The second most beautiful equation and its surprising applications 

I've gotten a few comments about this so just wanted to address it here. At the 14:00 min mark I start talking about negative curvature and how my floating globe stand actually has no curvature to which some have asked "what if you just rotate the principle curvatures like 45 degrees, then you'll have two segments with non zero curvature and you wouldn't get zero overall." I didn't mention this in this video but in the other one I did.... those principle curvatures can't just be any segments you want. At any given point you have to pick the segment with the most curvature and the one with the least then multiply those (and those two will always be perpendicular). So for the stand you can't rotate from the two I mentioned or else you wouldn't have the max and min values.

*This video is sponsored by Leonhard Euler*

my friend: how to solve this problem? me: use the euler formla him: *which one?*

I like this generalization of the Euler formula: F + V - E + 2G - 2B - H = 0 where F = number of faces V = number of vertices E = number of edges G = genus B = number of shells (discontiguous parts of the object) H = number of “holes” (actually number of interior boundaries on faces) Setting G = 0, B = 1 and H = 0 collapses it down to the simpler case that everyone knows.

20:10 quite wrong, the real reason the curvature is zero is because the tourus has the shape of a zero, that's what gauss actually meant

Vigorous proof mathematicians have a sense of humor That was good.

8:03 Technically, of all of the zero vectors we've seen, they all have an odd index. But the Euler Characteristic of a sphere is even. But, if you want to add any amount of odd numbers and get an even number, you have to use an even number of odd numbers to do that. So, on earth, there are at least *two* spots where there is no wind!

This is one of the coolest video on geometry and topology. I have a PhD in math (algebraic geometry) and nobody managed to explain geometry in an understandable way when I was taking grad courses. Thanks!

So the Poincaré-Hopf Theorem (Hairy Ball Theorem) explains why I can't comb my hair properly. I always knew something was up.

Your explaining of applications of different mathematical theories and formulas is extremely helpful and interesting. I wish my math teachers would have done this back in my school years, as at the time, I didn't quite "understand" the reason for the math because they never explained WHY mostly just stating that "THIS MUST BE LEARNED". This, on the other hand, is sparking my interest and I'm now following your channel. I will be checking your library for more good explanations. Thank you!

Euler when deriving the most beautiful equation: It's gonna help humanity soo much People in 2019: i dont know what it does but it sure is frickin beautiful!

*1m15s:* Or you could just use the pattern already built into that soccer ball/football, and count the vertices, edges, and faces on that [although it would take a lot more counting!]. There are 12 pentagons, each with 5 vertices, none of which is shared by any other pentagon, and there are no vertices that aren't found on a pentagon, so V = 12·5 = 60 Each pentagonal face has 5 edges all its own, plus 5 edges each connected to another pentagon, and each shared with that other pentagon. So E = 12·5 + 12·5/2 = 90 Finally, each pentagon borders 5 hexagons, each of which is shared with 2 other pentagons (3 in all), so F = 12(pentagons) + 12·5/3(hexagons) = 12 + 20 = 32 V - E + F = 60 - 90 + 32 = 92 - 90 = 2 *11m10s:* "For a circle...we just say that [the curvature] is 1/R" - no, you can actually *derive* that from your previous statement that, to paraphrase, (linear) curvature is the rate of turn of the tangent vector with path length: κ = dφ/ds. For a circle of radius R, centered at the origin, in polar coordinates, φ = θ + ½π; . . . = the angular direction of the tangent vector s = θR dφ/ds = dθ/ds = dθ/Rdθ = 1/R *17m12s:* Gauss-Bonnet Theorem [BTW, "Bonnet" is French, & is pronounced, "bo-NAY"]: This theorem applies not just to an entire surface, but to any simple closed curve on a surface: The total intrinsic linear curvature around the loop + the total Gaussian curvature enclosed by the loop = 2π; or, ∮ (intrinsic linear curvature) dl + ∫_A (Gaussian curvature enclosed by that curve) dA = 2π Where ∮ (intrinsic linear curvature) dl = Total change-of-direction around the curve. The 'intrinsic linear curvature' is defined wrt local geodesics, and I can understand if you didn't want to delve into that somewhat complicated explanation. G-BT, while formulated for 2-D manifolds, can be generalized to n-D manifolds, but let's not get into that. ;-) Hey - great summary of some great concepts. Thanks!! Fred

