Pie - oddly I lived in the same Florida, USA county as the airport with the symbol PIE for quite a while... PI and TAU etc wow - definitely interesting to think about the show business aspects of all of it - what people want & what is sustainable & will lead to sequels and so on... very different probably than working at an actual University or even Brilliant etc hahaha - I’ve liked some of your graphics work in the past and probably it has enhanced watch time etc - I liked the physical rolling stuff here but I’m wondering how it played with overall audience satisfaction!? - Physics girl did an appearance with The Science Asylum and if you haven’t yet I hope you do that and as much collaboration as possible too with him and other channels hahaha... Name Explain channel for a bunch of things including PI and TAU even... yeah I’ve mentioned the ABC conjecture to you multiple times perhaps and maybe you can come up with some zany take on all of that with regard to the human drama and so on yeah philosophy of mathematical knowledge and agreement etc that would somehow go super viral... Brady Haran’s podcast with David Eisenbud briefly mentioned it but yeah who knows many more in depth things to find even on UA-cam I’m sure before you decided to make an episode about it or similar things if you do! ⭕️ no emoji perhaps for hexagon etc!?!?!? 🎬🌈🗽🤯☮️💟🌎🌍🌏🚀🪐♾➰➿〰️...
I worked for a trucking company when I was younger and I remember the importance of matching tire circumference or tires when putting together duel wheels. If one of the tires is smaller it is dragged along and wears much faster as well as creating an increased load because of the drag.
"If one of the tires is smaller it is dragged along and wears much faster as well as creating an increased load because of the drag." Not if you're going in circles. It's called tire stagger and circle track racers have been using it forever. Also, in essence the same thing happens on a motorcycle when you lean the bike to turn. The tire on the motorcycle is rounded. When you lean the bike the smaller circumference outer edge of the tire is making contact as well as the larger circumference center and it creates a turning force. Anyway, the video and explanation is flawed. There does not have to slippage of the smaller wheel if the path of travel is circular. A good way to visualize this is with a dixie cup. The cup has a larger radius on top then on bottom. If you lie the cup on it side and roll it, it will roll in a circle without slippage.
@@stefanl5183 well the video was about the circumfence of circles so what benefit would you get in measuring them when going in cricles to solve this problem ? because when you go in circles the closer you are to your turning axle the smaller the distance you have to travel (thats why cars have differentials)so try rolling your cup with the bigger side as the tunring axle you will realise you get alot of slippage at the smaller side. Also if you try to roll the cup in a different sized circle than it would do naturally you would most definitly gets slippage
@@nokiarsxzmika The problem is the video is saying that you have to have slippage, and that it's inevitable. It isn't! You don't have to have slippage. Yes, you will go in a circle and the radius of the circle will depend on the difference in those circles and the distance between them, but the conclusion you must have slippage is wrong. Sometimes you want to go in circles. Take a look at a World of Outlaws Sprint car and notice the difference in size of the left and right rear tires. Maybe space and time isn't always straight and flat?
Three things I love about this video: 1. The explanations are clear and concise. 2. The tempo of the video is not too short and not too long. 3. The French husband, apparently half asleep, being such a good sport.
It was a really good video, but did she invent Aristotle’s Paradox? Why is that even a question? I don’t understand the problem. There’s only one wheel. The second is imaginary, and if you make it real you’ll have to roll it more quickly to travel the same distance in the same time. All work is done between the surface of the wheel and the part of the counter that it touches. It will take less energy to accelerate a smaller wheel. They will travel same distance.
@@gristly_knuckle My issue with your comments originates from my lack of understanding of practically every one of your observations. Clearly, your grasp of this concept is far beyond my own.
@@gristly_knucklethat's exactly what she's saying. your understanding of this problem comes from the modern interpreration of this problem and physics as well. take in account, greeks didn't have F = ma law, they didn't even know exactly what's acceleration or work. it seems odd because if you separate them enough to make the wheels spin at the same time attached to each other but on different surfaces, the paradox holds.
Poor Cantor. He was derided and ridiculed when his proposed his diagonal argument (countable and uncountable sets) only to be heralded as a pioneer of set theory decades after his death.
@Anirban Mondal Yes, one needs axioms. Constructivism is simply not accepting some of the axioms of modern math. Since math is nothing else but discovering all the implications of a given set of axioms, other sets of axioms do in fact lead to equally valid math (as long as they're not contradictory). So do it if it makes you happy. But be warned that not many modern mathematicians will bat an eye about your results, since the ZFC is widely accepted nowadays and has become the foundation of almost every field of mathematics.
@Anirban Mondal I'm not talking epistemology, I'm talking pragmatism. Since we already touched it, we might simply consider the old question: is mathematics invented or discovered? My answer: kind of both, but mostly the latter. The axioms are clearly an invention. They're made up stuff and there is no reason to be convinced that they resemble reality in any meaningful way. You're free to believe that they do or do not. But my point is: that's just a matter of personal worldview and nothing mathematics cares about. The moment you set your axioms down, the implications are there for you to unravel. That unraveling is what math is (and hence, math is discovered). But it's not a natural science; it does not claim that its axioms are the fundamental rules of the universe we live in or whatever. Therefore, if you want to choose a different set of axioms, that's fine by me. Either way there's no way to verify that either math tells us anything meaningful about the universe. We just can work our way through it, shunting symbols as you call it, and hope that we won't stumble over some ineluctable contradictions. And if we're lucky, there might be other people around who find our results useful. That's the best a mathematician can hope for. There is no absolute truth to be found. Btw, to my understanding, model theory is not an alternative approach to math opposing the axiomatic view. It's a theory of different axiomatic systems, but comes with its own set of axioms. Since you mentioned Kant we could as well go the one step that is an inevitable consequence of his epistemology but which he still chickened out to go himself: believing is more fundamental than knowing, in the sense that you have to believe something in order to be capable of knowing anything.
For the line drawing of the cycloid, the opposite situation gives a more intuitive picture to me. If you roll a wheel on an inner circumference, you’d see the line traced at the outer circumference to double back on itself in a short loop
One really cool thing about this problem is that it has many practical applications including in automotive design and engineering, one example being the slip of the inner and outer wheels during a turn.
Yeah. But more often than not they require a giant like Cantor. The greeks actually had kind of a habbit asking pretty tough questions you can rack your brain about for centuries.
For those wondering what the final answer was, here is what I thought she would say to tie it all together: The demo at 0:47 is not a _measurement_ of length. It is a _mapping_ of points. It indeed shows a 1:1 correspondence in points, but (as explained at length later in the video) a mapping of points does not always denote equality of length (7:20). Longer version: At 4:20-5:10 she explains that the small wheel must be slipping-and she wonders (more or less) if this implies that, for every 2 points the big wheel moves forward, the small wheel must be rolling 1 point forward and then sliding/skipping 1 additional point forward. (At least that’s how the rolling polygons worked.) But the small circle is _not_ being dragged in a traditional sense, because there is a one-to-one correspondence between each point of the wheels’ respective journeys. Our intuition says one-to-one correspondence means they are equal, and therefore the circumferences must equal, which they clearly can’t be - so she spends the rest of the video explaining that, for continuous quantities like geometric points, one-to-one correspondence does _not_ mean they are equal. Therefore there can indeed be one-to-one correspondence in points along their rolling motion while _not_ requiring the lengths to be equal, thus permitting a rather non-traditional kind of “slipping” in which no point is ever dragged. [Edited to add the short version at the top of this comment]
except that things such as "one-to-one correspondance" and "uncountable infinity" don't mean anything in the real life, because there aren't any continuity in the world safe for in our imagination (or fantasy). Mathematicians can play with their imagination as much as they want, but they should not mess it up with reality that's all
Great video - got my brain ticking along nicely. 🙂 For the circle paradox, this illusion that the circles travel the same distance is what happens when you move remove a dimension. The two lines appear the same length in one dimension, but it's clear they are not the same physical length. The linear distance is a projection into one dimension - Because you have to raise the platform up to meet the second circle, it is travelling in the same direction but on a different plane, not the same plane as the larger circle. (i.e. the distance travelled is only the same in one dimension) It's a bit like casting a shadow of a ball on the wall. Depending on how far away from the wall the ball is the shadow will have a larger or smaller area, but the ball itself remains constant. It also doesn't matter how far away you move the light from the ball, the shadow will remain constant. If you force the circles to travel on the same plane but connect them to each other centers so the rotational speed is the same, the system wants to move in a circle because the smaller cirlce physically can't keep up with the distance travelled by the larger circle. And then if you were to raise the floor on the side with the smaller circle it would happily roll forward alongside the larger circle.
I'm 56 years old and computer technician and thought I know a lot. But I never ever heard anything before about number lines or this wheel. This video was the most interesting thing I have ever seen AND I have ever seen in UA-cam and I'm totally impressed to find such a pearl between all the other crap in UA-cam.
Sorry but it is NOT up to you to decide and in fact NO ONE has the right to decide... Second no one asked for your opinion And third this comment shows how childish your way of thinking is and how foolish and arrogant a grown up can be
"Measuring tape doesn't exist, so what do you do?" Get yourself a ruler and a ling thin strip of cloth. Transfer the measurements from the ruler to the cloth and invent a measuring tape.
@@nssherlock4547 Any kind of ruler will do as long as its position stays constant during the measurement. I suggest removing the top portion to improve compliance.
I'm pretty sure it was a figure of speech. I mean, they were building Parthenons & Pyramids back then (not to mention Eratosthenes who calculated Earth's circumference), so I'm pretty sure they weren't using just "rulers".
This video is absolutely brilliant. You're one of the best teachers I've ever seen. You make everything so interesting and clear that it keeps the viewer not only interested, but on the edge of their seat, excited to learn more all the way to the end. It made me want to binge all of your videos. I always thought history was boring and difficult to remember. Out of all my history professors, only one ever was able to tell the story in such an interesting way that I remembered the story and did it without taking notes. He was so interesting that I didn't have to write anything down and I still remember his lectures all these years later. This is the quality you have achieved here. You made me excited to go and teach this to someone else because it was so interesting and I was able to understand it so well. Thank you so much!
@Wary of Extremes I remember reading about that, old manuscripts were being scanned or something and one of them showed really advanced stuff that had been erased and written over by a monk or priest.
@@Adenzel …, easy to correlate the terms (3+1) and (4) and in this context to define the word equal but my noodle doesn't want to broaden the definition of the word equal when pondering how the computer science terms (P) and (PN) are "equal." A better word perhaps would be 'correlate' rather than 'equal.'
Maybe it's just my knowledge base, but I had never heard of this paradox before and yet this video quickly became just a history lesson. The answer was just completely intuitive, and simpler than the video really made it out to be. One needs only consider the rotational axis of a wheel, which may be fixed at the center of the wheel, but when the wheel is rolled, it becomes a non-fixed point in space. This point has a circumference of zero, but still has the same velocity and rotational period as the rim of the wheel. Since it has no circumference, it does not experience rolling, and is dragged for 100% of its rotational period. Conversely, the rim of a perfectly rolling wheel is dragged for 0% of its period. And so the ratio between the axis and rim of the wheel equals the ratio of dragging vs rolling motion.
Hi Jade, your health is definitely top priority. It's no good burning out, and not being able to enjoy what you do. I did want to say that I really enjoy your content and you make the subjects you cover easily accessible to laymen such as myself. This is a testament to both your intelligence and charisma.
"I want you to imagine that you're an ancient mathematician" >Am 32-year-old junior engineer No need for imagination, I am within reasonable tolerances of the stated design goal
It's also funny how every science youtuber has no problem butchering German names (even though they have pretty consistent spelling), but at least she tries to make the French happy.
If you had made the two wheels gears, you could have seen how one was moving faster or slower (linearly) than the other (depending on which was moving on a fixed surface).
If you have ever parked too close to the curb, you have noticed the screech made by your hubcap as it slips (and rolls) on the curb while your tire merely rolls on the pavement. The smaller the small circle relative to the large circle, the more the small one slips. Of course the center of the two circles does not rotate at all, so it slides the whole way
Nice comparison. Although, of course in reality the curb rarely is the same height as the tire's sidewall. So the wheel rim might be hitting the side of the curb stone instead of rolling/sliding on top of it.
This isn't really an accurate description. The slipping would not be perceived, it's infinitely small. What you are hearing is the grinding because your hubcap is not perfectly aligned, and is usually grinding along the side, not the edge. The tire also deforms, it isn't circular.
