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59: A Good Source of Fibers (Fiber Bundles)
458 Plays3 years ago
Mathematics is full of all sorts of objects that can be difficult to comprehend. For example, if we take a slip of paper and glue it to itself, we can get a ring. If we turn it a half turn before gluing it to itself, we get what's called a Möbius strip, which has only one side twice the length of the paper. If we glue the edges of the Möbius strip to each other, and make a tube, you'll run into trouble in three dimensions, because the object that this would make is called a Klein flask, and can only exist in four dimensions. So what is a fiber? What can fiber bundles teach us about higher dimensional objects?
All of this, and more, on this episode of Breaking Math.
[Featuring: Sofía Baca, Meryl Flaherty]
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Transcript
Episode Licensing & Introduction
00:00:00
Speaker
This episode is distributed under a Creative Commons Attribution Share Like 4.0 international license. For more information, visit creativecommons.org.
Mobius Strip & Klein Flask Explanation
00:00:13
Speaker
Mathematics is full of all sorts of objects that can be difficult to comprehend. For example, if we take a slip of paper and glue it to itself, we can get a ring. If we turn it a half turn before gluing it to ourselves, we could get what's called a Mobius strip, which has only one side twice the length of the paper. If we try to glue the edges of the Mobius strip to each other and make a tube, it runs to trouble in three dimensions because this object that this would make is called a Klein flask and can only exist in four dimensions.
Understanding Fiber Bundles
00:00:40
Speaker
These are the type of objects that we're talking about.
00:00:43
Speaker
So what is a fiber? What can fiber bundles teach us about higher dimensional objects? And what is the Hopf vibration? All this and more on this episode of Breaking Math. Episode 59, a good source of fibers.
00:01:02
Speaker
I'm Sophia. And I'm Meryl. And you're listening to Breaking Math. Today, we're going to be talking about fibers. But before I talk to you about that, I'm going to talk to you about our plugs. If you want to get the poster, it's a cool poster, 24 by 36 inches. So it's a pretty big poster. It's matte. It's about tensors, which are like higher dimensional vectors kind of and also derivatives and also the cool.
00:01:24
Speaker
We talk about Einstein's general theory of relativity on the poster, and it's generally pretty cool. Get it for your nerdy friends, they can put it on their walls, your professors, whatever. If you want to buy that poster, you can buy it on Facebook for $19.65. That's $15.15 pie to the E dollars plus $4.50 shipping and handling.
00:01:43
Speaker
If you want to donate to us monthly, for $1 a month, you can have our outlines and our episodes without ads in them. If you also donate about $23, you can get the poster there, just as like a thank you.
Community Engagement & Episode Focus
00:01:56
Speaker
You can also find updates about this show on Facebook, on Twitter, at breakingmathpod, our website, breakingmathpodcast.app, and you can email us at breakingmathpodcast.gmail.com with ideas, comments, questions, and corrections.
00:02:09
Speaker
And also check out our sister podcast, Turing Rabbit Holes, and they're on YouTube. So today we're going to be studying fiber bundles, right?
00:02:17
Speaker
Yeah. So let's just give an overview of what we're going to be doing in this episode. We're going to give the motivation for studying fiber bundles. We're going to talk about some necessary concepts like maps, homeomorphisms, and how they relate to the meat of the episode, fiber, fiber bundles. And we're going to talk about what you can do with fiber bundles. And then finally at the end, we're going to talk about a really cool fiber bundle application called the HOPFibration.
00:02:45
Speaker
So Meryl, what do you think about fiber bundles? So fiber bundles are pretty much just a way of describing these complex, sometimes higher dimensional shapes in terms of products of lower dimensional shapes.
00:03:01
Speaker
So what is a product of a space?
Mathematical Concepts: Product Spaces & Set Theory
00:03:03
Speaker
For example, what would that be like? So if you have a space A or just a set A and a set B, then if you take a point from set A and a point from set B and you put them together, then so that's a coordinate, right? And so what the product is, it's the set of all of those coordinates together.
00:03:26
Speaker
Yeah, and so like, for example, the product of the real line, which is all numbers from negative infinity to infinity, pretty much, I mean, sometimes it doesn't include those two, with itself is the two-dimensional plane, right? Yeah, the Cartesian plane. And that's right, because it has two coordinates. So yeah, before we start getting too into fiber bundles, let's dive in. So before you already mentioned that you could study complicated objects in terms of simpler ones with fiber bundles, right?
