I think the problem with the "unphysical" springs can be solved by using weights instead. Just replace the springs with wires, which loop around A and B, and attach two weights of different masses to the ends of the wires (something like m = 1/(g * v)) and that will give you a constant pull!
+pacocsharp Here's a working setup: connect the weights by a string to the loop. Put one pulley at point A and the other at point B. Drape the string over the pulley so that each weight hangs below its pulley.
the thought-experiments described in Mark Levi's book "The Mathematical Mechanic" make use of two imaginary devices - the Constant -Force Spring and the Zero-Length Spring. imagine setting up one of Mark's physical models on a horizontal board with a hole drilled at each point where a spring is supposed to be fastened. the Constant -Force Spring is replaced by a mass hanging from a string passing through a hole. the string transmits a CONSTANT force towards the hole. the Zero-Length Spring is replaced by an assembly consisting of a real spring with a string attached to one end of the spring and a coat button to be attached to the other end of the string. a second horizontal board is mounted below the first board. the combined length of the unstretched spring plus the string is set equal to the vertical distance between the two boards. the free end of the spring is attached to the lower board directly below a hole in the upper board. the string passes up through this hole and the button is attached to the top end of the string to keep the string from falling back through the hole. now as the button is pulled away from the hole along the surface of the upper board the attached string will transmit a force towards the hole DIRECTLY PROPORTIONAL to the distance from the hole to the button.
At college I am a mathematics major, and I was also thinking about being a physics major. However, the two physics courses have been extremely disheartening for me, not because they are difficult, but because the professors teach in such a way where they never explain where any of the laws or theories we use come from! I ask them all the time if we could even get a general explanation, and they never do! It is such a frustrating experience and has really turned me away from physics altogether. I’m glad videos like yours exist that help give intuition to where these kinds of things come from!
All throughout high school I was compliant to the idea that I was supposed to hate math. My first semester of college when I had a pre calc professor that incorporated proof and theory in to lessons. Then a switch went off and since then I’ve had a new found appreciation for mathematics and the physical sciences as a whole. I love finding professors, videos creators and in this case People who share the intuitive thinking.
Having a poor physics professor can make or break your grade! I'm the same way, I can't learn to APPLY laws without fully (or at all..) explaining where they come from, why* they happen and how they are derived. I need to understand a concept inside, outside and upside down to really move forward applying it and comprehending it.
I am in the exact same situation, I am in the french equivalent of high school, and on monday I have to have chosen which one of my"speciality" subjects I want to drop. I know I will take math but between computor science and physics/chemistry(the subjects are combined here) I am really torn. I always thought I would go on learning maths and physics throughout high school and university but I really dont enjoy physics as much as I would like to because it just feels like memorising formulas and properties without understanding where they come from and why the univers functions the way it does. Whenever I ask, I am told that it doesnt make sense to ask why that is the case, and that scientists just observed that that was the case and developped theories and laws based on what they saw. I never feel satisfied with this answer because it feels like there should be a logical, somewhat intuitive reason as to why things work the way they do, like in math. On the other hand Computor science is really easy and quite fun, but I dont want end up the tech guy at some company I dont care about in 10 years. I'm clueless as to what I should do :(
I am sure it is a challenge for your professors to cover all the material necessary to fulfill the goals of each course without adding a historical component to them. You imply that the coursework is not that difficult for you. Yet, you cannot figure out where to learn the foundations underlying what you are being taught? Your complaint does not pass the test of common sense.
I remember back in my school days I would sometimes take a shortcut to class cutting diagonally across a couple of fields to get to class faster, and noticed partway there that one of the fields was firm and easy going and the other was wet and muddy and a bit of a slog to walk across. I distinctly remember wondering to myself while crossing that field one day about a hypothetical best path to trade off added distance to more of the walk spent on the firmer ground to traverse the fields as fast as possible. Some time later I learned about Snell's law and the idea immediately clicking in terms of the two fields, it was such a strange and rare instance of having somehow lived a thought experiment before learning about the concept it explains...
Here there is an alternative derivation of Snell Law (very simple), using the conservation of momentum of photons: If in medium 1 the index of refraction is n1 and in medium 2 the index of refraction is n2 and for the definition of index of refractions and wavelength we have: n1 = c / v1 , v1 = λ1 * f n2 = c / v2 , v2 = λ2 * f Dividing the above equations we obtain: n2 / n1 = λ1 /λ2 So if n2 is higher than n1, the wavelength in medium 2 is smaller than in medium 1. Now because the photon momentum is: p = h / λ Applying the conservation of momentum of the incident and transmitted photon along the parallel line of medium separation: p1 = p2 → h/λ1 *sin(θ1) = h/λ2 *sin(θ2) → n1 *sin(θ1) = n2 *sin(θ2) This is the Snell Law!! So the refraction is just a consequence of the conservation of momentum of photons! There are a lot of ways of derive this law!
Even though constant force springs are unphysical, there are objects we have that have constant force regardless of their how far they are extended. Take a piston with a vacuum inside and atmospheric pressure on the outside. Now, regardless of how far you pull the piston in or out, the total force would always be A*10^5, where A is the area of the top of the piston cylinder. This principle is actually used on the ISS to mimic earth gravity for astronauts to do all sorts of lifting exercises. So you could just replace the springs in your video with such pistons and the entire setup would be perfectly physical.
