside and outside

17:22 that remains clear even after you manipulate and massage

17:25 and contort the whole surface however you dream up.

17:29 The easiest way to do this I think is

17:31 going to be something familiar to any graphics programmers,

17:34 which is that you start by assigning a coordinate system to the sphere,

17:38 something like our usual notion of latitude and longitude.

17:42 The image you should have in your mind is that every point

17:44 of the sphere has a little label attached to it with a pair of numbers.

17:49 And importantly, these labels could follow along during

17:52 any motion or manipulation you apply to the sphere.

17:56 Around a given point,

17:57 consider the direction of increasing longitude and constant

18:01 latitude and draw a tangent vector in that direction,

18:04 and then draw a line of increasing latitude with constant

18:07 longitude and draw a tangent vector in that direction.

18:11 From here, the way we define orientation is

18:13 using what's known as the right hand rule.

18:16 You can point your index finger along that first

18:19 vector and your middle finger along that second vector,

18:21 and then when you stick out your thumb, it'll be perpendicular to both.

18:25 Notice using our right hand, the thumb is pointed outside.

18:29 And in fact, this is how we are going to define what

18:32 we even mean by outside with respect to the given coordinate system.

18:37 Doing this at every point,

18:38 you get what are known in the business as unit normal vectors.

18:42 As I referenced, these are very important in computer graphics,

18:44 where for example they let you compute how

18:46 light should reflect off of a given surface.

18:49 And for our story, the thing we care about is how,

18:51 no matter how you manipulate or warp the surface,

18:54 because that coordinate system you give to it

18:56 can kind of come along for the ride,

18:58 you can always play this game of pointing your index

19:00 finger along the direction where the first coordinate increases,

19:03 and your middle finger along

19:05 the direction where that second coordinate increases,

19:07 and sticking out your thumb.

19:10 It's otherwise surprisingly tricky to define what you mean by inside

19:13 and outside in a way that naturally follows along for any function.

19:18 So, why are we doing this?

19:20 Think now about that strange deformation induced

19:22 by a vector field on the sphere.

19:25 How can we conclude beyond any doubt that this must turn the sphere inside out?

19:30 Well, remember how each individual point starting at p ends up at negative p?

19:35 The much more straightforward way to get there, literally,

19:38 would be to reflect through the origin, like this.

19:42 So consider that picture that we just had of an example point,

19:45 together with the oriented lines of latitude and longitude passing through it.

19:49 Notice what it looks like if we let every

19:52 point in that diagram move over to its negative.

19:55 Again, you might imagine all the coordinate

19:57 labels of the surface riding along with it,

20:00 so when you play the same game of pointing

20:02 your index finger in the direction where that first coordinate increases,

20:06 and your middle finger in the direction

20:08 where that second coordinate increases, now, after everything has been negated,

20:12 notice that your thumb is pointing towards the origin instead of away.

20:17 This is all to say, the function that maps every point p

20:20 of a sphere to its negative necessarily turns the sphere inside out,

20:24 in the sense of reversing orientation the way we just defined it.

20:29 In particular, that very weird deformation

20:31 that we described in terms of a hypothetical

20:34 vector field must reverse orientation because it's

20:37 mapping each point p to negative p.

20:40 You can also see this effect with a much

20:42 simpler motion that gets us to the same final place.

20:45 Imagine rotating the sphere 180 degrees around the z-axis,

20:48 and then reflecting through the xy-plane.

20:51 That results in every point p landing on its negative,

20:54 and notice how all the unit normal vectors

20:57 that started pointing outward end up pointing inward,

21:00 and as it's rendered here with a blue exterior and a brown interior,

21:04 those two colors end up getting swapped.

21:07 And it's at this point that viewers of classic

21:10 math videos all might be bringing to mind an absolute

21:13 banger of a video that was produced in 1994

21:15 by the Geometry Center at the University of Minnesota.

21:18 This is really one of the true classics in all of math exposition.

21:22 It walks through this mind-blowing way to turn

21:24 a sphere inside out using a certain continuous deformation.

