Hello early people. A bunch of you are saying the video is low res. UA-cam is just taking its sweet time to process it. Soon it'll be available in glorious 3840 × 2160.
Works with chemistry too. Solubility, precipitation, adsorption, reactions and a bunch of other stuff is basically just doing complicated things with energy.
Pro-tip. As a professional career physicist, this is typically my first guestimate method of reasoning when coming across a new problem. Once it's given me an intuition for what's likely to be going on do I move on to more rigorious methods.
Me when I saw it curved inward : "it's cosh!" Me when I saw the equation : "oh maybe it's harder than I thought" Matt : "it's cosh!" Dammit. Never doubt good old cosh.
CV Boys, in his lectures given at the Royal Institution, drew a single *closed* bubble apart until it came very close to that state of instability that Matt found (around 21:32). Then he used that instability to read environmental phenomena in the room. Just as Matt was being very careful not to breath so that he wouldn't create a disturbance and cause the structure to divide prematurely, Boys took great care to reach that unstable place and to not disturb it ... until ... he allowed a minute amount of input to see if that could be measured by this delicate tool that he had caused the soap film to be. At one point, he put electromagnets on either side of the stretched bubble and, from across the room, when he threw the switch that caused the magnets to engage ....... the bubble pulled in two ... only because oxygen is slightly responsive to magnetism. The oxygen inside of the bubble responded to the magnetic charge and that response was instantly read by the soap film which was so unstable that that tiny disturbance caused it to pull into two separate bubbles. That was during his lectures that began in 1890.
@@yasheesinha8181 Mate, 1890 is Victorian times, so definitely not video.I don't know when his exact lecture was, but certainly pre-WW1, maybe there's a transcript with diagrams?
Oh, not going to lie, I thought this was modern and CV Boys was some kind of youtuber. In that scenario, the 1890 bit--in my head--seemed like a typo. But no, no. Not at all. No.
Note that b is always x/2. That's because b is just the horizontal translation of the cosh graph. If you plug that in, you only have one variable left to solve. The Newton method would work quite well too.
I was thinking when I saw Matt's Cosh equation "There has to be an x _somewhere_ or else it would be constant", and by simply analyzing the x=0 case, one could tell right away that a is not affected by x, so b must be a function of x, and the fact that the first b he calculated happened to be _extremely_ close to half of .5, my intuition made me suspicious.
In the Netherlands, most people program in English because the keywords are English anyway, which also means foreign people can read it... Though there're some who write their comments in Dutch, and there're some heretics that use Dutch words to name their variables/functions/classes
@@jtherrie I know. I just like saying people are heretics for using another language than the keywords in programming... And I like harmlessly confusing people before popping the bubble, so thanks for the reply :P
@@nicjansen230 It's similar in Germany. I prefer to keep everything English only, including comments. But sometimes the occasional German word slips through. And then there's people writing everything in German only. Or even a mix of both, which is even more infuriating. Most of the time that happens when multiple people with different naming schemes work on the same project.
I dunno what is more nerdy in this case: A man getting excited about playing with bubbles for 25 minutes, Or me who enjoyed watching a man get excited about playing with bubbles in the nerdiest way possible for 25 minutes. In either case, this was a blast.
Don’t care, bubbles are awesome. I used to literally take an hour to do the dishes because I was so engrossed in playing with the soap bubbles. I may be a nerd, but I’m a proud nerd, and anyone who thinks that’s a problem isn’t worth my time.
But actually he got it the wrong way around. The maths break down because reality breaks down, not the other way around. Which actually isn't supprising, because the maths were chosen to model reality.
@@schwarzarne He didn't say that maths breaking down was the cause of reality breaking down, he just said that when maths broke down is when reality broke down. He didn't talk about causality at all.
I had a similar situation involving numerical calculations I did during my PhD studies. At some point in my calculations, the more accurate I was trying to be, the less stable the results. Below that point, the calculations made perfect sense. Above that point, the results were inverted from what I expected. After a few weeks of debugging and headaches, my advisor prompted me to work out an analytical solution as an alternative (that is, an exact solution of a slightly simplified problem), and the exact solution showed there was an inversion point where the effect became opposite from what I expected. Then I was able to make sens of the experimental data, which had a high value, then a dip to almost zero, then a high value (I was measuring the square of the calculated value, that is why it was always positive even when it had reversed. Ok, as I write this, I realize how vague it is. Summary of the actual problem: I was working on tiny metal line grills to be used them as polarizers for ultraviolet light. With wire line polarizers, one expects that the photons with electrical fields oriented in the direction of the lines will be reflected, the ones with electrical fiels perpendicular will be transmitted. It turns out that as the wavelength becomes really close to the period of the wire grill, the effect reverses, photons with electrical field parallel to the wires actually makes it through, the one perpendicular is reflected. The calculation I was making involved matrices of infinite size, the numerical calculation was using finite size matrices to approximate the results, the bigger the matrices, the more accurate they were. For some values, say 7 x 7 matrices, I would get a positive value, then at 9x9 I get negative, at 11x11 positive again. That happens at the crossover point, where the answer is actually close to zero. The reason my experiments measured the square of the polarization is that there are no good UV polarizers, so I was using two of mine at 90 angles and measuring the extinction of light as a function of wavelength. Since I use two identical ones, it does not matter whether the parallel or perpendicular polarization makes it through, it will be blocked by the other polarizer, so extinction will always be higher, except at the crossover point since light is not polarized there.
That sounds really fascinating! Do you know why that change happens yet? At the moment I can imagine it breaking at equal wavelength and distance (guessing the light will end up scattered into an interference pattern). But why the narrower wires work differently eludes me. 🤔
@@widmermt I did do microwave polarization using grids too as a previous project. For UV, we used self-assembling materials (diblock copolymer thin films) which produce line patterns with 20 to 55nm period, way too small to make by "hand". Edit: by the way, inexpensive regular polarizers (for sunglasses, for example) are done similarly, electrically conductive material in a polymer film which is stretched in one direction to align the polymers, then died with materials conducting at the visible light frequencies. Works well for visible, but UV is too short for that (plus most substrates like the plastic or glass of sunglasses are opaque to UV, which is good for your eyes).
@@witerabid It has been a long time and I have not had to dabble in that area of physics since. Re-reading my thesis and trying to understand without going through the math all over again: First, I remembered wrong, the period of my material was much lower than that of the UV light, so that was not the cause. It has to do with the plasma frequency of the metal used (aluminum). All metals have a plasma frequency, above which the metal becomes essentially transparent to light, a phenomenon referred to as the "UV transparency of metals" since it usually happens in the UV range. Aluminum has one of the highest plasma frequencies, so that it becomes transparent "only" for wavelengths shorter than 99nm or so. However, when you have a grid of material instead of a solid piece, the grid's characteristics are that of the average of the metal and air. Thus, through some math, it is possible to show that for the E polarization (electric field parallel to wires) the plasma frequency goes down by the square root of 2, so the aluminum grid becomes transparent below 140nm or so (in theory for pure aluminum, which was not the reality of my experiment, as aluminum oxidizes quickly in air). It is also possible to show that the dielectric constant of the H polarization (perpendicular to the lines; I know I am adding new terms without defining them, sorry) is inversely proportional to that of the E polarization, so that when the grid becomes transparent to the E polarization, it becomes reflective for the H polarization. Then as one goes down in wavelength (up in frequency), the aluminum itself becomes transparent to light, as stated above, so then it does not matter, nothing is polarized.