Currently reviewing the laugranian method to calculate the equations of motion in my theoretical physics class. I hope for more videos like this. Very happy I came across your channel.
You saved my life. I am studying Snell's law, to give a physics final, and the proof with Fermat's Theorem seemed very mathematical to me than physical and it is like four spreadsheets which is boring. But this demonstration of Snell's law is more accurate and you can understand what is happening physically. Thanks for the video.
OMG The spring's and potential energy minimized scneario is also used to prove Stewart's theorem with springs and equilibrium! Mark Levi's clever proofs are really fascinating
Great video, as usual. May I ask you two questions? 1. Where did you come from? This may sound like a silly question, but youtube channels usually don't start out with high quality videos like yours did. If you have done any other similar mathematical explanations in the past, I would really like to see them. 2. How do you make your animations? They are unbelievably smooth, and are very satisfying to watch.
+nin10dorox I'm not quite sure how to answer the first question...Stanford, I guess, is where I "come from"? The videos on this channel are indeed the first videos I've put on youtube, though I'd done a few of casual lecture-type-talks before (none recorded). Thanks for your kind words, but I think there were more than enough faults with the first few videos I put up :) To your second question, all these animations are programmatically generated. I've never been much for artistic talent, so my general hypothesis here is that when you're animating math, it's not too hard to describe in code, then the computer takes care of making things look pretty.
+3Blue1Brown I am about to start learning Python and would be very curious what your code looks like. What environment do you use? Could you send me an example of one of your animation codes or is that asking too much?
+ElGrecoOB +3Blue1Brown Yes, I'm sure there are a lot of viewers that would be interested in seeing some of the code on github, but I can understand if you'd rather keep it closed :)
A key step in the proof is simple but ommitted : A central force of constant magnitude has a potential energy varying as distance from centre. (Hence, minimising the potential energy is same as minimising optical path length {equivalently time of travel}).
Actually, constant tension spring do exist, even though you could argue that they are not springs anymore : If you have a chamber with some vacuum in it and a piston that can go back and forth into that chamber, then the force on the piston is going to be the same regardless of the piston's position in that chamber because the vacuum density never changes. I guess that the force on the piston in such a chamber could be represented by : F=ax+b. Where a is the the numbers of moles of gas in that chamber and b some constant due to the outside pressure on the chamber. It's easy to see that when a goes to 0 the whole expression becomes constant. This property has actually been used for exercise machines in the ISS to reproduce the effect of weight : ua-cam.com/video/05oOst9kZXQ/v-deo.htmlm51s
There are constant force springs. They are made of flat ribbons of steel rolled into a coil. The only part of the ribbon that exerts a net force retracting the spring at any time is the part that is curved, but not quite laying on the coil. Since the geometry of that part is roughly constant, the force is roughly constant. Altering the geometry of the spring slightly along its length allows this to be tuned so that the force is constant within tolerances.
The springs are an interesting way to think about it, and might enhance understanding in some way, but with all the "finagling" involved to make the thought experiment work, I feel the calculus approach (which the video took just a few seconds to explain) is far more elegant.
i just got home from a 70 hour long work week to find TWO 3b1b videos in my sub pile i have never been so pleased if i hit play on this video and it's an april fools joke, "Hahaha you fools thought i would actually make any more videos" then i'm gonna cry don't make me cry man it's not cool
that assumption that F_(virtual spring of medium A)= 1/(v(in medium A)) is like using the assumption that triangles angle sum upto 180deg so we can say that α+β+γ = 180 in a triangle ABC where angle A=α angle B=β angle C=γ i would like to have a mathematical justification for that assumption
@@b.clarenc9517 It wasn't proven because we assume it's true in the video he says let F1= 1/v air and F2= 1/v water he used this method because it's less complicated and easier to understand than the calculase one I know I'm late but I hope I helped you a little :)
It is technically inversely proportional to velocity so that the force mimicks the time taken to travel a path. So by design, when we minimize force, we minimize time.
@@hassanakhtar7874 That's right, but the princple of minimum energy holds anyway. And, for those particular springs, minimum potential energy is the same as minimum travel time for light following the path depicted by the springs in equilibrium.
Grant always changes things for me... I thought I would focus a lot more on Math than Physics but, here you go... Such a beautiful proof of Snell's makes me think... Looks like I'm going to study a lot of Physics now! After Number Theory, Combinatorics and Geometry of course!😅
Mm... It makes sense, but I don't find it satisfactory as an explanation. Light finding this balance between time and energy seems more like a consequence of something else than the root cause of the behavior. Is this possible?
The subject of 3b1b's video is Fermat's Principle of Least Time. Fermat's Principle exemplifies the optics case of the Principle of Least Action. Action has dimensions ( length^2 / time ). It's often calculated with S = integral( L ) dt where S = action L = lagrangian = ( KE - PE ) t = time The Principle of Least Action observes that given a starting point and a destination point, a particle will always follow the trajectory with the least action. Newton's Three Laws of Motion can be derived from this principle. After the Double-Slit Experiment, physicists reframed this principle in terms of Feynman Paths. To ask why the universe behaves this way is to ask why the universe is quantum. Physicists don't know yet. In any case, it appears pretty fundamental. Finding the root cause of this principle is analogous to finding the root cause of the conservation of energy.
+the fance Thank you for sharing your thoughts and knowledge that relate to the subject of this video. It seems that we can only speculate about the root causes of physical phenomena of nature in our universe as we seem to know it.
