Embedding a Torus (John Nash) - Numberphile
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- Опубліковано 20 вер 2024
- 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....
Abel: www.abelprize.n...
Other mathematicians have done much work in this field (we have focused on Nash in these videos).
Great summary here: math.univ-lyon1...
And top paper on the corrugated torus: www.pnas.org/co...
James Grime: singingbanana.com
Edward Crane at the University of Bristol: www.maths.bris....
Twisted Torus video: • Topology of a Twisted ...
Torus Balloon at Maths Gear: bit.ly/TorusBal...
Support us on Patreon: / numberphile
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Numberphile is supported by the Mathematical Sciences Research Institute (MSRI): bit.ly/MSRINumb...
Videos by Brady Haran
Brady's videos subreddit: / bradyharan
Brady's latest videos across all channels: www.bradyharanb...
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Grime has to be my favorite Numberphile speaker.
Yeah. I really like him and the two guys from the old Graham's number video. So fun to hear them talk.
NowhereManForever He's the best explainer of things. And his voice is calming.
NowhereManForever Professors Grime and Moriarty, for me.
NowhereManForever Did you know he has his own channel? Look at singingbanana
nandafprado Did you read the other comments in this thread?
James Grime is so awesome, probably my favorite Numberphile professor.
Same here. Love his enthusiasm when he explains stuff!
Same. He's just the most fun to see. You can really tell he loves his job.
***** He's awesome
I am now telling people about how I got 59 likes on a UA-cam comment by saying that a specific mathematician is an amazing person. What's with all the confused looks?
***** 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.
You really should get Sharpie to sponsor your vids.
Mal Hubert do you know anyone in their marketing department!?
Mal Hubert That is an amazing idea
Mal Hubert true! i support that.
Numberphile Are you sponsored by the people that provide your brown paper?
+Numberphile I know someone who handles sponsorship for 3M, who have a rival range of markers pens. How brand loyal are you?
Big thumbs up for Dr James Grime, he's superb in his communication technique
BroadcastBro And for his next trick, here's a poodle XD
Yes, even I almost understood!
*Picture of the globe*
"This is flat"
WAIT A SECOND
A disc is topologically homologous to a sphere. So I guess flat earthers aren't that mad
@@maxwellsequation4887 it's not
@@themobiusfunctionIt is if you enforce the rules that going off the edge of the disc takes you to the point on the edge 180 degrees opposite of your exit point. Thing is, flat earthers don’t believe that. They believe that Antarctica is an ice wall that nobody can cross because it’s guarded by the combined militaries of the countries that signed the Antarctic Treaty.
"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.
Much more important is the question: Where do you get toric balloons?
PhilBagels I want to stick my nob init
Blow torical breaths
Maths Gear
Look for donut ballons
false.
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...
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.
Numberphile I salute your way of thinking!
It mentions it somewhat at 12:30 when it says John Forbes Nash, Jr 1928 - 2015
Numberphile , perhaps so, but the video still sais "John Nash is..." instead of "was"
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
I think his story of triumph over his schizophrenia is the most inspiring aspect of his achievements.
You know, I know a beautiful quote from John Nash
"It's just a theory, a game theory"
liar. George Washington said that
+Kbking16 No, you're both wrong.
Donald Trump said it
You know it's MatPat, right?
+Minna Rewers Psst, it was a joke, I know it was MatPat
And if the most efficient path was to treat everything a game, it would be the only theory.
Not just any theory, a GAME theory!
***** Thanks for watching !
***** Beat me to it
***** 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.
***** my thought :D
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.
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.
RIP John Nash. Your work helped inspire so many mathematicians and economists. May your legacy continue on for many generations to come.
I can tell that the months of absence have been invested in making wooshing-sounds while drawing.
An excellent use of time I would say
ahenryb1 I concur.
Social Walrus Me too, I always like the videos with James most
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.)
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.)
Nillie correct. ripples are deeper on the inside.
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?
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.
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.
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.
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.
