The second most beautiful equation and its surprising applications

This video is sponsored by Curiosity Stream. Home to over 2,500 documentary and non-fiction titles for curious minds. This is Euler's identity and many consider it the

most

beautiful

equation

in mathematics, since it relates e, i, Pi, 1 and 0 together in a very

surprising

way. But when you're looking for the

second

most

beautiful

equation

, well, there's really no specific answer. However, in a survey in which this question was asked of a group of mathematicians, many said it was their

second

favorite. This is Euler's characteristic. Well, a specific case where it is equal to 2. And this says that for any convex polyhedron the vertices minus the edges plus the faces will always be equal to 2.

Now, the

applications

of this and really the more general equation are deeper about you think. But first, to get a visual idea of ​​everything, let's first look at the five Platonic solids. A tetrahedron, for example, has four vertices or corners, six edges that connect them, and four faces. For the other shapes, I'll just put those values ​​on the screen and you'll notice that they all differ. However, when you subtract the edges from the vertices and add the faces, it all comes out to 2. This would be the case for any convex polyhedron. In fact, even for a sphere the Euler characteristic is 2.

Which many people will comment saying 'What are you talking about? There is only one face. It should be one. But consider this. If we draw big circles over here and then one over here that I'm going to represent with rubber bands, you'll notice that there are two points, one here and one at the back where the rubber bands intersect, so they're like two vertices. So the edges would be what would connect them. So one here, one here, two and three and I followed the four from behind. So there are four edges. And then there are four faces one here, two, three and four.

So when we apply the Euler characteristic, the value turns out to be two, as expected. In fact, even for a globe with all the lines of longitude and latitude, if you counted the vertices, edges, and faces of everything that Euler's characteristic implied, you would again get two. And this would always be the case when dividing the sphere. Basically, if the edges and faces are allowed to be curved instead of flat, the Euler characteristic remains the same as for convex polyhedra. So the most official rule for this is that if you take something with an Euler characteristic of two and transform it by continuous stretching, then the Euler characteristic will still be two.

As in the case of Euler, the Euler characteristic is conserved under homeomorphisms. We are not allowed to make holes or tear the object. But in general, if you can deform one shape into another continuously as if they were made of clay, then the Euler characteristic will remain the same. But not all shapes have the Euler characteristic of two, like a donut or a toroid. This has the zero Euler characteristic. To see why just like with the sphere we can draw circles around the torus in different sections. There are four points where all of these intersect, therefore four vertices.

There are eight edges that connect them. And then four faces. Sorry if it's hard to see, but we have two red ones at the top and two blue ones at the bottom. Then we simply apply the Euler characteristic and get zero. Another way we can define the Euler characteristic is to subtract twice the number of holes. The fact that we have an Euler characteristic of zero means that I definitely cannot transform a donut into a sphere or vice versa. Because to do so you would have to drill a hole or fill one which is not allowed when it comes to homeomorphisms.

And although Euler's characteristic began with polyhedra, it also applies to flat surfaces. Here we can see that there are two vertices, one edge and no faces. If we had another segment we would have three vertices, two edges and no faces. And once we connect two vertices we have three vertices, three edges and a face. So notice that the vertices minus the edges plus the faces in all cases are 1. However, if we claim that the outer section is also a face as if we were writing on a sphere, then all of these face values ​​increase by one.

And we have the Euler characteristic above that never changes no matter how many edges we draw assuming no intersections. Okay, now let's move on to the

applications

. Pretend for a second that instead of a sphere we live in a giant bull. And there is a constant wind that blows around the world that we can represent with this vector field. Ignore the cause of this, just realize that this would definitely be possible. And no matter where you are on the surface, you will feel that wind. But if you start closing the hole and keep the wind going, we notice something happens once it closes.

Now we have a zero vector here, a place where there is no wind. But remember, this is three-dimensional since we just closed the hole and a toroid as if we were looking at the top view right now. But if I just squeeze this into a sphere and keep that vector field, this is what we're actually looking at. You'll see that zero vector at the top like we just saw. But now from this view we can see that there is another one here at the bottom. So, for the sphere there are two points where there is no wind.

