Embedding a Torus (John Nash) - Numberphile

This videos features James Grime with a little bit of Edward Crane.
More links & stuff in full description below ↓↓↓
Ed's full discussion of Nash and embedding: • Nash Embedding Theorem...
Nash shared both the Nobel and Abel Prizes. Full details.
Nobel: www.nobelprize.org/nobel_prize...
Abel: www.abelprize.no/c63466/seksjo...
Other mathematicians have done much work in this field (we have focused on Nash in these videos).
Great summary here: math.univ-lyon1.fr/~borrelli/H...
And top paper on the corrugated torus: www.pnas.org/content/109/19/72...
James Grime: singingbanana.com
Edward Crane at the University of Bristol: www.maths.bris.ac.uk/~maetc/
Twisted Torus video: • Topology of a Twisted ...
Torus Balloon at Maths Gear: bit.ly/TorusBalloon
Support us on Patreon: / numberphile
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Numberphile is supported by the Mathematical Sciences Research Institute (MSRI): bit.ly/MSRINumberphile
Videos by Brady Haran
Brady's videos subreddit: / bradyharan
Brady's latest videos across all channels: www.bradyharanblog.com/
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Other merchandise: store.dftba.com/collections/n...

[@NowhereManForever]

Grime has to be my favorite Numberphile speaker.

[@The_Aleph_Null]

Yeah. I really like him and the two guys from the old Graham's number video. So fun to hear them talk.

[@CalvinHikes]

NowhereManForever He's the best explainer of things. And his voice is calming.

[@metalhusky]

NowhereManForever Professors Grime and Moriarty, for me.

[@nandafprado]

NowhereManForever Did you know he has his own channel? Look at singingbanana

[@NowhereManForever]

nandafprado Did you read the other comments in this thread?

[@Mallyhubz]

You really should get Sharpie to sponsor your vids.

[@numberphile]

Mal Hubert do you know anyone in their marketing department!?

[@DrGlickenstine]

Mal Hubert That is an amazing idea

[@chromatosechannel]

Mal Hubert true! i support that.

[@TheSentientCloud]

Numberphile Are you sponsored by the people that provide your brown paper?

[@johnlapage599]

+Numberphile I know someone who handles sponsorship for 3M, who have a rival range of markers pens. How brand loyal are you?

[@TheMaplestrip]

James Grime is so awesome, probably my favorite Numberphile professor.

[@CanariasCanariass]

Same here. Love his enthusiasm when he explains stuff!

[@devistnathan730]

Same. He's just the most fun to see. You can really tell he loves his job.

[@fade6827]

***** He's awesome

[@TheMaplestrip]

I am now telling people about how I got 59 likes on a KZbin comment by saying that a specific mathematician is an amazing person. What's with all the confused looks?

[@General12th]

***** Most people agree with you, methinks? I agree that James Grime is an excellent professor, although I prefer Dr. Simon Singh and wish he would make more videos.

[@BroadcastBro]

Big thumbs up for Dr James Grime, he's superb in his communication technique

[@IceMetalPunk]

BroadcastBro And for his next trick, here's a poodle XD

[@user-cl5il3fo5w]

Yes, even I almost understood!

[@1998wiwi]

*Picture of the globe* "This is flat" WAIT A SECOND

[@maxwellsequation4887]

A disc is topologically homologous to a sphere. So I guess flat earthers aren't that mad

[@themobiusfunction]

@@maxwellsequation4887 it's not

[@PhilBagels]

Much more important is the question: Where do you get toric balloons?

[@papayaman123]

PhilBagels I want to stick my nob init

[@anonomusanonomus1589]

Blow torical breaths

[@pupyfan69]

Maths Gear

[@eoagr1780]

Look for donut ballons

[@Triantalex]

false.

[@dominicaingui4246]

"There is no great genius without some touch of madness..." - Seneca A fitting tribute and explanation of John Nash and his innovative work in mathematics. It is a beautiful thing to be able to appreciate creativity in the harshest of disciplines and Nash truly defines thinking differently.

[@SalesforceUSA]

I think his story of triumph over his schizophrenia is the most inspiring aspect of his achievements.

[@PotatoChip1993]

Prof. Nash and his wife died in a car accident when coming back from receiving the Abel prize. It's weird to see this wasn't mentioned in the video...

