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.
"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.
@@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.
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.
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.
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 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.
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.)
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.
***** 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.
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 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.
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.
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!
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.
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
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.
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.
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?
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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 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.
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.
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.
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 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.
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...
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.
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.
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
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 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 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?
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.
+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..)
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?
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.
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.
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!
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?
"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.
*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.
RIP John Nash. Your work helped inspire so many mathematicians and economists. May your legacy continue on for many generations to come.
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.
I think his story of triumph over his schizophrenia is the most inspiring aspect of his achievements.
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.
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
Dr. Grime is so good at explaining complicated things in a simple way.
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
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.
'waves' ~~~~~ 'hand action' ~~~~~
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 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.
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.
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.
03:54, that joke went over most people's heads at a speed of light 😂
James Grime is my favorite mathematician who appears on this channel.
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
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 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.
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.
James Grimes explained it so much better than the other guy.
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!
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.
That’s exactly what I was thinking of as well - Perelman’s proof for the Poincaré Conjecture using waves to smooth out the discontinuities.
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
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.
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.
Brilliant, one of your best. I'm working on 3D printing Nash's embedded torus at the moment.
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?
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.
understanding a complicated theory is one thing, but explaining it in a simple way takes brains
Embedding a Torus: Subtitled "Why Mercator Projection is Horribly, Horribly Wrong"
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.
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.
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.
Rest In Peace John and Alicia Nash. You and your contributions will never be forgotten.
astonishingly I'm watching this because I have a very practical use for this information in computational-chemistry.
"ta sqeekz r extra" - singing banana 2015
5:03
It's good to see James again
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.
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.
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.
Imagine if this is the actual shape of our universe. That would mean it's endless and confined at the same time.
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.
Grime's wave hands are top notch.
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.
This is absolutely genuines idea, really really impressive
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 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.
James Grime didn't buy that balloon, he got it from his playground.
yay Dr. Grime's back!
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.
3:53 - "And for my next trick, here's a poodle" hahaha
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.
These videos are fantastic. Thanks, Drs. Grime and Crane.
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.
I now have a much fonder view of John Nash. what incredible things to think about.
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.
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.
So glad Nash did this. I don't know what I would do without it
Learned about Nash to found out in a Microeconomics lecture, impressed that he was also a pure mathematician :)
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.
I am slightly unable to discern this topic, yet it intrigues me none the less.
Welcome back, James Grime!
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...
Damn he explain this better than any of my calc or physics teachers.
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.
Damn... Video was filmed before Nash died, but released a week after... Weird watching it now
How are the Nash torus and regular torus related to each other? Are they homeomorphisms?
The torus has positive curvature in some parts negative in others and 0 in others.
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?
Well, the colors of green and red were switched, but otherwise: Dr Grime again the BEST!
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.
Prof.James Grime is the best, hands down😎🔥
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.
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
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 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?
4:58 finally, some numberphile ASMR
Wow! So much easier to understand this video over the other one! Better explanation!
This is so much better explained, then the other video, I loved it, thanks.
Much better explanation to this concept.
new video right as I checked the channel!
whoeveriam0iam14222 You know you could just subscribe, right?
Social Walrus I am subscribed.. but I came looking for the video on hyperbolic stuff and I saw this video 18 seconds old
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.
AFTER WATCHING THIS, OH MY GOD!
MATHEMATICS IS SO BEAUTIFUL!!!
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?
yes! James Grime is back!
@MathematicallyMindedPeople
I'm really sorry but I don't really understand something. Why is it a torus rather than a sphere?
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.
"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'...
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..)
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?
Ace! More of these on John Nash's work would be appreciated!
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.