21:28 The reason I bring this up for any of you who watched

21:30 that is to say that the context there was a little bit different.

21:33 The phenomenon that they were trying to avoid was

21:36 creating cusps and creases on the sphere during the process.

21:40 But over here, for our purposes, we don't really care about that.

21:43 However, there is one feature of our bizarre

21:46 vector field-induced deformation that really would be impossible.

21:52 No point of the sphere ever passes through the origin.

21:55 And there is a very beautiful way to see why turning

21:58 a sphere inside out without crossing the origin just could never happen.

22:03 Maybe the most fun way to illustrate this is with a physical model.

22:06 Imagine a fountain at the origin spewing out water uniformly in all directions,

22:11 say at a rate of one liter every second.

22:14 The way I'm animating it here is

22:16 with a bunch of droplets spewing away, but in principle,

22:19 I want you to think of this as a continuous flow of an incompressible fluid,

22:23 something uniform in all directions, and with that incompressibility,

22:27 we'll imagine that the density of water through all of space stays constant.

22:32 That's important.

22:33 If you have some oriented surface,

22:34 something like our sphere with its unit normal vectors,

22:37 you can measure how much water is flowing through that surface per unit time.

22:42 Physicists have a special name for this.

22:44 They call it the flux, where on a given patch of area,

22:47 the flux measures how much water passes through it per second,

22:50 and you count it as positive when the water flows from inside to outside,

22:54 aligned with the unit normal vectors of the surface,

22:57 whereas flux would be negative if it's going the other way,

23:01 going against those normal vectors.

23:03 When you add all of this up over the whole surface,

23:06 this gives you the total flux,

23:08 and the key observation is that this total flux has

23:11 to match the amount of water being produced inside the surface,

23:14 that one liter per second, and the cool part is that this will remain true even

23:19 if you warp or deform the sphere just a little bit.

23:22 That total flux stays at one liter per second,

23:25 even if the flux through a particular patch of area changes during the process,

23:29 and the basic reason is that we're treating the water

23:32 as an incompressible fluid with a constant uniform density through space,

23:36 so every little bit of water produced at the origin has

23:39 to be cancelled out by one that is exiting the surface.

23:44 Importantly, for what I just said to be true,

23:46 we have to be counting flux with a sign.

23:49 For example, let's say you warp the sphere so that it kind

23:52 of folds over itself like this, then you'll notice along a certain line,

23:56 the water goes out of the surface,

23:57 and then back into it, and then back out again.

24:01 So you would want to count this flux

24:03 as positive whenever it goes from inside to outside,

24:05 but negative when it goes from outside to inside,

24:08 so that you're not counting those particles three different times.

24:12 And from here, you can maybe see

24:13 the key point I'm getting at towards our contradiction.

24:16 The only way that you could ever change the net flux through

24:19 a surface like this is if part of that surface crosses through the origin.

24:24 For example, if you pull it over to the side

24:25 so that it doesn't include the source at all, the net flux would be zero.

24:29 There are as many water molecules flowing in as there are flowing out.

24:34 So with that in mind, think about everything we've been talking about.

24:38 If you could define this non-zero vector field on a sphere

24:41 that lets you create this bizarre deformation that turns the sphere inside out,

24:45 what happens to the flux?

24:48 Well, what we mean by turning the sphere inside out is

24:51 that all the unit normal vectors end up pointing inside instead of outside.

24:56 So for any particular patch of area,

24:57 at some point the flux through it transitions from being positive to negative,

25:01 and overall, at the end,

25:03 that total flux would have to end at negative one liters per second.

25:07 But at the same time, if it never crosses the origin,

25:10 the net flux can never change.

25:12 It starts at positive one, so it has to end at positive one.

25:15 This is the contradiction.

25:17 No deformation with these two properties can possibly exist.

25:21 And there you have it.

25:22 A non-zero continuous vector field on the sphere would be impossible.

25:26 You truly cannot comb a hairy ball.

25:29 I don't know about you, but I think that's so beautiful.