Nice explanation, yet it give the impression that nature knew the future in advance then decide to use the shortest path. I believe snell's law should be in terms of density to avoid confusion
I have heard similars explanation. But what I think is "wrong" is Fermat's Principle. I mean, you can get snell's law using Maxwell equations, and in my opinion that is more fundamental than Fermat's principle. I feel that I didn't explain properly. I think that the idea is similar as Pauli's Principle; Pauli's Principle is true, but it's a corollary of antisimetrization of fermions.
Snell's law follows also nicely from electrodynamics and maxwell's equations, in fact, it makes more sense given the properties of polarization. Interestingly, quantum-mechanics will tell that the shortest path explanation might actually harbor some accuracy.
I always felt unsettling about the explanation of shortest path. Always wondered whether thats what its actually doing! Do you have any pointers for the QM explanations?
I prefer to use the more physically correct boundary condition of waves and their derivatives being continuous perpendicular to the surface across the surface. It makes more sense to me and doesn't lead people down the wrong track of thinking that light picks the fastest path on purpose (I had an optics prof do that, a bona fide PhD). The fact that it happens to be the fastest is a coincidence, or more accurately a consequence of satisfying the boundary condition.
very nice ! i am a fan of your work ! I would like to start a similar kind of channel for the french youtube community, and I was wondering : which software do you use to make your animations ? they are more or less exactly what i am aiming for ! keep it up !
+MrGustave1er I'm glad you want to make math videos, the more there are in this world the better! I generate my animations programmatically, basically using a small library I've been building up. It's not the most friendly for others to learn (e.g. no tutorial), since it's mostly just been for my own use. You are welcome to take a look if you want at github.com/3b1b/manim, and I'll help you out if you have questions, but unless you really want full control for arbitrary new math animations, I would probably recommend looking into better-documented tools than mine.
Constant or nearly constant tension springs do exist. Ever notice how a tape measure has constant tension no matter how far you pull it out? With a quick google search you can find companies selling them.
I've heard about negative index metamaterials and I'm wondering how the thing about light taking the shortest path fits into that situation, great illustrative video btw.
take two points, one in water one out of it(as shown in the video) now if you want light to reach the second point, you need tom make light use the shortest distance(in time).
We were told at the university that saying light takes the shortest path is only half true. Light takes the extreme path, (if that's the right definition) which means that light could also take the longest path in a specific environment. I am not 100% sure but this can be an answer for you 6 years later lol
It's interesting that when you set up the problem this way it's not obvious that the answer should depend entirely on angles and not at all on distances.
Light need not take the fastest path....It could be the fastest or slowest or neutral, depending on the case, (here , in this case it is the fastest path) .Because, what we get is from the first derivative is ,what makes it an extremum.The second and third derivatives which are dependent on the situation are also required to find out wheather the path is a maxima ,minima or a neutral one.
Technically, this isn't Snell's law, but the law of refraction (which is true for all waves). Snell's law is the application of the law of refraction to light where we multiply both sides of the equation by the speed of light c, which allows us to write the equation in terms of the indices of refraction.
The problem is from experiment we know, that light not only gets refracted, but also reflected at a surface. This splitting of a ray at the surface cannot be expressed by Fermat's principle. Fermat's principle is really only a 0th order approximation to the real behavior of light.
hey why don't you use ropes on pulleys with hanging weights attached to them. You can change the weight and the tension stays constant no matter how much you pull on it.
Well, it is a nice stuff but it could have been interesting to developpe about the wave aspect of it, the explanation of the snell's law is because light is a wave, and we interpret it as a light ray. And then continue on the least action principle and the multiple paths formulation of feynman. But I guess it is suited for another channel^^
what do you mean the light's headed to B? The "landing" spot of a beam of light is not "known" to the particle of light. B is wherever the light ends up. How do you know it got there as fast as possible? How could it have gotten there slower? if you changed the direction it left A you'd change the location of B.
BraveLittIeToaster this also confused me at first but then I realized that the point is not to look at the rays that leave A, but the ray that end up at any arbitrary point B. The ray that end up at any arbitrary point b is going to be the light ray that takes the least amount of time. It's so strange and wonderful, right??!!
Sorry to be pedantic, but shouldn't the equations for the tensions be F_1 = k/v_air and F_2 = k/v_water, since the tensions are inversely proportional to the speeds of the light waves but not equal to the inverses of the speeds themselves. The k's would cancel once you set F_1 equal to F_2 and you would get the same result, but without "cheating."
Not that it means much in the bigger scheme of the tremendous value you bring, but since you seem interested in well versed criticism, I see three reasons why this animated explanation is less convincing than the standard you hold: 1) you introduce a physical representation which however does not exist 2) you say that it must be proportional to the inverse of the speed of light, but it could be the square inverse, or any power times constant 3) and force has unit of measurement Newton, which is not equal to s/m So the whole argument is just a bit too magical to be a credible alterntive for calculus. It sure delivers the same result but we only know that through the calculus backing up the argument, not because of the intrinsic truth of the argument being made apparent through your visualization. It's more trick than truth. But as said (and deviously hamburgering my criticism) your videos are always a treat.
i am wondering if you can actually measure the velocity or wavelength of the light in some meterials. dunk the photo-electric effct setup in some material, use the infra red light frequnecy , but its wavelength in the material is gamma ray wavlength. see if electrons will come out
Can gravity be viewed as something acting layer upon layer with varying "refractive gravity indices" perpendicular to its direction? Why is light in a hurry to trace the fastest route to destination. What is the underlying cause for this similarity in behaviour of the two seemingly uncorrelated events? I hope you get the question.