@@vincentpelletier57 Ok, that sounds even more fascinating. I never got into metals or light much during my physics degree. 😅 In the end it was mostly theoretical physics and math(s). So, all the experimental and material physics never seizes to baffle me. In theory, everything should be predictable with some calculation but I never know where to start. And metals do some really weird stuff which I guess I can now add one more mystery to. 😋 Thank you for sharing. 😊
@@Kokurorokuko it doesn't say "import . . . " but the library is called "math" but in the code he used "maths" which means he must've written somewhere "import math as maths"
And if nothing's happening I reckon it goes: Something something (Potential) energy Idk if that's correct. But it's clearly got potential. .... I apologise for nothing. 🎩👌
An interesting thing I noticed: The value of B was always equal to the height divided by two. So, if we assume this to be true, you might be able to get a more accurate equation, just using a single inverse cosh function.
I noticed this as well. When the fixed points have the same radius (distance from the _x_-axis) and one is located on the _y_-axis and the other is some distance out along the _x_-axis, the value of _b_ is just half that distance. If the radii of the fixed points are different, though, _b_ will be offset from that value.
The inverse cosh also won’t work since „something with a“ * cosh(„also something with a“) = something else generally cannot be solved for a algebraicly. Even the most simple a * cosh(a) is doomed to be solved numerically.
It is true that with symmetric boundary conditions for this zoomed in/out (by a) and x-shifted (by b) version of cosh(x) {which is [exp(x)+exp(-x)]/2 by the way} b is exactly in the middle, but I guess he used slightly different values for the ring and the basin.
@@lightningblender This kind of thing is something I've been working with recently, and yeah. The reason we can solve things like 3*a + 2*a = 15 is because we can say 3 * a + 2 * a = (3 + 2) * a always, and so combine our two 'a's into one. So in order to invert some `f (g a) (h a)` wrt a, we'd need to know some identity with f, g, and h to get some other already invertible function f' such that f' a = f (g a) (h a). In the case of distribution/factoring, we know that for f = (+), g = (n *), and h = (m *), we have f' = ((n + m) *), which has an inverse: inv f' = (1/(n+m) *). This is the same problem as with x * e^x, whose inverse is the Lambert W function, which can only be computed numerically. (Speaking of which, I bet we *could* write an inverse of a * cosh(a) in terms of the Lambert W)
@@MCLooyverse but this does not change the problem of needing to compute the result numerically. In fact, all trig functions including their inverses and hyperbolic variants, already can only be computed numerically (except for separately checked special cases). You are simply summing over the taylor expansion and then stopping at some point. These super general functions like Meijer G and Hyperbolic PFQ but also Bessel J or Gauß erf are just a name we gave to very useful functions. LambertW is no different: we had a problem and then decided to give it a name. Nevertheless, I don’t think Lambert W will yield a result. Normally, Mathematica is fairly skilled in solving such equations, but yet, it didn’t find any and my other reasoning is, that since cosh contains exp(-a) and exp(a) summed together they won’t combine into a single exp function, that’s required for Lambert‘s W function.
Me: "I can't see, but this feels like something where a catenary would come in..." Matt: "It's a catenary..." Me: "Go me!" Matt: "But you can't calculate a and b exactly." NO!!!!
I can't get over Matt's pronunciation of "catenary." Is that standard British pronunciation? In Philadelphia, electric trolleys get their power from CAT-en-air-ee wires. It's very satisfying to see that it's the same, and it feels intuitive. There's a tension force pulling the line to be shorter and an attraction "force" pulling the line towards the center of the circle - kind of like gravity in the case of a chain.
As an engineering student who has just completed a semester of studying heat conduction through cylinders, I would like to see better hoop sketching work from you in the future, Matt.
It looks a lot like the Science Center in Toronto. I guess since I grew up with such a place close to me, I just thought every major city would have something similar. I hope that's true. Hmm, just realizing most people haven't been to such extravagantly large science center like TO, I'd bet my family up north have never been, tiss a long way. I went like twice a year growing up. Guess I was science spoiled.
Damn, I thought this was going to be about the Euler-Lagrange equation and the calculus of variations at first, but then he skipped straight from the integral to the minimising function cosh, where all the interesting stuff actually happens in the maths. I suppose there was never going to be a whole degree level course in a 24 minute video.
@@cholten99 yes, it's exactly the same principle in that video. You are trying to find a path that minimises action in that case, and a path that minimises surface area in this case, and this is exactly where the E-L equation can be applied to solve the problem.
@@stanleydodds9 Out of interest, do you think this could be represented as a dynamic optimisation problem where you apply optimal control theory? I was wondering how the problem could be formulated.
I would guess that what "fell apart" there is just the assumption that cosh solves the Euler-Lagrange equation beyond this critical point, but I haven't dealt with this kind of math for too long to check it by myself
I love it when Matt does physics and calls it maths. I mean, seriously, he can enjoy whatever he likes and share that love with the world, without being constrained by superficial subject boundaries. That's wonderful
My physics teacher used to say that Physics is just "applied maths", and theoretical physics is just "can't yet be applied maths". Probably a good job he wasn't an English teacher.
@@danieljensen2626 By definition, P physics is using math to describe the physical world. Just as economics is using math to describe the economy and biology (or at least, part of it) is using math to describe biological systems. Again, I'm just happy he doesn't fall into this hole of "let's define everything", and just does what he likes
I think Maths City could benefit from having something like a whiteboard around all it's exhibits. Facilitate people who want to dive into the maths like you did here, and possibly leave some of that insight temporarily for people who follow to see. The only problem with this is the dicks that will be drawn and markers stolen.
Doesn't have to be a whiteboard. When I was at university there were movable chalkboards in the bigger halls so you could discuss whatever was at the top of your head with your peers whenever you liked. I think pieces of chalk might be cheaper than whiteboard markers and they have a nice old-school (ha!) touch IMO.
The chain catenary can also "choose" the same shape as the bubble. Imagine that you fix one end of a chain, and at the other end you put a pulley, the chain then hang vertically until it reaches the chain heap at the floor (which is at the axis of the bubble). One can see that if the two points are sufficiently far apart and the catenary is sufficiently long, it will outweigh the vertical segment, pull some chain, so the catenary is even longer and heavier, so it pulls more chain across the pulley ... BOOM! We've reached the same instability as Matt shown.
This! I tried to think of an analogy between the chain and the bubble but failed xD saw your comment, read it like 7 times to finally understand it, but seems to make sense.. I wonder if the chain collapses at the same point as the bubble.. Also, if anyone else tries to understand this comment, try to picture the chain 90° rotated in reference to the bubble ;)
@@hubi.92 Yes, it is modelled by the same variational problem (the one written by Matt: minimize the integral of y*sqrt(1+y'^2)dx). The chain minimizes its potential energy. For the potential energy (U=mgh) you may arbitrarily choose the initial level (where h=0). But by putting the chain heap at the "floor" we choose the zero level to be there. Then we do not need to account for the p.e. for the chain in the heap, where h=0, and the vertical segment gives constant summand in p.e., so it can also be ignored.