[@numberphile]

PotatoChip1993 that is true, and I think many people watching the video probably know that - but in a year or two I hope people still watch this video, and the fact he "died recently" might seem less important than his accomplishments. I think this video can be watched now in the context of his death, but later can just be a discussion of his work.

[@rentzepopoulos]

Numberphile I salute your way of thinking!

[@KrakenTheKode]

It mentions it somewhat at 12:30 when it says John Forbes Nash, Jr 1928 - 2015

[@LiviuGelea]

Numberphile , perhaps so, but the video still sais "John Nash is..." instead of "was"

[@johngalmann9579]

PotatoChip1993 It might have been filmed before his death I saw his lecture in Oslo, really weird when i heard he died just a few days later

[@marksmithwas12]

Not just any theory, a GAME theory!

[@loicoberle6156]

***** Thanks for watching !

[@jonahmioduszewski45]

***** Beat me to it

[@unvergebeneid]

***** Pretty sure Matthew Patrick knew about the actual field of game theory before he named his channel and in fact derived the channel name from mathematical economics. Apparently for many of his viewers it's the other way around.

[@1996Pinocchio]

***** my thought :D

[@zelivira]

Penny Lane Sucked when I was looking on youtube for game theory related videos a long time ago and a bunch of the results were from that channel.

[@KTC88]

RIP John Nash. Your work helped inspire so many mathematicians and economists. May your legacy continue on for many generations to come.

[@jOpaaD]

Dr. Grime is so good at explaining complicated things in a simple way.

[@alsoethan]

'waves' ~~~~~ 'hand action' ~~~~~

[@DustinRodriguez1_0]

This was a really interesting video, but also a bit sad. I had not heard that John Nash died yet. Major bummer. I'm from West Virginia, where he was from, and his work in game theory has always interested me.. comes in handy when working with simulations. His life was fascinating, too, in that he suffered from schizophrenia but after years and years of it he made a conscious decision to stop listening to the voices he heard, analyzing the things they said with reason and ignoring anything irrational. He was able to, essentially, think himself sane. That is, to me, absolutely astonishing. I am very sad to hear that after all of the things he survived in his life, a stupid car accident took him from us. At least he was able to receive the Nobel prize he so deserved before he left us.

[@diebydeath]

If you're playing the game of Asteroids, there's some interesting applications of this donut. If you're playing the game of Hemorrhoids, you're probably sitting on the donut.

[@4mathieuj]

I can tell that the months of absence have been invested in making wooshing-sounds while drawing.

[@ahenryb1]

An excellent use of time I would say

[@SocialWalrus]

ahenryb1 I concur.

[@4mathieuj]

Social Walrus Me too, I always like the videos with James most

[@CodyBenson13]

Dr. Grime is easily my favorite speaker on this channel. He is so excited to explain things. It really effects me when I watch. :D

[@strengthman600]

You know, I know a beautiful quote from John Nash "It's just a theory, a game theory"

[@kbking16]

liar. George Washington said that

[@MoonTheGoat]

+Kbking16 No, you're both wrong. Donald Trump said it

[@minnarewers3573]

You know it's MatPat, right?

[@strengthman600]

+Minna Rewers Psst, it was a joke, I know it was MatPat

[@godsadog]

And if the most efficient path was to treat everything a game, it would be the only theory.

[@bobbysanchez6308]

James Grime is my favorite mathematician who appears on this channel.

[@parkerd2154]

Brilliant, one of your best. I'm working on 3D printing Nash's embedded torus at the moment.

[@dharma6662013]

The torus has points of positive, zero and negative Gaussian curvature. The "outer" points are elliptic points (+'ve), the "inner" points are hyperbolic points (-'ve), and there are two circles of parabolic points (0) separating them.

[@mathoc5273]

This is my all time favorite numberphile video! I love how such a simple concept we're all so familiar with (asteroids) creates such a stunningly complex 3 dimensional shape!

[@thrillscience]

These videos are fantastic. Thanks, Drs. Grime and Crane.

[@gabrielsayers360]

James is surely the best Numberphile speaker, he explains it all really clearly, without being patronising and while maintaining the audience's interest through his own evident enthusiasm.

[@ragnkja]

The red line could also have been around the inside of the torus, which is the real problem. (I assume that's also what Nash's corrugated torus corrected.)