00:03:56
Speaker
Yeah, and so an example of that is, so one of topologists favorite things to talk about is the Mobius strip. And if you've ever been friends with a topologist, then they've taken you like over to their place after grabbing drinks, and they'll grab a piece of paper. And so they'll take this strip, they'll twist it in half and then put it back together.
00:04:20
Speaker
and show how you can draw a line on one side of it, and then it'll go to the other side, and both sides are the same sides, and that's the Mobius strip. Yeah, and it's kind of a cool thing, because I've heard that Mobius strips have been used as conveyor belts in some applications and stuff, but more to the point, though, they're just really cool mathematical objects, but they're a little elusive to study, right? How do we model these? And fiber bundles seem to give us a way of doing that.
00:04:50
Speaker
Yeah, because, so you'll notice the Mobius strip, it has a half twist in it, but you could also glue those ends of that piece of paper together such that it just makes a ring, a cylinder. So clearly there's some sort of difference and what we're exploring with fiber bundles is what that difference really is.
00:05:09
Speaker
Yeah, in some way, in somewhat, interestingly, we seem to have to reduce a Klein mobius strip to a cylinder and a Klein flask, for example, to a donut or a torus. But we're going to talk about that in the next few sections. All right, so we've got to start with the concept of a set. So what is
Maps & Functions in Mathematics
00:05:33
Speaker
a set? A set can be like, I mean, name some nouns.
00:05:37
Speaker
Yeah. So it's just a collection of things. So let's say that we had a set of fruits. We could have apple, pear, grape, orange, and that's a set of fruits. Yeah. And also you could have the set of all possible oranges, right? Yeah. Which would be every orange that could possibly be made. So it's, it's basically in the way that we think about sets often is by belonging, right? Yeah. So we could say, for instance, that grape belongs to the set fruits.
00:06:03
Speaker
Yeah, and fruits could be a subset of fruits, or vegetables even. I mean, not vegetables, fruits. Although vegetables, not a botanical term, apparently. So this is basically what a set is. So the real numbers, they're a set, right? Yeah. And so we can say that any number on the real line, let's say pi, belongs to the reals.
00:06:28
Speaker
It's something that you can belong to, a set of things or a consistent set of things. There are rules that you have to follow with this. With naive set theory, we get things like paradoxes. For example, the set of all sets that don't contain themselves both contains itself and doesn't contain itself.
00:06:48
Speaker
So obviously we can't just go crazy with it, but for what we're talking about today, sets are pretty well defined. We're going to be talking about mostly sets of continuous things. So we're going to be talking about sets that are called manifold.
Trivial Fiber Bundles & Trivializations
00:07:02
Speaker
Since we're going to be talking about it quite a bit, Meryl, can you tell us what a manifold is?
00:07:07
Speaker
So manifold is a particular kind of set that locally represents Euclidean space. So let's say that we have a two-dimensional manifold. This could be something like a sphere or a torus or even a Klein bottle or a Mobius strip.
00:07:24
Speaker
And so if you take any particular point on the set, then what you can do is you can map it to, I mentioned the product of the real numbers with itself, so you can map it to two dimensional real space. And a familiar example of a manifold might be just the space that we live in every day. I mean, at the large scale, it's largely non euclidean, right? But like, folded time and all that, but we exist in the manifold that like, we could consider it three dimensions, really.
00:07:54
Speaker
Yeah, so we call this locally Euclidean. And that just means that locally, this looks like Euclidean space. So locally, what that means is, so we look around us, we can call that a neighborhood. And in that neighborhood around us, we see three dimensional space. So the product of the real numbers with itself three times
00:08:16
Speaker
Yeah, because like we said, like, you know, the product of like, if we have to, if we have the set red and blue, and then we have the set apple and orange, then we, if we pick the product, then we'll get another set, which contains the pairs, red, orange,
00:08:35
Speaker
red apple, blue orange, blue apple. So if you think about it, the set that is all the real numbers, the product of the real numbers with themselves would have every combination of two real numbers. Take another product and it's every combination of three real numbers.
00:08:52
Speaker
And this can be done more and more to get into whatever space you want. And we use sometimes a notation. It's like an R with like a like two lines instead of one, a vertical line and then a number like an exponent saying how many times. Yeah. So we actually have a name for so like that R with a line drawn through it. It's actually called Blackboard bold.
