Originally I just wanted to explain how physics gets the soap film to attain a minimal surface configuration, but it turns out the process in which the soap film evolves is an active area of research that I accidentally stumbled onto while making this video. BTW: Once we got to the integral for the total force, I could have used Stokes' theorem, but this way is a lot more algebraic manipulation, while the only algebraic manipulation in this argument would be turning cross product into dot product. The Stokes' theorem method also doesn't intuitively explain why the divergence suddenly appears.
How do you not get bored and fed up and sick of math and frustrated and how can you spend 2p minutes on one topic? I am depressed enough as it is and find it hard ebough to concentrate..it should be fun and easy and enjoyable
I appreciate the intuition for divergence coming out. It felt so natural once I saw the flux and the area. Out of curiosity, may I ask what the manipulation for Stokes' theorem would be?
@@mathemaniac Oh wow that's actually pretty cool thanks! I now have a greater appreciation for your intuitive derivation by converting it to a dot product.
Minimal surfaces are such an interesting topic. I never really delved into the topic, but I remember playing with soap films as a kid. It's super interesting to see how 3D graphics play a role in all of this. Greatly looking forward to the next video!
As someone who studies stuff like this in my PhD-studies (not exactly minimal surfaces, but Willmore surfaces, which are closely related; as a side note, you should totally do a video on the Willmore energy, like conformal invariance, the Li-Yau inequality, minimization of the Willmore energy for fixed genus, there's so much to discover and even more which is still unknown!), this is such a great video! I actually didn't know that soap bubbles evolve by hyperbolic mean curvature flow rather than usual mean curvature flow.
@yak-i4c I've worked on this topic, which has been debated for more than 40 years now. We actually don't even agree on the exact nature of the forces and scales involved, and which framework should be used (hydrodynamics, non-equilibrium thermodynamics, statistical physics).
Cam you please tell me how the fuck can anyone get a ph d in anything..it seems wayntoo boring and frustrating and BORING sitting in front of s computer all day..how des anyone do it and why..hope you can respond
@@RunningRay9 well, it's that in a rather short duration of time "interface" was contrasted with "surface". For one, this made me realize that these two words are closely related. But more importantly, the soap film was just a thin layer between two volumes of air. Well they were actually the same air in this example, but you could imagine a rubber layer between two different liquids in an aquarium or something, so that it would take forces from one side, possibly transform them, and relay that to the other side. And an interface is basically exactly that - a layer between two things (human and a program, two different programs etc.) that enables passage of information between them, (usually?) with some transformations along the way to better suit the receiver.
@@dan-us6nk Just finished my first course which was precalculus with an A!! I had a very difficult time in HS basically stopped all education by 10th grade ended up getting my GED. On a whim and by the suggestion of a stranger I enrolled into a welding program at a local community college I got my associates where I took up to trigonometry. In my time at community college realized that I wanted to purse academics and mathematics by which i'm enthralled. I hate being able to conceptualize these higher level math concepts but not being able to solve them out completely on my own. I am not a traditional student and i'm not particularly gifted but I have passion and love for math. My goal with the degree is just to improve my quality of life, help me get more respect in the professional world, and for other various hobby ideas I have not to mention programming ideas. I think starting at later stage will benefit me because I feel as if it will keep math at a forefront for my whole life which is yet another thing I want from this degree to be able to help others learn concepts and topics and keep my mind sharp as I grow older.
For the many people who were wondering about the mean curvature flow in aluminum - no, it has nothing to do with viscosity. Annealing is a process in which metal which has undergone stress and permanent tranformation (like bending, rolling or forging) "heals up" again. Almost everything around us is composed of crystalline grains (not glass and most plastics, thats why you cann look through them). These grains are small volumes of atoms arranged in a lowest energy regular pattern. As they start forming simultaneously at different points there's a bunch of them laying side by side in different orientations. Their interfaces are whats called grain boundaries. When transformed, some of these crack up into smaller volumes, roll, yaw and pitch, and generally the amount of different orientations and dislocations increases. When we heat up the metal, it wants to minimise this increased surface of grain boundaries again. But as it doesn't follow from an outer pressure (like air), but from the minimisation of all the atoms laying perfectly ordered side by side the time dependence and law is different
I have to say, I was expecting some kind of calculus of variations approach, with a Lagrangian or two appearing maybe. However, looking at the forces involved directly is almost certainly more enlightening from a physical POV. I suspect that analytical solutions for most of these problems are out of the question, though.