[@dliessmgg]

Nillie There's really only two "good" lines that don't need correction. The common mathematical definition of a torus is that a circle with radius R is "coated" by circles of radius r. So you'd need to find the lines on the torus with the length 2*pi*r. In normal cases where R>r, those lines are closer to the inside. (Of course you could also have cases where R is so big that there are no "good" lines, or the case where R has the exact size that there's exactly one "good" line on the inside.)

[@kurtilein3]

Nillie correct. ripples are deeper on the inside.

[@palmomki]

kurtilein3 But, does it need a fractal structure? Wouldn't it suffice to make a "wavey" corrugation only on the "circular" direction and make it more accentuated towards the inside?

[@kurtilein3]

palmomki it is not a fractal structure, it just looks a bit like it. after the first set of waves, the green line is lengthened, but different parralels to the red line would have different length. the second set of waves fixes that, now all parralels to both the red and green line have same length. diagonals are still a bit off, the third set of ripples fixes all these. the 4th set of ripples is so shallow that its basically invisible even in a high resolution image.

[@palmomki]

kurtilein3 I would have personally tried adjusting the length of the green line by stretching the torus in the direction normal to the plane in which the torus "lies" (parallel to the red line). Maybe for some reason they wanted the torus to keep a "regularly circular" section? Sounds like making life more difficult. Or maybe, once he showed how to create a set of ripples, the first one was so simple that it didn't really make a difference.

[@AdrianRowbotham]

This corrugation technique appears to be related to the so-called "π = 4 paradox" - whereby constantly cutting corners out of a square (and out of the resulting shapes each step) gets you to an approximation of a circle where the perimeter is the same as the original square.

[@Oozes_Dark]

Ugh, I'm so happy I found this channel! As a high schooler who loves math, it's so exciting to look at these complex problems and be able to understand them on some level even though I haven't gone past somewhat basic calculus.

[@bleis1]

what I enjoy the most of your videos is that you take the time explain with paper and numbers in a way someone who as difficulty with math can still understand very clearly thank you for all your interesting videos, I am always looking froward to the next one thank you

[@josephhargrove4319]

Fascinating subject that appears to be able to unlock a lot of doors in applied mathematics. Glad to see James Grime back. We haven't seen him for a while and he was missed.

[@hermes_logios]

This is the coastline measurement paradox. The length of a coastline is infinite if you use a small enough unit of measurement. The more irregularity you ignore by using a longer unit of measurement, the shorter the final measurement will be. In other words, distance depends on granularity. It’s what Greek mathematicians called “exhaustion” (measuring geometric curves by dividing them into smaller and smaller units), and what algebraic mathematicians call “calculus.” In topology, the granularity is called “smoothness” of a surface.

[@ayasaki.pb_787]

For a long time a haven't seen you. It's good to see you again. =) I had watched the movie and it was really beautiful and motivated for math-lover.

[@Djole0]

This is so much better explained, then the other video, I loved it, thanks.

[@3snoW_]

3:53 - "And for my next trick, here's a poodle" hahaha

[@Daluxer]

Ace! More of these on John Nash's work would be appreciated!

[@sk8rdman]

This makes sense if you think about it, because if you took your flat surface and connected the sides in that torus shape, you could do it without stretching the surface if you could just crumple it up right. The inside would have a lot of ripples, and the outside wouldn't, and ultimately the surface area and distance between points doesn't change. It's no easy task with just a sheet of paper, but in theory, it should work.

[@sphakamisozondi]

03:54, that joke went over most people's heads at a speed of light 😂

[@vincentfiestada]

James Grimes explained it so much better than the other guy.

[@NikolajLepka]

yay Dr. Grime's back!

[@Fiendxz]

I designed this EXACT system, without being able to mathematically prove it (obviously) in my senior year of high school for a game that I was designing. Not that this had any significance, but it's really cool to see an idea that you had years ago re-appear with mathematical relevance. This is what learning is about.

[@elmoreglidingclub3030]

This is beautiful! I thoroughly enjoy your videos. I am 70 years old and just completed a PhD two years ago. I study AI. I wish I had years in front of me to immerse myself in maths, to hang out with guys like you.

[@wugsessed]

The terminology of corrugations and imagery of what that deformed torus looks like really reminded me of the process of sphere inversion. It's a fascinating topic, and there are some pretty good (but very old) KZbin videos on it.

[@thomasr5908]

It's good to see James again

[@PureAwesome33]

Yay, James Grime is my favourite numberphile contributor! :D

[@darwn977]

brilliant video. love the explanation.