0:35 You can't prove a law. A scientific law is merely a statement of what is happening. To explain the law, you can have hypotheses and theories, which uses experiments to support it. That doesn't mean they prove the theory, they just support it. And theories are subject to change.
Of courses, the fundamental assumptions/hypotheses of a theory might change over time, but as long as you assume these to be correct, ofc. you can prove statements and laws which follow from these axioms
+Alex Fox Thanks! I believe there is fan funding through UA-cam, and at some point here I might do a patreon thing. For now, though, probably the best form of support is to share it with any math enthusiast friends you have :)
+3Blue1Brown Absolutely. Even though I'm familiar with most of the math in your videos already, the way you present the concepts is really beautiful and creative. I especially enjoyed the episode on measure theory and music.
Ok, it seems to be nice. But why did you assume that the forces should be inversely proportional to the velocities? If you were trying to make the speed of sound to play the role of the speed of wave along the spring then it is not inversely proportional to force. It seems like you just made up the expression for the force just to get the right expression that looks like Snell's law. Can you justify this point?
You can image toatal time taken as a function and we want to calculate it's min value So at its minimum the slope will be 0 So the derivative at that point must be zero
Here you took the constant force in spring as 1/v but what if I take it as v then the Snell law will not be proved...so don't you think that this method is useful only if we already know what the law is which basically is not that useful as proof are done in order to make new laws out of already known knowledge
Hello, excuse me, but the snell's law states n1 sin(i1) = n2 sin(i2) whenn n1 is top, i1 is top, n2 is bottom i2 is bottom. The snell's law you wrote states n1/sin(i1) = n2/sin(i2) Are you wrong on this one or do I miss something ?
Except the basis of this refuted by a simple mirror. If you have light shining onto a mirror, reflecting back and hitting a detector, it takes quite a long path. A much shorter and quicker path is for light to simply curve back around. If the fundamental basis for this proof was actually true, laser ranging systems and radar would not work. The fact that they do, and that light takes a different amount of time depending on the distance to the object rather than always taking the fastest path, shows that the fundamental basis for this proof is wrong.
Any Law named after me has to be a good law.
😂
Snell's biggest law: 1=2
🍻
lol
I think the problem with the "unphysical" springs can be solved by using weights instead. Just replace the springs with wires, which loop around A and B, and attach two weights of different masses to the ends of the wires (something like m = 1/(g * v)) and that will give you a constant pull!
+pacocsharp Good point! That's pretty clever.
+pacocsharp Really cool! This would make a really great physics demo.
I was going to suggest the mass on a string as well.
+pacocsharp Here's a working setup: connect the weights by a string to the loop. Put one pulley at point A and the other at point B. Drape the string over the pulley so that each weight hangs below its pulley.
the thought-experiments described in
Mark Levi's book "The Mathematical Mechanic"
make use of two imaginary devices -
the Constant -Force Spring and the Zero-Length Spring.
imagine setting up one of Mark's physical models on a horizontal board
with a hole drilled at each point where a spring is supposed to be fastened.
the Constant -Force Spring is replaced by a mass
hanging from a string passing through a hole.
the string transmits a CONSTANT force towards the hole.
the Zero-Length Spring is replaced by
an assembly consisting of a real spring
with a string attached to one end of the spring
and a coat button to be attached to the other end of the string.
a second horizontal board is mounted below the first board.
the combined length of the unstretched spring plus the string
is set equal to the vertical distance between the two boards.
the free end of the spring is attached to the lower board
directly below a hole in the upper board.
the string passes up through this hole and
the button is attached to the top end of the string
to keep the string from falling back through the hole.
now as the button is pulled away from the hole
along the surface of the upper board
the attached string will transmit a force towards the hole
DIRECTLY PROPORTIONAL to the distance
from the hole to the button.
At college I am a mathematics major, and I was also thinking about being a physics major. However, the two physics courses have been extremely disheartening for me, not because they are difficult, but because the professors teach in such a way where they never explain where any of the laws or theories we use come from! I ask them all the time if we could even get a general explanation, and they never do! It is such a frustrating experience and has really turned me away from physics altogether. I’m glad videos like yours exist that help give intuition to where these kinds of things come from!
Read Feynman
All throughout high school I was compliant to the idea that I was supposed to hate math. My first semester of college when I had a pre calc professor that incorporated proof and theory in to lessons. Then a switch went off and since then I’ve had a new found appreciation for mathematics and the physical sciences as a whole. I love finding professors, videos creators and in this case People who share the intuitive thinking.
Having a poor physics professor can make or break your grade! I'm the same way, I can't learn to APPLY laws without fully (or at all..) explaining where they come from, why* they happen and how they are derived. I need to understand a concept inside, outside and upside down to really move forward applying it and comprehending it.