00:09:12
Speaker
Oh, cool. Yeah. So it's black, a blackboard bolded R with an exponent of the space that you're talking about. And if you want to talk about a general idea of space, you can use a variable there like N or new or whatever. Yeah. So we use like R for the reals Q for the quotients. And also if any of you out there are using latex, then you can, you know, write this as backslash math BB.
00:09:36
Speaker
Yeah, LaTeX is a typesetting system that we used in our first paper that's on the site. We have not gotten to doing any of the other papers, but hopefully that'll change. But anyway, now that we have the idea of a set, let's talk about what a map is.
00:09:56
Speaker
So a map, the simplest way to think of it is that you take an element of one set and you effectively you map it to another element in another set. So you take in a thing, you get out a thing in another set.
00:10:12
Speaker
Yeah. So like, for example, let's say everyone has a favorite food, right? The set
Exploring the Klein Bottle & Hopf Vibration
00:10:16
Speaker
that everyone's favorite food belongs to would belong to the set of foods, right? Yeah. So a map from the set of people to the set of foods could describe someone's, everyone's favorite food or their least favorite food or whatever. Yeah. So like, for example, this map would take, let's say I put myself into the map and so the output of this map would be, let's say steak.
00:10:39
Speaker
Yeah, and so let's call this function f. So f of Meryl is steak, where f is the function that gives your favorite food. So let's say we have a map from the real numbers to the real numbers, right? We could do something as simple as doubling things, right? So we take all the real numbers and we double them. So we have a real number 2 and that maps to 4. 13 maps to 26, etc. Pi maps to 2pi. It maps to tau actually. Jokes on you my birthday's tau day.
00:11:07
Speaker
I think the joke is on people who use pi. There's a debate in math of where 2 pi is written as tau, and it makes certain trigonometric stuff a lot less annoying to deal with. You don't have to constantly divide by 2. We might do an episode about tau actually sometime.
00:11:24
Speaker
But we're going to talk about maps a little more. So what's the difference between a function and a map? So a function is a type of map. So we put a few restrictions on it. A map can map an object to any number. So a map can take something as an input and give you any number of outputs. A function puts a few restrictions on that. For one thing, we have that for, let's say we have a function f from a to b.
00:11:53
Speaker
then we know that this is well-defined, which means that f takes in anything in A. So its input is the whole set of A, and we know that for any item in A that we have exactly one output in B. Yeah, and so from this definition, we don't know that everything in B can be reached, right?
00:12:15
Speaker
Let's say everybody's favorite food is steak, right? Let's say, so we have the set of all the foods, but then in this function,
Three-Sphere & Stereographic Projection Discussion
00:12:22
Speaker
it would only map to steak, right? Yeah. And so let's say that there were some food that was nobody's favorite food. Yeah. What is nobody's favorite food? Pretty much everything could be somebody's favorite food. Like I was trying to think liverwurst sandwiches. You know, let's just pretend that nobody's favorite food is liverwurst sandwiches. Yeah. Okay.
00:12:42
Speaker
So we can also have that there is something in B the set of foods that is nothing in a maps to it.
00:12:49
Speaker
Yeah, and if we're doing functions that are like, what an example of this is a sine function, right? The sine function is a function that if you go around a circle, it tells you, and the circle is centered at zero. So if you go around the circle and you go around the point zero, zero, this tells you the height. So, and it only goes from negative one to one, right? So there's plenty of numbers in the reals that it doesn't map to.
00:13:15
Speaker
Yeah, exactly. So we have different properties that we could talk about with functions. We have injectivity, surjectivity, and bijectivity, right? Yeah, also known as one-to-one, onto, or bijective would be one-to-one and onto. Yeah, so one-to, and a lot of times I see that called bijective because it's the only one that isn't kind of like confusing in terminology. So injective functions are one-to-one functions. Do you want to describe these?
00:13:42
Speaker
So a one to one or injective function is a function that so for any two things in our set, let's say set A is the set of people in a town and set B is everyone's bank account number. And we'll just assume there's no joint accounts. Everyone has their own bank account. Then if you have any two people in the town and they're not the same person, they'll have different bank account numbers.
00:14:12
Speaker
Yeah, so if you see two bank account
Deep Dive into Fiber Bundles & Complex Three-Spheres
00:14:14
Speaker
numbers that are different you know that the people associated with them are different people.
00:14:17
Speaker
Yeah, or if you get two bank account numbers and next to each other, you see they're the same number, they belong to the same one person. So the logical way of saying this is let's say that we have x and y in the set A, and we have a function f, then f is injective. If for a different x and y, we have different f of x and f of y.