This video made my day.... people like you really do good!! Explaining complex stuff accurately yet simply

Literally Zach's best video ever. I read that book before watching this. First off, I immediately recognized the source material, lol. I'm also a passionate pure math undergrad at a good research university. I can honestly say Zach faithfully captured most of what that book said, and he did it without any distortion of the actual technical math behind this subject. Thank you so much for bringing this material to a wide audience with such a quality production. The media of video works really well for this. The book already did a terrific job of expositing beautiful math in a way that's somehow casual and understandable, but also accurate, and yet this video will reach way more people. I read that book twice and yet at no time did I feel like this video was redundant. In fact, this is a great and memorable summary! Wow. What an impactful video. I'm just so glad this exists. I actually ran a math club talk dedicated to ideas in this book. I'll just go ahead and share this video with them now. Wish I saw this last semester!

This is an amazing video. It definitely clarified the Gauss/Bonet theorem that I plan to use in a project. Great job.

This video is amazing, I wish I could find more explanations like this on yt with a little more math. Like the surface integrals that was omitted.

Ive hated maths before but now, you've made me love mathematics as much as my love towards physics

This might be my favorite video of yours so far.

I used it for my PhD thesis, to evaluate the topology of complex materials, ie. percolating structures. Thanks for bringing back nice memories :-)

Bravo. You succinctly, coherently and completely explained a ton of info in this presentation in such a way that anyone could follow along. Well done. 👏✌️

Thank you for the surface curvature talk, geodesics and all, I didn't know that about the excess angle, and it's mighty useful. For example, one could tell a convex from concave (in a discrete topology) simply by comparing angles against the ones predicted for the 0 surface curvature. This is really handy. If that topology was terrain, a game (i.e.) could tell whether any two vertices connect through the air or through the ground without having to resort to ray casting.

Great video! thanks so much for such a detailed explanation!

You have an amazing talent to explain math to all sorts of people, regardless of their background. I've learned a ton of stuff just from this video. Thanks a lot!

This is one of your best videos in my opinion... I tend to either get lost as the videos go on, but I didn't feel that way with this one. Love it

Well worth thinking about, thank you for the opportunity.

Great video, sir! Love your content.

Awesome! Thanks a lot for this video

Wow, this is so awesome. Thank you! New way of seeing the world is always good.

Thank you sir MajorPrep and CuriosityStream sir for your good teachings

Great video and I even signed up for Curiosity Stream due to your suggestion.

Great video, thanks!

That was probably the greatest video I've ever seen!

I love your videos man

You're amazing. I recommend your channel to all my friends! Love from India 🇮🇳

Love, love, love all your videos! Have BS in Applied Mathematics (many moons ago!) but you've reinvigorated my interest. One thing: at the 12:30-ish mark you mention the 'this is the amount a car would have to turn' and mention 60 degrees... it's actually 32 degrees for the pentagon a = b = c = d = e = 72 degrees; 72 x 5 = 360. Unless I'm really, really missing something.

Wow this video blew my mind!

great video as usual

This video is exactly what I needed, my next class with my geometry teacher will be about that. Thanks RU-vid for recommending this to me

I am happy that I subscribed your chanel. You managed to clear up my confusions in just 20 odd minutes. By the way, you may have heard the word "topological insulators". It's effectively an application of Gauss-Bonet theorem in real materials.

Great! Thank you! It pays to be a Patron. :-)

Such a great content

Great video!

Very clear exposition. Suggestion to Major Prep: check out - if you haven’t already - R B Fuller’s ‘Synergetics’ Vols. 1 & 2 - a lot of this stuff treated there within a comprehensive geodetic theory of nature. For instance - that pebble ball could be reduced to all hexagons except at the poles where each pole would have to be a pentagon. (The hexagon of pebbles at the pole shown is actually arbitrary and an error.) Pentagons ‘add curvature’ - simple enough: cut one triangle out from a hexagon and join the edges - the figure no longer exists on a flat plane. All that goes to the head of the ‘Polar 2’ - which Euler is describing.