Better example is when Feynman talked about the shape of a train wheel. It's about concentricity along the length of a cone and the speed at which each circle moves relative to each other. If you were to put a cone and try to roll it forward (all points on each circle moving at the same speed), it would spin in circles around its central point (How fixed-axle train wheels manage bends in the track, they are slightly conical). They travel the same DISTANCE, but at faster and slower RATES. Bicycle gears are another good example of an expression of this.
@@mr.bennett108 No, they travel different DISTANCES but at the same (rotational) RATE. The outer track in a curve is longer than the inner track. The outer wheel can go longer because it is riding on a part of its conical perimeter with a larger radius than the inner wheel does. They have to go at the same rate since they are fixed to the same axle...
I didn't / don't get why it was even a mystery to anyone. The edge of the smaller circle isn't the circumference in contact with the measuring device (the table)...
@@georgeruck7797 I think you missed the part where a board was also place on the table and the inner circle still traveled the length of the bigger circle when attached. see 3:58
@@captainwasabi13 and then you take the smaller circle off of the bigger one to discover that the "length of the circumference" of that smaller circle has become demonstrably smaller than when it was attached to the bigger circle. And you arrive at the same conclusion as Galileo, the circle is skipping or slipping along with the larger one.
@@mariamedicinewheel9414 so you are saying circumference + height of elevation = the enlarged circumference of the smaller circle? Would you argue that the higher one holds any wheel, the circumference increases?
@@captainwasabi13 Well yeah, its slipping, of course it travel the same lenght. Replace the small circle with a point at the center of the bigger circle, and you end up with a 100% slipping object. While the bigger circle is 100% rolling. I agree, it's strange that it took 1600 years to figure it out.
I love that he's in bed! Like she's up and working and Frenchy is sleepy butt haha! Maybe he works nights or something. If that were me I would not be happy with wifey
I'm on the fence about it being set up... I mean it may have been, but there may have also been a good 10 minutes between when she first asked and when the shot was taken. Just try to imagine the sequence of events. She tip toes into the bedroom quietly with her camera rolling and wakes him up, "Honey can you say this?" He groggily tries to figure out what's going on. Pleasantries are exchanged as are good morning affections. She gets back to the point, but he's still like, "what are you talking about?" She has to explain and he's like, "You woke me up for that? Can I at least get out of bed first?" She says "No" coyly and asks him to say again. He says "I love you" and then we get the money shot.
She Just wants to make sure that all the guys out there that are falling for her know that she’s married! So they won’t send her messages hitting on her….I’m not kidding either.
4:20 not slipping but floating. Slipping suggests contact intermittent grip whereas floating is what the inner circle appears to the out circle and us viewers. Circumference = 2 Pi × radius
I just saw this channel for the first time and now I have to ask: why the hell it has only 200k subscribers?! I'm telling EVERYONE about this one, thanks for the great explanation!
My 7 year old looked over and said," wait is that an optical illusion" , she then proceeded to listen to you explain it. Bravo and cheers from a dad who has struggled to get her interested in math. :)
I was about to say the same thing. Just because there is no closest rational number to 0 does not mean the unit interval consisting of just the rational numbers is uncountable. The difference is the set of rationals is listable while the set of reals are not.
When I was a kid I was intrigued by this so I built a model with meccano to see how fast the wheel would rotate. In fact it locked up solid, so I'd solved the first bit. Years later as an undergraduate I studied Cantor's maths as a precursor to Turing. More years later I applied countability to digital television, using it to demonstrate that the list of transforms of an image was countably infinite. This then showed how to do transforms with minimum hardware. Great video, Jade. It's real world useful, keep it up.
I knew about this since I was like 18 and always asked my self "how can a person explain this without questioning reality as a whole? "... how does this not raise extremely profound philosophical questions? since that first day, I believe we live in the paradox of infinity.
I really don't understand what any of this kuffuffle is all about and I am shocked this held the attention of any mathematician. Everyone amazed by this is being fooled by a "magic trick". If you were to start the wheel roll and only look at the axle as it went from point A to point B, no one would have any problem understanding it's linear travel as the wheel rolls from point A to point B. If the wheel is 2 feet in circumference and it rolls 2 feet the axle has also moved 2 feet even though technically ( if it was just a point ) it hasn't rolled at all. It's no difference at any other radius up to the actual radius of the wheel. Yes the other radius's also roll with the wheel but they are also being moved linear fashion. So it's 2 forms of motion for them.
@@VicMikesvideodiary i ment to point the paradox that diferent infinities are 'bigger' than others... seems to me that that should be imposible... I understand the proces through which we arrive at the conclusion but I can't accept the result as a logical one. cause it doesn't make sense, I just think that infinity is a impossible thing and just like at singularities, things just stop working at that point
@@TBT.Stories I agree that the universe is built on a paradox but this wheel thing is no paradox. Yes it's true, there are what is called "sets" of infinities. For instance an infinite number of boxes could hold an infinite number of apples. Etc. etc. etc. As for infinity itself, yes it is real, any thing less than infinity as far as the total scale of the universe goes is what is impossible.
The thing is you do not need to rotate a circle to move it. The smaller circle is in slipping motion. Meaning that its centre is moving forward while it is also rotating
Okay I'm just gonna propose a solution before watching the whole thing (I'm at 02:11 right now) So my understanding kinda goes like this: •good morning! the two circles are _not_ covering the same distance, they're undergoing the same _displacement_ •thus, any point ∆ on the circumference of the larger circle, while experiencing the same _displacement_ (overall, through a full 360° rotation) as any point Ω on the circumference of the smaller circle will travel a greater distance in the process, since it traces out a larger path •so our friend ∆ moves through space _at a greater rate_ than Ω but travels a proportionally greater distance, which explains why, even though Ω and ∆ experience the same _displacement_ they do not travel the same _distance_ , which also explains why the circumference traced out by ∆ > circumference traced out by Ω. Also now that I think of it, that probably also explains why the equator seems to spin faster than the poles. Huh, cool stuff. P.S. okay so having watched the whole thing through, I'm pretty satisfied with my explanation, but I'd be lying if I said I wasn't thoroughly impressed with the kinds of amazing insights that these mathematical geniuses throughout history were able to draw from such a basic problem that ignorant morons like me would've just dismissed as an easily explained dumb question 😂😂😂 P.P.S. The production value of this video totally made it worth the wait, looking forward to the next one!
I would just like to add that I think the connection between this problem and the spin of the equator is not the best. The whole philosophic-mathematical points aside, this problem in physics is solved by considering the difference between a movement solely made of rotation versus a movement consisting of rotation and translation. The different velocities depending on where you are on the earth come from the fact, that every point on earth rotates with the same angular velocity (360° in 24h), but you get the real velocity when multiplying the radius of a given point on earth (shortest way to the axis of rotation, so r=0 on the poles) with the angular velocity. There is no translation involved.
Yes! Finally someone who thought of the same thing as me. It's a matter of trying to collapse a 3-dimensional problem into a 2-dimensional frame of reference. By collapsing what is essentially a cone into a plane, you lose the "discrete-ness" that comes with the 3rd dimension and get stuck with an infinity in the 2nd dimension. It's kinda how String Theorists create new dimensionalities to eliminate those pesky nulls and infinities when they try to merge Relativity with the Standard Model. What're they up to these days, anyway? 12? 13? Anyway. Same idea. Compressing the dimension (from 2 concentric circles along the length of a cone) to a 2 D problem (2 concentric circles embedded in each other) you get stuck messing around with infinities because you don't have enough dimensions to represent what is causing the change (speed).
"This paradox leads Galileo to conclude that a belief in atoms, in the sense that matter is "composed of infinitely many unquantifiable atoms" is sufficient to solve the problem of the wheel" - Wikipedia
I think Everett is calling...on that phone of his. Seriously though, if this wasn't backed up by the insanity of quantum mechanics/wave-particle duality it sounds like such a cop out. Unless I'm just an idiot and don't get what is being said here lol.
but are there an infinite number of particles making up the circumference of the wheel? or is any wheel that we can build actually made up of a countable set of points
I just had a thought. If you got two wheels with rubber tyres and attach them together, then run it along the table but have a piece of wood anchored to the table the same length at the circumference of the lager wheel and the smaller wheel touching that piece of wood, you should be able to see the smaller tyre skidding along that bit of wood. It might even act as a brake of sorts.
As a non-native English speaker, I appreciate the lack of grandiloquent words that litter most of the texts about stuff like this. Yes, I used a dictionary several times while writing this comment.
All of us married folks understand Jade's poor husband as he obligingly says a word for his wife's project despite being all wrapped up in his blankets. It's another paradox! I can love you always, but just not so much right at this one moment.
This video was extremely well laid out. You went through the information from least to most complex to ensure understanding is reached before moving on to the next concept. Bravo!
@@tamim4963 I agree somewhat like at 7:39 do you agree with me that I don't see why she finds it strange or surprising at all that a line segment is not equal to the entire line..obviously i.mewn to me..soni don't see what the big insight..I guess this took so long to figure out because nonone was drawing clear enough diagrams? Otherwise indont see why it would take so long..
Jade, that was AWESOME! I'm 62 years old, and back in 7th grade (1973 or so), I had a math teacher that was VERY cool. He was talking about infinite numbers one day. One of the other students said, "What about a million, million, million?" I said, "Plus 1". The teacher pointed at me and said, "Exactly! Frank gets it." We went on for a bit about that, but I felt like I grasped it. This explains it perfectly. Thank you.
Actually, while I have not seen this example before, I've seen the 'proof' conducted with a known-circumference wheel marked by 'distance in inches' as it rolls...it makes matching contact with the path-line and it's linear distance. Compared to the 'smaller circle', if you mark it by 'distance in inches', you will find that 'an inch of travel' on the linear is shorter than an inch-of-travel depicted on the wheel circumference...in effect, you could consider the wheel to be 'slipping' along the linear path...because you are traveling (linearly) at the rate of the outer circumference, but trying to match it to the inner wheel's 'inch-markings'...you have, for all purposes, created a simple planigraph, which is used in artwork for scaling drawings and models. You can even control this 'scaling ratio' by the diameter of the outer wheel divided by the diameter of the inner wheel. Your 'marker inserted' tool you use here is actually not what you think it is...it is not a 'rounder or flatter path', it is an X-Y chart showing cumulative scaling as X approaches '1' (1-rotation)! It becomes a 'scaling distribution chart'!
True, the paradox vanishes when you realise s non-slipping wheel's contact point is stationary, by painting 2 circumferences at different levels at the same height one or both wheels must be slipping. Using toothed wheels and tracks of different sizes, would show a track being dragged or pushed back to accommodate the differing radii. There's a 1:1 if you just measure the angular rotation and scale your distance measurement to the wheel radii. The video seems to miss completely the change of choice of coordinate systems, with for instance the triangle showing a fixed proportion of circumference travelled for varying radii.
*Please read my Explanation:-* Take the time of the each rotation T. The outer and the inner circle should have taken the same amount of time to complete one rotation. Here's the point come,if the circumfarence of the outer and inner circle is same then the both have taken the same time and rotated with the same "velocity" But,they didn't.The outer and inner circle have taken same time but the "velocities" are different.As being more distanced outside the outer circle's velocity must have a higher magnitude and on the other hand the small circle being less distanced must have a less velocity. So the conclusion is that the time taken for each rotation is same in both cases but the velocities are different which proves that they traveled different distances in equal amount of time. Edit: To know the velocity(angular),you should follow the angular velocity formula.which is V=W×r [v=velocity,W=angular velocity,r=radius]
*Please read my Explanation:-* Take the time of the each rotation T. The outer and the inner circle should have taken the same amount of time to complete one rotation. Here's the point come,if the circumfarence of the outer and inner circle is same then the both have taken the same time and rotated with the same "velocity" But,they didn't.The outer and inner circle have taken same time but the "velocities" are different.As being more distanced outside the outer circle's velocity must have a higher magnitude and on the other hand the small circle being less distanced must have a less velocity. So the conclusion is that the time taken for each rotation is same in both cases but the velocities are different which proves that they traveled different distances in equal amount of time.
Yes But here we are considering the actual magnitude of distance not the time. Yes your explanation is reasonable but actually not for the essence the concept. Sorry but keep it up Please understand the concept which she is trying to explain
This clearly differentiates the distance and displacement concept....The displacement of a point on the inner and outer circle are same. But the distances travelled by them are different.