There are already at least one video using the calculus of variations approach on Dr Trefor Bazett's channel, which is why I'm not doing this. For simple problems, e.g. the one in the beginning with two circles, it is simple enough (with the cylindrical symmetry) that there is an analytical solution, but for most of the others, you definitely need numerical approaches.
The case for two circles (with centers forming perpendicular bisectors) is fairly straightforward to solve with the Euler-Lagrange Equation with the answer being aCosh(x/a-c). Eulur-Lagrange (or Beltrami's identiy) yield the diffeq (y')^2 - yy'' + 1 = 0
One very satisfying thing about this problem is using the Euler-Lagrange equation to find the minimum surface area. It almost feels like magic setting up and solving the differential equations, which in this case gives us cosh(x). Physics is filled with similar examples such as minimizing the energy of a system or using the Lagrangian formulation!
If you're interested in a real deep dive on this topic, I recommend Jacob Israelachvili's book, "Intermolecular Forces". It's a dense read, but it will give you a very strong background in how molecules interact, how surfaces interact, and how a surface may configure itself to minimize its total surface energy, which then drives deformation to the lowest surface area.
The main building of the Faculty of Applied Sciences and Engineering of Ghent University (Belgium) is in a street named after this Plateau guy. I never realized he was kind of an engineer himself!
Y'know, I was always confused by why shrink wrap made this shape around empty edges (assuming the wrap going straight from edge to edge would be minimal), and assumed it was something to do with how it was heat treated, and the air cooled / pressure dropped in the item which caused the walls of the wrap to get sunken. But this actually makes a lot of sense, as to why that shape would actually be minimal.
That's my suspicion too, but I don't know enough fluid mechanics to say that. There is a link in the description on the original paper that suggests the annealing aluminium follows mean curvature flow, if you want to know more.
A simple intuition about why nature forms these surfaces of minimal curvature (and thus minimal surface area as explained in the video): The shape of a soap bubble is determined by surface tension, which is analogous to a "regular," 1d tension applied in all directions to a point on a 2d manifold. Imagine taking a piece of rope and throwing it into the air so that it lands with a random curvature. If you grab both end of the rope and pull them away from each other, the rope will lose all of its curvature and straighten out. In other words, the tension you apply to the rope causes it to assume a minimal curvature. This is happening on the surface of a soap bubble. Thanks to cohesive molecular forces, every point on the bubble's surface is experiencing a tension in every direction - which eventually causes the curvature to be minimized in every direction.
I was surprised when you showed the concave surface being a minimal surface being curved surfaces have more surface area then a smooth area due to being curved.
This is already Riemannian geometry in some sense. Probably said this before, but even if I am passionate about general relativity or differential geometry, I need to find a topic unique enough to make a video on it; or a unique enough angle to approach that topic. Otherwise, it simply isn't creatively fulfilling to make a video almost regurgitating what others have said on RUclips already. But I'll never say never, maybe some day I can find a unique enough spin on the topics surrounding GR.
This is a great video and I really enjoyed learning about this problem from the perspective of forces and vectors! I'm curious to know if there's also a way to approach this from the energy POV? For instance, I know that systems in nature always "want" to be in the lowest energy configuration. I always then assumed that the configuration of soap bubbles that minimized surface area also were the ones that minimized energy, which explains why they are the ones nature prefers. Is this reasoning valid?
Would you approach a minimal surface if you iteratively move all the points to the average of their neighbours while having Dirichlet boundary conditions at the borders?
I know you probably understand what goes on between steps, but by the tone of your voice you also assume we do too. College math was a few years ago for me, and I would like to see the inner details of each step rather than just assuming equivalency. No intention to bash your work, I just find myself confused when something like this is not explained in the excruciating level of detail.
I loved the video but I am a bit confused about the representation at 9:45, should not deltan be in the same direction as b? Should it not be parallel to the force vector? Is there a reason why it is written like this?