I am in the exact same situation, I am in the french equivalent of high school, and on monday I have to have chosen which one of my"speciality" subjects I want to drop. I know I will take math but between computor science and physics/chemistry(the subjects are combined here) I am really torn. I always thought I would go on learning maths and physics throughout high school and university but I really dont enjoy physics as much as I would like to because it just feels like memorising formulas and properties without understanding where they come from and why the univers functions the way it does. Whenever I ask, I am told that it doesnt make sense to ask why that is the case, and that scientists just observed that that was the case and developped theories and laws based on what they saw. I never feel satisfied with this answer because it feels like there should be a logical, somewhat intuitive reason as to why things work the way they do, like in math. On the other hand Computor science is really easy and quite fun, but I dont want end up the tech guy at some company I dont care about in 10 years. I'm clueless as to what I should do :(
I am sure it is a challenge for your professors to cover all the material necessary to fulfill the goals of each course without adding a historical component to them. You imply that the coursework is not that difficult for you. Yet, you cannot figure out where to learn the foundations underlying what you are being taught? Your complaint does not pass the test of common sense.
I remember back in my school days I would sometimes take a shortcut to class cutting diagonally across a couple of fields to get to class faster, and noticed partway there that one of the fields was firm and easy going and the other was wet and muddy and a bit of a slog to walk across. I distinctly remember wondering to myself while crossing that field one day about a hypothetical best path to trade off added distance to more of the walk spent on the firmer ground to traverse the fields as fast as possible. Some time later I learned about Snell's law and the idea immediately clicking in terms of the two fields, it was such a strange and rare instance of having somehow lived a thought experiment before learning about the concept it explains...
Here there is an alternative derivation of Snell Law (very simple), using the conservation of momentum of photons:
If in medium 1 the index of refraction is n1 and in medium 2 the index of refraction is n2 and for the definition of index of refractions and wavelength we have:
n1 = c / v1 , v1 = λ1 * f
n2 = c / v2 , v2 = λ2 * f
Dividing the above equations we obtain: n2 / n1 = λ1 /λ2
So if n2 is higher than n1, the wavelength in medium 2 is smaller than in medium 1.
Now because the photon momentum is: p = h / λ
Applying the conservation of momentum of the incident and transmitted photon along the parallel line of medium separation:
p1 = p2 → h/λ1 *sin(θ1) = h/λ2 *sin(θ2) → n1 *sin(θ1) = n2 *sin(θ2)
This is the Snell Law!!
So the refraction is just a consequence of the conservation of momentum of photons!
There are a lot of ways of derive this law!
GUd shiZ
👌
Cancel out the 'p'-s in 'p1=p2' and you get '1=2'
Another way:
1. 0=0, 2. 1/0=2/0, 3. 1=2😂😂😂
How did you get theta on your keyboard?
@@rr1218-w8s Greek alphabet
See here συθηβμφΩ
"But we have something better than calculus"
Lies! There is nothing better than calculus.
that's what i was thinking haha
*angry algebraists have entered the chat*
geometers be ragin lol
Arithmetists are offended
Solved this using set theory lmao
yea but... who tf doesnt love calculus
+John Doe IKR :D
Calc 1 students.
I'm a tutor. I know this shit first hand.
John Doe calculus is beautiful
Plebs who can't calc
John Doe Multi var calc anyone?
Even though constant force springs are unphysical, there are objects we have that have constant force regardless of their how far they are extended. Take a piston with a vacuum inside and atmospheric pressure on the outside. Now, regardless of how far you pull the piston in or out, the total force would always be A*10^5, where A is the area of the top of the piston cylinder. This principle is actually used on the ISS to mimic earth gravity for astronauts to do all sorts of lifting exercises. So you could just replace the springs in your video with such pistons and the entire setup would be perfectly physical.
Great video! I also like the proof using geometry and picturing where each beam of light is at a certain time.
This is the most beautiful and intuitive solution I've seen to prove the Snell's law. Thank you for posting this.
A rolling metal tape measure can be viewed as a constant tension spring, they are often called negators
Currently reviewing the laugranian method to calculate the equations of motion in my theoretical physics class. I hope for more videos like this. Very happy I came across your channel.
You saved my life. I am studying Snell's law, to give a physics final, and the proof with Fermat's Theorem seemed very mathematical to me than physical and it is like four spreadsheets which is boring. But this demonstration of Snell's law is more accurate and you can understand what is happening physically.
Thanks for the video.
OMG The spring's and potential energy minimized scneario is also used to prove Stewart's theorem with springs and equilibrium! Mark Levi's clever proofs are really fascinating
Roy Long absolutely
최단 시간 경로의 법칙인 스넬의 법칙을 수학적으로 표현하는 것이 아닌 실생활에서 쉽게 볼 수 있는 스프링이라는 사물로 표현해주셔서 많은 사람들이 실질적으로 와닿게 이해할 수 있을 것 같다는 생각이 듭니다. 항상 좋은 영상 감사합니다.
Great video, as usual. May I ask you two questions?
1. Where did you come from? This may sound like a silly question, but youtube channels usually don't start out with high quality videos like yours did. If you have done any other similar mathematical explanations in the past, I would really like to see them.
2. How do you make your animations? They are unbelievably smooth, and are very satisfying to watch.
+nin10dorox I'm not quite sure how to answer the first question...Stanford, I guess, is where I "come from"? The videos on this channel are indeed the first videos I've put on youtube, though I'd done a few of casual lecture-type-talks before (none recorded). Thanks for your kind words, but I think there were more than enough faults with the first few videos I put up :)
To your second question, all these animations are programmatically generated. I've never been much for artistic talent, so my general hypothesis here is that when you're animating math, it's not too hard to describe in code, then the computer takes care of making things look pretty.