In 10:40, while it is true that the set of real numbers between 0 and 1 is uncountable, the argument that 'since there is always a number between 0 and epsilon, the set is uncountable' is false. it would imply that rational numbers are uncountable, but thay are not. Countability doesn't mean that you have to count them in ascending order, instead we just need a one-to-one correspondence with natural numbers. We can enumerate the rationals for example by spiraling out in Z^2 (look it up, there are nice visualizations, but it's not worth it trying to explain in a comment), and by taking that and adding the rule 'if it's not between 0 and 1, we skip it' we get an enumeration for the rationals between 0 and 1. This infinite sequence will contain every rational in the interval, including any that you could name by saying 'ha, there is one between this one and 0'. on the other hand to show that real numbers are uncountable, you would need something like Cantor's diagonal argument. However, what we can't do is count one infinite set and then go 'now that those are out of the way, let's add this other set', unlike the animation at 8:55 leads to believe, since the first list would never end and therefore we would never get to the second one. Instead we could do 1,-1,2,-1,... and so on, since for any specific element, the list gets there in a finite number of steps. I know this all sounds very pedantic, but you kinda have to be when you are dealing with such an abstract concept as infinity. :)
While it is true that one can formulate both countable and uncountable infinities, the example you gave doesn't automatically lend itself to being uncountable. Remember, the rationals are countably infinite. Therefore the finite decimals you provided (e.g. adding X more zeroes) are also countably infinite.
Fun trivia for everyone. Car people talk about this as slip angle. In the case of cars, this means a difference between two physical actions involving the wheel. Such as tire speed vs road speed (locking breaks), wheel angle vs turning angle (slipping and sliding) or spinning your wheels. In the math you can work out an angle between where one frame of reference is vs the other. Intuitively this led me to the conclusion of slipping pretty quickly, not quite feeling I'm completing the connection though.
The connection is that they assumed a motion which appears to be smooth must be similar to the hexagon, where it lifts from the track then drops back down on the flat points. So, if your circle was made of a million individual sides, each point would still theoretically be acting the same as a hexagon, lifting at each corner and setting back down on each flat. The concept was that a circle must have infinite sides, which would explain why the little gaps weren't found. Slippage wasn't actually proven to be the issue until much later. Mainly, once we were able to spin something very precisely as very high speeds that was milled extremely precisely so that we could eliminate other potential variables, and discovered that heat was created in the process. But heat is also an effect of compression, so the experiment had to use a minimally compressive material, of excessive smoothness, and do the "two-wheel" experiment she did, then recreate it with only the large wheel touching and then only the small wheel touching (while also adjusting the weight placed on the wheel, to cancel out variables in compression) , then compare the thermal increase in both of the single wheel experiments against the thermal increases of the double wheel experiments...... In the end finding that the difference was slippage, where the contact surface was continually changing as the rotation happened, but that on a molecular level there was drag happening also. Our ability to intuitively understand the phenomenon by looking at a wide cylinder rolled across the ground, seeing how turning on a point drags the edges of the cylinder, then experimenting to learn how the different speeds between the inner and outer ends of that cylinder still exist even at higher rotations and smaller turns..... that is merely a different manner of approaching the problem, from a more modern viewpoint really. The whole problem of the wheel slippage wasn't fully recognized until the invention of powered vehicles with wheels, when it was definitely shown that having a straight axle for your power to the ground made it harder to turn the vehicle, and where the axle had enough flexibility (along with the wheels themselves having some flex) so that the slippage finally presented as one wheel jumping in position vs the other, instead of the continual slippage she showed in the video. As with many things we see daily.... intuitively making a hypothesis is one thing, but definitively proving a theory is much harder.
lol i missed the part where it was a paradox to begin with. its like saying "theres a car on a trailer, how is it moving when the engine isnt running!?"
@@ICKY427 Lol it's not like that at all. There's either an engine or not. But if there is, it's running "both" circles. And when you solve a paradox, it's not really a paradox anymore. That's the point.
Yes! Thank you for some comradery... i was beginning to think the world had gone crazy and i was all that was left to bring reason back to society, haha
When I was younger I had a similar idea to this, where I imagined a long rope attached to the earth that stretched out for thousands of miles into space, and wondered how it was possible for one end of the same object to be traveling at totally different speeds without tearing itself apart
I believe the solution is that the wheel is slipping; because the question is why are the two circumferences equal? Then you go into infinity to show why that is the answer. If you wanted the answer on how to measure circumference if an inner circle from an putter circle, hold a marker still on the circle, rotate the circle so it makes a ring inside the wheel, grab a piece of string and measure it. 😅
I find the 8:04 statement to be quite thought provoking: "These objects are continuous, uninterrupted you can't split them up" I've always viewed it the opposite way; continuous means you can split it up, as many times as you want, in fact you're never done splitting, you can split forever and ever
I guess this is a characteristic that separates mathematicians from engineers. It was intuitive that one-to-one correspondence between the circumference of 2 circles must be done with equal length units simply because you're measuring lengths. You can't treat them as continuous if you are measuring their discreet length. Only a mathematician could get this confused, lol. Really great video, though.
You know the difference between an engineer and a philosopher? Once an engineer and a philosopher died at the same moment and found themselves being tortured side by side. A demon put a beautiful woman at the end of a field and told the two each time he clapped his hand the woman would come half the distance towards them. After a few claps the philosopher collapsed, tearing at his hair crying and shouting in torment. He realized through all eternity the woman would never reach him. The engineer, meanwhile, seemed quite content. The demon was frustrated and confronted the engineer. "Don't you realize this woman will never reach you?" The engineer smiled and retorted, after a few more claps she will be close enough for all intents and purposes."
@Grace Jackson Neither do I. Furthermore, I think their demonstration is flawed. They show the "skipping" or slippage" by the smaller wheel, but that only happens because they are forcing the 2 to follow a straight path. If the 2 wheels are allowed to follow a circular path they can do so without slippage. That's the principle behind tire stagger in circle track racing.
Actually, you definitely can treat them as continuous. Obviously, atoms and molecules were not discovered at the time Galileo or Aristotle tackled the problem. But also, physical objects are usually considered to be continuous until the model fails even today. You seem to have a misunderstanding about the nature of discreeteness and of continuity: you absolutely can make the assumption that the objects are continuous and not discreet. The error arised later, when they tried to reason about the notion of measurement of continuous objects using intuition and not mathematical rigor.
7:33 Okay, there's some nuance here that I feel I need to add, since the video at this point becomes misleading in my opinion. There's two different notions of size at play here: cardinality and measure. Cardinality is where the "different types of infinities" are relevant. Cardinality refers to the size of sets, and if there is a bijection (one-to-one correspondence) between two sets, then they have the same cardinality. So the two lines are the same size in this sense; they have the same cardinality. Measure represents the "length" of the curves. What we generally think of when we think of "length" and "measurement" corresponds to the Lebesgue measure. In this sense, the two lines are different sizes. The bottom line has a greater Lebesgue measure than the upper line. What this demonstrates is that having the same cardinality does not imply having the same Lebesgue measure.
In fact, THIS is (almost) the actual answer to the problem, since rational numbers are countable, yet have the same "problem" where any interval has the same cardinality as any other. The "almost" comes from the fact that thinking of measure here is a very overpowered idea (though correct). It can be thought in terms of distance (metric, rather than measure), which is a simpler tool, closer to geometry in roots.
Measure is not an "overpowered" idea, just different (than either cardinality or metric), but metric does seem to be more intuitive notion in this context.
@@Czeckie Because cardinality has nothing to do with this problem. You can encode that same problem with rational numbers rather than real numbers, and you'll have the same problem.
Another thing I found strange is when she explains about drawing a line coming from the top of the triangle passing through both lines, and saying that there's always one-to-one points. But if you draw a line passing at all points of the smaller line, there will be empty spots on the bigger line. Or whatever I'm just overly confused
I think I've found my new favorite UA-cam channel. I regularly watch 3Blue1Brown, Mathologer, MindYourDecisions, Infinite Series (which also addressed this subject during it's run), etc. and I've read [probably too much] philosophy of mathematics about Russell, Cantor, Hilbert, Wittgenstein, etc. This stuff is just fascinating to me and always has been. Very well done!
"In fact, the question, 'what is the next number after zero?' is impossible to answer because for any number we choose there will be always a number that is closer to zero. So, we can't even count the numbers between zero and one ..." This same argument can be applied to the rational numbers which is a countable set. The correct way to argue that the real numbers are uncountable (or the real numbers between zero and one) is to use Cantor's diagonal argument which goes as follows: Assume you have a complete list (of one-to-one correspondence with the natural numbers) of the decimal representation of the numbers, and show that for any such a list you can get a number that is not on the list. The way to find this number, is to pick its first digit to be different of that of the first number, and the second digit to be different of that of the second number ..... and so on. You can do that forever, as most of them have an infinite decimal representation (either due to being irrational or being a rational with a denominator which is coprime with 10). As for those whose decimal representation terminate, you can add infinitely many zeros. This shows a contradiction that there's such a list, and hence the set must be uncountable.
I can't explain just how good this video is. The methods, the shapes, the examples, the explanation... Everything is amazing. Thank you for this awesome video!
Thank God we're all different. I can't explain just how irritating this video is. The only thing that's amazing is that she can't even pronounce the word _wheel._
@@HappyBuddhaBoyd Shhh!! You're spoiling everything. The video is AWESOME and EXCELLENT. Look how _cheerful_ she is! She makes it so fun and cool! And the blue whill is so nice! This is what science is meant to be all about. 😃
A deceivingly advanced topic that is broken down into easy to understand pieces presented in a fun and engaging way. I could show this to a 5, 10, 15, or 65 year old and they all could learn from it. You earned my subscription today.
Here is one thing I can say about numbers: Our species is in desperate need of more instructors like her, speaking in many different languages. The number that is currently needed is less than 8 billion. The number of languages in which the instructions are needed may be more than 6000 (at this point). Do keep them coming, and fast! (I like to think that millions of my contemporaries feel the same way --even in Afghanistan.)
0:50 The inner wheel would be similar to slipping against the ground in this scenario, that's why the 2 distances are the same. If you actually shrink the wheel to the smaller size, it would also travel a shorter distance, provided that it doesn't slip against the ground.
@Anon Ymous Wrong about slippage, otherwise mostly correct. *Smaller radius = smaller circumference = shorter distance per rotation.* But in this example both wheels travel the SAME distance (they have to, because they share the same center point), therefore at least ONE of the wheels MUST be slipping, at all times while moving.
Another thing to point out is that this paradox is only confusing when the big wheel rolls on the ground and the small wheel "magically" keeps up. If, instead, the small wheel rolls along a ledge and the big wheel hangs over the ledge, it is easy to see what happens. The big wheel moves faster and the points on its edge actually move backwards at the bottom.
The two “wheels” that are leaving their paint marks on their respective surfaces as the big circle is rolled: rig a system that allows the surface that the smaller wheel is in contact with to slide, like on a linear bearing. The surface will be pushed “backward” the distance that is the difference of the circumferences.
I think there are 2 reasons behind this.... 1. Here the centre of both the circle is not static in place and keep shifting while rolling on surface.... 2. Also, length of the centre(focus) from the string is different in both circles.... Thats why it is happening....
I love that i learn something almost everyday that bends and warps my mind to a point were i cant look at the world the same, not necessarily from this channel but in general.
By the argument of there always being a number closer to zero, the rationals would also be uncountable. One has to show that there is NO way to try to count the set, which is of course much harder to demonstrate.
Excellent point. When counting a set of numbers, there's no requirement that you count them *_in order,_* just that you'll eventually get to any specific number.
1:30 "...I'm going to get my French husband to say it." Did you hear that noise? It's the sound of nerds' hearts breaking all over the world. So sorry...
Thanks for this lovely lecture. I have some additional quanta of points 1.) Numbers are like points with spaces between them. There can be intermediate points in between two terminal points. The number of intermediate points between let's say 0-100 can be equal to the number of intermediate points between 0-1 if the size of the space between each point is different; the latter being higher than the former. The space between each point is called an interval. The size of the interval is the unit, and it is a measure of the amount of discreteness or discontinuity between each point. Larger the interval (in other words the higher the unit), the more discontinuous the points are, and vice versa 2.) In terms of set theory, the smaller hexagon (B) is like a smaller subset of the larger hexagon (A) such that all elements of B are in A the lines are like lists of all the elements in both sets. When the elements the of both sets are being listed (B above A), they are aligned, and identical elements are positioned in the same region like an intersection. Because A has more elements than B, an amount of it will not intersect with B. The gaps in the B list are the elements in A that aren't in and therefore do not intersect with B. 3.) From a geometric point of view, the distances between the an angle of A and B is like a hypotenuse, and the distance between a side of A and B is like an opposite. The hypotenuse is longer than the opposite, and the difference between the hypotenuse and the opposite is the gap in the line of B. As the number of sides and angles in the polygon increases the difference between the hypotenuse and opposite decreases and becomes less obvious. A circle is like a polygon with infinite number of sides and angles, in which the hypotenuse equals the opposite.
