The Most Remarkable Theorem Part I (Theorema Egregium)

Part I of the remarkable theorem investigates curves and curvature in 2 and 3 dimensional spaces.
#theoremaegregium #gauss #artofproblemsolving #mathematics #calculus #algebra #vectors #tangentvectors
#integral #integration #vectorcalculus #physics #speed #velocity #acceleration #curvature #parametrizedcurves
#parametricequations #parameter #multivariablecalculus #multivariable #vectoralgebra #helix

[@sackixfilms8950]

If anyone is wondering the footage from the start of the video is from a movie called “Die Vermessung der Welt”, which is about Gauss and Alexander von Humboldt

[@vtrandal]

I absolutely love this video: content, backdrop, music, etc all of it. Thank you!

[@lanimulrepus]

Excellent start to a subject that caused many sleepless nights for me 60 years ago... Look forward to the follow-up editions..

[@tonygeorgiou6939]

I love the way you present . Please keep doing more

[@talalabdou]

Very nicely presented. However, I have to mention that the background music was very loud and distracting which diminished the value of the very good work

[@ffggddss]

Nice intro to the subject of curvature. As you were going along, I thought there would be some discussion of so-called Gaussian, or surface curvature, which for a 2-D manifold embedded in Euclidean 3-space, is specified by a single parameter, but for higher-D manifolds is a tensor (of rank 2? or 4? The Riemann curvature tensor? I'm a little rusty on my differential geometry). But your presentation of linear curvature was spot-on. Anyway, I was hoping the title was referring to the Gauss-Bonet Theorem, a truly remarkable relation between linear and surface curvature! Fred

[@ZeddZeeee]

sweet video! well done and interesting. I love the math vids like numberphine or 3blue1brown etc but i hadnt thought of this before. keep up the great work and cant wait to see what comes next!

[@shruggzdastr8-facedclown]

Michael Penn also has some good videos on vector geometry and topology wherein he goes into the actual nuts and bolts math, itself

[@shruggzdastr8-facedclown]

Just as a curve can be defined (whether geometrically, trigonometrically, via calculus, or any other higher math) in terms of a line, a line can in turn be defined in terms of a curve, describing it as the perimeter (or segment of that perimeter) of a circle whose radius is of infinite length

[@kevincampbell1395]

Nice!

[@notu483]

The music is too loud and distracting. Otherwise, good video!

[@KaliFissure]

I think you'll find that if you apply a similar flow to this form, Shirley's Surface, it will both describe the flow of the universe but also demonstrate a topology for renormalization Why it requires 4pi to complete Surface Surface(cos(u/2)cos(v/2),cos(u/2)sin(v/2), sin(u)/2),u,0,2pi,v,0,4pi

[@michaelblankenau6598]

Nice graphics and animations . Very enlightening .

[@JojiThomas7431]

Very nice video. Very, very good. It got me back to math video watching in KZbin. Thanks.

[@brettmoore3194]

It still took till 1861 before gleason created a accurate map. A azimuthal equidistant map, as it is, is used today by all Ariel and nautical engineers🎉

[@leif1075]

Why exactly did it take so long, is that known?

[@brettmoore3194]

@@leif1075 well most of the previous map makers were under the impression the globe is a sphere. Indoctrination is one hell of a drug. Previous ancient maps showed correct land and sea but soon as the cult of Pythagorean confused people. Its very interesting old myths and legends get verified as we get more advanced they can't keep all the little lies going... Could you imagine if all space and military funding went to bettering humanity... This is why the world is the way it is. Some people know the truth and use it as a weapon. Because if you believe in lies you will never be able to come to a concrete solution for problems facing individual and collective lives

[@drslyone]

Looking forward to part 2.

[@stochasticks]

Fantastic video. Thanks!

[@zagrosqazy3798]

This is an amazing video thank you for teaching us about Gaws

[@Ruktiet]

I don’t know if you’re joking or not, but the name is Gauss.