@@alexeyklimenko4387 yes now i see the connection.. with the bubble you try to minimize the surface area dependent on radius and heigth at each point and with the chain you minimize the p.e. with the heigth of each "infinitesimal mass element" and the number of these elements..
@@alexeyklimenko4387 but i think you can't consider the chain heap in a mathematical formulation, more like that the chain is generated from nothing at the end of the vertical segment.. you can see this by putting your reference plane for the p.e. at another height, then the solution would be dependent on the chain reserve in the heap and that can't be..
The function is dependent on the variables for the force generated by the surface area tension pulling sideways, and gravity pulling down. So in physics, it's a little bit more complicated than pure 3d geometry.
Excellent video, I find it really counter intuitive that the solution is not always the "Goldschmidt" solution of just two circles at each end. But it makes sense when you think of huge circles very close to each other.
watching this makes me wonder where i would have been in life, if i had spent alot more time with maths, and learning more about it. i feel math is something that could help you achive many things in life.
How to make a great math video: 1. derive a cool equation. 2. wave your hands and say «very difficult, links below» 3. Write down the solution, wave your hands, say «impossible. Here is some code I wrote.» Great video as allways!
Maths (is) was my favorite topic in school. A lot of content like this is already over my head, but I still get a kick out of hearing "this is pretty standard [over my head[, but THIS, is where it gets interesting [so far over my head I barely understand it]" :)
That last bit where the bubble collapses into two, just as your maths predicted, completely blew my mind. Awesome. BTW, gravity should make the bubble sag ever so slightly. Luckily, the mass of the soap film is negligible.
Technically yes. The shape locally-minimizes the combined surface energy plus gravitational potential energy, but a soap film weighs so little that gravity can be mostly ignored.
Yeah, it being a catenary sort if gives this away (shame he just pulled the formula out of thin air to be honest) A catenary is the shape of a rope under gravity, so exactly two forces acting on it. The tension (latterally) and gravity vertical. The soap bubble being modeled as a catenary shows that one force is used inwards to minimise surface area and the other force is the tension along the soap bubble. No room for gravity or air pressure :(
the force exerted b the surface tension of soapy water ain't much ... but a typical soap film is usually thinner than wavelengths of light. Not much mass ...
That was so cool!. I was watching on my way out of the office, so paused part way through for my drive home. During my drive, I kept thinking: "certainly two discs with an infinitesimally small connector string in the middle has to take over for least surface area at some point, right?!?!" ... and then wondered if some variable for surface tension needed to play a role--- but was so pleased when I finished the video at home!
That knot at 12:24! My Scout Leader soul is tormented firstly by the two half-hitches forming a lark's head rather than a clove hitch and secondly by the cut end being left to fray.
Blue Peter did a similar experiment back in the 1970s. They solved the traveling salesman problem with it. Nails in a wooden board dipped in bubble solution sideways - when pulled out it hows the shortest path between the nails. They said if the nail locations corresponded to city locations that it meant the bubbles reveled the shortest path on which you could build roads connecting all the cities. Also, why is it (Dx^2 * y'^2) and not (Dx * y')^2 as the second term in the root formula? I guess I need a slightly more basic foundation than you're providing here.
The square of a product equals the product of the squares, which means (x^2 * y'^2) is exactly the same as (x * y')^2 . For example 2^2 * 3^2 = 4 * 9 = 36, which is the same as (2 * 3)^2 = 6^2 = 36.
That's not the traveling salesman problem. In that problem you can't add new intersections (known as Steiner points) but the soap film will. Also the traveling salesman problem is about planning a tour of the nodes in an optimal order. The soap film doesn't say anything about how to traverse the network optimally, it "finds" a network of minimum total length. The problem it solves is known as the Steiner Tree problem.
6:00: This would have been a great point at which to introduce calculus of variations. As it stands simply going from the equation, to the solution, to saying the solution can't be evaluated analytically is sort of disappointing. The "reveal" later than the solution was only a local optima is kind of hollow when it came out of the blue in the first place.
Yeah I had hoped that he'd at least gone until the lagrangian formulation and maybe do something crazy like solving the differential equation in Excel or Python numerically (which you totally can do!). But just giving the solution was very underwhelming.
I get that. To be fair, there isn't really an entertaining/non-confusing way to present the calculus of variations to the layperson. Or at least if there is, I'm not creative enough to think of it.
I think that would have turned off a big portion of the audience (though a much lower portion of those who comment) but that kind of thing would be really cool to see on a companion channel for those who would be hyped about it. Edit: Actually, I'd be really interested to see him upload a video with all the gritty details you would have wanted here, and a follow-up with watch time statistics to see how many viewers left at that point compared to a similar length video. I may be totally wrong, who knows?
I thought someone might comment about it before me! :D I've been to the Mathematikum several times actually, with family twice and once with the school. And in one year they had an exhibition that travelled around germany and also to my city
Matt I think you've redeemed yourself with this video. Finally a problem where your results are more highly correlated to the actual result than a random guess.
This video has so many physics insights! One is that some physicists think our universe, in the Higgs field, is like this bubble, and it may be in the state between the two ratios (local minimum) which would be a metastable state. A high enough energy event may raise the hoop so to speak and cause our universe, like the bubble, to collapse (into a global minimum state). I also like how the Principle of Least Action ties into this. The PLA states all paths in physics must minimize a value called Action, which is the integral over the whole path of the lagrangian, which in classical mechanics is the kinetic minus potential energy. The way to minimize this integral and find the path is through the Calculus of Variations, the exact same method used to minimize the integral derived at the beginning! It all relates because the PLA itself is why the bubble minimizes surface area since you can show that minimizing surface area minimizes the action due to all the surface tension forces added up over all points in the bubble.
I like that you are taking organic substances and turning it into a computer generated render. Quite a game asset genius. I would like to see many more of these types of videos with other organic substances.
In 3d modeling software such as sketchup you can create a torus by extruding a circle along a larger circular path but this wouldn't apply to a system trying to minimize surface area. I suspect it could be done if you used Centrifugal force while making it but it would quickly collapse in to a sphere once you stopped applying the force and trap the air in the hole as another bubble. [Edit] Yep watch?v=9ZoQLk61v88
I reckon that the precision you measured the bubble width was almost as good as the one in the Earth’s Radius video. If only Hannah Fry was there to help you out...
"Let's figure out a formula for the shape the soap makes!" "This gets ugly, so we'll just skip to the end. OK, here at the end we find out that there's no way to get that formula we were after!"
I don't like how he writes the multiplication dot. He uses \ldot …. for multiplication and \cdot · for decimal points. I'm the opposite. I write 2.5 · 8.7, whereas he writes 2·5 . 8·7. What the heck lol. I first noticed this on another (older) video he did about calculating π by hand. I'm triggered! :D Is it an Australian thing?