And for the donut that was zero. And if you remember, the donut had the Euler characteristic of zero, while the sphere was two. Hmm. I wonder if there is any relationship there. Well, there is, but one fact is simple. And that's because these zero vectors come in different flavors. There are some so-called centers, which is what we just saw with the sphere or closed torus. And the wind spirals to a zero point. But there are also saddle sources and sinks. These are not the only examples, but in general I just realized that continuous vector fields can behave differently near their zero vectors.

Now each of these vector fields has an associated index that isn't too hard to find. To find it for the sink, let's say you simply draw a circle around the zero vector and place a dial somewhere on the circle that points to the direction of a vector at that point. Then we're going to move the dial around the circle just once counterclockwise while always having a point in the direction of the vector at its location to see what the dial does. You'll notice that it will make a counterclockwise spin during that time, as shown here. A counterclockwise rotation of that dial corresponds to an index of plus one.

Turn the dial twice, then the next one would be plus two and so on. Then, if we look at the font and do the same thing again with the circle and the dial, you will notice that the dial will again make a single counterclockwise rotation. Which means its index is also plus one. And if we go to the center again, we see the same thing with a counterclockwise turn. So so far we have only seen +1 ratings. However, for the saddle, when we do the exact same thing, you will notice that this time the dial will make a single turn clockwise.

And a clockwise rotation corresponds to a negative index. So, in summary, these four types of continuous vector fields with a zero point, all but the saddle have an index value of +1. Now, the reason we need to do all this is so we can understand the Poincaré-Hopf theorem. This simply says and yes, I'm going to oversimplify that for a continuous vector field with isolated zeros on a surface, some of those indices that we just looked at are going to be equal to the Euler characteristic of that surface. So, like the sphere we saw, it had two centers, one at the top and one at the bottom, each of which has an index of one and that adds up to the sphere's Euler characteristic of two.

This means we could also have a sphere with the source like the one seen here at the north pole. Since this is an index of +1, it means that there must be at least another 0 vector somewhere, even if it exists. In this case it would be the south pole where we will find a sink also with an index of +1. And these added together still result in 2 as expected. So we can say that for any sphere with a continuous tangent vector field there must be at least one 0 vector. Because if there were none, the index values ​​would add up to 0, contradicting the Poincaré-Hopf theorem.

This means that on Earth right now there is at least one place where there is no wind. It could be an extreme case like the center of a violent cyclone, but it doesn't have to be. There is a more official theorem for this and I'm not kidding. It's called the hairy ball theorem. Yes. That's the thing. The good thing is that this was the first rigorous proof that mathematicians have a sense of humor. But in reality it is just a special case of the Poincaré-Hopf theorem. Rather than referring to winning, the hairy ball theorem gets its name from the fact that you cannot comb hair over a sphere without obtaining a strand, also known as a zero vector, where the hairs on the comb can be considered tangent vectors. .

Also, if you Google the theorem, you'll find an old Reddit post with this image. Which isn't even a math-related subreddit, but rather the main comment that mentions the theorem. Anyway, here you see the sculpture created with rocks, which fit together very well, like a smooth vector field. But we see that an exception had to be made at the pole, which is like a zero vector or a tuft. This is to be expected since the sculpture is spherical. And although this may not exactly reflect the theorem, you can see the similarities. Now, going back to the torus from before, it now makes more sense why there were no zero vectors.

According to the Poincaré-Hopf theorem, the left side will be zero since there are no index values ​​to add. Which corresponds to the Euler characteristic to the right of zero for the torus. So everything is fine. But a torus could have a source and a sink as seen here with the source at the top and the sink at the bottom. That was a little difficult to draw here, but note that by adding a source and a sink we also get two mounts here and here. And since each of them has negative indices, they cancel the source and sink indices, which are both positive, producing a total value of zero, the Euler characteristic of the torus.

And with that we're now going to switch gears slightly to the topic of curvature. I talked about some of this in a previous video on general activity, but there's a lot I didn't get to. For a plane curve, the physical intuition of curvature is how much the tangent vector rotates as we move along the curve. Which means that this straight line has zero curvature as expected, since the tangent vector never has to rotate as we move down the segment. However, for this curve the tangent vector rotates between nearby points. Basically, if you start somewhere and move the tangent vector a little bit along the curve, you'll have to rotate it a little to maintain that tangency.