[@muzzletov]

@@Ruktiet yes, exactly, Frodorik Gows

[@Ruktiet]

@@muzzletov this hurt my eyes so bad reading this, I’m going to need some gauze

[@JohnDlugosz]

1:00 I figured that out as a kid, too, in the context of playing Yahtze. The upper-section bonus needs a certain number of each count, on average. I don't recall the exact chain of reasoning, but I figured out the pattern when trying to rapidly ascess the progress and make decisions on how to fill in the upper section. 1..6 is not as labor-saving as 1.100, but once I realized this, 1..100 was the next thing I did.

[@KaueLopesMoraes]

GOAT explanation

[@PKINGU]

This channel finna pop off

[@mastershooter64]

yooo let's gooo more differential geometry content!!

[@leif1075]

Am i the only.one who had never heard of the constructabiloty of the 17 gon..? Or was everyone else new to this also?

[@fatalheart7382]

You can also get that first answer by halving the number you're continuing to, adding .5, then multiplying by the last number again.

[@Killer_Kovacs]

This video isn't about spacetime?

[@ohgosh5892]

great

[@lehuytien]

Beautiful video ❤

[@venkybabu8140]

Time creates curved surfaces. Though they are same anywhere. Powers create time singularity.

[@maxim7718]

9:00 given that there is no "neat" equation to know the perimeter of an ellipse, do we/can we use this ?

[@ewg6200]

He prooved it

[@sahhaf1234]

music is a bit intrusive. otherwise, excellent.

[@charlesbrowne9590]

Fifty years ago I used the textbook ‘Differential Equations with Applications and Historical Notes’. The historical notes included biographies of relevant mathematicians, and I recall reading there for the first time about Gauss, Euler, LaPlace, Bernoulli etc. Gauss is - arguably - the greatest mathematician of all time. I never liked the story about his adding the first 100 natural numbers. Common sense tells us that this sum is very simple, and there probably isn’t a classroom without at least one student who could not pull off the same trick.

[@nicolasreinaldet732]

Just now you crashed my pride for having done that when I was 10 :(

[@vlix123]

Thats not true. No 10 year old, hwoever mathematically gifted, could’ve pulled that off without seeing a trick like that beofre. Its easy enough to say something is easy once you have seen an elegant and simple solution of it. Common sense would tell you, however, that it is surprisingly difficult to be original and do it without seeing an example first.

[@ericrawson2909]

I did it by adding the numbers in pairs: 100+1=101=99+2.......=51+50. So, fifty lots of 101=5050. I didn't need to introduce a second series of numbers.

[@nicolasreinaldet732]

@@ericrawson2909 I did it in a similar maner, but I used the zero so I had one 50 pairs of 100 and one pareless 50 ( this one got me at first and I took loonger to get it )

[@JojiThomas7431]

But I think the idea is similar, not exactly the same, but similar. Hats off to you for such a nice trick.@@ericrawson2909

[@keeperofthelight9681]

Bros genius is on another level ngl xdd

[@starguy2718]

Gauss, Newton, Archimedes are considered the 3 greatest mathematicians of all time. Maybe add in Euclid, as well.

[@leif1075]

Who says we today can't do the same thing??

[@CharlieVegas1st]

​@@starguy2718Euler..?

[@genomicmaths]

The story attributed here to Gauss, about the sum of integer number is not from Gauss but form Leonard Euler

[@kenderpl]

While the story might very well be apocryphal it is still attributed to Gauss. 5 seconds of googling should convince you of that.

[@zimriel]

@@kenderpl Probably why there were 15 downdings, from historians of maths who ragequit this vid and went to look up something else about the ACTUAL theorem

[@pixelapse9613]

1:20 What? this is what I thought in my Highschool

[@Eyalkamitchi1]

One of the best math videos I've seen, I've noticed I got distracted by the music a lot and had to rewatch some parts...