The point where math as well as reality fell apart reminded me of the one time I tried to use the Lorentz factor in an excel sheet to calculate how long a spaceship journey would take subjectively. The maths would break as soon as the speed exceeded the speed of light.
It seems to me that when you turned the chart and made the vertical axis horizontal, you neglected the effect of GRAVITY. Although the weight of the bubble is small, I guess the real curve is not symmetrical but the minimum diameter is BELOW the middle height.
If the answer to the question of why the equation "stopped working" is that it's actually behaving in a way that is a perfect predictor of reality, then that's where the biggest satisfaction kicks in. I love it.
I believe an easier to interpret equation is using (R/c) = cosh(D/c) where R is the radius of a disk and d is half the distance btwn them. Letting R=1 meter and plotting c you can see that there are certain c values that will not satisfy larger distances at which point the bubble collapses. Keep in mind that values on the right side of the cosh plot of c give results more accurate to real life. If R = 1 then if D > 2/3 there is not c to make the system stable.
You can make your terrible Python slightly better by using a better initial estimate. Take the second-order Taylor series approximation of cosh with the same parameterisation: p(x) = a + (x-b)^2 / 2a . Set p(0) = p(l) = r and p'(l/2) = 0. Then b = l/2, and a can be found by finding the positive solution of the quadratic 2a^2 - 2ar + b^2. This gives quite good initial estimates which you can then feed to your solver. (Fun fact: I did essentially the same trick once with modelling the shape of cross-country electrical transmission lines, where the span length was typically about 1km. It was a couple of years before we realised that, due to a bug, we were using the initial estimates instead of the output of the solver in one particular application. Nobody noticed because the parabola was that close!)
Hello early people. A bunch of you are saying the video is low res. UA-cam is just taking its sweet time to process it. Soon it'll be available in glorious 3840 × 2160.
Its so weird, I just happened to go to check your channel for the first time in a while and you had this video just uploaded!
360p gang poggers
I finished the video in low res but had to go back to watch the bubble breaking in HD it looks so good!
I just refreshed to get the 1080p and not only do i get it, I get this comment too 🙃
Glorious 8,294,400?
"Something something, energy" is how I will explain all physics from now on.
Works with chemistry too. Solubility, precipitation, adsorption, reactions and a bunch of other stuff is basically just doing complicated things with energy.
I laughed out loud at this awesome line. He just "yada-yada-yada"-ed all of science and engineering.
this is exactly how i graduated high school
Pro-tip. As a professional career physicist, this is typically my first guestimate method of reasoning when coming across a new problem. Once it's given me an intuition for what's likely to be going on do I move on to more rigorious methods.
F=MA everything else is decoration.
"Something something energy"
This is how mathematicians do physics.
And they call physicists for making approximations....
@@maxwellsequation4887 Just assume the physics approximates math.
"...something something energy" I see Matt knows his physics.
I would say his response to how the maths broke down predicted when the physical world broke down corroborates your statement fully.😉
It's called Parker's First Law. Look it up.
Me when I saw it curved inward : "it's cosh!"
Me when I saw the equation : "oh maybe it's harder than I thought"
Matt : "it's cosh!"
Dammit. Never doubt good old cosh.
CV Boys, in his lectures given at the Royal Institution, drew a single *closed* bubble apart until it came very close to that state of instability that Matt found (around 21:32). Then he used that instability to read environmental phenomena in the room. Just as Matt was being very careful not to breath so that he wouldn't create a disturbance and cause the structure to divide prematurely, Boys took great care to reach that unstable place and to not disturb it ... until ... he allowed a minute amount of input to see if that could be measured by this delicate tool that he had caused the soap film to be.
At one point, he put electromagnets on either side of the stretched bubble and, from across the room, when he threw the switch that caused the magnets to engage ....... the bubble pulled in two ... only because oxygen is slightly responsive to magnetism. The oxygen inside of the bubble responded to the magnetic charge and that response was instantly read by the soap film which was so unstable that that tiny disturbance caused it to pull into two separate bubbles.
That was during his lectures that began in 1890.
Thats awesome!
Are these lectures accessible somewhere?
@@yasheesinha8181 Mate, 1890 is Victorian times, so definitely not video.I don't know when his exact lecture was, but certainly pre-WW1, maybe there's a transcript with diagrams?
Oh, not going to lie, I thought this was modern and CV Boys was some kind of youtuber. In that scenario, the 1890 bit--in my head--seemed like a typo.
But no, no. Not at all. No.
@@hareecionelson5875 there's a book ... with diagrams
'Mattbook Pro', this is the high comedy I subscribe for
+
yep, I expect no less
+
Thanks for pointing it out; I didn't even notice (and I *did* read the code)! :D
And yet, Mathsbook Pro was left right there. xD
9:58 and 10:27
"where math stops working is where reality stops working"
That sentence makes me very happy :D
I need this on a t-shirt
kind of crazy to think about
Didn't you mean "maths stops working"
Just asking.
@@terryshippee4996 FYI, in North America, there is no "s" in math. But I watch enough UK TV that I say it both ways in my head.
reminds me of black holes.
Note that b is always x/2. That's because b is just the horizontal translation of the cosh graph. If you plug that in, you only have one variable left to solve. The Newton method would work quite well too.
Yeah, I think that's because he set both end diameters to the same, but otherwise it could be something else.
@@sly1024 True
I wonder if he knew this and choose to use the Newton method because he wanted to highlight the instability effect
That is if the rings have equal diameters
I was thinking when I saw Matt's Cosh equation "There has to be an x _somewhere_ or else it would be constant", and by simply analyzing the x=0 case, one could tell right away that a is not affected by x, so b must be a function of x, and the fact that the first b he calculated happened to be _extremely_ close to half of .5, my intuition made me suspicious.
"import math as maths"
I've always wondered how people program in foreign languages =p
In the Netherlands, most people program in English because the keywords are English anyway, which also means foreign people can read it... Though there're some who write their comments in Dutch, and there're some heretics that use Dutch words to name their variables/functions/classes
@@nicjansen230 I was making a joke about how English people say 'Maths' and everyone in America says 'Math'. Thanks for the insight though.
@@jtherrie I know. I just like saying people are heretics for using another language than the keywords in programming... And I like harmlessly confusing people before popping the bubble, so thanks for the reply :P
I never thought of doing that. I'll try that next time!
@@nicjansen230 It's similar in Germany. I prefer to keep everything English only, including comments. But sometimes the occasional German word slips through. And then there's people writing everything in German only. Or even a mix of both, which is even more infuriating. Most of the time that happens when multiple people with different naming schemes work on the same project.
I dunno what is more nerdy in this case:
A man getting excited about playing with bubbles for 25 minutes,
Or me who enjoyed watching a man get excited about playing with bubbles in the nerdiest way possible for 25 minutes.
In either case, this was a blast.
I see you all the time in curiosity show comments, nice to see you here too lol
I think the nerdy part might be when video man does calculus
@@biquinary thank you. Been a fan of Matt Parker before I found Curiosity Show. But I do enjoy them both.
