The second most beautiful equation and its surprising applications
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- Опубліковано 12 вер 2019
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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.
As it is homeomorphic to a sphere, it must have a total curvature of 4 pi. Where is it gone? The answer is easy, I don't give it.
Major Prep, I have a question:
What career in the field of math and physics for someone who hates programming and computer simulations?
Can you do a video on computer science vs math major?
@@massecl Well, sort of easy; it has 8 cube corners, each carrying an integrated Gaussian curvature, G = ½π, for a total of 8·(½π) = 4π.
But it also carries some positive G along its 2 convex edges, and some negative G along its 2 concave edges, which cancel each other.
Fred
PLEASE REPLY...
1. Which shape has negative curvature on each and every point on it?
2. For an ellipsoid the total curvature is 4*pi, what about the curvature at a particular point on it, As you said for sphere it was one over R squared?
3. If flat planes have zero curvature, how could a shape(polyhedron) made out of those flat planes have a non-zero curvature. {I mean shouldn't the sum of curvatures all flat surfaces that make up the polyhedron be zero again?}
*This video is sponsored by Leonhard Euler*
No more liking this comment,it has reached 69 likes which is a constant in another Euler's formula
but why if this identity was discovered before Euler?
The real Euler's identity is sum(1/n^2=pi^2/6)
@@jlopez47 Because Euler...
@@nikmustaqim6885 the ASCII character "E" has a numerical value of 69 (hexadecimal 45)
my friend: how to solve this problem?
me: use the euler formla
him: *which one?*
I could be wrong about this but I think I read somewhere that Euler came up with so many theorems/formulas that they stopped naming things after him to prevent even more confusion (although there already is a ton of stuff with his name on it). Instead they named certain theorems after the first person to use his work or prove it more rigorously (or something like that, I'm not completely remembering).
I think it was made in jest but it could have a partial truth that if had not happened, would actually happen such that there would be too much confusion in mention of "Euler's Theorem/Formula"
that's mah boi yooler for ya
me: yes
To top this off, there are things called “Euler operators”, used in CG as one technique for constructing mesh models, that Euler himself had nothing to do with-they were merely named in his honour.
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
Or.. it is how Aryabhatta got the shape of the number zero, as in the shape '0' will have curvature equal to 0.
Is this a joke ?
@@macmarc6661 of course it is, a pretty bad one seeing it in retrospective (but I got a heart from Zach :D)
It's a great joke. Cant see how it can possibly be interpreted as anything else
Vigorous proof mathematicians have a sense of humor That was good.
account hairy balls
@@account5223 It's left as an exercise for the viewer
*rigorous
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!
PhD in math!! sir can you please suggest some books for university level math? i like am just gonna go into a college and way too interested in math, so yes i am pursuing BSc- economics, math and statistics, so can you pleaseeeeeeeeeeeeeee suggest some books on math >_
So the Poincaré-Hopf Theorem (Hairy Ball Theorem) explains why I can't comb my hair properly. I always knew something was up.
Wait, that theorem only applies to a complete sphere. So is your whole head full of hair?
@@maulob1523 Yes
Presumably, it was your hair
_ba dum tss_
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!
Good thinking, but not necessarily. Actually, a vector field can have zeros with arbitrarily high indices, they just weren't shown in the video. It is possible to have a vector field with only one zero of index 2 on a sphere. This possibility is shown (and animated) at the top of Wikipedia's entry for the Hairy Ball Theorem, for example.
Ive hated maths before but now, you've made me love mathematics as much as my love towards physics
Physics is mostly math though. You need to understand math to understand physics.
@@Exachad Yeah, my thought also. Albert Einstein had to upgrade his math skills before he was able to complete his General Relativity theory. And it was long ago. All that came after that was even more math.
Nick Dolgy xd upgrade
@@Exachad he just said he hated math
Wait how did you love phys but hate math?
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!
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 might be my favorite video of yours so far.
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.
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. 👏✌️
Great video! thanks so much for such a detailed explanation!
*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
The best response to the video....infact the best comment the Zach channel has ever got
@@priyadarshinibarath2935 Aw, shucks!
Fred
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!
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.
Well worth thinking about, thank you for the opportunity.
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
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
Have you become his Patreon yet?
@@NickDolgy No, but I might soon
Great video and I even signed up for Curiosity Stream due to your suggestion.
That was probably the greatest video I've ever seen!
Great video, sir! Love your content.
Thank you!
Right 🎵🎶🎵
Great video, thanks!
I love your videos man
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.
great video as usual
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 :(
Wow this video blew my mind!
This video is exactly what I needed, my next class with my geometry teacher will be about that. Thanks UA-cam for recommending this to me
Great video!
Such a great content
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.
You're amazing. I recommend your channel to all my friends! Love from India 🇮🇳
Thnx for sharing
Great video
Great! Thank you! It pays to be a Patron. :-)
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.
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!
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'...
Really wish this video came out last year when I was taking math physics. This basically just summed up 80% of that class.
In this field, unfortunately, I rarely see anyone who explains the concepts of topology and geometry in such a creative way. Keep it up!
Dude you are amazing.
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.
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!
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.
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 :)
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!
19:22 So the Gauss-Bonnet Theorem can be used to prove that the piramid does, indeed, have exactly 4 corners. Groundbreaking.