In the illustration, the surface was actually a sphere, so b and delta n in this case actually is in the same direction, but that's not true in general, so I just want to make sure that it doesn't come across as always pointing in the same direction.
This will make a great Python__code{ }{ } .format(gama) + str(' ') problem.😮🎉 great video.. got lost a bit in following the math and materials science.
Good video, but could you please turn of forced subtitles. The viewer preference should be taken not a Video just turning them on it you click the video. People who want subtitles would've them turned on before clicking the video. And people who have them turned off probably don't need and want them.
If you don't mind me asking: I don't recognize your accent. Were you born in the UK? If so, which part? Nothing about you: I'm just into accents, but know little about those outside my country.
Originally I just wanted to explain how physics gets the soap film to attain a minimal surface configuration, but it turns out the process in which the soap film evolves is an active area of research that I accidentally stumbled onto while making this video.
BTW: Once we got to the integral for the total force, I could have used Stokes' theorem, but this way is a lot more algebraic manipulation, while the only algebraic manipulation in this argument would be turning cross product into dot product. The Stokes' theorem method also doesn't intuitively explain why the divergence suddenly appears.
How do you not get bored and fed up and sick of math and frustrated and how can you spend 2p minutes on one topic? I am depressed enough as it is and find it hard ebough to concentrate..it should be fun and easy and enjoyable
I appreciate the intuition for divergence coming out. It felt so natural once I saw the flux and the area. Out of curiosity, may I ask what the manipulation for Stokes' theorem would be?
See the second link of the "/Physical/ side of things" in the description.
@@mathemaniac Oh wow that's actually pretty cool thanks! I now have a greater appreciation for your intuitive derivation by converting it to a dot product.
you could have just used a tube and two plates 😂
Minimal surfaces are such an interesting topic. I never really delved into the topic, but I remember playing with soap films as a kid. It's super interesting to see how 3D graphics play a role in all of this. Greatly looking forward to the next video!
I love your videos ‼️
Same!
These vids derivation vids are the most satisfying hope the follow up will meet out expectations
As someone who studies stuff like this in my PhD-studies (not exactly minimal surfaces, but Willmore surfaces, which are closely related; as a side note, you should totally do a video on the Willmore energy, like conformal invariance, the Li-Yau inequality, minimization of the Willmore energy for fixed genus, there's so much to discover and even more which is still unknown!), this is such a great video! I actually didn't know that soap bubbles evolve by hyperbolic mean curvature flow rather than usual mean curvature flow.
@yak-i4c I've worked on this topic, which has been debated for more than 40 years now. We actually don't even agree on the exact nature of the forces and scales involved, and which framework should be used (hydrodynamics, non-equilibrium thermodynamics, statistical physics).
Cam you please tell me how the fuck can anyone get a ph d in anything..it seems wayntoo boring and frustrating and BORING sitting in front of s computer all day..how des anyone do it and why..hope you can respond
I'm a programmer, and around 2:30 was a lightbulb moment for me - "_that's_ why interfaces are called that!"
why? what did you realize?
@@RunningRay9 well, it's that in a rather short duration of time "interface" was contrasted with "surface". For one, this made me realize that these two words are closely related.
But more importantly, the soap film was just a thin layer between two volumes of air. Well they were actually the same air in this example, but you could imagine a rubber layer between two different liquids in an aquarium or something, so that it would take forces from one side, possibly transform them, and relay that to the other side.
And an interface is basically exactly that - a layer between two things (human and a program, two different programs etc.) that enables passage of information between them, (usually?) with some transformations along the way to better suit the receiver.
A correction, Pressure is not a vector, although we use the words "pressure points inside or outside" quite freely.
I was going to comment about the pressure too
Lovely video incorporating the intuition and the physical aspect of the maths of surfaces and planes
this is amazing 26 year old welder here going to school for bachelors in math
Good luck! Hope you enjoy it, math is beautiful
@@dan-us6nk Just finished my first course which was precalculus with an A!! I had a very difficult time in HS basically stopped all education by 10th grade ended up getting my GED. On a whim and by the suggestion of a stranger I enrolled into a welding program at a local community college I got my associates where I took up to trigonometry. In my time at community college realized that I wanted to purse academics and mathematics by which i'm enthralled. I hate being able to conceptualize these higher level math concepts but not being able to solve them out completely on my own. I am not a traditional student and i'm not particularly gifted but I have passion and love for math. My goal with the degree is just to improve my quality of life, help me get more respect in the professional world, and for other various hobby ideas I have not to mention programming ideas. I think starting at later stage will benefit me because I feel as if it will keep math at a forefront for my whole life which is yet another thing I want from this degree to be able to help others learn concepts and topics and keep my mind sharp as I grow older.