@@upandatom Your demonstration is flawed though. There doesn't have to be slippage of the smaller wheel. You only got slippage because you were forcing them to follow a straight path. If you allow them to follow a circular path slippage isn't necessary. This can be demonstrated by rolling a dixie cup on a table top. The top of the cup has a larger radius than the bottom. If you place it on the table on it's side and roll it, it will roll in a circular path without slippage.
Yes exactly! Am I stupid, or why is it a problem that different sized wheels, have different lengths of circumference?? Then why go into triangles and infinity??
Honestly, Jade, this is one of the most gorgeous maths videos I've seen on UA-cam. Everything remains crystal clear as we dive into the topic, while maintaining some mystery to encourage us to follow you deeper. Brilliant!
The fact that you cannot pick "the smallest number after zero" (with the usual order), does not imply that the set of numbers you are working with is uncountably infinite. The same argument holds with the rationals if you consider (again) the usual order. But anyway, very nice video, I really like your content!
By the same argument, all whole numbers would also be uncountably many, since there is no smallest whole number to start, but that’s ridiculous: you can just do (0, 1, -1, 2, -2, 3, …).
@@felipevasconcelos6736 I see what you mean but the point is that within the integers and its usual order, there is in fact the smallest number after 0, namely 1.
@@DiegoMathemagician yes, but there are infinite numbers smaller than 0 that have to go after it on the list, so it’s really easy to show that countable sets don’t need to have an increasing bijection to the naturals. All they need is to have a bijection at all. The whole point of asking for “the smallest number after 0” was precisely to answer the question “if we list all reals between 0 and 1 in an increasing manner, what number do we put after 0?”. The equivalent question for the integers is “if we list all whole numbers in an increasing manner, what number do we put first?”, and asking for “the smallest number” is equally irrelevant.
I've never heard of the paradox and thought of slipping right in the beginning of the video as soon, as innercircle was mentioned and it baffles me, why it took centuries to figure it out.
An incredibly pedantic critique: everything you said to 'suggest' that the reals are uncountably infinite could be applied to the rational numbers too (because every statement you made is true of rationals as well, and all your examples were rationals (0.01 = 1/100). And yet, the rationals ARE countable!
Proving as such is a fun exercise for the reader. (hint 1: don't worry about including the same *value* more than once) (hint 2: try a grid arrangement and a space filling curve)
I remember seeing this before. A simpler conclusion is that the smaller wheel moves slower then the larger wheel because it has a smaller size (causing the skips). This can be seen very clearly in a very large model. In one rotation of a wheel 4 feet in diameter, the small wheel moves very slowly, while the large wheel moves very fast for the same distance
@@mikehunt1528 Not gears. Imagine you are holding a umbrella ☔, you twist the handle in your hands. The outside edge of the umbrella will rotate faster then the handle to keep up with the rotations. The farther away the edge is, the faster it has to move to keep up. Same idea as this video.
I hope your life unfolds exactly how you want it to. The passion you have for maths is captivating. Thank you for contributing to the betterment of human kind with your wholesome content.
@@supernova5618 Galileo didn't discover calculus. He reasoned the same as Archimedes and others did many years before his time, but never developed a theory. At best you could say he successfully used indivisibles in his work. Besides, the history of the discipline has had contributions from many different people, and there isn't a sole 'inventor' on it.
It's hard to believe that it took them centuries to figure out the inner parts of the wheel are travelling at a different speed compared the outer parts.
It gets complicated once the question of force is introduced. For example the outer circle travels a further faster gait, but the inner circle travels with more force relative to the distance.
I have solved the "youtube learning paradox" where I watch a video to learn something new and finish the video feeling even more confused. Conclusion = I am a dumbass
Nah don't be too hard on yourself. Paradoxes like these are supposed to leave you confused due to the nature of challenging assumptions that are mistakenly made. Once you find the assumption that is erroneous, the paradox becomes more clear to understand.
Nope. You're actually smart. Smart enough to learn AND ALSO smart enough to realize that even when you thought you've learned you don't know it all. When we don't get confused we think we know it all quickly.
Mathematics is nothing more than cleverly disguised nonsense. Numbers themselves are nothing more than meaningless abstractions that require some additional language in order to mean anything at all. Everything about mathematics is purely illusory. For instance, 0.999_=1, 1.999_=2, which goes on ad infinitum. So a whole number is equal to the whole number less an infinitely small quantity. That is impossible, yet it has been mathematically proven. So whole numbers are equal to not so whole numbers. It becomes even more strange when you question what the functions of mathematics actually mean in reality. How do you add one apple to another apple in reality? Does pointing at them and saying one apple, two apple really mean you have added them? Now, how do you subtract an apple from an apple? How about multiplication and division? Can you multiply and divide apples? How would one go about doing that? Is one apple truly equal to another apple? Absolutely not. They may appear similar, but they are different. One may be green, one may be red. One may be older or spoil quicker than the other even if they appear identical at first glance, etc. There are always numerous differences which are often neglected by mathematics in order to proclaim equality of things in reality. Oddly enough, the same thing is typically done in science as well. It is easy to proclaim equality when all of the differences are arbitrarily ignored. That is what we do so we can pretend we know more than we know and feel smart about what minute amount we do know. Don't beat yourself up. We are all idiots pretending to be smarter than we are.
Yeah, I saw that pretty easily. The bigger wheel drags the smaller wheel along a line faster than it would naturally roll, proportional to the difference between them. So instead of each point touching the surface instantaneously, then rotating away and off, that touch point slides slightly before and after direct radial contact. It's similar to Zeno's paradox in that it confused people due to dissimilar infinities regarding numbers sets.
I cant help but appreciate how nice this video is made. You think you saw every kind of math content on youtube but here you are and paint your handmade hexagon to show how it skips and I just love it.
@@badradish2116 i dont know if i misspelled something or did not make myself clear but you really never had a crush on some one older when you were a kid?
This video was wheely fun to make!
I'll show myself out...
But you changed the icon/profile picture.. you can’t scam me.. I am extremely observant
damn I am foiled!
I'll never get tired of these puns. Keep them rolling.
Pie - oddly I lived in the same Florida, USA county as the airport with the symbol PIE for quite a while... PI and TAU etc wow - definitely interesting to think about the show business aspects of all of it - what people want & what is sustainable & will lead to sequels and so on... very different probably than working at an actual University or even Brilliant etc hahaha - I’ve liked some of your graphics work in the past and probably it has enhanced watch time etc - I liked the physical rolling stuff here but I’m wondering how it played with overall audience satisfaction!? - Physics girl did an appearance with The Science Asylum and if you haven’t yet I hope you do that and as much collaboration as possible too with him and other channels hahaha... Name Explain channel for a bunch of things including PI and TAU even... yeah I’ve mentioned the ABC conjecture to you multiple times perhaps and maybe you can come up with some zany take on all of that with regard to the human drama and so on yeah philosophy of mathematical knowledge and agreement etc that would somehow go super viral... Brady Haran’s podcast with David Eisenbud briefly mentioned it but yeah who knows many more in depth things to find even on UA-cam I’m sure before you decided to make an episode about it or similar things if you do! ⭕️ no emoji perhaps for hexagon etc!?!?!? 🎬🌈🗽🤯☮️💟🌎🌍🌏🚀🪐♾➰➿〰️...
We're in your house, mate. You don't need to show yourself out for that.
_[yeets myself out a window]_
I worked for a trucking company when I was younger and I remember the importance of matching tire circumference or tires when putting together duel wheels. If one of the tires is smaller it is dragged along and wears much faster as well as creating an increased load because of the drag.
"If one of the tires is smaller it is dragged along and wears much faster as well as creating an increased load because of the drag."
Not if you're going in circles. It's called tire stagger and circle track racers have been using it forever. Also, in essence the same thing happens on a motorcycle when you lean the bike to turn. The tire on the motorcycle is rounded. When you lean the bike the smaller circumference outer edge of the tire is making contact as well as the larger circumference center and it creates a turning force. Anyway, the video and explanation is flawed. There does not have to slippage of the smaller wheel if the path of travel is circular. A good way to visualize this is with a dixie cup. The cup has a larger radius on top then on bottom. If you lie the cup on it side and roll it, it will roll in a circle without slippage.
Thank you Barry and Stefan !
I guess if you did have duel wheels they would be fighting each other. Dual wheels on the other hand would work well together.
@@stefanl5183 well the video was about the circumfence of circles so what benefit would you get in measuring them when going in cricles to solve this problem ? because when you go in circles the closer you are to your turning axle the smaller the distance you have to travel (thats why cars have differentials)so try rolling your cup with the bigger side as the tunring axle you will realise you get alot of slippage at the smaller side. Also if you try to roll the cup in a different sized circle than it would do naturally you would most definitly gets slippage
@@nokiarsxzmika The problem is the video is saying that you have to have slippage, and that it's inevitable. It isn't! You don't have to have slippage. Yes, you will go in a circle and the radius of the circle will depend on the difference in those circles and the distance between them, but the conclusion you must have slippage is wrong. Sometimes you want to go in circles. Take a look at a World of Outlaws Sprint car and notice the difference in size of the left and right rear tires. Maybe space and time isn't always straight and flat?
Three things I love about this video:
1. The explanations are clear and concise.
2. The tempo of the video is not too short and not too long.
3. The French husband, apparently half asleep, being such a good sport.
It was a really good video, but did she invent Aristotle’s Paradox? Why is that even a question?
I don’t understand the problem. There’s only one wheel. The second is imaginary, and if you make it real you’ll have to roll it more quickly to travel the same distance in the same time. All work is done between the surface of the wheel and the part of the counter that it touches. It will take less energy to accelerate a smaller wheel. They will travel same distance.
@@gristly_knuckle My issue with your comments originates from my lack of understanding of practically every one of your observations. Clearly, your grasp of this concept is far beyond my own.
@@stoneymcneal2458 crazy, ok. Well the chick probably wants some Eviler dude, a French guy.
@@gristly_knucklethat's exactly what she's saying. your understanding of this problem comes from the modern interpreration of this problem and physics as well.
take in account, greeks didn't have F = ma law, they didn't even know exactly what's acceleration or work. it seems odd because if you separate them enough to make the wheels spin at the same time attached to each other but on different surfaces, the paradox holds.
@@juxx9628 perhaps my understanding is grounded in modern physics, so that the problem appears irrelevant to me.
Poor Cantor. He was derided and ridiculed when his proposed his diagonal argument (countable and uncountable sets) only to be heralded as a pioneer of set theory decades after his death.
such is the case with many great minds
@@sumdumbmick future people will judge
@@sumdumbmick you're reading something into her comment which just isn't there, which is really really annoying
@Anirban Mondal Yes, one needs axioms. Constructivism is simply not accepting some of the axioms of modern math. Since math is nothing else but discovering all the implications of a given set of axioms, other sets of axioms do in fact lead to equally valid math (as long as they're not contradictory). So do it if it makes you happy. But be warned that not many modern mathematicians will bat an eye about your results, since the ZFC is widely accepted nowadays and has become the foundation of almost every field of mathematics.
@Anirban Mondal I'm not talking epistemology, I'm talking pragmatism.
Since we already touched it, we might simply consider the old question: is mathematics invented or discovered? My answer: kind of both, but mostly the latter. The axioms are clearly an invention. They're made up stuff and there is no reason to be convinced that they resemble reality in any meaningful way. You're free to believe that they do or do not. But my point is: that's just a matter of personal worldview and nothing mathematics cares about. The moment you set your axioms down, the implications are there for you to unravel. That unraveling is what math is (and hence, math is discovered). But it's not a natural science; it does not claim that its axioms are the fundamental rules of the universe we live in or whatever.
Therefore, if you want to choose a different set of axioms, that's fine by me. Either way there's no way to verify that either math tells us anything meaningful about the universe. We just can work our way through it, shunting symbols as you call it, and hope that we won't stumble over some ineluctable contradictions. And if we're lucky, there might be other people around who find our results useful. That's the best a mathematician can hope for. There is no absolute truth to be found.