Thanks!
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.
ThankU, wonderful.
you bring the most complex to the average person - always amazed
Very very very interesting. Thank you for your effort in explaining this.
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!
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).
Dude I wish you would do a video about telecommunication engineering... Thanks
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? 😂
Hell yea
commenting for the alghoritm, great video!
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 .
What helped me only after years of programming and thinking in abstractions is that math has its own language, and inevitably you will not understand some of the words people use (which represent specific concepts). It is a skill that anyone can master, moving from A to B using black boxes for the stuff you don't understand, and then going back and diving into them at a later point. Which itself is a form of A to B with further abstractions to learn. Honestly I'm still on the same dive of my very first A to B, but I'm pretty surprised how much I know now a couple of years later. I had a nice conversation with a legitimate mathematician at the beach a couple months ago, and she asked if I was a graduate student at the local university. I'm not, I actually got an AA in Liberal Arts from the community college many years ago lol but it was interesting to think that I'd come far enough in my understanding to talk in their version of a foreign language. I do genuinely believe that anybody can learn high level math concepts, if they're willing to proceed through reasoning with oversimplifications and black boxes that they will come back to tackle later on.
L0j1k You, my friend, are a good person, and you give great advice and motivation. Good on you for taking the time to help a stranger out, and good luck on you quest from A to B
@@L0j1k thank you and i certainly will take your advice , i'll keep at it im hooked , again thank you have a wonderfull day :)
@@L0j1k excuse me, you said programming so if it's possible can you please tell me what to do so as to start programming and usderstand it? I mean I don't know how to use the computer, I know almost nothing so any recommendations? I don't know books, sites, apps? Whatever comes to your mind
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?
Luvv from 🇮🇳 India
21:00 please do share that particular Up and Atom's video link.
Im happy that i can eat my pizza by bending the edge even if its sloppy. Thank you Euler.
"Euler's Gem" is an awesome book!
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.
I liked this video so much. It links so many things. Torus is not curved at all! :)
thank you :-)
waaaaiit .. could this relate to particle physics ? … in the sense of subatomic particles ? … like the particles , their spin and how they interact ?
An interesting extension of Euler Characteristics from Euclidean space to Non-Euclidean space.
On the cover you have a torus which has genus = 1 and the Equation V + F - E = 2 only work for genus = 0 manifolds.
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"
It makes you wonder whether the Angle of the Dangle is directly or inversely proportional to the Hairy Ball Theorem.
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)
9:37 that's a surprised face
nice lecture and handsome lecturer ..from east asia
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.
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.
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
The continuous deformation has to be a bijection. In particular then, it must be one-to-one : different points on the line or curve have to also be sent to different points. To map a line or curve to a point means every point on the line or curve is sent to the same point which violates the one-to-one condition.
@@raythomas8267 Thank you sir. What you said is fine for homeomorphism. But for squeezing there is no bijection but we still talk about squeezing. must squeezing still be a bijection?
@@DrIlyas-sq7pz , In topology two shapes are considered the same if they are homeomorphic meaning that there must be a continuous bijection between them.
"Squeezing" is just an informal term not a mathematical one. You may think of a homeomeorphism is a continuous bijective squeezing. There is another topological mapping which does consider a line and a point to be equivalent: it is called a homotopy and we can say that a line and a point are homotopically equivalent. Homotopy is a much weaker
topological concept than a homeomorphism. You may google these concepts :-)
@@raythomas8267 Thank you sir. I will study more from this point of view. I studied homotopy and fundamental group to some extent, I did take squeezing as something essential in topology but i got it now when you said it is something weaker.
May be squeezing should not be taken as actual squeezing but just keep in mind to understand that how two shapes are different.
Regards
@@DrIlyas-sq7pz You're welcome! Your last line about squeezing is correct. It is just a figure of speech :-)
Really cool, I love it!Is this how you come out with great circles and latitude and longitude on the Earth 🌍
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!
How many extra courses would you have to take if you double majored? Could you minor in CS&E to access the courses that you want without switching your major?
Double majors and majors with minors can indicate your strong interest in the fields that you majored/minored in. For related subjects there is usually enough overlap to make the extra work reasonable, or in some cases trivially small.
I feel like I have a better handle on topology from watching this short video than in my entire graduate topology class (which I really did poorly in).
0:30 1 point - 2 line - 3 plane/circle = (+/-/0)2.
3:10 4 faces + 1 outside + 1 inside. A torus is a line.
e=mc squared or cubed depending on the internal curvature of the axiom co efficent and if the are hermaphebic
I wonder if the vector field and zero vectors thing can be used in black holes and magnetic monopoles
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.
Astrophysics
Those vector fields remind me of complex numbers, split complex numbers, and dual numbers, respectively. Is there a similarly associated number system similar to the source one? Or would that just be reals in this analogy?
Bonnet, as in Gauss-Bonnet theorem, is named after Pierre Bonnet, a Frenchman, and his name is pronounced more like "buh-NAY" ([bɔnɛ]) than "BONN-it".
Learning topology without an integral in sight. Fantastic.
i want you to be my teacher. i love you learn from you. you teach so amazingly.
What is a Mathematical Torus with an Imaginary Hole (Center) of a Point. A section view would look like a pair of circles conjoined.
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.