So you write programs for the animations? That's really cool!
May I ask what programming language you use?
+nin10dorox python
+3Blue1Brown
I am about to start learning Python and would be very curious what your code looks like. What environment do you use? Could you send me an example of one of your animation codes or is that asking too much?
+ElGrecoOB +3Blue1Brown Yes, I'm sure there are a lot of viewers that would be interested in seeing some of the code on github, but I can understand if you'd rather keep it closed :)
A key step in the proof is simple but ommitted : A central force of constant magnitude has a potential energy varying as distance from centre. (Hence, minimising the potential energy is same as minimising optical path length {equivalently time of travel}).
Actually, constant tension spring do exist, even though you could argue that they are not springs anymore :
If you have a chamber with some vacuum in it and a piston that can go back and forth into that chamber, then the force on the piston is going to be the same regardless of the piston's position in that chamber because the vacuum density never changes.
I guess that the force on the piston in such a chamber could be represented by : F=ax+b.
Where a is the the numbers of moles of gas in that chamber and b some constant due to the outside pressure on the chamber. It's easy to see that when a goes to 0 the whole expression becomes constant.
This property has actually been used for exercise machines in the ISS to reproduce the effect of weight :
ua-cam.com/video/05oOst9kZXQ/v-deo.htmlm51s
thats sick
Fun to think about things in different ways. This was fun. Made me smile. Thanks
There are constant force springs. They are made of flat ribbons of steel rolled into a coil. The only part of the ribbon that exerts a net force retracting the spring at any time is the part that is curved, but not quite laying on the coil. Since the geometry of that part is roughly constant, the force is roughly constant. Altering the geometry of the spring slightly along its length allows this to be tuned so that the force is constant within tolerances.
The springs are an interesting way to think about it, and might enhance understanding in some way, but with all the "finagling" involved to make the thought experiment work, I feel the calculus approach (which the video took just a few seconds to explain) is far more elegant.
i just got home from a 70 hour long work week to find TWO 3b1b videos in my sub pile
i have never been so pleased
if i hit play on this video and it's an april fools joke, "Hahaha you fools thought i would actually make any more videos" then i'm gonna cry
don't make me cry man
it's not cool
Just what I needed. Thank you UA-cam algorithm
that assumption that F_(virtual spring of medium A)= 1/(v(in medium A)) is like using the assumption that triangles angle sum upto 180deg so we can say that α+β+γ = 180 in a triangle ABC where angle A=α
angle B=β
angle C=γ
i would like to have a mathematical justification for that assumption
Same here.
Why were the forces equal to the inverse velocity in each medium?
Yep, I was going to point out that was not proven.
@@b.clarenc9517 It wasn't proven because we assume it's true
in the video he says let F1= 1/v air and F2= 1/v water
he used this method because it's less complicated and easier to understand than the calculase one
I know I'm late but I hope I helped you a little :)
It is technically inversely proportional to velocity so that the force mimicks the time taken to travel a path. So by design, when we minimize force, we minimize time.
Not real springs.
@@hassanakhtar7874 That's right, but the princple of minimum energy holds anyway. And, for those particular springs, minimum potential energy is the same as minimum travel time for light following the path depicted by the springs in equilibrium.
Grant always changes things for me...
I thought I would focus a lot more on Math than Physics but, here you go...
Such a beautiful proof of Snell's makes me think...
Looks like I'm going to study a lot of Physics now! After Number Theory, Combinatorics and Geometry of course!😅
Mm... It makes sense, but I don't find it satisfactory as an explanation. Light finding this balance between time and energy seems more like a consequence of something else than the root cause of the behavior. Is this possible?
en.wikipedia.org/wiki/Principle_of_least_action#Fermat
Thanks for the link... but it does not explain much, still.
The subject of 3b1b's video is Fermat's Principle of Least Time. Fermat's Principle exemplifies the optics case of the Principle of Least Action. Action has dimensions ( length^2 / time ). It's often calculated with
S = integral( L ) dt
where
S = action
L = lagrangian = ( KE - PE )
t = time
The Principle of Least Action observes that given a starting point and a destination point, a particle will always follow the trajectory with the least action. Newton's Three Laws of Motion can be derived from this principle. After the Double-Slit Experiment, physicists reframed this principle in terms of Feynman Paths. To ask why the universe behaves this way is to ask why the universe is quantum. Physicists don't know yet. In any case, it appears pretty fundamental. Finding the root cause of this principle is analogous to finding the root cause of the conservation of energy.
+the fance Thank you for sharing your thoughts and knowledge that relate to the subject of this video. It seems that we can only speculate about the root causes of physical phenomena of nature in our universe as we seem to know it.
iirc the best explanation of it is geodesics? or perhaps that only explains gravity, we forget...
Nice explanation, yet it give the impression that nature knew the future in advance then decide to use the shortest path. I believe snell's law should be in terms of density to avoid confusion
Yeah right. I wouldnt really call it a proof tbh
I have heard similars explanation. But what I think is "wrong" is Fermat's Principle. I mean, you can get snell's law using Maxwell equations, and in my opinion that is more fundamental than Fermat's principle. I feel that I didn't explain properly. I think that the idea is similar as Pauli's Principle; Pauli's Principle is true, but it's a corollary of antisimetrization of fermions.