1:57, gentlemen, I recognize this surface!
Men of culture, unite.
Really? What is that dark blue surface called?
boobs
𝐢𝐧𝐝𝐞𝐞𝐝.
Everything reminds me of her
For the many people who were wondering about the mean curvature flow in aluminum - no, it has nothing to do with viscosity. Annealing is a process in which metal which has undergone stress and permanent tranformation (like bending, rolling or forging) "heals up" again. Almost everything around us is composed of crystalline grains (not glass and most plastics, thats why you cann look through them). These grains are small volumes of atoms arranged in a lowest energy regular pattern. As they start forming simultaneously at different points there's a bunch of them laying side by side in different orientations. Their interfaces are whats called grain boundaries. When transformed, some of these crack up into smaller volumes, roll, yaw and pitch, and generally the amount of different orientations and dislocations increases. When we heat up the metal, it wants to minimise this increased surface of grain boundaries again. But as it doesn't follow from an outer pressure (like air), but from the minimisation of all the atoms laying perfectly ordered side by side the time dependence and law is different
Very cool stuff! I've encountered minimal surfaces while studying droplets in aerosol physics
I have to say, I was expecting some kind of calculus of variations approach, with a Lagrangian or two appearing maybe. However, looking at the forces involved directly is almost certainly more enlightening from a physical POV. I suspect that analytical solutions for most of these problems are out of the question, though.
There are already at least one video using the calculus of variations approach on Dr Trefor Bazett's channel, which is why I'm not doing this. For simple problems, e.g. the one in the beginning with two circles, it is simple enough (with the cylindrical symmetry) that there is an analytical solution, but for most of the others, you definitely need numerical approaches.
@@mathemaniacThanks for sharing. Hope you can respond to my other co.ment when you can.
The case for two circles (with centers forming perpendicular bisectors) is fairly straightforward to solve with the Euler-Lagrange Equation with the answer being aCosh(x/a-c). Eulur-Lagrange (or Beltrami's identiy) yield the diffeq (y')^2 - yy'' + 1 = 0
Great video! And I love that you put so much resources in the description
One very satisfying thing about this problem is using the Euler-Lagrange equation to find the minimum surface area. It almost feels like magic setting up and solving the differential equations, which in this case gives us cosh(x).
Physics is filled with similar examples such as minimizing the energy of a system or using the Lagrangian formulation!
Amazing video! A physics phd here. I love how the physical concepts make it easier to understand complicated geometrical problems.
I've had the pleasure of working with one of the authors of the paper! Thanks for the great video :)
If you're interested in a real deep dive on this topic, I recommend Jacob Israelachvili's book, "Intermolecular Forces". It's a dense read, but it will give you a very strong background in how molecules interact, how surfaces interact, and how a surface may configure itself to minimize its total surface energy, which then drives deformation to the lowest surface area.
The main building of the Faculty of Applied Sciences and Engineering of Ghent University (Belgium) is in a street named after this Plateau guy. I never realized he was kind of an engineer himself!
Y'know, I was always confused by why shrink wrap made this shape around empty edges (assuming the wrap going straight from edge to edge would be minimal), and assumed it was something to do with how it was heat treated, and the air cooled / pressure dropped in the item which caused the walls of the wrap to get sunken. But this actually makes a lot of sense, as to why that shape would actually be minimal.
Great approach - subscribed :)
Another great video, this guy doesn't miss.
more like: how we can mathematically calculate a physics phenomenon
that thing with first time derivative in aluminium look suspiciously like motion of highly viscous medium
That's my suspicion too, but I don't know enough fluid mechanics to say that. There is a link in the description on the original paper that suggests the annealing aluminium follows mean curvature flow, if you want to know more.