Great video

I found an interesting relation between Euler's characteriatic formula and the "phase rule" wich tells you that the degrees of freedom, minus the amount of substances plus the amount of physical phases computes 2. (Freedom degree-Substances+Phases=2). The formula it's used to determine how many intensive independent properties you must find to determine a thermodinamic state un a given system.

I have more positive curvature than negative curvature for this explanation. Great work! Clear and easy to digest explanation. Fascinating that the torus has a sum of zero curvature.

In this field, unfortunately, I rarely see anyone who explains the concepts of topology and geometry in such a creative way. Keep it up!

Thnx for sharing

19:22 So the Gauss-Bonnet Theorem can be used to prove that the piramid does, indeed, have exactly 4 corners. Groundbreaking.

Really wish this video came out last year when I was taking math physics. This basically just summed up 80% of that class.

Great Video! Nice laymen explanations of a lot of topology! I'm sure others have said this before me but for the Gauss-Bonnet Theorem, Bonnet is pronounced "Bon-ay" as they're french. Great video though! Really awesome how simple it is to understand when it's such a major theorem in Differential Geometry that you only tend to learn at university and you might never even encounter it on a mathematics course!

This vid is excellent and has helped me with a physics problem. thanks for post. Euler was always my favourite scientist; on his deathbed he observed a final datum: 'I die'...

Dude you are amazing.

I remember when i was in grade 4 i noticed this on my own and just thought it was a cool coincidence. showed my teacher and then i had to take AP classes for the rest of my school career :(

if I found this channel earlier - I bet I wouldn't've flunked multi-variable calculus. But that was back in 2010. The world's gotten more info since then. It's a relief to look at 3d objects in 3d than a book!

my brain's always exploded with intelligence after watching this channel - always above my head for me to catch up. this is what I look for in a youtube channel!

1:18 who else had to think about the One Piece world with the Red Line and the Grand Line when the ball with the two rubber bands showed up? 😂

Very cool video. You could have also mentioned that on a soccer ball you always have the same number of pentagons even if you make it with a different number of hexagons :)

ThankU, wonderful.

Thanks!

That is so beautiful!

Sir, Thanks for many of your videos. Can you do a video on Sequence, Series and Mathematical Induction. How it is related to each other and how it could be applied in Real Life scenarios. Thanks in advance.

you bring the most complex to the average person - always amazed

It certainly is most useful. and it keeps getting 'usefuler'. Since about the mid 1990s comb geom has had a bit of a stunning run. one of the instigating papers by shahrokhi, sykora, szekely and vrto, "crossing numbers: bounds and applications" includes a stunningly simple proof of bounds for crossing numbers based on the planar graph version of this using just a probabilistic approach: taking the 'average' for all planar graphs. dunno whether this type of argument works for higher dims, but it is remarkable as it works for all graphs, not just straight line graphs (geometry).

9:37 that's a surprised face

I wish my brain could keep up with this stuff lol , im not sure if im a builder or a deconstructer when it comes to math .

Dude I wish you would do a video about telecommunication engineering... Thanks

My great uncle (who is Gary Flandro, who happened to discover the alignment in the gas giants that allowed for the Grand Tour for both voyager 1 and 2. The flight path which involved the gravity assist around the gas giants) he an 11th generation pupil of Euler himself going through Euler, Lagrange and Fourier!

commenting for the alghoritm, great video!

10:21 it has negative curvature if you rotate a little the perpendicular vectors. It does not sustain the zero curvature if you rotate those vectors. In a flat sheet of paper or a sphere it sustains no matter in which part of the surface you place those perpendicular vectors.

Very very very interesting. Thank you for your effort in explaining this.

Luvv from 🇮🇳 India

"Euler's Gem" is an awesome book!