Btw, to my understanding, model theory is not an alternative approach to math opposing the axiomatic view. It's a theory of different axiomatic systems, but comes with its own set of axioms. Since you mentioned Kant we could as well go the one step that is an inevitable consequence of his epistemology but which he still chickened out to go himself: believing is more fundamental than knowing, in the sense that you have to believe something in order to be capable of knowing anything.
For the line drawing of the cycloid, the opposite situation gives a more intuitive picture to me. If you roll a wheel on an inner circumference, you’d see the line traced at the outer circumference to double back on itself in a short loop
This was so great, I love that you made the shapes. Animations are cool but practical demos are awesome
Inspiration to practically recreate your plane shenanigans?
Damn. 3blue1brown better protecc his necc around this bigboi!
Agreed!
Dude you're everywhere
One really cool thing about this problem is that it has many practical applications including in automotive design and engineering, one example being the slip of the inner and outer wheels during a turn.
I guess you could say that ancient problems require ancient solutions.
Yeah. But more often than not they require a giant like Cantor.
The greeks actually had kind of a habbit asking pretty tough questions you can rack your brain about for centuries.
OUTSTANDING MOVE !!
Whats that in uncountable formations of legionnaires?
Math is eternal!!!
For those wondering what the final answer was, here is what I thought she would say to tie it all together:
The demo at 0:47 is not a _measurement_ of length. It is a _mapping_ of points. It indeed shows a 1:1 correspondence in points, but (as explained at length later in the video) a mapping of points does not always denote equality of length (7:20).
Longer version:
At 4:20-5:10 she explains that the small wheel must be slipping-and she wonders (more or less) if this implies that, for every 2 points the big wheel moves forward, the small wheel must be rolling 1 point forward and then sliding/skipping 1 additional point forward. (At least that’s how the rolling polygons worked.) But the small circle is _not_ being dragged in a traditional sense, because there is a one-to-one correspondence between each point of the wheels’ respective journeys. Our intuition says one-to-one correspondence means they are equal, and therefore the circumferences must equal, which they clearly can’t be - so she spends the rest of the video explaining that, for continuous quantities like geometric points, one-to-one correspondence does _not_ mean they are equal. Therefore there can indeed be one-to-one correspondence in points along their rolling motion while _not_ requiring the lengths to be equal, thus permitting a rather non-traditional kind of “slipping” in which no point is ever dragged.
[Edited to add the short version at the top of this comment]
except that things such as "one-to-one correspondance" and "uncountable infinity" don't mean anything in the real life, because there aren't any continuity in the world safe for in our imagination (or fantasy). Mathematicians can play with their imagination as much as they want, but they should not mess it up with reality that's all
they have an equal amount of points on them , infinite, but the circumference's are not equal. Isn't the triangle just a ratio?
Her version was way easier to follow. :)
I thought she was done when she did the drawing tracing the two circles, but then she went on to explain it in infinite detail :-)
@@ruadrift lol
Great video - got my brain ticking along nicely. 🙂
For the circle paradox, this illusion that the circles travel the same distance is what happens when you move remove a dimension. The two lines appear the same length in one dimension, but it's clear they are not the same physical length. The linear distance is a projection into one dimension - Because you have to raise the platform up to meet the second circle, it is travelling in the same direction but on a different plane, not the same plane as the larger circle. (i.e. the distance travelled is only the same in one dimension) It's a bit like casting a shadow of a ball on the wall. Depending on how far away from the wall the ball is the shadow will have a larger or smaller area, but the ball itself remains constant. It also doesn't matter how far away you move the light from the ball, the shadow will remain constant.
If you force the circles to travel on the same plane but connect them to each other centers so the rotational speed is the same, the system wants to move in a circle because the smaller cirlce physically can't keep up with the distance travelled by the larger circle. And then if you were to raise the floor on the side with the smaller circle it would happily roll forward alongside the larger circle.
Never seen more smile of a human talking math beside this.
@@FriendlyDiscourse well it's definitely working 🤤
Seriously. I always feel like she's flirting with me.
@@philochristos that's creepy
@@philochristos waitresses do the same thing to get more tips.
3Blue1Brown
I'm 56 years old and computer technician and thought I know a lot.
But I never ever heard anything before about number lines or this wheel.
This video was the most interesting thing I have ever seen AND I have ever seen in UA-cam and I'm totally impressed to find such a pearl between all the other crap in UA-cam.
thank you Peter, this comment made my day!
With more videos like this I would not be worried about my kids spending hours on UA-cam! Kudos!
Sorry but it is NOT up to you to decide and in fact NO ONE has the right to decide...
Second no one asked for your opinion
And third this comment shows how childish your way of thinking is and how foolish and arrogant a grown up can be
@M. K. Jones
Just as a footnote Peter, I'd strongly recommend "Philosophical Devices" by David Papineau, which covers the same topics (without the dazzling smile).
"Measuring tape doesn't exist, so what do you do?"
Get yourself a ruler and a ling thin strip of cloth. Transfer the measurements from the ruler to the cloth and invent a measuring tape.
Can you use a dead ruler, or do they need to be living ? My ruler of choice would be QE2. She is a long ruler.
@@nssherlock4547 Any kind of ruler will do as long as its position stays constant during the measurement. I suggest removing the top portion to improve compliance.
Still wondering why pi is magic...
you'll be filthy rich if you patent it
I'm pretty sure it was a figure of speech.
I mean, they were building Parthenons & Pyramids back then (not to mention Eratosthenes who calculated Earth's circumference), so I'm pretty sure they weren't using just "rulers".
This video is absolutely brilliant. You're one of the best teachers I've ever seen. You make everything so interesting and clear that it keeps the viewer not only interested, but on the edge of their seat, excited to learn more all the way to the end. It made me want to binge all of your videos.
I always thought history was boring and difficult to remember. Out of all my history professors, only one ever was able to tell the story in such an interesting way that I remembered the story and did it without taking notes. He was so interesting that I didn't have to write anything down and I still remember his lectures all these years later. This is the quality you have achieved here. You made me excited to go and teach this to someone else because it was so interesting and I was able to understand it so well. Thank you so much!
When they said reinvent the wheel, I didn't think they meant this.
Galileo was so close to discovering calculus.
@Wary of Extremes I remember reading about that, old manuscripts were being scanned or something and one of them showed really advanced stuff that had been erased and written over by a monk or priest.
The word equal can be so misleading.
@@MrBurgerphone1014 @Wary of Extremes I think that that is mentioned in a VSauce video, I think it's called "mistakes"
@@MrBurgerphone1014 www.theguardian.com/books/2011/oct/26/archimedes-palimpsest-ahead-of-time#:~:text=Using%20multispectral%20imaging%20and%20an,19th%20century%2C%20and%20anticipated%20calculus.
@@Adenzel …, easy to correlate the terms (3+1) and (4) and in this context to define the word equal but my noodle doesn't want to broaden the definition of the word equal when pondering how the computer science terms (P) and (PN) are "equal." A better word perhaps would be 'correlate' rather than 'equal.'
I dig the "Deploy French Husband" move.
But, I despise the "having a husband" move all together... lol
@Rob Allen SIMP!
First Last lol. Yeah if this COVID nonsense keeps up kids of the future will ask “what’s a party”.
“Clever girl....”
I am not french, but a Latin lover as well
Maybe it's just my knowledge base, but I had never heard of this paradox before and yet this video quickly became just a history lesson. The answer was just completely intuitive, and simpler than the video really made it out to be.
One needs only consider the rotational axis of a wheel, which may be fixed at the center of the wheel, but when the wheel is rolled, it becomes a non-fixed point in space. This point has a circumference of zero, but still has the same velocity and rotational period as the rim of the wheel. Since it has no circumference, it does not experience rolling, and is dragged for 100% of its rotational period. Conversely, the rim of a perfectly rolling wheel is dragged for 0% of its period. And so the ratio between the axis and rim of the wheel equals the ratio of dragging vs rolling motion.
Excellent description 👍
I had something like this in mind from intuition but couldn't figure out how to put it into words. Very eloquently put!
Even simpler explanation, great!
Good one! 👍
Yeah but misses the point of the video which is about understanding infinity.
That is marvellously constructed video, really excellent.
Welcome back Jade. We've missed you.
I've missed you guys too! I took a 2 week break because stress and burn out, but now I feel better than ever!
@@upandatom I'm glad to hear that you're feeling better. Take care of your self :D
Indeed
@@upandatom Please always take your time to relax - health is very important.
Hi Jade, your health is definitely top priority. It's no good burning out, and not being able to enjoy what you do. I did want to say that I really enjoy your content and you make the subjects you cover easily accessible to laymen such as myself. This is a testament to both your intelligence and charisma.
"I want you to imagine that you're an ancient mathematician"
>Am 32-year-old junior engineer
No need for imagination, I am within reasonable tolerances of the stated design goal
Get gray heir leather skin finish your schooling and lose your nose and you're already there
i think the correct wording was ... "parameters." sounds more profesh. thumbs up #74
Ok lets test this. Is the glass half empty or half full?
@@jamesperfect2027 the glass was not designed to proper specifications
Lol
I love how the husband is just glad to teach correct French even if you wake him up at 1am. That's very French.
It's also funny how every science youtuber has no problem butchering German names (even though they have pretty consistent spelling), but at least she tries to make the French happy.
Very Fronch!
@@ralf391 well, maybe if she had a German husband that wouldn't be the case
ouai
Frenchsplaining. 😏
If you had made the two wheels gears, you could have seen how one was moving faster or slower (linearly) than the other (depending on which was moving on a fixed surface).
If you have ever parked too close to the curb, you have noticed the screech made by your hubcap as it slips (and rolls) on the curb while your tire merely rolls on the pavement. The smaller the small circle relative to the large circle, the more the small one slips. Of course the center of the two circles does not rotate at all, so it slides the whole way
really cool example, thanks for sharing 😃
Nice comparison. Although, of course in reality the curb rarely is the same height as the tire's sidewall. So the wheel rim might be hitting the side of the curb stone instead of rolling/sliding on top of it.
This isn't really an accurate description. The slipping would not be perceived, it's infinitely small. What you are hearing is the grinding because your hubcap is not perfectly aligned, and is usually grinding along the side, not the edge. The tire also deforms, it isn't circular.
Better example is when Feynman talked about the shape of a train wheel. It's about concentricity along the length of a cone and the speed at which each circle moves relative to each other. If you were to put a cone and try to roll it forward (all points on each circle moving at the same speed), it would spin in circles around its central point (How fixed-axle train wheels manage bends in the track, they are slightly conical). They travel the same DISTANCE, but at faster and slower RATES. Bicycle gears are another good example of an expression of this.
@@mr.bennett108 No, they travel different DISTANCES but at the same (rotational) RATE. The outer track in a curve is longer than the inner track. The outer wheel can go longer because it is riding on a part of its conical perimeter with a larger radius than the inner wheel does. They have to go at the same rate since they are fixed to the same axle...
My explaination: the smaller circle is taking a ride on the bigger one.
I didn't / don't get why it was even a mystery to anyone. The edge of the smaller circle isn't the circumference in contact with the measuring device (the table)...
@@georgeruck7797 I think you missed the part where a board was also place on the table and the inner circle still traveled the length of the bigger circle when attached. see 3:58
@@captainwasabi13 and then you take the smaller circle off of the bigger one to discover that the "length of the circumference" of that smaller circle has become demonstrably smaller than when it was attached to the bigger circle. And you arrive at the same conclusion as Galileo, the circle is skipping or slipping along with the larger one.
@@mariamedicinewheel9414 so you are saying circumference + height of elevation = the enlarged circumference of the smaller circle?
Would you argue that the higher one holds any wheel, the circumference increases?
@@captainwasabi13 Well yeah, its slipping, of course it travel the same lenght. Replace the small circle with a point at the center of the bigger circle, and you end up with a 100% slipping object. While the bigger circle is 100% rolling.
I agree, it's strange that it took 1600 years to figure it out.
I’m kind of new to your channel but the two infinites explanation has got me hooked. Well done!
"Hey French husband, can you say this dudes name in your native tongue for my youtube video?" "But I am le tired"
oh man, that le tired is an oldie.
Take a nap, then fire ze mizzles
Bout that time, ey chaps
Zut alors!
Australia can come too!
I love how she woke up her husband just to make him pronounce a name lol.
I love that he's in bed! Like she's up and working and Frenchy is sleepy butt haha! Maybe he works nights or something. If that were me I would not be happy with wifey
@@burtan2000 - I think it was probably a set up as he didn't seem sleepy!!