00:14:43
Speaker
Yeah, that makes sense. And so now we have, now we want to talk about surjective or onto maps. So surjective maps map at least one thing for everything that they can be mapped to. So let's say we have a map of people to addresses, right? Yeah. So let's just assume that we have a town where there are no vacant homes then. So what we have is that so we can have that for every address in the town. There's a person who lives at that address.
00:15:13
Speaker
Yeah, and let's say people live in the same house, right? They cohabitate, then there are multiple people. If you have one address, you might have the same address for two different people. So that means it's not one-to-one, right?
00:15:27
Speaker
Right. So one to one and onto don't necessarily mean each other. And so what we mean by onto or surjective is that so for everything in a set B, so let's say that we have just a set B, a set A and a function f, then for every y in set B, we have some x in a such that f of x equals y.
00:15:54
Speaker
Yeah. So if a function is a one to one, but not onto, we can think about the bank accounts example, right? So everybody in the town obviously has a bank account number or because we're going to, or at least we're only considering the people who have bank account numbers. We'll assume that's the case, that everyone has a bank account number. Yeah. And it's a different account. Yeah. But if you've seen bank account numbers, they're like 10, 20, whatever digits long, right?
00:16:19
Speaker
Yeah, so there might be some bank account numbers that aren't used. So not every bank account number has a person associated with it. So it's not onto. Yeah, whereas the houses, every house has at least one person living in it. So a bijective function is something that is both surjective and injective. Or both onto and one to one. Yeah.
00:16:41
Speaker
So let's say that there's a classroom where there's 30 people in the class and there's 30 seats. Right. Yeah. So then, so bijective function would be that every person has their own seat and for every seat in the classroom, um, there's a person in it. And so what's cool about a bijective function is that you can say that it's injective or surjective both ways.
00:17:07
Speaker
Oh, yeah, right. You're meaning like, if you, if you take the inverse of the function, which reverses the way that it goes, well, bijective function, first of all, can be inverted, right? Yeah. And the inverse of that function is also bijective. Yeah, which is pretty cool. And finally, let's talk about the preimage of something.
00:17:25
Speaker
So first, we'll talk about, I guess, what an image is. So let's say that we have the set A. And so we'll call the image f of A. So what this is, is it's the collection of all sets in B, such that there's an x and a such that f of x is in that set.
00:17:44
Speaker
So basically just so first you just take a collection right of a certain amount of points or like elements from the set that is mapping from right and then you apply the map to all of them and then you collect them right.
00:17:57
Speaker
Yeah. And so that's an image. So a pre-image then is sort of going the other way. And it's, let's say that we have a point B in Y. So we can have a whole set of elements we call A that F maps over to B. And so that whole set of elements X is called the pre-image. So that's basically what we're going to talk about maps and functions. All right. So you want to describe fibers?
00:18:25
Speaker
Yeah, so we're going back to our idea of maps and functions. So what I want to say is that we have two manifolds. Let's say that one is in two dimensions, the other is in one, the most basic definition of a fiber is that it's the pre image of a function at a particular point.
00:18:46
Speaker
Yeah, and so let's break this down a little bit. Alright, so now we're going to talk about fiber bundles, and before we talk about that, we're going to have to talk about what fibers are. So let's say we have an infinitely large cylinder, right? That's infinitely long. Describe a point on the cylinder in terms of the angle it is from the center, so we could call a certain point on the cylinder, we could say that zero angle.
00:19:11
Speaker
And we could say that that is zero distance too because right if we have a point on the cylinder we could describe it in terms of how far it is down the cylinder and then how much we have to go around the cylinder to get to that point, right? Right. And so if we take a function that takes a cylinder that takes a point on the cylinder and remember this described in terms of the angle and the distance and we take that and we map it to just the angle we're on our way to describing a fiber
00:19:38
Speaker
Yeah, because what will happen is, so let's say that we map a cylinder to a circle. Let's say that we do that by saying, however far around this point is on the cylinder is how far around it'll be on the circle. And then, so let's call that map pi for projection, then. So let's talk about what pi inverse of this point on the circle will be. So it won't just be a single point on the cylinder, will it?
00:20:08
Speaker
Yeah, so it'll be every point on a line that is that angle on the cylinder. Yeah, so it would be a point that goes straight down the cylinder, right? Yeah, it'll be a straight line on the cylinder.