Don’t care, bubbles are awesome. I used to literally take an hour to do the dishes because I was so engrossed in playing with the soap bubbles. I may be a nerd, but I’m a proud nerd, and anyone who thinks that’s a problem isn’t worth my time.
That conclusion at 22:21 is why we need maths, why we need patience, and why we need you. :-) Thanks for the great work Matt!
But actually he got it the wrong way around. The maths break down because reality breaks down, not the other way around.
Which actually isn't supprising, because the maths were chosen to model reality.
@@schwarzarne He didn't say that maths breaking down was the cause of reality breaking down, he just said that when maths broke down is when reality broke down. He didn't talk about causality at all.
I had a similar situation involving numerical calculations I did during my PhD studies. At some point in my calculations, the more accurate I was trying to be, the less stable the results. Below that point, the calculations made perfect sense. Above that point, the results were inverted from what I expected. After a few weeks of debugging and headaches, my advisor prompted me to work out an analytical solution as an alternative (that is, an exact solution of a slightly simplified problem), and the exact solution showed there was an inversion point where the effect became opposite from what I expected. Then I was able to make sens of the experimental data, which had a high value, then a dip to almost zero, then a high value (I was measuring the square of the calculated value, that is why it was always positive even when it had reversed.
Ok, as I write this, I realize how vague it is. Summary of the actual problem:
I was working on tiny metal line grills to be used them as polarizers for ultraviolet light. With wire line polarizers, one expects that the photons with electrical fields oriented in the direction of the lines will be reflected, the ones with electrical fiels perpendicular will be transmitted. It turns out that as the wavelength becomes really close to the period of the wire grill, the effect reverses, photons with electrical field parallel to the wires actually makes it through, the one perpendicular is reflected.
The calculation I was making involved matrices of infinite size, the numerical calculation was using finite size matrices to approximate the results, the bigger the matrices, the more accurate they were. For some values, say 7 x 7 matrices, I would get a positive value, then at 9x9 I get negative, at 11x11 positive again. That happens at the crossover point, where the answer is actually close to zero.
The reason my experiments measured the square of the polarization is that there are no good UV polarizers, so I was using two of mine at 90 angles and measuring the extinction of light as a function of wavelength. Since I use two identical ones, it does not matter whether the parallel or perpendicular polarization makes it through, it will be blocked by the other polarizer, so extinction will always be higher, except at the crossover point since light is not polarized there.
Vincent, I remember doing an undergrad lab using wires to polarize microwaves. That's amazing that they can be made fine enough to work for uv!
That sounds really fascinating! Do you know why that change happens yet? At the moment I can imagine it breaking at equal wavelength and distance (guessing the light will end up scattered into an interference pattern). But why the narrower wires work differently eludes me. 🤔
@@widmermt I did do microwave polarization using grids too as a previous project. For UV, we used self-assembling materials (diblock copolymer thin films) which produce line patterns with 20 to 55nm period, way too small to make by "hand".
Edit: by the way, inexpensive regular polarizers (for sunglasses, for example) are done similarly, electrically conductive material in a polymer film which is stretched in one direction to align the polymers, then died with materials conducting at the visible light frequencies. Works well for visible, but UV is too short for that (plus most substrates like the plastic or glass of sunglasses are opaque to UV, which is good for your eyes).
@@witerabid It has been a long time and I have not had to dabble in that area of physics since. Re-reading my thesis and trying to understand without going through the math all over again:
First, I remembered wrong, the period of my material was much lower than that of the UV light, so that was not the cause.
It has to do with the plasma frequency of the metal used (aluminum). All metals have a plasma frequency, above which the metal becomes essentially transparent to light, a phenomenon referred to as the "UV transparency of metals" since it usually happens in the UV range. Aluminum has one of the highest plasma frequencies, so that it becomes transparent "only" for wavelengths shorter than 99nm or so.
However, when you have a grid of material instead of a solid piece, the grid's characteristics are that of the average of the metal and air. Thus, through some math, it is possible to show that for the E polarization (electric field parallel to wires) the plasma frequency goes down by the square root of 2, so the aluminum grid becomes transparent below 140nm or so (in theory for pure aluminum, which was not the reality of my experiment, as aluminum oxidizes quickly in air). It is also possible to show that the dielectric constant of the H polarization (perpendicular to the lines; I know I am adding new terms without defining them, sorry) is inversely proportional to that of the E polarization, so that when the grid becomes transparent to the E polarization, it becomes reflective for the H polarization.
Then as one goes down in wavelength (up in frequency), the aluminum itself becomes transparent to light, as stated above, so then it does not matter, nothing is polarized.
@@vincentpelletier57 Ok, that sounds even more fascinating. I never got into metals or light much during my physics degree. 😅 In the end it was mostly theoretical physics and math(s). So, all the experimental and material physics never seizes to baffle me. In theory, everything should be predictable with some calculation but I never know where to start. And metals do some really weird stuff which I guess I can now add one more mystery to. 😋 Thank you for sharing. 😊
"If people are familiar with my back catalog . . ."
He's talking about the Parker Square, isn't he?
Or the more recent video with Hannah Fry where he fails to estimate the size of the Earth.
In 9:58, "import math as maths" xD, btw it was really nice that moment when your calculations were that close to the actual real value
lol he just had to correct it
where?
@@Kokurorokuko You don't see that line specifically, but you do see all the maths., which can only happen if he imported math as maths
@@Kokurorokuko it doesn't say "import . . . " but the library is called "math" but in the code he used "maths" which means he must've written somewhere "import math as maths"
@@ayrtonsenna6311 got it
All scientific explanations end with "something something energy"
Yadda yadda yadda. Etcetera, and so on...
Sometimes
Not all of the often it's something/something (angular) momentum.
And if nothing's happening I reckon it goes:
Something something (Potential) energy
Idk if that's correct.
But it's clearly got potential.
....
I apologise for nothing.
🎩👌
An interesting thing I noticed: The value of B was always equal to the height divided by two. So, if we assume this to be true, you might be able to get a more accurate equation, just using a single inverse cosh function.
I noticed this as well. When the fixed points have the same radius (distance from the _x_-axis) and one is located on the _y_-axis and the other is some distance out along the _x_-axis, the value of _b_ is just half that distance. If the radii of the fixed points are different, though, _b_ will be offset from that value.
The inverse cosh also won’t work since „something with a“ * cosh(„also something with a“) = something else generally cannot be solved for a algebraicly. Even the most simple a * cosh(a) is doomed to be solved numerically.
It is true that with symmetric boundary conditions for this zoomed in/out (by a) and x-shifted (by b) version of cosh(x) {which is [exp(x)+exp(-x)]/2 by the way} b is exactly in the middle, but I guess he used slightly different values for the ring and the basin.
@@lightningblender This kind of thing is something I've been working with recently, and yeah. The reason we can solve things like 3*a + 2*a = 15 is because we can say 3 * a + 2 * a = (3 + 2) * a always, and so combine our two 'a's into one. So in order to invert some `f (g a) (h a)` wrt a, we'd need to know some identity with f, g, and h to get some other already invertible function f' such that f' a = f (g a) (h a). In the case of distribution/factoring, we know that for f = (+), g = (n *), and h = (m *), we have f' = ((n + m) *), which has an inverse: inv f' = (1/(n+m) *).