'waves' ~~~~~ 'hand action' ~~~~~
Dr. Grime is so good at explaining complicated things in a simple way.
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.
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.
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.
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
James Grime is my favorite mathematician who appears on this channel.
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.
3:53 - "And for my next trick, here's a poodle" hahaha
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.
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.
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.
James Grimes explained it so much better than the other guy.
Brilliant, one of your best. I'm working on 3D printing Nash's embedded torus at the moment.
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.
Embedding a Torus: Subtitled "Why Mercator Projection is Horribly, Horribly Wrong"
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!
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?
03:54, that joke went over most people's heads at a speed of light 😂
"ta sqeekz r extra" - singing banana 2015
5:03
I feel like I may be missing something, but wouldnt just making the torus taller also make the two distances equal?
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) UA-cam videos on it.
Doesn't the infinite number of corrugations form a fractal surface?
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..
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.
Yes, it was, like Weistrauss's "pathological" function an example of a fractal before that concept entered mathematics
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.
6:27
Doesn't it have positive curvature only on the outside? And negative on the inside? Or is it positive everywhere? Can someone help?
Yeah, by curvature I also understood "Gaussian Curvature" and there are different points in the torus that have positive, zero and negative curvature.
I'd advise you to watch Cliff Stole's videos on gaussian curvature, or topology in general. he explains a torus.
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.
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.
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
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.
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.
Grime's wave hands are top notch.
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.
In fusion science we call the two directions of the torus toroidal (along the torus) and poloidal (across the hole). In tokamaks we use axisymmetric tori (the cross section is symmetric around the axis, so it's a revolution solid,) while in the stellarators they are non-axisymmetric (twisted and with changing cuvature along the toroidal direcction.) The designs of the latter are really mind-boggling, since the toroidal axis are also twisted along the toroidal direction. They are an engineering nightmare, but very interesting. The trick is to keep plasma [charged] particles confined within a toroidal volume using magnetic fields.
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.
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.
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.
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?
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.
Rest In Peace John and Alicia Nash. You and your contributions will never be forgotten.
Could someone explain to me, why the torus instead of a sphere? I think the torus comes from connecting one line together by curving the plane, then connecting the second line later by curving the new plane in on itself, resulting in two circumferences. But since we started with a square, the two lines were identical in length in the beginning and in the torus, one of the lines was stretched because you cannot make the second curving without ending up with two different circumferences. In short, it is impossible to have a torus with equal "x" and "y" circumferences, assuming these circumferences are perfect circles and not any other ellipse shape. This part I understand. A workaround for the torus circumferences is to skew the "y circumference" taller resembling an oval in cross-section, enough to make the perimeter equal in length to the "x" circumference, but the problem will be, of course, that then only the "x" circumference is a true circumference and the "y" is no longer a circle, and the 3D shape still resembles a tube and not a torus.
Anyway, instead of curving the square plane one way first, and then in on itself after, could we curve both simultaneously? And if so, could it result in a sphere? If it were a sphere, the two lines would be identical in length in a 2D plane and a 3D model. This is very intriguing to me.
***** I'll try to answer your questions and ideas as best I can.
First of all, you just realized the same problem that troubled mathematicians, specifically that you can't simply bend your square around like that, since that will stretch some parts more than other, meaning that lines that are the same length on the square aren't necessarily the same length on the torus. In fact, it's impossible to make _any_ smooth surface in three dimensions such that any two lines of the same length on the square, will have the same length of the surface. (A smooth surface in this case is a surface where curvature is well-defined everywhere.)
Your proposed workaround will not solve the problem, because of the aforementioned fact that no smooth surface will preserve lengths, and simply bending the surface like that still means that it's still a smooth surface.
To answer your idea about curving the square into a sphere, imagine the following. Draw two lines at 90° angles that intersect in some point. As I'm sure you will agree, both these lines will have a length of 2πr (as they both go around the sphere once). However, they intersect at another point on the other side, exactly 180° from the first point of intersection. However, on the square (and the torus), two lines at 90° angles will only intersect once, so we must conclude that a square cannot be wrapped into a sphere.