00:20:23
Speaker
And that straight line that goes from negative infinity to infinity or however long the cylinder is would be a fiber. And we can also reverse the way that we're talking about fiber bundles in this cylinder example because it's so simple, right? Like for example, we can have the line, we can draw an arbitrary line that goes down the cylinder.
00:20:39
Speaker
And we can map points on the cylinder because remember we're describing them in terms of distance and angle. We can map that to just distance. And then in this case, our fibers would be the little circles that go around the cylinder at any given point. Right. And in this particular case, either example works. So how do we use fibers to construct fiber bundles? Like what is the difference between a fiber and a fiber bundle?
00:21:03
Speaker
So a fiber bundle is described as, so you have four things that make up a fiber bundle. The first is our total space. So that's pretty much the space that we're working with. So it's a complex object, right? That we want to simplify. Yeah. And we'll call that E. So we have that we are breaking it down into two things, a base space and a fiber.
00:21:29
Speaker
Yeah, and so before in our cylinder example, so let's go with the first example. We can imagine the total space as the surface of the cylinder, right? Yeah, so E is the cylinder. And the base space, it's just an arbitrary kind of circle that goes around, that can be mapped to around the cylinder, right? And so the base space goes from 0 to 2 pi, because that's one rotation of a circle. A lot of times people use degrees, but 2 pi or tau.
00:21:52
Speaker
Yeah, and then so let's think, because this is an infinitely long cylinder, that means that we have that one of our coordinates is that point on the circle. And then so we still have something left. And that's the fiber, which would be our our copy of the real line from negative infinity to infinity. And that's our other coordinate.
00:22:13
Speaker
Yeah, and the projection basically takes the distance and the angle and spits out just the angle. And the angle is from negative infinity to infinity, right? So it is a part of the fiber. Because remember, the fiber is the pre-image. So if the projection maps everything to this singular ring, then the fiber at a certain point on the ring is every point on the cylinder that maps to that point on the ring.
00:22:41
Speaker
Yeah, and something important to know about fiber bundles is that every single fiber is the same. So we're saying that for every point on the circle, we have an entire copy of our fiber of this real line, negative infinity to infinity. And so together, we get that those things are called a product space
00:23:02
Speaker
And because our total space E is the same thing as our product space, which is a product of our base and our fiber, we get that this is called a trivial fiber bundle.
00:23:16
Speaker
Yeah, and so the product space, remember, we took out the angle with our projection, right? We're kind of putting the angle back in by multiplying the two sets, taking the product of them. Because remember, the product of the real line with itself is anywhere on the two-dimensional plane, right?
00:23:35
Speaker
Exactly. And remember, the base space is lower dimensional than what we had before. We go from a higher dimensional space to a lower dimensional space. And it's apparent on the ring, right? Because an angle around the ring, that's only one number, versus a coordinate on the tube that requires two dimensions to describe like well.
00:23:57
Speaker
Yeah, makes sense. And I mean, of course, you can have space-filling curves and stuff, but those are not continuous when you move smoothly. And we talked about that on a previous episode, if you'd like to go listen to that. So what's some examples of some fiber bundles? And we might be using the term vibration, but technically a fiber bundle is a type of vibration, but not the other way around. But we're just going to use them interchangeably for this podcast only.
00:24:22
Speaker
Yeah, so, and in case you're wondering, a vibration is just a generalization of a fiber bundle where not all the fibers necessarily have to be identical. So what is the trivial vibration? Yeah, so, and we'll just call it the trivial fiber bundle. So it's kind of like we talked about with the cylinder. So we'll have four things again, E, the total space, B, the base, F, the fiber, and pi, the projection.
00:24:50
Speaker
So what this is, is that it's a fiber bundle where because pi is what projects the E down to be the base. And then so what will happen is that we'll have for every point on the base will have a fiber. And so the product between the base and the fiber will be homeomorphic, or it'll be the same space as the total space itself.
00:25:16
Speaker
Yeah, so these are fiber bundles that don't really require fiber bundles, right? Yeah, they're about the simplest objects that could possibly use those sets of parameters that they do. Yeah, and I mean another one might be like a line in 2D space and the lines, and then take the shortest distance between any point on the 2D space and the line, and that would be another trivial vibration, right? Exactly.
00:25:45
Speaker
But then we have things like the Mobius strip. Where the Mobius strip, do you want to describe that in terms of fibers? Yeah. So just like the cylinder. So if we draw a line at the middle of the Mobius strip and we just keep going and looping around, then eventually we'll still loop back to ourselves. So we still have a circle in one dimensional space. So we're going to call this circle at the center of the Mobius strip our base space.