This is the same problem as with x * e^x, whose inverse is the Lambert W function, which can only be computed numerically. (Speaking of which, I bet we *could* write an inverse of a * cosh(a) in terms of the Lambert W)
@@MCLooyverse but this does not change the problem of needing to compute the result numerically. In fact, all trig functions including their inverses and hyperbolic variants, already can only be computed numerically (except for separately checked special cases). You are simply summing over the taylor expansion and then stopping at some point. These super general functions like Meijer G and Hyperbolic PFQ but also Bessel J or Gauß erf are just a name we gave to very useful functions. LambertW is no different: we had a problem and then decided to give it a name. Nevertheless, I don’t think Lambert W will yield a result. Normally, Mathematica is fairly skilled in solving such equations, but yet, it didn’t find any and my other reasoning is, that since cosh contains exp(-a) and exp(a) summed together they won’t combine into a single exp function, that’s required for Lambert‘s W function.
"Let's use maths to calculate this cool shape!" 8 minutes later: "Just kidding, it can't be done."
"something, something energy" .. these sure are math videos :D
:kek:
Yes, that's G. H. Hardy level mathsiness right there.
I do agree 😂
With appropriate hand-waving gestures
Ah, the something something energy theorem - one of physic's greatest discoveries...
Me: Hey that curve kinda looks like a catenary!
Matt: It's a catenary!
Me: YAY!
Me: "I can't see, but this feels like something where a catenary would come in..."
Matt: "It's a catenary..."
Me: "Go me!"
Matt: "But you can't calculate a and b exactly."
NO!!!!
This is toxic Mathsculinity.
I can't get over Matt's pronunciation of "catenary." Is that standard British pronunciation? In Philadelphia, electric trolleys get their power from CAT-en-air-ee wires.
It's very satisfying to see that it's the same, and it feels intuitive. There's a tension force pulling the line to be shorter and an attraction "force" pulling the line towards the center of the circle - kind of like gravity in the case of a chain.
As an engineering student who has just completed a semester of studying heat conduction through cylinders, I would like to see better hoop sketching work from you in the future, Matt.
Mathscity sounds like a wonderful place.
I would love to be the librarian in that city!
It looks a lot like the Science Center in Toronto. I guess since I grew up with such a place close to me, I just thought every major city would have something similar. I hope that's true. Hmm, just realizing most people haven't been to such extravagantly large science center like TO, I'd bet my family up north have never been, tiss a long way. I went like twice a year growing up. Guess I was science spoiled.
@@Krumm420 The San Francisco version is the Exploratorium
@@Krumm420 Halifax has something similar too (Discovery Centre) including a bubble room. Absolutely fantastic place.
This is toxic Mathsculinity.
Damn, I thought this was going to be about the Euler-Lagrange equation and the calculus of variations at first, but then he skipped straight from the integral to the minimising function cosh, where all the interesting stuff actually happens in the maths. I suppose there was never going to be a whole degree level course in a 24 minute video.
Maybe 3b1b will get around to a 12-part series in 25 years....
PBS Space Time touched on Euler-Lagrange for about 10 seconds in their latest video (all way over my head)
@@cholten99 yes, it's exactly the same principle in that video. You are trying to find a path that minimises action in that case, and a path that minimises surface area in this case, and this is exactly where the E-L equation can be applied to solve the problem.
@@stanleydodds9 Out of interest, do you think this could be represented as a dynamic optimisation problem where you apply optimal control theory? I was wondering how the problem could be formulated.
I would guess that what "fell apart" there is just the assumption that cosh solves the Euler-Lagrange equation beyond this critical point, but I haven't dealt with this kind of math for too long to check it by myself
Seeing you do "import math as maths" in the Python code got a nice chuckle out of me. Gotta love that Python lets you do stuff like that.
"Seeing" or "deducing" tho? :-B
I love it when Matt does physics and calls it maths. I mean, seriously, he can enjoy whatever he likes and share that love with the world, without being constrained by superficial subject boundaries. That's wonderful
Physics is just maths with silly restrictions like 'reality'
My physics teacher used to say that Physics is just "applied maths", and theoretical physics is just "can't yet be applied maths".
Probably a good job he wasn't an English teacher.
I mean most of physics is math.
@@danieljensen2626 By definition, P
physics is using math to describe the physical world. Just as economics is using math to describe the economy and biology (or at least, part of it) is using math to describe biological systems.
Again, I'm just happy he doesn't fall into this hole of "let's define everything", and just does what he likes
Intelligence sometimes warps the reality field :)
I think Maths City could benefit from having something like a whiteboard around all it's exhibits. Facilitate people who want to dive into the maths like you did here, and possibly leave some of that insight temporarily for people who follow to see. The only problem with this is the dicks that will be drawn and markers stolen.
Doesn't have to be a whiteboard. When I was at university there were movable chalkboards in the bigger halls so you could discuss whatever was at the top of your head with your peers whenever you liked.
I think pieces of chalk might be cheaper than whiteboard markers and they have a nice old-school (ha!) touch IMO.
Or if that doesn't work the Royal Armouries Museum is close; about 10mins drive.
wouldn't the bubble rather pop maths?
No it burst maths open
Immer diese Klugscheisser :p
Wie geht's, Papa?
Can you make a video explaining the solution to this minimizing problem (explaining all the identities used and so on)
Love your work!
nono, it's the math that pops the bubble
“The math stopped working, is exactly the same place where reality stopped working….”
Mind BLOWN!!!
Awesome video Matt 👍✅👍
Mathematician: "My calculated value is within about 10% of the real value, awesome!"
Engineer: O_o
Nothing speaks to me more than when doing a measurement and just saying "You know, if we aggressively round then I'm completely correct here."
The chain catenary can also "choose" the same shape as the bubble. Imagine that you fix one end of a chain, and at the other end you put a pulley, the chain then hang vertically until it reaches the chain heap at the floor (which is at the axis of the bubble). One can see that if the two points are sufficiently far apart and the catenary is sufficiently long, it will outweigh the vertical segment, pull some chain, so the catenary is even longer and heavier, so it pulls more chain across the pulley ... BOOM! We've reached the same instability as Matt shown.
This! I tried to think of an analogy between the chain and the bubble but failed xD
saw your comment, read it like 7 times to finally understand it, but seems to make sense.. I wonder if the chain collapses at the same point as the bubble..
Also, if anyone else tries to understand this comment, try to picture the chain 90° rotated in reference to the bubble ;)
@@hubi.92 Yes, it is modelled by the same variational problem (the one written by Matt: minimize the integral of y*sqrt(1+y'^2)dx).
The chain minimizes its potential energy. For the potential energy (U=mgh) you may arbitrarily choose the initial level (where h=0). But by putting the chain heap at the "floor" we choose the zero level to be there. Then we do not need to account for the p.e. for the chain in the heap, where h=0, and the vertical segment gives constant summand in p.e., so it can also be ignored.