Therefore there is curvature. We define that this section has positive curvature. While this section has negative curvature. Sort of like a concavity, but more officially, if the curve bends in the direction of the normal vector, then that is a positive curvature and the curvature away would be negative. Now, it can be difficult to determine the value of this curvature, since we need a sofa that I am not going to fit into. However, for a circle it is easy. We simply say that the curvature of any point is 1 over the radius of that circle. So when the radius is large, we get 1 over a large number, which means small curvature.

Which makes sense because if you walk around a very large circle you don't curve as much. To find the total curvature of a circle, simply multiply 1 over R times the circumference, which gives 2 pi. Which will always be true regardless of the size of the circle. But the cool thing is that when you warp a circle into any non-intersecting curve, the total curvature is still 2 pi. And the positive regions will be completely canceled out by the new negative regions that maintain that 2 pi curvature. This also means that any polygon has a total curvature of 2 pi since we can warp the circle into it.

A polygon is flat on its edges, but has its curvature at the vertices. And as most of you probably know, the exterior angles of any polygon will add up to 360 degrees or 2 pi radians, regardless of the number of sides. So this matches the total curvature we just found. A visual way to look at this is to first realize that an external angle like here measures how much a car would need to turn if it went around that corner. So if a were 60 degrees, then the car would have to turn 60 degrees to go around that curve. Then for any polygon that loops, it corresponds to a 360 degree rotation, as the car will rotate 360 ​​degrees or 2 pi radians to return to the original configuration.

So that total curvature is something like how much the car will turn in total, which explains the two pi curvatures for shapes homeomorphic to a circle. Since you always turn 360 degrees when going through a single loop. So what we have seen so far is that the total curvature is not determined by thegeometry nor by things like angles and lengths. And it actually depends on the topology of the shape. Since they are homeomorphic to each other, they have the same topology and therefore the same total curvature. But where things get more interesting is with surfaces because now we have to do with Gaussian curvature, which I mentioned before.

In that video I mentioned that, as expected, a flat sheet of paper has zero Gaussian curvature. However, the reason for this is that at any point a straight line segment does not curve at all. That is, it has zero curvature. And the same thing happens with another perpendicular segment. Then when you multiply them, you get zero, of course, which is the actual Gaussian curvature at that point. The two individual segments are each the principal curvatures, but for a Gaussian curvature, it is necessary to multiply them. So, on a sphere, at any point, a small segment curves outward a little, which we say has positive curvature.

And the same can be said of a segment perpendicular to that. Therefore, the sphere has positive Gaussian curvature at any point because it has two positives multiplied. And for Negative Curvature I use my razor because at one point it curved outward in one direction but inward when going perpendicular to that. So you get negative curvature times positive, which is negative overall. And that part generated some comments. Four people said, 'Why didn't you use that thing behind you?' But this does not have a negative Gaussian curvature. In fact, although it is curved, it has zero Gaussian curvature almost everywhere.

And that's because yes, here at this point we could say that it is curved, this is negative curvature. But we have to take into account that perpendicular segment. And when I move perpendicular to that, it doesn't curve at all. So this is zero curvature. And zero for anything is zero. That's why this Gaussian curvature is zero pretty much anywhere. But what I didn't talk about in that last video was the real value of the Gaussian curvature of these surfaces. So, just like for a sphere, the value of Gaussian curvature at any point is 1 over the radius squared.

Then, to find the total curvature, we can integrate the entire surface or simply multiply it by the surface area, which gives us 4 pi again, regardless of the size of the sphere. Okay, now that we know what I did was add a third rubber band to our soccer ball. And what you'll find is that there are now a bunch of geodesic triangles along the surface. What I mean by this is to rule triangles with 3 edges: 1, 2, 3 and 3 vertices. But those edges run along geodesic curves. Basically, the fastest route from point A to point B. Which for a sphere is always part of a great circle or the largest circle you can draw around the sphere.

Now there are a total of eight of those geodesic triangles here, four at the top and the same at the bottom. And since we know that the total curvature of this is four pi, then the total curvature enclosed by each of those geodesic triangles should be greater than eight or pi greater than two. Which is it. And I'll keep that number on the screen. Now, if we analyze these triangles a little more, you will notice that their angles do not add up to 180 degrees. In fact, if we look at any angle like this here, it's actually 90 degrees.