00:26:14
Speaker
Yeah, and so if you're on the Mobius strip, right, you have this line now going through the middle of it. And so you could say that if you have another point somewhere else on the Mobius strip, the point that is the shortest distance between any point and that point or like basically the line that goes parallel to it is one fiber, right? I mean perpendicular to it, right?
00:26:35
Speaker
Yeah, so, and what we'll say is that for any point on the Mobius strip that the projection from our Mobius strip to our base space, the circle, is wherever along that middle line, the circle, that point is.
00:26:51
Speaker
Yeah, so now if you take the product of the base space and the fibers, remember each fiber is a line segment, we can say it goes from negative one to one, right? And we can say the base space is like a circle, so it goes to zero to two pi. So the product of those two turns out to be a cylinder, which is not homeomorphic to a Mobius strip, right? Yeah. So before we continue talking about the Mobius strip and what we can do with it, let's talk about trivializations. So what is a trivialization, what does that term mean?
00:27:22
Speaker
So we know that we have trivial fiber bundles, and that means we don't have to do anything to explain how the total space is related to our product space. But for something with the Mobius strip where we get a product space that is the cylinder, clearly we have to do something to explain how that works. So we look at our base space, we take a point on it, and we take a neighborhood around that point.
00:27:49
Speaker
And a neighborhood is like all the points that are like kind of like glued and immediately like if you take a point and like let's say you have like an inky pen and like the surface that we're drawing on is really fibrous and you just kind of touch it and let the ink roll out and that's that ink will be a neighborhood, right?
00:28:05
Speaker
Yeah. And so for any neighborhood, just like for any point, you can still have a pre-image. And in this case here, so the pre-image of a neighborhood on the circle will be a slice of the Mobius strip. Right. And since the space that we're dealing with is the base space, which is a circle times a line segment, right? That it almost locally looks like a cylinder. It seems like that's the power of this.
00:28:30
Speaker
Yeah and so a slice of the Mobius strip looks like or is even homeomorphic to the slice of a cylinder is what's important here. Yeah and I think you could kind of imagine this as like let's say you have a cylinder and it has some like drawings on it or something and you turn it around 180 degrees and you see more drawings. You keep turning around so it's a full circle and all of a sudden you start seeing the same things upside down and then only once you turn it around turn it around twice do you get back to the same place.
00:29:00
Speaker
Yeah, and so I think at this point, I can safely say what a trivialization is. And so if we take a set of neighborhoods that cover our base space, then so their pre images will cover the total space. And at that point, for every neighborhood, we can have a homeomorphism between the pre image, the slice of that Mobius strip, and the slice on the cylinder.
00:29:26
Speaker
Right, and this is like some kind of imaginary cylinder, obviously. It's like if we were a tiny, tiny ant crawling along this Mobius strip, kind of basically in two dimensions. If we were a completely flat little ant, right, we go around the Mobius strip and it looks pretty much like a cylinder to us locally, right?
00:29:43
Speaker
Yeah, so what we would see on the Mobius strip and what we would see on the cylinder would look the same. So I want to construct a trivialization for the Mobius strip now. And so the way I'll think about it is that I want four neighborhoods on the base space. And let's talk about those as semicircles.
00:30:04
Speaker
And the base base, remember, is that ring that is the circle. Yeah. So let's say that we have a left and right half of the circle. And then, so that covers everything except the North and South poles. So what will we do if we want to have the other? So what will we do if we want to have everything else covered? Oh, yeah. We could get another two semi-circles and then orient them parallel, perpendicular, right?
00:30:32
Speaker
Yeah. So if we put those all together, so the union of two sets, we don't put copies of things in it. So the union of all four of these together is just the circle. Yeah. And although each segment looks a lot like a cylinder, so it's kind of, it's, I mean, you could kind of imagine this as like gluing manifolds together, right?
00:30:53
Speaker
Yeah, and so we need to think then about what happens between a semicircle and its pre-image on the Mobius strip. So we know that this is going to be homeomorphic to some strip of the cylinder, but what we also need to think of is what's happening on the Mobius strip itself, because that tells us how the Mobius strip is different from the cylinder.
00:31:17
Speaker
Yeah, so you almost have to be cognizant of this from the get-go, which is obviously the point, right? You're dealing with these complex objects and simplifying them. And so one thing that obviously you know about going in, so let's say we have some fiber, right? Let's say it corresponds to angle that we're calling zero. And on each side of the fiber, on each edge of the Mobius strip, in the same fiber you have, so a blue racer on the left and a red racer on the right.