Snell's law follows also nicely from electrodynamics and maxwell's equations, in fact, it makes more sense given the properties of polarization. Interestingly, quantum-mechanics will tell that the shortest path explanation might actually harbor some accuracy.
I always felt unsettling about the explanation of shortest path. Always wondered whether thats what its actually doing! Do you have any pointers for the QM explanations?
WHAATTTT THIS IS BLOWING MY MIINDDDD
I ACTUALLY LOVE THIS CHANNEL
I prefer to use the more physically correct boundary condition of waves and their derivatives being continuous perpendicular to the surface across the surface. It makes more sense to me and doesn't lead people down the wrong track of thinking that light picks the fastest path on purpose (I had an optics prof do that, a bona fide PhD). The fact that it happens to be the fastest is a coincidence, or more accurately a consequence of satisfying the boundary condition.
very nice !
i am a fan of your work !
I would like to start a similar kind of channel for the french youtube community, and I was wondering : which software do you use to make your animations ?
they are more or less exactly what i am aiming for !
keep it up !
+MrGustave1er I'm glad you want to make math videos, the more there are in this world the better! I generate my animations programmatically, basically using a small library I've been building up. It's not the most friendly for others to learn (e.g. no tutorial), since it's mostly just been for my own use. You are welcome to take a look if you want at github.com/3b1b/manim, and I'll help you out if you have questions, but unless you really want full control for arbitrary new math animations, I would probably recommend looking into better-documented tools than mine.
Constant or nearly constant tension springs do exist. Ever notice how a tape measure has constant tension no matter how far you pull it out? With a quick google search you can find companies selling them.
I've heard about negative index metamaterials and I'm wondering how the thing about light taking the shortest path fits into that situation, great illustrative video btw.
take two points, one in water one out of it(as shown in the video) now if you want light to reach the second point, you need tom make light use the shortest distance(in time).
We were told at the university that saying light takes the shortest path is only half true. Light takes the extreme path, (if that's the right definition) which means that light could also take the longest path in a specific environment. I am not 100% sure but this can be an answer for you 6 years later lol
Mark Levi! Love his book on applied physics to solve math problems, and this video too!
It's interesting that when you set up the problem this way it's not obvious that the answer should depend entirely on angles and not at all on distances.
Dude your videos are sooo nice
Light need not take the fastest path....It could be the fastest or slowest or neutral, depending on the case, (here , in this case it is the fastest path) .Because, what we get is from the first derivative is ,what makes it an extremum.The second and third derivatives which are dependent on the situation are also required to find out wheather the path is a maxima ,minima or a neutral one.
Technically, this isn't Snell's law, but the law of refraction (which is true for all waves). Snell's law is the application of the law of refraction to light where we multiply both sides of the equation by the speed of light c, which allows us to write the equation in terms of the indices of refraction.
What a brilliant analogy!
Seeing what people have done in the past makes me feel like a piece of shit
Tan Onay Try having my last name and hearing about the guy, and I teach myself calculus out of boredom.
it's been 5 years, enough time to do something that would make you feel a bit less like a piece of shit :)
@@alejrandom6592 But still Piece of shit.
Cant agree with the feeling more
The problem is from experiment we know, that light not only gets refracted, but also reflected at a surface. This splitting of a ray at the surface cannot be expressed by Fermat's principle. Fermat's principle is really only a 0th order approximation to the real behavior of light.
Can you please teach me how F of both the springs equals to 1/speed of light in that medium?
Can any one help me understanding this part of the proof?
How come F equals the inverse speed of the light? (F=1/v)
Hi, where can I find the video about angular momentum illustrated by prof. Mark Levi :)
hey why don't you use ropes on pulleys with hanging weights attached to them. You can change the weight and the tension stays constant no matter how much you pull on it.
How about trying Snells law considering Elastic Collisions between a rest mass and a moving mass, "Wave-Particle Duality"?
Well, it is a nice stuff but it could have been interesting to developpe about the wave aspect of it, the explanation of the snell's law is because light is a wave, and we interpret it as a light ray. And then continue on the least action principle and the multiple paths formulation of feynman. But I guess it is suited for another channel^^
Could you substitute indices of refraction for the spring constant instead?
what do you mean the light's headed to B? The "landing" spot of a beam of light is not "known" to the particle of light. B is wherever the light ends up. How do you know it got there as fast as possible? How could it have gotten there slower? if you changed the direction it left A you'd change the location of B.
BraveLittIeToaster this also confused me at first but then I realized that the point is not to look at the rays that leave A, but the ray that end up at any arbitrary point B. The ray that end up at any arbitrary point b is going to be the light ray that takes the least amount of time. It's so strange and wonderful, right??!!
@@ningshanma1094 Interesting perspective but still why? Why does it behave like that. Feels strange.
Why exactly F = 1/V? And does that analogy work with not equal distances between A and the axis, and B and the axis?
Neat. I remade this in Algodoo so you can play arround with the springs and target points
Awesome. Can you see the snells law concept ?
Okay... But why does light change its speed in different mediums? I thought the speed of light was constant?
Thank You so much for this one!! :D... and all the other ones
calculus: left out
video: amazing
mind: blown
hotel: trivago
There are two different people talking, I thought I was crazy.
Sorry to be pedantic, but shouldn't the equations for the tensions be F_1 = k/v_air and F_2 = k/v_water, since the tensions are inversely proportional to the speeds of the light waves but not equal to the inverses of the speeds themselves. The k's would cancel once you set F_1 equal to F_2 and you would get the same result, but without "cheating."