@@alexeyklimenko4387 yes now i see the connection.. with the bubble you try to minimize the surface area dependent on radius and heigth at each point and with the chain you minimize the p.e. with the heigth of each "infinitesimal mass element" and the number of these elements..
@@alexeyklimenko4387 but i think you can't consider the chain heap in a mathematical formulation, more like that the chain is generated from nothing at the end of the vertical segment.. you can see this by putting your reference plane for the p.e. at another height, then the solution would be dependent on the chain reserve in the heap and that can't be..
The function is dependent on the variables for the force generated by the surface area tension pulling sideways, and gravity pulling down. So in physics, it's a little bit more complicated than pure 3d geometry.
22:32 "the math stopped working exactly at the same place reality stopped working."
Simulation hypothesis confirmed.
Excellent video, I find it really counter intuitive that the solution is not always the "Goldschmidt" solution of just two circles at each end. But it makes sense when you think of huge circles very close to each other.
Not gonna lie, that expected jump of the bubbles tot he other axis was so epic & satisfying
watching this makes me wonder where i would have been in life, if i had spent alot more time with maths, and learning more about it. i feel math is something that could help you achive many things in life.
Well, it can also crush one's soul, js. ;-)
"You can't do it algebraically, you have to do it... numerically"
Omg! You mean you have to use numbers instead of letters in your maths? Disgusting!
Filthy reality... ugh..
Love that your mac book is called MattbookPro ;)
Wow 21:50 was so satisfying to watch having the background knowledge needed to understand what was going on, great video!
How to make a great math video:
1. derive a cool equation.
2. wave your hands and say «very difficult, links below»
3. Write down the solution, wave your hands, say «impossible. Here is some code I wrote.»
Great video as allways!
When the bubble film separates into 2 discs is my new favorite moment in the history of UA-cam.
When I get married I demand that this video is played at the wedding.
“Something something energy” is the best summary of thermodynamics I’ve ever heard.
Today i've learned that cosine stands for compliment sine.
Thanks, Matt!
Or rather complement [to] sine. (as wiki says, The word cosine derives from a contraction of the medieval Latin complementi sinus.)
I rarely comment on youtube, but this video was so exciting, I had to come up and say thank you for your effort in making it.
Marvellous work Matt
Maths (is) was my favorite topic in school. A lot of content like this is already over my head, but I still get a kick out of hearing "this is pretty standard [over my head[, but THIS, is where it gets interesting [so far over my head I barely understand it]" :)
That had a very satifying ending. Your accuracy or precision was also especially on point this episode.
Wow not going to lie the 2nd half with the errors was really interesting!
That last bit where the bubble collapses into two, just as your maths predicted, completely blew my mind. Awesome.
BTW, gravity should make the bubble sag ever so slightly. Luckily, the mass of the soap film is negligible.
Does gravity not have an effect on the shape of the curve? I would have thought that it would mean the curve wouldn't be exactly symmetrical.
Technically yes. The shape locally-minimizes the combined surface energy plus gravitational potential energy, but a soap film weighs so little that gravity can be mostly ignored.
Yep, you can also see the air pressure change the shape as Matt moves
Yeah, it being a catenary sort if gives this away (shame he just pulled the formula out of thin air to be honest)
A catenary is the shape of a rope under gravity, so exactly two forces acting on it. The tension (latterally) and gravity vertical. The soap bubble being modeled as a catenary shows that one force is used inwards to minimise surface area and the other force is the tension along the soap bubble. No room for gravity or air pressure :(
the force exerted b the surface tension of soapy water ain't much ... but a typical soap film is usually thinner than wavelengths of light. Not much mass ...
Everyone seems to ignore that Matt heats the air around himself, that's why we have an upward airflow that tries to pull air "inward" at the bottom.
That was so cool!. I was watching on my way out of the office, so paused part way through for my drive home. During my drive, I kept thinking: "certainly two discs with an infinitesimally small connector string in the middle has to take over for least surface area at some point, right?!?!" ... and then wondered if some variable for surface tension needed to play a role--- but was so pleased when I finished the video at home!
Obviously, they should rename it to Matt city
So we can spot Matts in their natural environment
They are generally known as Parker Towns
That knot at 12:24! My Scout Leader soul is tormented firstly by the two half-hitches forming a lark's head rather than a clove hitch and secondly by the cut end being left to fray.
Blue Peter did a similar experiment back in the 1970s. They solved the traveling salesman problem with it. Nails in a wooden board dipped in bubble solution sideways - when pulled out it hows the shortest path between the nails. They said if the nail locations corresponded to city locations that it meant the bubbles reveled the shortest path on which you could build roads connecting all the cities.
Also, why is it (Dx^2 * y'^2) and not (Dx * y')^2 as the second term in the root formula? I guess I need a slightly more basic foundation than you're providing here.
Surely that could settle on a local minimum, not necessarily the global minimum?
The square of a product equals the product of the squares, which means (x^2 * y'^2) is exactly the same as (x * y')^2 .
For example 2^2 * 3^2 = 4 * 9 = 36, which is the same as (2 * 3)^2 = 6^2 = 36.
That's not the traveling salesman problem. In that problem you can't add new intersections (known as Steiner points) but the soap film will. Also the traveling salesman problem is about planning a tour of the nodes in an optimal order. The soap film doesn't say anything about how to traverse the network optimally, it "finds" a network of minimum total length. The problem it solves is known as the Steiner Tree problem.
@@Quantris thanks for the clarification. 40 years makes the memory of exactly what they said a little fuzzy
Congrats on the Parker Bubble; you gave it a go and it appeared to not work at some point but that actually matched reality.
Watching Matt try to measure and hold the cord makes me realize I want more physical challenge math problems.
I appreciate how you're just in shock the entire video that all mathematical predictions you made bore out.
when you said "there's only one true parabola", I waited for your head to spin...
Eyebrow raised.
if you happen to be around Dresden in Germany we got a similar exhibit that is always there
Gießen has the 'Mathematikum' which is also very similar
6:00: This would have been a great point at which to introduce calculus of variations. As it stands simply going from the equation, to the solution, to saying the solution can't be evaluated analytically is sort of disappointing. The "reveal" later than the solution was only a local optima is kind of hollow when it came out of the blue in the first place.
Yeah I had hoped that he'd at least gone until the lagrangian formulation and maybe do something crazy like solving the differential equation in Excel or Python numerically (which you totally can do!). But just giving the solution was very underwhelming.
Counterpoint: It is never a good time to introduce calculus of any kind
@@PatrickZysk Counter^2point: It’s *always* a good time for calculus
I get that. To be fair, there isn't really an entertaining/non-confusing way to present the calculus of variations to the layperson. Or at least if there is, I'm not creative enough to think of it.
I think that would have turned off a big portion of the audience (though a much lower portion of those who comment) but that kind of thing would be really cool to see on a companion channel for those who would be hyped about it.
Edit: Actually, I'd be really interested to see him upload a video with all the gritty details you would have wanted here, and a follow-up with watch time statistics to see how many viewers left at that point compared to a similar length video. I may be totally wrong, who knows?
