The most important theorem in differential geometry: Gauss-Bonnet theorem

[@mathemaniac]

To try everything Brilliant has to offer-free-for a full 30 days, visit brilliant.org/Mathemaniac/. You’ll also get 20% off an annual premium subscription. Next video should be about a genius proof of hairy ball theorem (in fact, proof of the more general Poincaré-Hopf theorem). That will be out next year (I haven't made the video yet)!

[@PickleRickkkkkkk]

This video was beautiful, especially when it all comes together at the end. I literally gasped at 18:41 and just had to write a comment. Your animations are honestly extraordinary and I love your videos.

[@mathemaniac]

Thank you so much!

[@mmmao0630]

The last point (generalisation to higher dim) is extraordinarily important in theoretical physics. The Euler characteristic of the manifold is closely contingent to the conformal anomaly when coupling a conformal field theory to this certain background. The anomalous characteristic monotonically decreases as the energy scale goes down. Such statement is proved in 2D(1989) and 4D(2011) which reflects the thermodynamic property of quantum field theory. However, the statement cannot be established in 3D precisely because of the vanishing Euler characteristic in odd dimension, and a lot of our condensed matter models are in 3D. If mathematicians find a way to generate non trivial topological characteristics in 3D, it might open the door to resolving such conundrum . We live in an amazing era of rapidly advancing mathematics and theoretical physics, with so much more yet to be discovered!

[@mathemaniac]

Actually if you asked me my favourite theorem of all time a year ago, I would have put Gauss-Bonnet as a close second (Theorema Egregium is the first), but because of the relation to physics, this is now my favourite theorem of all time!

[@TheoriesofEverything]

How the heck you can produce content this quickly, of this quality, is remarkable. Loved your Euler char video regarding its range (rather than the strict equality). Keep it up!

[@Aiden-xn6wo]

Best Christmas present.

[@mathemaniac]

Glad you like it!

[@ryancantpvp]

This was a super satisfying follow-up to both the video on Theorema Egregium and the one on Euler characteristic!

[@ffs55]

A wonderful holiday gift, bravo @Mathemaniac

[@mathemaniac]

Hope you like it :)

[@jackwilliams1468]

Very good description!

[@draziraphale]

A really good set of explanations, thank you and well done!

[@mathemaniac]

Thanks!

[@kylebowles9820]

Differential geometry is badass

[@jellybabiesarecool4657]

Amazing video as always! Differential geometry is the perfect topic to do videos like this for so I don't know why we don't get more people doing so. You are definitely my favourite maths KZbinr.

[@mathemaniac]

Thank you so much!

[@IsekainiKami]

Beautiful, as someone who has not yet studied Differential Geometry I could still follow along the intuition. Really loved how it all came together at the end with the euler characteristic bit, it was like a well written movie script!!❤

[@DeathSugar]

I don't understand one thing - if we add some nipple to a torus on the side, the surface area certainly will grow, so why doesn't it affect the formula? Do we speak about some kind of regularity or maybe local euclideanness to Gauss-Bonnet to work properly? Like stellarator form is fine, but the cup isn't though both are just weird torii in topological terms? What are the prequisites for a surface?

[@mathemaniac]

Surface area grows, but when you stick something to the torus, then the site at which you stick to the torus will have negative curvature. "Surface" already means locally Euclidean, and we do require differentiable and orientable surfaces in order for curvature to even be defined.

[@DeathSugar]

@mathemaniac nipple on the side doesn't prevent diffs or normals, it will just "spike" curvature at certain regions. But locally euclidean wasn't obvious - methinks that was never mentioned in previous videos, but I might be wrong. Also a followup - how do we distinguish valid locally eucledean from invalid ones? Will it just loop to "must have area according to gauss-bonne".

[@drdca8263]

Very nice! Thank you!

[@publiconions6313]

Wonderful video as always.. lol . And as usual, ill probably need to watch it 3 times cause large chunks go over my dense head unless i pause repeatedly

[@stellarstarmie8221]

This basically is the reason you can eat pizza holding the crust with merely a finger. Good video!

[@mathemaniac]

You are confusing with the other theorem called Theorema Egregium, which is the theorem that is often introduced with pizza-eating.

[@stellarstarmie8221]

🤣🤣 It has been a little while since I have read a book by Presley. Fun read while I got the chance for an independent study.

[@charlievane]

how about a proof that we can't split the sphere into less than 8 triangles. Or can we ?

[@garylouderback4338]

Is it negative though

[@dougdimmedome5552]

Ok, now do the generalized version. Edit: oh you discussed it

[@garylouderback4338]

It looks like an interlude and the riddler

[@johnsmith1474]

It's been 50 years since I aced the Math SAT, though I could still if I had to duck chalk thrown at me by a Jesuit. Though I fought to get some sense of the magic, this was nearly incomprehensible, and I had to settle for admiring the brilliance of the production. When I think Euler it's Emanuel Handmann's fantastic 1753 portrait that comes to mind. Gauss exists in my head via anecdotes I have read of him as an especially brilliant child. I am left to wonder about the utility of this vein of mathematics, is it for making money, or designing weapons? Seems that those two areas of endeavor eat up all the engineering talent.

[@garylouderback4338]

Is this a version of king of the hill

[@garylouderback4338]

What is delta