[@MrofficialC]

This just seems wrong. Imagine if all the internal space of the toroid was filled with really long strings. Would only the strings along the edge wrap around to their end points of would strings in the middle also wrap around to their endpoints. Basically it seems like it should be a sphere or infinite space or a physical barrier along the edge because at the center of the ring on the toroid-let me rephrase this point. If you took a donut with a hole in the middle of it and put in on the table and took a knife and cut it perfectly in half you would see that you now have two circles where the ring crossed through to the other half of the donut. Now in the very center of one of those two circles would space wrap around to itself in a tighter and tigger circle to closer you got to the center of one of the two circles (in the toroidal model of spacetime)? Probably not. What I'm suggesting is that there would be a sorties paradox there to be seen if conceptualize correctly that on one hand you have the edge of the circle which wraps around to itself right? And on the other hand, in the center of the circle you either have just space without any wrapping around onto itself, or a tighter and tighter string that wraps around to its own end getting smaller and smaller the closer you get to the center of the ring of the toroid. It's really hard to explain without being able to point physically at what I'm talking about but it makes more sense that there is a physical wall somewhere out there in the universe or there is just endless space without bounds. Just a thought

[@muhammad1502]

“Ant Hill high school” 😂

[@jamesraymond1158]

Somewhere out there is a 12-year-old who, after seeing this video, will realize that he wants to be a mathematician.

[@xbzq]

The music is terrible. I downloaded this video so i could watch it undisturbed. I regret it. 7012 different songs, super loud. Can't hear you talk. What were you thinking.

[@leif1075]

I love the music..to each hus own..shows hiw varied musical taste can be

[@sumeetsharma7256]

Karl Gauss inside doughnut......that was nice

[@mnada72]

I am missing the point where derivative is creeped into the formula

[@radix133]

It's all pretty obvious if you think about it.

[@Sapientiaa]

How did Gauss know that he had to divide by 2?

[@bazsnell3178]

Because he could easily see that he wrote every number down twice, so had to half the result.

[@Sapientiaa]

@@bazsnell3178 That’s my question. How could he see that?

[@Sapientiaa]

@@eratous4477 Again, HOW did he know that he wrote every number down twice. How did it even come to that in the first place. The video doesn’t give details. My question is if Gauss had an intuitive feeling as a child. It clearly appears to have been an intuitive feeling for the kid; that’s insane intuition for a child like Gauss. Obviously 100 values don’t check out for us.

[@Sapientiaa]

@@eratous4477 I understand that you eventually hit 51/50 therefore it caps at 50. The boys intuition about adding these values backwards and then realizing it caps at 50 is extremely impressive for his age.

[@dschai0220]

28 × 3 + 63 × 4 = 336

[@agrajyadav2951]

23:56 🗿

[@scottstensland]

Please no background music ... Totally distracting and ruines your presentation

[@MultiBrad777]

You can’t bend a shadow…..nor can you bend space

[@anilkumarsharma8901]

This is called pagadi. Millions of indian🇮🇳 use this🎉🎉🎉

[@thegreatestdemon1288]

These inventions were already done by several hindu mathematicians way before than european christian mathematicians

[@walterbrownstone8017]

This is silly. You can't curve space.

[@shruggzdastr8-facedclown]

Two words: black holes

[@marksordahl8872]

Einstein showed that gravity is the curvature of space-time.

[@walterbrownstone8017]

@@marksordahl8872 The Great health robbery of 2020-22 showed me how little people understand about the world around you. Just primates like any other. Space is nothing. You can't bend nothing. Why would you be so silly. What you are doing is practicing a religion.

[@walterbrownstone8017]

@@marksordahl8872 The fake epidemic clearly showed how easily fooled most people are. What you believe to be real is a religion.

[@motherisape]

Wut you believe in flat earth 😊

[@jiioannidis7215]

Interesting video ruined by the idiotic background music

[@figur3itout307]

Your input, although both True and useful, lacks Kindness. I appreciate constructive criticism when I see others trying to lift me up. It might be useful for you to consider that. Either way I appreciate all the support my channel gets. Thanks for watching.

[@SharpObserver1A]

Crap.

[@esteralfiah4861]

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