00:31:41
Speaker
It works. It works. So let's say these two drive at the same speed down the Mobius strip. They're going to continue being on the same fiber at any given point if they have the same speed because they're the same kind of racer.
00:31:53
Speaker
Yeah, so let's say that they're starting at one point of the semicircle on the base space. So let's think about what happens as they make it across that semicircle. So what happens is they'll make it around the Mobius strip, but the blue racer will be where the red started and the red will be where the blue started.
00:32:12
Speaker
And so this, uh, this base space that we're talking about, it seems to always be twice the length of a normal circle because we go halfway around it and it's almost like going all the way around the circle. But remember that the racers now will be quote unquote on the other side. Yeah. Cause the red is where blue was started and the blue is where red started.
00:32:31
Speaker
And if these are two-dimensional erasers, you'll notice that they're flipped. If one had a decal that said E on it and another decal that said P on it, an uppercase P, when they get back to the original point, it'll be a weird 3 and a strange Q.
00:32:47
Speaker
All right. So remember that when you go halfway around the Mobius strip, you're kind of going the quarter way around the circle. So that's just important to, uh, of the base space circle. Yeah. So going around the base space of the circle is going around the Mobius strip all the way, but consider that halfway around the Mobius strip, you cross through the same point once, but you're facing the other way.
00:33:13
Speaker
Yeah, it's like if you put your finger on the Mobius strip, the tip of your finger would make a circle going halfway around. Yeah, so you go all the way around the Mobius strip, but you're facing the other way, so you haven't gone all the way around yet. You have to go around one more time before you're facing the same way. Then you've gone all the way around the Mobius strip, which means you've gone all the way around the base space, the circle.
00:33:40
Speaker
Yeah, exactly. That makes, yeah. So how does this relate to the Klein bottle? So, and this is something that I think we could leave more as an exercise, but so let's think about what the Klein bottle really is. And so we're going to think about it in the relation, in relation to a Taurus. So what is the Taurus? Like, what's it a product of?
00:34:07
Speaker
The torus seems to be a product of two circles. Yeah, so S1 cross S1 as we would write it. Yeah, and the reason why I'm, my reasoning here is you go around the circle and that's one circle and then for each point on the circle you can have like a little slice of the doughnut. Yeah, and an important caveat here is that, so this doesn't work with the sphere does it?
00:34:29
Speaker
Um, no, it doesn't. It seems like, uh, the sphere would be, um, cause I mean, I could imagine like going around, like having each great circle with respect to one great circle on the sphere.
00:34:41
Speaker
Yeah. Cause so let's think about it this way is that so the sphere goes from, so you have one set of coordinates that goes around. So you could say from zero to two PI, but so you have another set. So that's from the North, you have another set of coordinates and that goes from the North pole to the South pole. And so that would be from 90 degrees or PI over two to negative 90 or negative PI over two. Oh, so it'd be a semicircle obviously, which is a line segment.
00:35:10
Speaker
Yeah, so it's not quite the same thing as a Taurus in this case. Yeah, so yeah, that makes sense.
00:35:16
Speaker
So what I want to leave to you, the listener, as an exercise is we talked about how to get a fiber bundle out of the Mobius strip and how could we do that the same way and how could we construct a trivialization with the Klein bottle such that we can get a relationship between the Klein bottle and the Taurus.
00:35:43
Speaker
Yeah. And I mean, a little bit of a hint and kind of talking about what a Klein bottle is. Remember, a Klein bottle is what happens if you stitch together the edges of a Mobius strip, which can only exist in 4D because you'll get to a point where you're kind of trying to stitch into itself. Yeah. And the way it'll look if you try to immerse it in three dimensions. Kind of like a shadow, right? Yeah. It looks, so it'll look like a bottle that is intersecting itself, sort of.
00:36:08
Speaker
Yeah, you can imagine this, like, imagine a Chianti bottle, like a wine bottle with a white base, and imagine that the top of the bottle is extended and the tube goes into itself and then it connects to the bottom and then becomes the bottom. Yeah. In fact, there's even a limerick that I heard once that describes the Klein bottle. And it goes something like, there once was a mathematician named Klein who found the Mobius strip divine, said, he, if you glue the edges of two, you'll get a neat bottle like mine.