***** Hopefully this can help. In the sheet of paper, the green and red lines intersect at exactly 1 point. On the torus, these two lines still intersect at exactly 1 point.
If you were to form a sphere in the way you are proposing, the red and green lines would intersect at 2 distinct points.
(After typing this, I realize that frxstrem also included this point! Excellent!) Here's another way to think about it too:
To form a sphere in the way you are doing it, you are collapsing the entire boundary of the square into a single point. That means that anywhere on the boundary of the square is exactly the same point as anywhere else on the boundary. In this sphere view of the paper, if you decided to go off the bottom of the paper, you could change directions and appear on the right side of the paper or the left side of the paper.
This is slightly different from the flat sheet of paper, since adjacent sides are not the same with this flat torus. Only opposite points of the boundary are glued together, not every point in the boundary. So with the flat torus view, if you went off the bottom of the paper, even if you changed directions, you could only appear on the top of the paper or the bottom of the paper.
frxstrem Thank you for the thorough answer!
Where'd you get the weird balloon?
Ryan N why would you want that? ( ͡° ͜ʖ ͡°)
Ryan N It's in the description.
mamupelu565 save you a fortune on females
Ryan N From a fetish website.
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
The flat square torus can also be seen as a straight tube where the height is half the circumfence and both the inside and outside make up the surface.
ie fold the square paper along the red line(the equator), then make it into a cylinder. Works in videogames...
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.
I am not a mathematician... and yet, I get so much joy out of watching these kinds of videos on UA-cam. In fact, I can't stop watching them! I'm completely addicted! What is wrong with me?
The problem of curvature around 9:30 in the video is very obvious in model trains: if you have fixed track pieces and join a straight with a curve your train goes from infinite radius (=straight) to 20cm radius in an instant. It results in a jerking of the cars going around the curve at exactly that point, very annoying looking.
That's why we use "flex track" and add curve "easements" to avoid that visible jerking.
"The squeaks are extra" That's what i was told when i bought my apartment in Hells Kitchen...i dont think the Russian dude meant Torus'...
What if we just take a cuboid shape like a Book a very thick book and cut a cuboid hole at its centre & just make sure that the thickness of book is large enough to make sure that the length of green line on flat torus is equal to the length of red line ?
Why would you want to keep the lines the same? As it stands it gives you a focus and a decompressed space of which allows the mind to string many different circuits together throughout it as well.
4:58 finally, some numberphile ASMR
Been watch Numberphile for years, and only just seen one with someone I know in it. Hi Ed!
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.
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?
On a graph where one dimension is time and the other is speed, no single point describes acceleration (not only the points where speed starts/stops changing.
Acceleration is the change of speed between TWO points on the graph.
Ah now THIS is straightforward, unlike the other video. Now I see what he was trying to say. Now with the ripples on the torus, you're essentially creating a fractal surface. After an infinite number of iterations of the waves, wouldn't the dimension of the torus be a hair less than 3 (or rather, less than 2 since we're talking about the surface of a torus)? Why wouldn't this affect the theorem, or is the theorem referring to any generically higher dimensional space embedding, not a specific dimension?
yay Dr. Grime's back!
Just a quick side note to those who still think that Brady should have made it clearer that Mr Nash sadly died about a week before this video was released, I'd like to point out that he actually explained his reasoning for not doing so in Episode #39 of his podcast "Hello Internet", just after the half-hour mark. Not talking about his death was a conscious decision. I encourage you to listen to the podcast if you are interested in Brady's decision.
I don't understand this... Maybe I'm crazy but you could surely stretch the Torus into a shape that allows for both line to be the same distance. Wouldn't inflating the top and bottom but keeping the middle circumference the same allow both lines to be the same? Consistently?
I didn't understand why you needed to "run waves" over it so many times. Why can't you just make the green path (through the hole) wavy so it becomes longer? It would look like the top left image at 10:42. Wouldn't that be enough? Why are the extra waves necessary?