I'm on the fence about it being set up... I mean it may have been, but there may have also been a good 10 minutes between when she first asked and when the shot was taken. Just try to imagine the sequence of events. She tip toes into the bedroom quietly with her camera rolling and wakes him up, "Honey can you say this?" He groggily tries to figure out what's going on. Pleasantries are exchanged as are good morning affections. She gets back to the point, but he's still like, "what are you talking about?" She has to explain and he's like, "You woke me up for that? Can I at least get out of bed first?" She says "No" coyly and asks him to say again. He says "I love you" and then we get the money shot.
She Just wants to make sure that all the guys out there that are falling for her know that she’s married! So they won’t send her messages hitting on her….I’m not kidding either.
He’s a lucky man.
So the answer is "you need to develop measure theory and calculus."
A rather large number of ancient paradoxes have that answer.
This is what Isaac Newton also realized and he invented calculus.
In order to make an apple pie from scratch, you must first create the universe.
@ゴゴ Joji Joestar ゴゴ help me out my bizarre friend, what does velocity have to do with the distance they traveled?
@@anthonynorman7545 maybe its like, one traveled for the same time but it went slower, therefore less distance. I dont know just guessing
@@aleblanc6904 woah how deep but true
4:20 not slipping but floating. Slipping suggests contact intermittent grip whereas floating is what the inner circle appears to the out circle and us viewers.
Circumference = 2 Pi × radius
I just saw this channel for the first time and now I have to ask: why the hell it has only 200k subscribers?! I'm telling EVERYONE about this one, thanks for the great explanation!
It's now 305k
My 7 year old looked over and said," wait is that an optical illusion" , she then proceeded to listen to you explain it. Bravo and cheers from a dad who has struggled to get her interested in math. :)
Hopefully, s/he then proceeded to explain it with no/limited upspeak.
aww how wonderful!
*East or west Jade explains the best*
This is hands down my favorite math/science channel. Thanks for the good work you do
Just beware, there is no "first rational right after 0", but the rationals are countable anyway!
@Jason Gnosaj Definitely!
I was about to say the same thing. Just because there is no closest rational number to 0 does not mean the unit interval consisting of just the rational numbers is uncountable. The difference is the set of rationals is listable while the set of reals are not.
oh shit that blew my mind
*listable!
Can rationals be put in biunivocal relationship with naturals?
I was expecting to learn how Pi was derived.
For that my friend you have this beautiful channel: ua-cam.com/video/dBoG4eRSWG8/v-deo.html
PI was "derived" from the mating of his parents.
It is the ratio of the radius to the circumference of any circle.
I also thought she would say for both circl: circumference/diameter pi ...hence it was same length
@@Twas-RightHere thank you for recommending this
When I was a kid I was intrigued by this so I built a model with meccano to see how fast the wheel would rotate. In fact it locked up solid, so I'd solved the first bit.
Years later as an undergraduate I studied Cantor's maths as a precursor to Turing.
More years later I applied countability to digital television, using it to demonstrate that the list of transforms of an image was countably infinite. This then showed how to do transforms with minimum hardware.
Great video, Jade. It's real world useful, keep it up.
I'm going to pretend I know what you're talking about.
Good job! Amazing contraptions. Intuitive problem solving! Televisions are amazing!
I knew about this since I was like 18 and always asked my self "how can a person explain this without questioning reality as a whole? "... how does this not raise extremely profound philosophical questions?
since that first day, I believe we live in the paradox of infinity.
I really don't understand what any of this kuffuffle is all about and I am shocked this held the attention of any mathematician. Everyone amazed by this is being fooled by a "magic trick". If you were to start the wheel roll and only look at the axle as it went from point A to point B, no one would have any problem understanding it's linear travel as the wheel rolls from point A to point B. If the wheel is 2 feet in circumference and it rolls 2 feet the axle has also moved 2 feet even though technically ( if it was just a point ) it hasn't rolled at all. It's no difference at any other radius up to the actual radius of the wheel. Yes the other radius's also roll with the wheel but they are also being moved linear fashion. So it's 2 forms of motion for them.
@@VicMikesvideodiary i ment to point the paradox that diferent infinities are 'bigger' than others... seems to me that that should be imposible... I understand the proces through which we arrive at the conclusion but I can't accept the result as a logical one. cause it doesn't make sense, I just think that infinity is a impossible thing and just like at singularities, things just stop working at that point
@@TBT.Stories I agree that the universe is built on a paradox but this wheel thing is no paradox. Yes it's true, there are what is called "sets" of infinities. For instance an infinite number of boxes could hold an infinite number of apples. Etc. etc. etc. As for infinity itself, yes it is real, any thing less than infinity as far as the total scale of the universe goes is what is impossible.
The thing is you do not need to rotate a circle to move it. The smaller circle is in slipping motion. Meaning that its centre is moving forward while it is also rotating
Okay I'm just gonna propose a solution before watching the whole thing (I'm at 02:11 right now)
So my understanding kinda goes like this:
•good morning! the two circles are _not_ covering the same distance, they're undergoing the same _displacement_
•thus, any point ∆ on the circumference of the larger circle, while experiencing the same _displacement_ (overall, through a full 360° rotation) as any point Ω on the circumference of the smaller circle will travel a greater distance in the process, since it traces out a larger path
•so our friend ∆ moves through space _at a greater rate_ than Ω but travels a proportionally greater distance, which explains why, even though Ω and ∆ experience the same _displacement_ they do not travel the same _distance_ , which also explains why the circumference traced out by ∆ > circumference traced out by Ω.
Also now that I think of it, that probably also explains why the equator seems to spin faster than the poles. Huh, cool stuff.
P.S. okay so having watched the whole thing through, I'm pretty satisfied with my explanation, but I'd be lying if I said I wasn't thoroughly impressed with the kinds of amazing insights that these mathematical geniuses throughout history were able to draw from such a basic problem that ignorant morons like me would've just dismissed as an easily explained dumb question 😂😂😂
P.P.S. The production value of this video totally made it worth the wait, looking forward to the next one!
I would just like to add that I think the connection between this problem and the spin of the equator is not the best. The whole philosophic-mathematical points aside, this problem in physics is solved by considering the difference between a movement solely made of rotation versus a movement consisting of rotation and translation.
The different velocities depending on where you are on the earth come from the fact, that every point on earth rotates with the same angular velocity (360° in 24h), but you get the real velocity when multiplying the radius of a given point on earth (shortest way to the axis of rotation, so r=0 on the poles) with the angular velocity. There is no translation involved.
Yes! Finally someone who thought of the same thing as me. It's a matter of trying to collapse a 3-dimensional problem into a 2-dimensional frame of reference. By collapsing what is essentially a cone into a plane, you lose the "discrete-ness" that comes with the 3rd dimension and get stuck with an infinity in the 2nd dimension. It's kinda how String Theorists create new dimensionalities to eliminate those pesky nulls and infinities when they try to merge Relativity with the Standard Model. What're they up to these days, anyway? 12? 13? Anyway. Same idea. Compressing the dimension (from 2 concentric circles along the length of a cone) to a 2 D problem (2 concentric circles embedded in each other) you get stuck messing around with infinities because you don't have enough dimensions to represent what is causing the change (speed).
Confusing the concept of distance for length is a mistake not a paradox.
Congruence of a line segment and similarity?
If you count with food it's a piece of cake
"This paradox leads Galileo to conclude that a belief in atoms, in the sense that matter is "composed of infinitely many unquantifiable atoms" is sufficient to solve the problem of the wheel"
- Wikipedia
I think Everett is calling...on that phone of his.
Seriously though, if this wasn't backed up by the insanity of quantum mechanics/wave-particle duality it sounds like such a cop out. Unless I'm just an idiot and don't get what is being said here lol.
There it must be "atoms of space".
but are there an infinite number of particles making up the circumference of the wheel? or is any wheel that we can build actually made up of a countable set of points
Found myself shouting at my screen “its slipping!” and getting excited when i was right. Made me feel like a kid again watching dora lol
I just had a thought.
If you got two wheels with rubber tyres and attach them together, then run it along the table but have a piece of wood anchored to the table the same length at the circumference of the lager wheel and the smaller wheel touching that piece of wood, you should be able to see the smaller tyre skidding along that bit of wood. It might even act as a brake of sorts.
This is where my mind went as well. That would have been a cooler demo imho!
A wonderful girl showing us complicated science in simple words!
As a non-native English speaker, I appreciate the lack of grandiloquent words that litter most of the texts about stuff like this.
Yes, I used a dictionary several times while writing this comment.
All of us married folks understand Jade's poor husband as he obligingly says a word for his wife's project despite being all wrapped up in his blankets. It's another paradox! I can love you always, but just not so much right at this one moment.
paradox is also an oft misused word. I suggest reading Willard O.V. Quine "The Ways of Paradox And Other Essays". Brilliant.
@@QuizmasterLaw Sometimes the word "paradox" is used identically to the word "theorem" in math
@@JeffreyMarshallMilne1 so you agree, paradox is a polysemic oft misused term.
It ought to be avoided by using more precise or exact words.
This video was extremely well laid out. You went through the information from least to most complex to ensure understanding is reached before moving on to the next concept. Bravo!
I don't think she came to an actual conclusion that made the whole point clear, though.
@@tamim4963 I agree somewhat like at 7:39 do you agree with me that I don't see why she finds it strange or surprising at all that a line segment is not equal to the entire line..obviously i.mewn to me..soni don't see what the big insight..I guess this took so long to figure out because nonone was drawing clear enough diagrams? Otherwise indont see why it would take so long..
Jade, that was AWESOME!
I'm 62 years old, and back in 7th grade (1973 or so), I had a math teacher that was VERY cool. He was talking about infinite numbers one day. One of the other students said, "What about a million, million, million?" I said, "Plus 1". The teacher pointed at me and said, "Exactly! Frank gets it."
We went on for a bit about that, but I felt like I grasped it. This explains it perfectly. Thank you.
Actually, while I have not seen this example before, I've seen the 'proof' conducted with a known-circumference wheel marked by 'distance in inches' as it rolls...it makes matching contact with the path-line and it's linear distance. Compared to the 'smaller circle', if you mark it by 'distance in inches', you will find that 'an inch of travel' on the linear is shorter than an inch-of-travel depicted on the wheel circumference...in effect, you could consider the wheel to be 'slipping' along the linear path...because you are traveling (linearly) at the rate of the outer circumference, but trying to match it to the inner wheel's 'inch-markings'...you have, for all purposes, created a simple planigraph, which is used in artwork for scaling drawings and models. You can even control this 'scaling ratio' by the diameter of the outer wheel divided by the diameter of the inner wheel. Your 'marker inserted' tool you use here is actually not what you think it is...it is not a 'rounder or flatter path', it is an X-Y chart showing cumulative scaling as X approaches '1' (1-rotation)! It becomes a 'scaling distribution chart'!
True, the paradox vanishes when you realise s non-slipping wheel's contact point is stationary, by painting 2 circumferences at different levels at the same height one or both wheels must be slipping.
Using toothed wheels and tracks of different sizes, would show a track being dragged or pushed back to accommodate the differing radii.
There's a 1:1 if you just measure the angular rotation and scale your distance measurement to the wheel radii. The video seems to miss completely the change of choice of coordinate systems, with for instance the triangle showing a fixed proportion of circumference travelled for varying radii.
*Please read my Explanation:-*
Take the time of the each rotation T.
The outer and the inner circle should have taken the same amount of time to complete one rotation.
Here's the point come,if the circumfarence of the outer and inner circle is same then the both have taken the same time and rotated with the same "velocity"
But,they didn't.The outer and inner circle have taken same time but the "velocities" are different.As being more distanced outside the outer circle's velocity must have a higher magnitude and on the other hand the small circle being less distanced must have a less velocity.
So the conclusion is that the time taken for each rotation is same in both cases but the velocities are different which proves that they traveled different distances in equal amount of time.
Edit: To know the velocity(angular),you should follow the angular velocity formula.which is
V=W×r [v=velocity,W=angular velocity,r=radius]
*Please read my Explanation:-*
Take the time of the each rotation T.
The outer and the inner circle should have taken the same amount of time to complete one rotation.
Here's the point come,if the circumfarence of the outer and inner circle is same then the both have taken the same time and rotated with the same "velocity"
But,they didn't.The outer and inner circle have taken same time but the "velocities" are different.As being more distanced outside the outer circle's velocity must have a higher magnitude and on the other hand the small circle being less distanced must have a less velocity.