1:04 'the bubble film wants to minimise its surface area; something something, energy.' - Matt haha
in germany there is one in the city Gießen called "Mathematikum" where i went as a kid (with school and with my dad ) :)
Jaaaa Gießen!
I thought someone might comment about it before me! :D
I've been to the Mathematikum several times actually, with family twice and once with the school.
And in one year they had an exhibition that travelled around germany and also to my city
Mal sehen wie viele Deutsche jetzt die Kommentarspalte fluten xD
I also haven't heard a more self-deprecating description of machine learning. I appreciate it.
This video was beautiful, shows just how amazing maths is.
"maths are" , it's plural!
Matt I think you've redeemed yourself with this video. Finally a problem where your results are more highly correlated to the actual result than a random guess.
Sound like you had a great time there!
This bubble surface area problem is one of the first examples I had for learning functionals! Love the video
“..something something energy..”
Did Matt just skip a load of math?
Matt are you okay?
He just "yadda yadda"'d maths! lol
He skipped the boring, gross, practical physics.
"a load of maths"?
This video has so many physics insights! One is that some physicists think our universe, in the Higgs field, is like this bubble, and it may be in the state between the two ratios (local minimum) which would be a metastable state. A high enough energy event may raise the hoop so to speak and cause our universe, like the bubble, to collapse (into a global minimum state).
I also like how the Principle of Least Action ties into this. The PLA states all paths in physics must minimize a value called Action, which is the integral over the whole path of the lagrangian, which in classical mechanics is the kinetic minus potential energy. The way to minimize this integral and find the path is through the Calculus of Variations, the exact same method used to minimize the integral derived at the beginning! It all relates because the PLA itself is why the bubble minimizes surface area since you can show that minimizing surface area minimizes the action due to all the surface tension forces added up over all points in the bubble.
"I wrote some terrible python code" is a catchphrase now.
Always has been.. 🙂
I like that you are taking organic substances and turning it into a computer generated render. Quite a game asset genius. I would like to see many more of these types of videos with other organic substances.
Those designs in the background made me wonder... is it possible to have a torus-shaped bubble?
In 3d modeling software such as sketchup you can create a torus by extruding a circle along a larger circular path but this wouldn't apply to a system trying to minimize surface area.
I suspect it could be done if you used Centrifugal force while making it but it would quickly collapse in to a sphere once you stopped applying the force and trap the air in the hole as another bubble.
[Edit] Yep watch?v=9ZoQLk61v88
I cannot believe how much better this video got towards the end.
"MattBook-Pro"
If I wasn't subscribed already I would have done it exactly at 10:42
I like how Matt measures the "breaking-point" height to be 71 cm, but the zero of the scale is dangling in the air.
This reminds me of the NEMO science museum in Amsterdam, which also has all these cool hands-on science things
Searched for "average bubble size" (don't ask) ended up here. Love your content Matt! Glad I rediscovered you!
I reckon that the precision you measured the bubble width was almost as good as the one in the Earth’s Radius video.
If only Hannah Fry was there to help you out...
"Let's figure out a formula for the shape the soap makes!" "This gets ugly, so we'll just skip to the end. OK, here at the end we find out that there's no way to get that formula we were after!"
PS: The best part of this video was realizing that Matt named is computer a MattBook-Pro!
Matt is the most engineer of all mathematicians out there
Blah blah, something something energy and so on, etcetera... Yadda yadda yadda.
He is still surprised when math models reality. Mathematicians, that's what math is for.
12:35 when Stand-up Math said "I'm gonna stand up" and stands all over that bubble, truly one of moments of all time in math history
I don't like how Matt writes integral signs. They're so bent over and take up so much horizontal space.
I DEMAND A REFUND! :)
I don't like how he writes the multiplication dot. He uses \ldot …. for multiplication and \cdot · for decimal points. I'm the opposite. I write 2.5 · 8.7, whereas he writes 2·5 . 8·7. What the heck lol. I first noticed this on another (older) video he did about calculating π by hand. I'm triggered! :D Is it an Australian thing?
@@robertjenkins6132 As near as I can tell this is a Matt thing. At least he doesn't mix commas in there somehow lol
The point where math as well as reality fell apart reminded me of the one time I tried to use the Lorentz factor in an excel sheet to calculate how long a spaceship journey would take subjectively. The maths would break as soon as the speed exceeded the speed of light.
It seems to me that when you turned the chart and made the vertical axis horizontal, you neglected the effect of GRAVITY. Although the weight of the bubble is small, I guess the real curve is not symmetrical but the minimum diameter is BELOW the middle height.
If the answer to the question of why the equation "stopped working" is that it's actually behaving in a way that is a perfect predictor of reality, then that's where the biggest satisfaction kicks in. I love it.
Your new videos do anything but burst my bubble
That is so cool. The math just breaking and its relation to reality is a super awesome visual, and a nice bridge between maths and the physical world.
When he said his Python code is terrible, was Matt speaking hyperbolically?
It's Python code, no one actually writes Python code that isn't terrible.
You’re going off on a tangent there.
@@Colopty haha - that brightened my day. Cuz I don’t have to write python. :)
@@Colopty Dean's joke is that the whole video is about a hyperbolic function.
One of your best videos in my opinion. Well done!
Maybe this video should be titled "the math that breaks (or pops?) the bubble"
"It's about 93 centimeters!" - exlaiming enthusiastically at 13:00 - and holiding the measurement tool visibly not perpendicular to the floor :D
10:11, love the use of the mathS module instead of the standard math module : - )
I believe an easier to interpret equation is using (R/c) = cosh(D/c) where R is the radius of a disk and d is half the distance btwn them. Letting R=1 meter and plotting c you can see that there are certain c values that will not satisfy larger distances at which point the bubble collapses. Keep in mind that values on the right side of the cosh plot of c give results more accurate to real life. If R = 1 then if D > 2/3 there is not c to make the system stable.
The next time Matt breaks maths, he will break the fabric of reality.
OK so, you know all those "Future Matt" appearances in his videos..?
Videos like this remind me just how much physics and math relate to each other
The One True Parabola reference was for true fans
If that place wasn't a 13 hour flight away I'd be all over it. I hope it does great!
If the two hoops are the same size, since cosh is symmetric, b has to be L/2
Thanks, I was wondering about about.
You can make your terrible Python slightly better by using a better initial estimate.
Take the second-order Taylor series approximation of cosh with the same parameterisation: p(x) = a + (x-b)^2 / 2a
. Set p(0) = p(l) = r and p'(l/2) = 0. Then b = l/2, and a can be found by finding the positive solution of the quadratic 2a^2 - 2ar + b^2. This gives quite good initial estimates which you can then feed to your solver.
(Fun fact: I did essentially the same trick once with modelling the shape of cross-country electrical transmission lines, where the span length was typically about 1km. It was a couple of years before we realised that, due to a bug, we were using the initial estimates instead of the output of the solver in one particular application. Nobody noticed because the parabola was that close!)
At 21:00, serious question, can you “inch” your way up if you are using the metric system to measure? 🤔
Same thoughts. I was partially expecting him to say he'd centimeter his way up.
So close to 1 million! Keep up the great work ❤️