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.
John Nash didn't really "recover" from his mental illness, and I think this is an important thing that should be talked about.
A lot of the time, when someone with a disability, whether it be physical or mental, does something really important, there's this narrative that people take to that says that they "overcame" their illness, or that their success was "despite" their disability.
However, this is ignoring a lot about disability. To say that someone's disability is a hurdle that must be jumped over is saying that their disability is totally removed from them as a person.
Monet's water lilies are painted with such brilliant blue hues because of his failing sight. Beethoven's symphonies are noted by critics as beautiful for their time because of the lack of high notes, which he couldn't feel the vibrations of. He was deaf. Frida Kahlo, born with a deformed foot, was inspired to start painting after a car accident put her in a wheelchair. Kahlo led the Mexicanidad movement, which encouraged Mexican cultural self-appreciation and rejected European beauty standards. To say that these people, and I am just listing artists, overcame their disabilities is disingenuous at best, and prejudiced at worst.
John Nash's schizophrenia is, of course, radically different from Frida Kahlo's disability. But, he didn't overcome it. John Nash actually stopped taking his medication, because he didn't like the side effects. John Nash was actually homeless for a while, living on the streets. But he managed to die in a car crash on the way back from winning an Abel Prize instead of in a shelter or a prison. He was good at mathematics, whether it was because of his schizophrenia, or whether it had nothing to do with it. He didn't overcome his schizophrenia, he dealt with it every day of his life, unmedicated.
I don't know how to end this rant, but I just want to say that that story of someone "overcoming" their disability to something is harmful. It marginalizes the disabled who don't accomplish incredible things, and ignores the actual difficulty of living with a disability day-to-day. Think about it.
The difference between this video and the Edward Crane video (linked in the description) on this same topic is a perfect microcosm of the sad state of university education, particularly for beginning students:
My reaction to Edward Crane, representing the vast majority of professors: "I know he's knows what he's talking about, but I sure don't. Is this what I'm going to have try to make sense of at some point if I continue doing this? Is that possible? Is it worth my time to see if I ever can?"
My reaction to James Grime, representing a select few among professors: "I know he knows what he's talking about, and while I don't understand it completely, I kind of get it. I bet if I studied this a little more with the right tutelage, it would make sense and eventually I'd even understand more like a pure mathematician."
My experiences with CS, EE, ChemE, physics, math, and many other technical professors when I was in school bore this out. I've heard plenty of other stories of the same thing and seen lots of smart people ditch courses of study simply because they couldn't find any connection to it to keep them going. James Grime seems like the kind of guy that actually wants to help more people become mathematicians. Most professors seem like they don't care or actively want to discourage students from continuing, and it shows in their pitiful communication of ideas to beginners.
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!
It's not just Asteroids! Games like Chrono Trigger, Secret of Mana, and Final Fantasy VII are also played on a torus. In fact, prior to last year's Geometry Wars 3: Dimensions, I had only encountered ONE game that is played on a topologically-correct sphere: E.T. the Extra-Terrestrial for the Atari 2600!
How are the Nash torus and regular torus related to each other? Are they homeomorphisms?
This is probably wrong but I don't know why:
If you are allowed to make discontinuities in your surface, why don't you just first fold the paper in half along the green line. That way you join the edges and thus join the red line. Granted, it curves infinitely much in two points but I gather that is allowed. Finally you make the cylinder like shown in the video, joining the green line.
***** If I have understood the concept of the video correctly: First of all, we're not allowed to make discontinuities in the surface. Having discontinuities would mean that there were part of the surfaces that were not connected, which is not the case here.
Secondly, sharp edges, while not being discontinuities, are still not allowed, since we want to keep the surface differentiable at every point, while the points along the edges have no derivative.