25:32 Very often topology is this game where seemingly intuitive

25:35 facts have these surprising but kind of frustrating counter-examples,

25:38 but the real insights and the creativity often

25:41 comes from the other side of the coin,

25:43 where you take something that seems intuitive,

25:45 but you find the construction that really justifies why it's fundamentally true.

25:50 Now there is more to say about making

25:52 the argument I just showed you fully rigorous,

25:54 and also more to say about how this whole

25:56 thing does and does not generalize to other dimensions.

25:59 But before that, if you'll indulge me

26:01 in shifting gears entirely for a minute here,

26:03 I want to tell you about an experimental

26:05 new thing that I'm starting up this year,

26:06 which I'm thinking of as a kind of virtual career fair.

26:09 If you go to 3b1b.co/talent,

26:11 what you'll find is a set of companies that have two things in common.

26:16 The first one, essentially by definition,

26:18 is that they are interested in recruiting from this audience.

26:22 They value the kind of mathematical and technical

26:24 curiosity clearly to be found in someone like you,

26:27 who's watching a video like this for fun.

26:29 If you go and explore the page and see the kind of puzzles

26:32 and challenges and technical work that each one has chosen to share with you,

26:35 you'll pretty quickly get a sense of the shared values here.

26:39 The second thing they have in common is

26:40 that the people working there really love what they do.

26:43 And this one's important to me.

26:44 I was pretty careful about it while setting this whole thing up,

26:47 because all of this only makes sense

26:49 to do if it's actually valuable to the audience.

26:51 So I took some time to sit down and chat with the technical teams at each group,

26:55 and inclusion only really made sense if they clearly like what they do.

26:58 Pretty universally, a core reason that the people enjoyed their work

27:01 was out of a very sincere respect for all of their teammates.

27:05 In the hopes of making this relevant to you, whoever you might be,

27:08 there is a range of job types available across a range of industries,

27:12 including senior roles, new careers,

27:14 internships, or even part-time tutoring gigs.

27:17 For broader context, if you're curious,

27:18 I recorded a whole video on the second channel

27:21 explaining why I started this, what the deal is.

27:24 One thing that I mention over there is

27:25 how I'm looking to make a few hires myself, and whenever this is the case,

27:29 I will include my own page up on this virtual career fair,

27:31 where you can go and find the description

27:33 of what I'm looking for and the applications.

27:36 The whole page will stay updated as time goes

27:38 on, so even if you're not looking for a job now,

27:40 but you are sometime in the future, be sure to check it out.

27:44 Alright, so back to the hairy ball theorem.

27:46 The argument at the end rested on this whole idea of flux,

27:49 which admittedly is a little hand-wavy without further details,

27:52 so I'll leave up on screen a set of exercises outlining one way that you could

27:56 make this more rigorous if you have

27:58 a background with multivariable calculus and the divergence theorem.

28:01 There is another, deeper way to get at the same basic idea,

28:04 using something called a homology group,

28:06 but that one is certainly beyond the scope for today.

28:09 The last point I want to leave you

28:11 pondering on is the nature of other dimensions.

28:13 It's not hard to see that you can comb down the hairs on a fluffy circle,

28:18 even though you can't do it for a sphere,

28:20 and in general the rule is that spheres

28:22 in all the even dimensions can be combed down,

28:25 but those in all the odd-numbered dimensions cannot.

28:28 Now what I like about the argument that we just talked about for this whole

28:32 video is that it offers a pretty direct clue for why that's the rule,

28:35 at least if you're comfortable with the notion of orientation.

28:39 In all the even dimensions, the function that maps a point

28:42 to its negative is an orientation-preserving function,

28:46 whereas in all the odd dimensions, that's an orientation-reversing function.

28:50 You might enjoy taking a moment to pause and ponder on why

28:54 that means the proof we just outlined works in all of the odd dimensions,

28:59 and even though it doesn't explicitly tell you

29:02 that nothing could work in the even dimensions,

29:05 it's a fun puzzle to see if you can construct an explicit

29:09 example of a non-zero vector field in all of those even dimensions.

29:13 For example, how do you comb down the hairs on a hypersphere in four dimensions.