[@ImmaterialDigression]

I haven't watched a lot of these videos but this presenter is awesome! MORE OF THIS GUY! He is slightly mad, but only slightly, which makes it really interesting.

[@vector8310]

Love your enthusiasm

[@GaryMarriott]

I think what you just described is an Origami Torus, something whose surface is flat except for a large number of folds where curvature has no meaning. It is something tedious but no way impossible to make.

[@reubenfrench6288]

Welcome back, James Grime!

[@bjrnvegartorseth9028]

Grime's wave hands are top notch.

[@jonathonsanders1844]

Wow! So much easier to understand this video over the other one! Better explanation!

[@technopoke]

Been watch Numberphile for years, and only just seen one with someone I know in it. Hi Ed!

[@TheKlawyify]

Great as always

[@lucidlactose]

It's been quite a while since I have seen Grime in a recent Numberphile video. I was actually shocked when I first saw a video without him when I first found this channel with every video with him and then suddenly without. Now that I think about it, why have I not subbed to the Singing Banana yet?

[@smoosq9501]

This is absolutely genuines idea, really really impressive

[@willdeary630]

I'd think it would be really good if you went more into the maths behind these topics for those with a higher maths level, e.g. I'd like to know how partial differential equations are applied to this situation.

[@YCLP]

Do all equal-length lines on the flat square surface have equal length on the 'bumpy' torus? Or does this only hold for the green and red line?

[@ColossalZonko]

"ta sqeekz r extra" - singing banana 2015

[@hubert6943]

5:03

[@JNCressey]

When I was a kid I had a play mat that was a flat torus (although it was a slightly longer rectangle, not a square). It had an aerial representation of roads and buildings on it; where the roads went off the sides they lined up with the roads going off the opposite sides.

[@vimalgopal5873]

I am not a mathematician... and yet, I get so much joy out of watching these kinds of videos on KZbin. In fact, I can't stop watching them! I'm completely addicted! What is wrong with me?

[@mueezadam8438]

4:58 finally, some numberphile ASMR

[@thanosAIAS]

the only video that made me get it.. bravo and thanx!!!

[@pietvanvliet1987]

I like numberphile video's, but normally I can hold on for a minute or two (doesn't keep me from watching the full video though). This time, I feel like I sort of got this. Which makes me conclude that Dr. Grime either did an excellent job explaining, barley scratched the surface of this topic in order to avoid scaring people like me, or a combination of the two. I'll carry on believing the first one is true. Thanks.

[@pranav24299]

Prof.James Grime is the best, hands down😎🔥

[@whoeveriam0iam14222]

new video right as I checked the channel!

[@SocialWalrus]

whoeveriam0iam14222 You know you could just subscribe, right?

[@whoeveriam0iam14222]

Social Walrus I am subscribed.. but I came looking for the video on hyperbolic stuff and I saw this video 18 seconds old

[@MegaBacon77]

Learned about Nash to found out in a Microeconomics lecture, impressed that he was also a pure mathematician :)

[@e7540]

Nice sound effects, James Grime.

[@CobaltYoshi27]

Rest In Peace John and Alicia Nash. You and your contributions will never be forgotten.

[@NoriMori1992]

I can't remember if I've watched this one before. No matter! I just finished watching Cédric Villani's RI lecture on Nash's work in geometry and partial differential equations, so this should be easy to grasp!

[@JasonOlshefsky]

Another way to look at it-correct me if I'm wrong: you could corrugate the paper and get it to approach a toroidal shape where the red line and green line were continuous, but you could never quite corrugate it enough.

[@Anonymous71575]

I wonder... why is it only Mr. Grime that can make me understand and not bored during a Numberphile video? Maybe every Numberphile video that doesn't have Mr. Grime should have a reupload with the version whose speaker is Mr. Grime.

[@muffinproductions1626]

yes! James Grime is back!

[@TheGodlessGuitarist]

So glad Nash did this. I don't know what I would do without it

[@callumgilfedder9097]

Brilliant explanation

[@Matt-pr1xv]

Embedding a Torus: Subtitled "Why Mercator Projection is Horribly, Horribly Wrong"

[@moevyas4822]

great James

[@rhysappa]

I love James

[@Richard_is_cool]

Well, the colors of green and red were switched, but otherwise: Dr Grime again the BEST!