So the conclusion is that the time taken for each rotation is same in both cases but the velocities are different which proves that they traveled different distances in equal amount of time.
Clever
Yes But here we are considering the actual magnitude of distance not the time.
Yes your explanation is reasonable but actually not for the essence the concept. Sorry but keep it up
Please understand the concept which she is trying to explain
Nailed it bro....
This clearly differentiates the distance and displacement concept....The displacement of a point on the inner and outer circle are same. But the distances travelled by them are different.
In 10:40, while it is true that the set of real numbers between 0 and 1 is uncountable, the argument that 'since there is always a number between 0 and epsilon, the set is uncountable' is false. it would imply that rational numbers are uncountable, but thay are not. Countability doesn't mean that you have to count them in ascending order, instead we just need a one-to-one correspondence with natural numbers.
We can enumerate the rationals for example by spiraling out in Z^2 (look it up, there are nice visualizations, but it's not worth it trying to explain in a comment), and by taking that and adding the rule 'if it's not between 0 and 1, we skip it' we get an enumeration for the rationals between 0 and 1. This infinite sequence will contain every rational in the interval, including any that you could name by saying 'ha, there is one between this one and 0'. on the other hand to show that real numbers are uncountable, you would need something like Cantor's diagonal argument.
However, what we can't do is count one infinite set and then go 'now that those are out of the way, let's add this other set', unlike the animation at 8:55 leads to believe, since the first list would never end and therefore we would never get to the second one. Instead we could do 1,-1,2,-1,... and so on, since for any specific element, the list gets there in a finite number of steps.
I know this all sounds very pedantic, but you kinda have to be when you are dealing with such an abstract concept as infinity. :)
While it is true that one can formulate both countable and uncountable infinities, the example you gave doesn't automatically lend itself to being uncountable. Remember, the rationals are countably infinite. Therefore the finite decimals you provided (e.g. adding X more zeroes) are also countably infinite.
Fun trivia for everyone. Car people talk about this as slip angle.
In the case of cars, this means a difference between two physical actions involving the wheel. Such as tire speed vs road speed (locking breaks), wheel angle vs turning angle (slipping and sliding) or spinning your wheels. In the math you can work out an angle between where one frame of reference is vs the other. Intuitively this led me to the conclusion of slipping pretty quickly, not quite feeling I'm completing the connection though.
The connection is that they assumed a motion which appears to be smooth must be similar to the hexagon, where it lifts from the track then drops back down on the flat points. So, if your circle was made of a million individual sides, each point would still theoretically be acting the same as a hexagon, lifting at each corner and setting back down on each flat.
The concept was that a circle must have infinite sides, which would explain why the little gaps weren't found.
Slippage wasn't actually proven to be the issue until much later. Mainly, once we were able to spin something very precisely as very high speeds that was milled extremely precisely so that we could eliminate other potential variables, and discovered that heat was created in the process.
But heat is also an effect of compression, so the experiment had to use a minimally compressive material, of excessive smoothness, and do the "two-wheel" experiment she did, then recreate it with only the large wheel touching and then only the small wheel touching (while also adjusting the weight placed on the wheel, to cancel out variables in compression) , then compare the thermal increase in both of the single wheel experiments against the thermal increases of the double wheel experiments......
In the end finding that the difference was slippage, where the contact surface was continually changing as the rotation happened, but that on a molecular level there was drag happening also.
Our ability to intuitively understand the phenomenon by looking at a wide cylinder rolled across the ground, seeing how turning on a point drags the edges of the cylinder, then experimenting to learn how the different speeds between the inner and outer ends of that cylinder still exist even at higher rotations and smaller turns..... that is merely a different manner of approaching the problem, from a more modern viewpoint really.
The whole problem of the wheel slippage wasn't fully recognized until the invention of powered vehicles with wheels, when it was definitely shown that having a straight axle for your power to the ground made it harder to turn the vehicle, and where the axle had enough flexibility (along with the wheels themselves having some flex) so that the slippage finally presented as one wheel jumping in position vs the other, instead of the continual slippage she showed in the video.
As with many things we see daily.... intuitively making a hypothesis is one thing, but definitively proving a theory is much harder.
I came here because i couldn't sleep and now i can't sleep 😭😭
I totally missed when you solved the Wheel Paradox.
lol i missed the part where it was a paradox to begin with. its like saying "theres a car on a trailer, how is it moving when the engine isnt running!?"
@@ICKY427 Lol it's not like that at all. There's either an engine or not. But if there is, it's running "both" circles.
And when you solve a paradox, it's not really a paradox anymore. That's the point.
Yes! Thank you for some comradery... i was beginning to think the world had gone crazy and i was all that was left to bring reason back to society, haha
When I was younger I had a similar idea to this, where I imagined a long rope attached to the earth that stretched out for thousands of miles into space, and wondered how it was possible for one end of the same object to be traveling at totally different speeds without tearing itself apart
I believe the solution is that the wheel is slipping; because the question is why are the two circumferences equal?
Then you go into infinity to show why that is the answer.
If you wanted the answer on how to measure circumference if an inner circle from an putter circle, hold a marker still on the circle, rotate the circle so it makes a ring inside the wheel, grab a piece of string and measure it. 😅
I find the 8:04 statement to be quite thought provoking:
"These objects are continuous, uninterrupted you can't split them up"
I've always viewed it the opposite way; continuous means you can split it up, as many times as you want, in fact you're never done splitting, you can split forever and ever
I guess this is a characteristic that separates mathematicians from engineers. It was intuitive that one-to-one correspondence between the circumference of 2 circles must be done with equal length units simply because you're measuring lengths. You can't treat them as continuous if you are measuring their discreet length. Only a mathematician could get this confused, lol. Really great video, though.
You know the difference between an engineer and a philosopher? Once an engineer and a philosopher died at the same moment and found themselves being tortured side by side. A demon put a beautiful woman at the end of a field and told the two each time he clapped his hand the woman would come half the distance towards them. After a few claps the philosopher collapsed, tearing at his hair crying and shouting in torment. He realized through all eternity the woman would never reach him. The engineer, meanwhile, seemed quite content. The demon was frustrated and confronted the engineer. "Don't you realize this woman will never reach you?" The engineer smiled and retorted, after a few more claps she will be close enough for all intents and purposes."
@Grace Jackson Neither do I. Furthermore, I think their demonstration is flawed. They show the "skipping" or slippage" by the smaller wheel, but that only happens because they are forcing the 2 to follow a straight path. If the 2 wheels are allowed to follow a circular path they can do so without slippage. That's the principle behind tire stagger in circle track racing.
@@stefanl5183 Her mapping doesn't work at planck length scale.
@@joebombero1 That is brilliant!
Actually, you definitely can treat them as continuous. Obviously, atoms and molecules were not discovered at the time Galileo or Aristotle tackled the problem. But also, physical objects are usually considered to be continuous until the model fails even today. You seem to have a misunderstanding about the nature of discreeteness and of continuity: you absolutely can make the assumption that the objects are continuous and not discreet. The error arised later, when they tried to reason about the notion of measurement of continuous objects using intuition and not mathematical rigor.
7:33 Okay, there's some nuance here that I feel I need to add, since the video at this point becomes misleading in my opinion. There's two different notions of size at play here: cardinality and measure. Cardinality is where the "different types of infinities" are relevant. Cardinality refers to the size of sets, and if there is a bijection (one-to-one correspondence) between two sets, then they have the same cardinality. So the two lines are the same size in this sense; they have the same cardinality. Measure represents the "length" of the curves. What we generally think of when we think of "length" and "measurement" corresponds to the Lebesgue measure. In this sense, the two lines are different sizes. The bottom line has a greater Lebesgue measure than the upper line. What this demonstrates is that having the same cardinality does not imply having the same Lebesgue measure.
In fact, THIS is (almost) the actual answer to the problem, since rational numbers are countable, yet have the same "problem" where any interval has the same cardinality as any other. The "almost" comes from the fact that thinking of measure here is a very overpowered idea (though correct). It can be thought in terms of distance (metric, rather than measure), which is a simpler tool, closer to geometry in roots.
Measure is not an "overpowered" idea, just different (than either cardinality or metric), but metric does seem to be more intuitive notion in this context.
you are right, but why are you saying the video is misleading? This is precisely what's shown in the video.
@@Czeckie Because cardinality has nothing to do with this problem. You can encode that same problem with rational numbers rather than real numbers, and you'll have the same problem.
Another thing I found strange is when she explains about drawing a line coming from the top of the triangle passing through both lines, and saying that there's always one-to-one points. But if you draw a line passing at all points of the smaller line, there will be empty spots on the bigger line. Or whatever I'm just overly confused
I think I've found my new favorite UA-cam channel. I regularly watch 3Blue1Brown, Mathologer, MindYourDecisions, Infinite Series (which also addressed this subject during it's run), etc. and I've read [probably too much] philosophy of mathematics about Russell, Cantor, Hilbert, Wittgenstein, etc. This stuff is just fascinating to me and always has been. Very well done!
Numberphile, Stand Up Maths
Tibees
"In fact, the question, 'what is the next number after zero?' is impossible to answer because for any number we choose there will be always a number that is closer to zero. So, we can't even count the numbers between zero and one ..."
This same argument can be applied to the rational numbers which is a countable set. The correct way to argue that the real numbers are uncountable (or the real numbers between zero and one) is to use Cantor's diagonal argument which goes as follows:
Assume you have a complete list (of one-to-one correspondence with the natural numbers) of the decimal representation of the numbers, and show that for any such a list you can get a number that is not on the list. The way to find this number, is to pick its first digit to be different of that of the first number, and the second digit to be different of that of the second number ..... and so on. You can do that forever, as most of them have an infinite decimal representation (either due to being irrational or being a rational with a denominator which is coprime with 10). As for those whose decimal representation terminate, you can add infinitely many zeros. This shows a contradiction that there's such a list, and hence the set must be uncountable.
I can't explain just how good this video is. The methods, the shapes, the examples, the explanation... Everything is amazing. Thank you for this awesome video!
Thank God we're all different. I can't explain just how irritating this video is. The only thing that's amazing is that she can't even pronounce the word _wheel._
for a first grader sure. For anyone older than 8.... it is insanely self explanatory and offensive to anyone who can do more than basic math.
@@HappyBuddhaBoyd Shhh!! You're spoiling everything. The video is AWESOME and EXCELLENT. Look how _cheerful_ she is! She makes it so fun and cool! And the blue whill is so nice! This is what science is meant to be all about. 😃
I am French, and was satisfied by your husband pronunciation. Oh, by the way, the video is géniale !
merci!
My condolences
f.. I've read it as genitale and was like what da f.? for about 10 seconds
A deceivingly advanced topic that is broken down into easy to understand pieces presented in a fun and engaging way. I could show this to a 5, 10, 15, or 65 year old and they all could learn from it. You earned my subscription today.
Here is one thing I can say about numbers: Our species is in desperate need of more instructors like her, speaking in many different languages. The number that is currently needed is less than 8 billion. The number of languages in which the instructions are needed may be more than 6000 (at this point). Do keep them coming, and fast! (I like to think that millions of my contemporaries feel the same way --even in Afghanistan.)
0:50 The inner wheel would be similar to slipping against the ground in this scenario, that's why the 2 distances are the same.
If you actually shrink the wheel to the smaller size, it would also travel a shorter distance, provided that it doesn't slip against the ground.
Idk why but to me your comment was more clear than her explanation. Thanks
@Anon Ymous Wrong about slippage, otherwise mostly correct.
*Smaller radius = smaller circumference = shorter distance per rotation.*
But in this example both wheels travel the SAME distance (they have to, because they share the same center point), therefore at least ONE of the wheels MUST be slipping, at all times while moving.
Sion, to price or disprove slipping, she should have put pins on the edges of each wheel instead of paint. What would happen then?!
*This is (unintentionally) an EXCELLENT demonstration of what I mean:*
ua-cam.com/video/b-nU21YgXTg/v-deo.html
Another thing to point out is that this paradox is only confusing when the big wheel rolls on the ground and the small wheel "magically" keeps up. If, instead, the small wheel rolls along a ledge and the big wheel hangs over the ledge, it is easy to see what happens. The big wheel moves faster and the points on its edge actually move backwards at the bottom.
The two “wheels” that are leaving their paint marks on their respective surfaces as the big circle is rolled: rig a system that allows the surface that the smaller wheel is in contact with to slide, like on a linear bearing. The surface will be pushed “backward” the distance that is the difference of the circumferences.