That will be the case for any of these angles. So for any of these triangles, we have 90 plus 90 plus 90, which is 270 degrees. 90 more than a typical triangle in a plane. So we can say that there is an excess of 90 degrees or pi by 2 radians. And that excess in radians coincides with the total curvature enclosed by that geodesic triangle. And it turns out that, in fact, it will always be that way. As in the total curvature enclosed by any geodesic triangle will always be equal to the excess angle of that triangle. We have just seen a sphere that has positive curvature.

But for a surface with negative Gaussian curvature, the angles will add up to less than 180 degrees. Which means that there is an angle deficit that can be considered as an excess of negative angle that coincides with the negative curvature enclosed by the triangle. And on a flat surface the angles will always add up to 180 degrees. Therefore, there is no excess or Gaussian curvature since we are again on a flat surface. But where does Euler's characteristic come into play in all this? Well, this is something called the Gauss-Bonnet theorem. To oversimplify this again, we say that the total curvature of a surface, also known as the integral of Gaussian curvature, is equal to 2 pi times the Euler characteristic of that surface.

This theorem is interesting because it links two branches of mathematics. Curvature is more of a geometric property that depends on local features such as angles or arcs. and how the form bends in space. While the sum of them is what results in the total curvature. But Euler's characteristic has to do with the topology of the object, a global property that will not change due to these continuous deformations. So geometry and topology or the local and global properties of the Gauss-Bonnet theorem link them in a rather simplistic way. So for a sphere that has an Euler characteristic of two, the total curvature according to the above equation would have to be multiplied by 2 pi or 4 pi, which is exactly what we just found.

And if we deform the sphere into something else, the Euler characteristic and the total curvature do not change. Any new regions that do not curve as much are now balanced by other regions that curve even more. This also means that all Platonic solids have a total curvature of 4 pi since they also have an Euler characteristic of 2. These are flat on their faces, but maintain all their curvature at the corners. Now, if we look at the cube, you will notice that the angles at any corner add up to 270 degrees. Then I'll subtract that from 360, which gives us 90 degrees or pi over 2 radians.

Because this is considered the angle deficit. Unlike the previous triangles, this deficit tells us to what extent the angles do not add up to 360 degrees or two pi radians. If we multiply this deficit of pi over two in each corner by the eight corners we obtain 4 pi, which is the total curvature of the surface. For a tetrahedron, the angles at any corner add up to 180. Therefore, the angle deficit is 108 degrees or pi radians. Then, if we multiply that deficit by the four corners, we again get 4 pi the total curvature. This will be the case for the other solids as well.

And I just wanted to highlight that there is a direct relationship between the deficit or excess of angle and the total curvature. Then, if we look again at the Gauss-Bonnet theorem, you will notice that if the Euler characteristic of a surface is 0, as in the case of a torus, then the total curvature must also be 0. If we look at a torus, this has a lot of sense. Its outer part has positive curvature because at any point those main curvatures curve outward. But inside, at some point, one would curve inward while the other curves outward, generating a negative Gaussian curvature.

But when you sum it all up, the positives and negatives will cancel out, giving us a total curvature of zero, just like the Gauss-Bonnet theorem says. The most famous application of curvature probably has to do with general relativity and the curvature of our universe. But again, you can find all of that in the video link below. Now moving on from curvature, we have a more random example: the Euler characteristic can be used to prove the image theorem. A formula that finds the area of ​​any polygon on a grid of equally spaced points. Which only depends on the number of points along the boundary and inside the polygon.

So the area of ​​this is 15, which you can find by simply counting the red and blue dots. Or another example: Euler's characteristic can be used to prove the five-color theorem which states that only five colors are needed to cover an entire map, so no two bordering regions have the same color. Although this was surpassed by the four color theorem, which is much more difficult to prove. Up and Atom made a great detailed video on this, which I'll link below if you want to see the actual test. And there are many more examples that I'm just not going to get to because of how long this video already is.

But if you enjoyed learning about topology, graph theory, and the Euler characteristic, I highly recommend checking out this book. Euler's Gem covers pretty much everything I talked about and much more, from the history of the Euler characteristic to knot theory in higher dimensions, etc. So I'll link it below for anyone interested. And then, for anyone looking to learn about anything else, from science to technology, history and more, you can head over to today's sponsor video trivia stream. Curiosity Stream is a streaming service that hosts thousands of documentaries and non-fiction titles covering topics such as from physics in the universe to technology, nature, etc.

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