Bernard Eisberg what is K I’m sorry
Not that it means much in the bigger scheme of the tremendous value you bring, but since you seem interested in well versed criticism, I see three reasons why this animated explanation is less convincing than the standard you hold:
1) you introduce a physical representation which however does not exist
2) you say that it must be proportional to the inverse of the speed of light, but it could be the square inverse, or any power times constant
3) and force has unit of measurement Newton, which is not equal to s/m
So the whole argument is just a bit too magical to be a credible alterntive for calculus. It sure delivers the same result but we only know that through the calculus backing up the argument, not because of the intrinsic truth of the argument being made apparent through your visualization. It's more trick than truth.
But as said (and deviously hamburgering my criticism) your videos are always a treat.
i am wondering if you can actually measure the velocity or wavelength of the light in some meterials.
dunk the photo-electric effct setup in some material, use the infra red light frequnecy , but its wavelength in the material is gamma ray wavlength. see if electrons will come out
Brilliant!
Really clever! I love that so much!
Can gravity be viewed as something acting layer upon layer with varying "refractive gravity indices" perpendicular to its direction?
Why is light in a hurry to trace the fastest route to destination.
What is the underlying cause for this similarity in behaviour of the two seemingly uncorrelated events?
I hope you get the question.
Your second question is what made me uncomfortable the day I heard it first in school.
Why are springs a good analogy? I should go over analytical mechanics again...
Edan Coll He probably should have used ideal strings or ropes
0:35 You can't prove a law. A scientific law is merely a statement of what is happening. To explain the law, you can have hypotheses and theories, which uses experiments to support it. That doesn't mean they prove the theory, they just support it. And theories are subject to change.
Of courses, the fundamental assumptions/hypotheses of a theory might change over time, but as long as you assume these to be correct, ofc. you can prove statements and laws which follow from these axioms
You can prove them from other laws though
But, how does the light know it has to reach B? Silly question, but I want an answer.
This is way better than the "marching ants" analogy which really doesn't make any sense at all.
ironically there is a Fermat's law which talks about when the derivative is zero [f'(x)=0]. He want's us to work that way
Smart solution.
Grant.. Thanks for making this
Hey, I love your videos! How can I help support you?
+Alex Fox Thanks! I believe there is fan funding through UA-cam, and at some point here I might do a patreon thing. For now, though, probably the best form of support is to share it with any math enthusiast friends you have :)
+3Blue1Brown Absolutely. Even though I'm familiar with most of the math in your videos already, the way you present the concepts is really beautiful and creative. I especially enjoyed the episode on measure theory and music.
what about horizontal forces how will you show that it balances itself?
Thank you so much.
Ok, it seems to be nice. But why did you assume that the forces should be inversely proportional to the velocities? If you were trying to make the speed of sound to play the role of the speed of wave along the spring then it is not inversely proportional to force. It seems like you just made up the expression for the force just to get the right expression that looks like Snell's law. Can you justify this point?
Daniyar Saparov something about distance's relationship to speed and energy's relationship to force.
That’s exactly what im a thinking mate
ok....that impressed me
Thanks!
So I understand derivatives but why is the fastest path where the derivative=0?
You can image toatal time taken as a function and we want to calculate it's min value
So at its minimum the slope will be 0 So the derivative at that point must be zero
Awesome videos, but sound quality could be a lot better
+lexvi I agree, this awesome animations can be improved significantly with quality audio.
But I don't understand why are we allowed to say that the concept behind the spring analogy is the same as with light
Fascinating!
2 new videos!!!!!!
Here you took the constant force in spring as 1/v but what if I take it as v then the Snell law will not be proved...so don't you think that this method is useful only if we already know what the law is which basically is not that useful as proof are done in order to make new laws out of already known knowledge
wow....great thinking !
Maximizing entropy
Fantastic
1:54 "but we got something better than calculus"
Newton: *triggered*
I approve of this law. 👍
Hello, excuse me, but the snell's law states n1 sin(i1) = n2 sin(i2) whenn n1 is top, i1 is top, n2 is bottom i2 is bottom.
The snell's law you wrote states n1/sin(i1) = n2/sin(i2)
Are you wrong on this one or do I miss something ?
ok, mb, I found out: , v = c/n
sure that's nice and all, but calculus is so much more fun! :3
Can someone explain to me why the horizontal components of the tension in the springs have to balance out?
if they didn't the ring wouldn't stop, it would keep sliding
+Adam Khan That is the static equilibrum where it stops moving
Amazing
how such force in a medium depends on the velocity of light ?
Sandeep Dubey more force proportional to change in velocity
F=1/v how was this relation found
Love it.
Except the basis of this refuted by a simple mirror.
If you have light shining onto a mirror, reflecting back and hitting a detector, it takes quite a long path.
A much shorter and quicker path is for light to simply curve back around.
If the fundamental basis for this proof was actually true, laser ranging systems and radar would not work.
The fact that they do, and that light takes a different amount of time depending on the distance to the object rather than always taking the fastest path, shows that the fundamental basis for this proof is wrong.
Thats nice! Talking about light the other way feels weird.
0:47 is there some other reason?
Reminds me of law of sines
Great!
What’s the music used in this video
Sweet Jesus the Pi creatures were horrifying