Im happy that i can eat my pizza by bending the edge even if its sloppy. Thank you Euler.

0:30 1 point - 2 line - 3 plane/circle = (+/-/0)2. 3:10 4 faces + 1 outside + 1 inside. A torus is a line.

Thank you sir. I read this book you mentioned; Euler's Gem. I have an easy question i would be grateful if clarified. In topology two shapes are considered the same if they can be continuously deformed to each other. In that case a line or a curve should be equal to a point as we can squeeze it. even the compact disk or filled cylinder would be a line or even a point but they are not, why? Is it like the difference of taking boundary or not taking it? Thanks

On the cover you have a torus which has genus = 1 and the Equation V + F - E = 2 only work for genus = 0 manifolds.

It makes you wonder whether the Angle of the Dangle is directly or inversely proportional to the Hairy Ball Theorem.

An interesting extension of Euler Characteristics from Euclidean space to Non-Euclidean space.

nice lecture and handsome lecturer ..from east asia

Euler : " My head's under water, but I'm breathing fine" us to euler : " You're crazy and I'm out of my mind" maths to human : "'Cause all of me Loves all of you Love your curves and all your edges"

Actually, Euler's Identity is also called Euler's Formula And, I hesitate to say it, but Euler's Characteristic is also called Euler's Formula!

21:00 please do share that particular Up and Atom's video link.

I am interested in studying the topological characteristics of words in languages and apply such findings to the fabric of societal space time. Any suggestions on how to go about it with those that might be interested in helping? Thanks

I liked this video so much. It links so many things. Torus is not curved at all! :)

thank you :-)

Hey Major Prep, this is a bit of an off topic question but here we go. I am currently in my 3rd year of community college, majoring in computer science, and will transfer next fall semester (next year) to UC Davis. I was planning on just obtaining a bachelors in computer science but I heard they have a bachelors in computer science and engineering. I've done some research and saw that the difference is CS majors have more electives to choose from than CS&E majors, but which do you think will look better for when I graduate and am looking for jobs? I already took ENGR 401 (Intro to electrical circuits) and the whole PHYS 410/421/431 (the 9 series, except for 9D) series and I enjoyed it as much as I enjoyed learning about data structures, linear algebra, and the programming languages, but I would be more interested in someones opinion, especially one who had studied in STEM. Thank you and keep up the good work! These videos really do come in handy in undergrad!

At 14:23 wouldn't there be a positive Gaussian curvature if you would rotate the cross of lines at that point by 45 degree?

i want you to be my teacher. i love you learn from you. you teach so amazingly.

waaaaiit .. could this relate to particle physics ? … in the sense of subatomic particles ? … like the particles , their spin and how they interact ?

this really feels like a rubix cube - as with rubix cubes - the centers all stay the same, but it feels like with this equation - it's the poles that stay the same. I'm just guessing trying to wrap my head around figuring this.

I wonder if the vector field and zero vectors thing can be used in black holes and magnetic monopoles

13:58 - this is how engineers accomplish simple, everyday tasks - studying the math behind it (through overlaying) and overcomplicating it. I literally did that watching grass grow as a kid.

Question: I’m interested in mathematics related to astronomy, something like general relativity, string theory, wormhole equations e.t.c… what would I call this specific group of math? I want to find other cool formulas similar to these since I’ve studied them through and through but don’t know where to start.

What about two perpendicular lines that intersect at the bottom point of the inner face of that magnet, but neither of which is the straight line along its width? Won't that give a different Gaussian curvature than zero?

7:06 The saddle variation went anti-clockwise same as the others. What is the necessity to make a circulation and why is it called an index?

Really enjoyed this video! Admittedly, you completely lost me at about 16:00, but I enjoyed being lost. (Math, while neat... is not my best subject lol)

Learning topology without an integral in sight. Fantastic.

This seems to tie into the Maxwell equations. And the Navier-Stokes equation.

Sorry for asking for a video but I have a question. What should I do if I want to handle environmental issues and sustainability? Any type of engineering? A comparison would be great. Thanks a lot beforehand.