[@CylonDorado]

Reminds me of how the distance of the borders of a country (or whatever) on a map changes depending on how much you zoom in and account for every nook and cranny.

[@michaelmoran9020]

astonishingly I'm watching this because I have a very practical use for this information in computational-chemistry.

[@maxisjaisi400]

Ah, back to the old feel of Numberphile videos which made me fall in love with them in the first place. I enjoyed the previous videos, especially with James Simons, but baseball just didn't go well with hyperbolic geometry.

[@jopaki]

I now have a much fonder view of John Nash. what incredible things to think about.

[@eladnic]

Very interesting. Thank you

[@Nykstudio]

understanding a complicated theory is one thing, but explaining it in a simple way takes brains

[@NothingMaster]

In pure mathematics you could theoretically define a space just about any way you like, and even pump it full of straight edges, singularities and other niceties to which you might even be able to find some cooked up, albeit internally coherent, solutions. That’s the inherent power of a purely/mathematically conceptual creation. Now, whether such a made up creation could translate into anything physically meaningful or not is subject to a philosophical debate or an empirical observation. In a Universe where pure mathematics gives you wings to fly a fantasy plane you might as well take to the skies, even if you never actually leave the ground.

[@jpopelish]

Another way to look at this solution is that the corrugations make the flat, inelastic paper stretchy and compressible, so after you form it into a cylinder, the cylinder is elastic enough to bend round on itself, into a torus, without actually having to stretch or compress the paper along its surface. The stretching and compression only alter the shape of the waves.

[@2Cerealbox]

Where'd you get the weird balloon?

[@mamupelu565]

Ryan N why would you want that? ( ͡° ͜ʖ ͡°)

[@MetaKaios]

Ryan N It's in the description.

[@balsham137]

mamupelu565 save you a fortune on females

[@gorgolyt]

Ryan N From a fetish website.

[@rchandraonline]

sounding similar to the recently discussed hyperbolic space, where the shortest distance between two points is no longer a straight line, but a curved line aldo seems to be hinting at another Brady Haran video which showed any image could be represented as a series of combined sine waves

[@probiner]

UV Mapping :) By the way the corrugations would have more amplitude in the hole of the torus than on the outer edges right? In the pictures they look uniform, or you slide the geometry outwards?

[@urbanninjaknight]

The torus has negative Gaussian curvature as well. Also, the curve which seems to be a quarter-arc of a circle plus a straight line does have curvature defined everywhere but it's discontinuous.

[@atkmachinei]

Much better explanation to this concept.

[@WillToWinvlog]

So much calculus involved here!

[@bimsherwood7006]

Why was Nash not allowed to take the paper, fold it into a tube, flatten the tube by creasing it along two of its lengths, and then fold the result into a new, very squat cylinder with a double-thick walls? If you permit points with indefinite curvature, why not also permit sharp creases?

[@AkiSan0]

Finally Dr. Grime again! =)

[@__malte]

YES! Grime is back!

[@zh84]

Doesn't the infinite number of corrugations form a fractal surface?

[@elijahgardi7501]

zh84 As I commented, it reminds me of a Fourier series. By that I mean, if a particle were represented as an EM toroidal vortex, the corrugations (Fourier series) begins to define the size of the torus. Maybe..

[@louis1001]

But the point in fractals isn't defining an infinite number of corrugations, is it? The torus example probes that a finite number and depth of corrugations would eventually get the lengths in the lines to be equal sized. That was my guess, though.

[@markcarey67]

Yes, it was, like Weistrauss's "pathological" function an example of a fractal before that concept entered mathematics

[@cyr-9564]

Luis González is correct, a fractal means that there is infinite perimeter. In this case, a finite number of grooves works because you have a finite distance set in mind.

[@cliumay9]

AFTER WATCHING THIS, OH MY GOD! MATHEMATICS IS SO BEAUTIFUL!!!

[@ChinaPrincessDoll]

You better feel special that I watched the whole thing! ☺️

[@pinsfast4165]

This is amazing!

[@Twentydragon]

I didn't notice any part of the rules here stating you couldn't "crease" your embedded torus, so to preserve distances in every direction (not just those two), I would "fold" the flat torus along the red line and then attach the "ends" together. I am, of course, also assuming that you could fold it in such a way that the thickness is 0, but that each "side" of the ring was still separate from the other.

[@ddmarsh21]

Good video. A nice way to honor the late John Nash