Correct, except that it will be pushed "forward" in the same direction that the wheel is moving.
@@philcarlson326 Yes! Thank you.
I think there are 2 reasons behind this.... 1. Here the centre of both the circle is not static in place and keep shifting while rolling on surface.... 2. Also, length of the centre(focus) from the string is different in both circles.... Thats why it is happening....
When she said “You can always go lower.”
*I felt that*
Good one! I laughed. :)
1:54 vsauce brachistochrone flashbacks
or is it?
Hey, VSauce!
@@heyandy889 Michael here!
So......are we really humans....?
And what are humans?
More importantly what are we?
**drops Vsauce music
I love that i learn something almost everyday that bends and warps my mind to a point were i cant look at the world the same, not necessarily from this channel but in general.
Why don't this channel have 1M subs already, this entire generation is behind entertainment only... They'll net get how cool science is
I don’t trust a lot of this geometry stuff, the logic seems a bit... circular. (Just kidding, I love educational content)
why must you go off on a Tangent?
Good one
@@QuizmasterLaw I thought it was a refreshing angle on an old line of reasoning.
@@SoAboutThat i keep trying to see your Point but it seems so small as to be nigh inexistent
By the argument of there always being a number closer to zero, the rationals would also be uncountable. One has to show that there is NO way to try to count the set, which is of course much harder to demonstrate.
Thank you, that was very annoying
I should have read your comment before writing my own one xD
Excellent point. When counting a set of numbers, there's no requirement that you count them *_in order,_* just that you'll eventually get to any specific number.
1:30
"...I'm going to get my French husband to say it."
Did you hear that noise? It's the sound of nerds' hearts breaking all over the world. So sorry...
There’s a reason she put her husband in the video
Why?
Well she got the French husband, if she ever needs a husband from another country, I'm available lol
You can just adulterate 😂 nothing stopped my ex from doing so when I left her alone for 3 days, so it must be fine.
Thanks for this lovely lecture. I have some additional quanta of points
1.) Numbers are like points with spaces between them. There can be intermediate points in between two terminal points. The number of intermediate points between let's say 0-100 can be equal to the number of intermediate points between 0-1 if the size of the space between each point is different; the latter being higher than the former. The space between each point is called an interval. The size of the interval is the unit, and it is a measure of the amount of discreteness or discontinuity between each point. Larger the interval (in other words the higher the unit), the more discontinuous the points are, and vice versa
2.) In terms of set theory, the smaller hexagon (B) is like a smaller subset of the larger hexagon (A) such that all elements of B are in A the lines are like lists of all the elements in both sets. When the elements the of both sets are being listed (B above A), they are aligned, and identical elements are positioned in the same region like an intersection. Because A has more elements than B, an amount of it will not intersect with B. The gaps in the B list are the elements in A that aren't in and therefore do not intersect with B.
3.) From a geometric point of view, the distances between the an angle of A and B is like a hypotenuse, and the distance between a side of A and B is like an opposite. The hypotenuse is longer than the opposite, and the difference between the hypotenuse and the opposite is the gap in the line of B. As the number of sides and angles in the polygon increases the difference between the hypotenuse and opposite decreases and becomes less obvious. A circle is like a polygon with infinite number of sides and angles, in which the hypotenuse equals the opposite.
This demonstration is really cool. Love your new set!
Thank you Vanessa! Nothing on your Sleeping With Friends set tho ;)
Y
@@upandatom Your demonstration is flawed though. There doesn't have to be slippage of the smaller wheel. You only got slippage because you were forcing them to follow a straight path. If you allow them to follow a circular path slippage isn't necessary. This can be demonstrated by rolling a dixie cup on a table top. The top of the cup has a larger radius than the bottom. If you place it on the table on it's side and roll it, it will roll in a circular path without slippage.
This just gave me the feeling of finding a solution to a non-existent problem yet interesting to see it explained with math
Yes exactly! Am I stupid, or why is it a problem that different sized wheels, have different lengths of circumference?? Then why go into triangles and infinity??
I don’t think you got it, but I might be wrong.
@@JavierBonillaC no I get it but it really is finding a solution to a non-existent problem
I could have watched a video about how rocks stand still on their own, and would have been just as impressed.
Honestly, Jade, this is one of the most gorgeous maths videos I've seen on UA-cam. Everything remains crystal clear as we dive into the topic, while maintaining some mystery to encourage us to follow you deeper. Brilliant!
Her saying "now let's do something" and coming closer made me shiver...
“Measuring tapes didn’t exist”
Me: I’m pretty sure rope existed..
🤨
The fact that you cannot pick "the smallest number after zero" (with the usual order), does not imply that the set of numbers you are working with is uncountably infinite. The same argument holds with the rationals if you consider (again) the usual order. But anyway, very nice video, I really like your content!
Thank you! This was the one part of the video which irked me and I was hoping someone had already covered it off in the comments.
By the same argument, all whole numbers would also be uncountably many, since there is no smallest whole number to start, but that’s ridiculous: you can just do (0, 1, -1, 2, -2, 3, …).
@@felipevasconcelos6736 I see what you mean but the point is that within the integers and its usual order, there is in fact the smallest number after 0, namely 1.
@@DiegoMathemagician yes, but there are infinite numbers smaller than 0 that have to go after it on the list, so it’s really easy to show that countable sets don’t need to have an increasing bijection to the naturals. All they need is to have a bijection at all.
The whole point of asking for “the smallest number after 0” was precisely to answer the question “if we list all reals between 0 and 1 in an increasing manner, what number do we put after 0?”. The equivalent question for the integers is “if we list all whole numbers in an increasing manner, what number do we put first?”, and asking for “the smallest number” is equally irrelevant.
@@felipevasconcelos6736 Sure, sure. To be fair I have not watched the video again when you commented.
I literally had goose bumps at this end of the video. This demonstration is so, so beautiful
That means so much thank you!
@@upandatom Speaking of bumps, your prints made me think about relativity.
Do you know if a link between space time curvatures and cycloids exists ?
I've never heard of the paradox and thought of slipping right in the beginning of the video as soon, as innercircle was mentioned and it baffles me, why it took centuries to figure it out.
An incredibly pedantic critique: everything you said to 'suggest' that the reals are uncountably infinite could be applied to the rational numbers too (because every statement you made is true of rationals as well, and all your examples were rationals (0.01 = 1/100). And yet, the rationals ARE countable!
Proving as such is a fun exercise for the reader.
(hint 1: don't worry about including the same *value* more than once)
(hint 2: try a grid arrangement and a space filling curve)
I remember seeing this before. A simpler conclusion is that the smaller wheel moves slower then the larger wheel because it has a smaller size (causing the skips). This can be seen very clearly in a very large model. In one rotation of a wheel 4 feet in diameter, the small wheel moves very slowly, while the large wheel moves very fast for the same distance
Thank you!
It's like time and distance have a correlation
I like your answer better. It was what I thought she was going to say... but she made it more complicated than it needed to be.
Gears ?
@@mikehunt1528 Not gears. Imagine you are holding a umbrella ☔, you twist the handle in your hands. The outside edge of the umbrella will rotate faster then the handle to keep up with the rotations. The farther away the edge is, the faster it has to move to keep up. Same idea as this video.
I hope your life unfolds exactly how you want it to. The passion you have for maths is captivating. Thank you for contributing to the betterment of human kind with your wholesome content.
If you were my math teacher back in the day, I probably would have understood more. Great video.
Very interesting and well-presented concept. I love the fact that even the first number after zero cannot be determined.
Huh, I didn't know that Galileo was so close to discovering calculus.
He was not close he already discovered it, you should do some research on the topic. No offence.
@@supernova5618 No offense taken. I should do some research on the topic.
and Archimedes too, he was working on areas and infinitesimals for a long time, tho not quite as rigours as actual integrals
Kinda makes me wonder if his "genius" was legit. Good thing he came up with other things because this one alone, pretty sad 🤣
@@supernova5618 Galileo didn't discover calculus. He reasoned the same as Archimedes and others did many years before his time, but never developed a theory. At best you could say he successfully used indivisibles in his work. Besides, the history of the discipline has had contributions from many different people, and there isn't a sole 'inventor' on it.
Jade, you’re just the best : )
Aww thanks BlueTigerDan :)
@@upandatom really must say that you are adorable and your videos are awesome, keep up the good work
This is the closest I've come to understanding the difference between discrete and continuous maths. You are an amazing teacher, thank you!
07:14 All lines are equal but some are more equal than others!
Watching these videos makes me wish I were still learning at this level. Great stuff!
tonpal Hey, I am on the verge to b labeled rand parent and I am still learning at this level! It is really rest stuff.
If you'd drill in the center of the wheel then it would have made a straight line :0
that would baffle ancient mathematicians even more. how can a dot and a circle have the same circumference?
@@ubikledek distance traveled is not circumference, just the unit of measurement
@@DVXCine To measure distance traveled, you must use a unit of measure.
Great insight. That would mean the "circle" will not roll but 'drag' along.
@@DVXCine from what i understand, they didnt know that. Yet. Thats the baffling part.
It's hard to believe that it took them centuries to figure out the inner parts of the wheel are travelling at a different speed compared the outer parts.
It gets complicated once the question of force is introduced.
For example the outer circle travels a further faster gait, but the inner circle travels with more force relative to the distance.
@@sirdeadlock Just considering centripetal force, and henceforth angular velocities and linear velocities should do the trick.
You’ve made mathematical matters become interesting, which all the math textbooks I went through during schooldays couldn’t.
I have solved the "youtube learning paradox" where I watch a video to learn something new and finish the video feeling even more confused.
Conclusion = I am a dumbass
Nah don't be too hard on yourself. Paradoxes like these are supposed to leave you confused due to the nature of challenging assumptions that are mistakenly made. Once you find the assumption that is erroneous, the paradox becomes more clear to understand.
Nope. You're actually smart. Smart enough to learn AND ALSO smart enough to realize that even when you thought you've learned you don't know it all. When we don't get confused we think we know it all quickly.
Mathematics is nothing more than cleverly disguised nonsense. Numbers themselves are nothing more than meaningless abstractions that require some additional language in order to mean anything at all. Everything about mathematics is purely illusory. For instance, 0.999_=1, 1.999_=2, which goes on ad infinitum. So a whole number is equal to the whole number less an infinitely small quantity. That is impossible, yet it has been mathematically proven. So whole numbers are equal to not so whole numbers. It becomes even more strange when you question what the functions of mathematics actually mean in reality. How do you add one apple to another apple in reality? Does pointing at them and saying one apple, two apple really mean you have added them? Now, how do you subtract an apple from an apple? How about multiplication and division? Can you multiply and divide apples? How would one go about doing that? Is one apple truly equal to another apple? Absolutely not. They may appear similar, but they are different. One may be green, one may be red. One may be older or spoil quicker than the other even if they appear identical at first glance, etc. There are always numerous differences which are often neglected by mathematics in order to proclaim equality of things in reality. Oddly enough, the same thing is typically done in science as well. It is easy to proclaim equality when all of the differences are arbitrarily ignored. That is what we do so we can pretend we know more than we know and feel smart about what minute amount we do know. Don't beat yourself up. We are all idiots pretending to be smarter than we are.
Welcome to the elite society of people with just enough skills of deduction to reach that enlighted conclusion. We are not ignorant of the truth.
@@mennovanlavieren3885 We are all ignorant in our own way.
Yeah, I saw that pretty easily.
The bigger wheel drags the smaller wheel along a line faster than it would naturally roll, proportional to the difference between them.
So instead of each point touching the surface instantaneously, then rotating away and off, that touch point slides slightly before and after direct radial contact.
It's similar to Zeno's paradox in that it confused people due to dissimilar infinities regarding numbers sets.
I cant help but appreciate how nice this video is made. You think you saw every kind of math content on youtube but here you are and paint your handmade hexagon to show how it skips and I just love it.
What an amazing video, I liked how it was made by explaining not only the correct answers but the process to get to them.
Love the different style and the jumpsuit!
We all missed you ❤️
She looks like every kids show host that i had a crush as a child
as a child... yeah... definitely wouldnt feel that way as a fully grown adult man. i mean that would be silly. what are you talking about? ridiculous.
@@badradish2116 ,
Well, he's just being a *_child!_*
@@badradish2116 i dont know if i misspelled something or did not make myself clear but you really never had a crush on some one older when you were a kid?
@@TheFrancardAdventure he is just flustered and is agreeing with you
Why do I like you so much. I just came across this channel and I'm hooked. Subscribed, waiting, and will dig into the past.