The curvature of the surface is connected to the second derivative of the surface, or rather, the radius of the sphere that approximates the surface best around a particular point. At the sharp edges, there is no well-defined second derivative, so there's no well-defined curvature either. So there's no such thing as "infinite curvature".
understanding a complicated theory is one thing, but explaining it in a simple way takes brains
when I looked at his work in game theory I was frankly just surprised that anyone ever had to say that. it all seemed absurdly obvious to me. the other stuff he did was absolutely more important.
but that's also why it was possible to make a film around the game theory stuff, because it's so simple that you can sort of explain it a bit to movie goers in the midst of an entertaining story about people and their relationships and dramas. can't do that with anything of real substance. so if you think I'm just bragging by saying his game theory work was trivial, no... it's a demonstrable fact that it was trivial. the movie is that demonstration.
One other problem with the torus mapping is that the green line isn't the same length in all parts of the torus. Does Nash also introduce corrugations inside the torus to match those up? My intuition says that you'd only need a finite number of corrugations to solve that problem, but I thought it was worth mentioning.
astonishingly I'm watching this because I have a very practical use for this information in computational-chemistry.
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.
Even with the speed example, isn't this where Fourier Analysis comes in? When you speak of compounded waves as well - what is the connection here?
Actually curvature is defined for that second graph. As long as the line is not vertical then there is acceleration.
These videos are fantastic. Thanks, Drs. Grime and Crane.
As far as I understood, the equal lengths are preserved by making ripples, which decreases the width of the torus while the circumference along the green line is as long as along the red line since the green line is not the direct way (it goes in serpentines).
What has this to do with curvature or maintaining a curvature of zero, or differentiation?
Well, the colors of green and red were switched, but otherwise: Dr Grime again the BEST!
But that's just a theory, a game theory, thanks for watching!
As soon as he said Game Theory I instantly thought of Matt!
+lewis brearley Matt actually took the name 'game theory' because he thought it'd be funny if it meant video games. (Which in some cases I guess it could..)
If the objective is to make the green axis the same length as the red axis, why the waves? Why not just make the height of the doughnut (as it lies on the table at the end) really high so that the green line axis stretches as long as the red line axis? Such that, if you cut a cross section of the doughnut, it'd be an oval instead of a circle.
DJSkunkieButt waves look cooler. (although there's probably some actual mathematical reason behind it meaning that ovals will cause the universe to collapse in on itself)
Because then the flat 2D plane that you've turned into a torus would not be a square as in the video.
DJSkunkieButt You could. But the point is to start with a flat square and fold it into a torus with the same lengths. The objective is not to make the green axis the same length as the red axis, it's to make the ratio the same as it was in the original flat torus.
DJSkunkieButt Ah, but you have to do it for all possible red lines. So your method would only work for one of them (the one shown in the video), but wouldn't work for others. For example, the red line could be on the inside of the torus, instead of on the outside.
Social Walrus Samuel Fitzsimmons So you're saying that these small waves maintain the ratio + shape of the original 2D plane?
So.... Why can't you just make the doughnut taller and not wider? You're already artificially lengthening it by introducing waves to it so I'm not sure I see the difference between that and just making it taller
(Edit) The taller you make it the longer the green line would get until it matched the length of the red line
This is the same concept as measuring shorelines then right? Depending on how much detail you include (how many ripples on the 3d object) the longer your measurements end up being.
Maybe I am stupid, but can't this be achieved by simply connecting the ends of the flat sheet and then folding it flat?
In this case, points along the folds would have undefined curvature, everywhere else would be flat, and distances would be preserved. Am I missing something in this hypothesis?
I HAVE NO IDEA IF ANYONE HAS SAID THIS......BUT THE EPISODE IN THE SIMPSONS WITH FERMANT LAST, THEIR WERE A SET OF NONCHALANTLY PLACED DONUTS. WHICH MATHEMATICALLY SPEAKING, THEY'RE TUROS SPIRALS.
really make a flat cylinder (just the sides) and you got a torus with equal length on both axes
just make the cylinder's circumference the same as half the height right?
no complicated ripples
if you can make that with paper it works
It's good to see James again