00:36:39
Speaker
So finally, we're going to talk about a cool thing that you could do with fiber bundles. And it's a fiber bundle called the Hopf vibration. Right. And of course, vibration is a generalization of fiber bundles. It just so happens the Hopf vibration is also a fiber bundle. So what is a three-sphere? We've talked about this before on a few episodes, but it's good to review.
00:37:01
Speaker
Yeah. So we can kind of think of it as a shape in four dimensional space where, so we think of the sphere as in three dimensional space, all of the points that are equally distant to the origin, right? Yeah. So, and, and so this would be, uh, called a two sphere, right? If it were like in three dimensions, because it's like a two dimensional skin.
00:37:27
Speaker
Yeah. So in four-dimensional space, we then have a three-dimensional skin and we call this the three-sphere. Yeah. And this is almost like a space. Like if you were in the space as a three-dimensional creature, you would not know you were in it until you went in a certain direction and then got back to where you started by going in a straight line. Right. So it's kind of similar to how if you're a flat lander on the sphere, you don't necessarily know it's a sphere.
00:37:53
Speaker
Yeah, or like even if you're a human being on Earth, it takes until people invent astronomy until you realize you're not on a large plane. You're actually on a sphere.
00:38:03
Speaker
Yeah, and some people would still like to argue otherwise. So what is the Hopf vibration? So basically what it is, is you can imagine a three sphere. A three sphere would also have great circles, right? Great circles being the circles that bind the four dimensional sphere. And it turns out that we can map each great circle on the three sphere to a point on the two sphere.
00:38:27
Speaker
Yeah, so what we got then is a fiber bundle where we're thinking of the total space as the three sphere and the base space as a two sphere. So all that's left is then each fiber is a circle or a one sphere.
00:38:46
Speaker
And if you're interested, the formula for the Hopf vibration given 40 coordinates a, b, c, and d is a 3D point, which is a squared plus b squared minus c squared minus d squared. The y-coordinate, or the second one, is 2 times quantity a, d plus b, c in quantity.
00:39:04
Speaker
And the Z, or the last one, is 2 times quantity BD minus AC. And if you check this out on your own, you'll see that if A, B, C, and D are, for example, the distance between them and 0, 0, 0, 0 is 1, then the point that you get here is also equidistant from 0, it is 1 as well. So what does the vibration look like when it's projected into 3D?
00:39:27
Speaker
So we can get some sort of idea of what this three-sphere looks like by using a technique called stereographic projection. We can use this to show what a three-sphere would look like in three-dimensional space, or a two-sphere in two-dimensional, or even just a one-sphere of the circle in one-dimensional space. Yeah, you gotta think of this almost like a shadow. Like a shadow takes a three-dimensional object and projects it into two-dimensional space on the wall.
00:39:52
Speaker
Yeah. And so what kind of happens then is that, so for any point on the two sphere, we get this whole ring of the three sphere. And so as we move about the two sphere, we get that these rings sort of are linked together. Yeah, which is pretty cool. And it turns out that they fill all space.
00:40:14
Speaker
Yeah, because the three-sphere is a whole much more complicated beast than we might think of it being just a sphere. Yeah, it's really kind of mind-boggling to get your head around the sphere because, I mean, the simplest way to really describe a three-sphere is like a balloon that you blow up and then you kind of slow down blowing it up and then you, like, let it go. And over time that's a three-sphere.
00:40:38
Speaker
Something like that, yeah. And that's assuming that a fourth dimension is time, I imagine. Oh, yeah. Yeah, I should have mentioned that. Three spheres are complicated. I mean, there is a problem that was unsolved for 102 years that we're not going to quite talk about what they call the Poincare conjecture. Yeah, it involves three spheres.
00:40:55
Speaker
Yeah, it's a little terrifying to think about. Fiber bundles let us know what shapes are different by relating their coordinates. In this way, they're a way of interacting with tricky mathematical objects. We learned how the three-sphere can be described using a sphere of circles. Their beauty, however, can surely be said to lie in the fact that they show us both how simple space can be regarded as while also letting know of their inherent complexities.
00:41:23
Speaker
I'm Sophia. And I'm Meryl. And this has been Breaking Math. Want to support the show? Buy a poster. Facebook.com slash Breaking Math podcast. Also Patreon, we really appreciate Patreon patrons. One dollar or more gets you the show free of ads. And we're going to start adding a little more, a few more incentives to that. So stay tuned for that. And as always, I would like to give a shout out to the University of New Mexico Graduate Workers Union. Awesome. And keep on breaking math and breaking