A simple intuition about why nature forms these surfaces of minimal curvature (and thus minimal surface area as explained in the video):
The shape of a soap bubble is determined by surface tension, which is analogous to a "regular," 1d tension applied in all directions to a point on a 2d manifold.
Imagine taking a piece of rope and throwing it into the air so that it lands with a random curvature. If you grab both end of the rope and pull them away from each other, the rope will lose all of its curvature and straighten out. In other words, the tension you apply to the rope causes it to assume a minimal curvature.
This is happening on the surface of a soap bubble. Thanks to cohesive molecular forces, every point on the bubble's surface is experiencing a tension in every direction - which eventually causes the curvature to be minimized in every direction.
Great content!
I was surprised when you showed the concave surface being a minimal surface being curved surfaces have more surface area then a smooth area due to being curved.
Good video!
Thank You
Wohoo a Joseph Plateau (1801 - 1883) video!
man im never gonna figure out how to self derive these minimization formulas lmao
Delighted man. It is wonderful topic, gateway to General relativity?, Riemann geometry?
This is already Riemannian geometry in some sense. Probably said this before, but even if I am passionate about general relativity or differential geometry, I need to find a topic unique enough to make a video on it; or a unique enough angle to approach that topic. Otherwise, it simply isn't creatively fulfilling to make a video almost regurgitating what others have said on RUclips already. But I'll never say never, maybe some day I can find a unique enough spin on the topics surrounding GR.
True to that!!
This is a great video and I really enjoyed learning about this problem from the perspective of forces and vectors! I'm curious to know if there's also a way to approach this from the energy POV? For instance, I know that systems in nature always "want" to be in the lowest energy configuration. I always then assumed that the configuration of soap bubbles that minimized surface area also were the ones that minimized energy, which explains why they are the ones nature prefers. Is this reasoning valid?
Oooh nice topic
Would you approach a minimal surface if you iteratively move all the points to the average of their neighbours while having Dirichlet boundary conditions at the borders?
I guess that's why when you use loft in Softwares such as solidworks, they go so thin towards the middle
Guessing catenary's it's always hyperboloids
I know you probably understand what goes on between steps, but by the tone of your voice you also assume we do too.
College math was a few years ago for me, and I would like to see the inner details of each step rather than just assuming equivalency.
No intention to bash your work, I just find myself confused when something like this is not explained in the excruciating level of detail.
What is the relevance of minimal surfaces to black hole physics?, in particular their entropy.
What's the physical reason for velocity (instead of acceleration) being proportional to the pressure here? Is this somehow related to drag forces?
I suspect it is due to viscosity, but I am not well-versed enough in fluid mechanics to explain it further.
NO PLEASE.
I WANNA LIVE
I loved the video but I am a bit confused about the representation at 9:45, should not deltan be in the same direction as b? Should it not be parallel to the force vector? Is there a reason why it is written like this?
In the illustration, the surface was actually a sphere, so b and delta n in this case actually is in the same direction, but that's not true in general, so I just want to make sure that it doesn't come across as always pointing in the same direction.
Can you do a course on Markov / Semi Markov / Hidden Markov / Semi Hidden Markov models please.
This will make a great Python__code{ }{ } .format(gama) + str(' ') problem.😮🎉 great video.. got lost a bit in following the math and materials science.
The universe is the fastest supercomputer
3:40 isn't it more correct to write this as 2 gamma Int(n × dl)
I don't get it, can someone tl;dr how a curved surface has less surface area than a cylinder?
What was the 3D graphics problem that this physics solved?
Can you share video code ?
You can download all the files in the link in the description (with the password). I don't use coding to make videos.
@@mathemaniacthanks You 🫸🏼🫷🏼
Manim Code ?
This is Blender devs
Good video, but could you please turn of forced subtitles.
The viewer preference should be taken not a Video just turning them on it you click the video. People who want subtitles would've them turned on before clicking the video. And people who have them turned off probably don't need and want them.
Can I marry you❤
If you don't mind me asking: I don't recognize your accent. Were you born in the UK? If so, which part?
Nothing about you: I'm just into accents, but know little about those outside my country.
Third~
First!
First comment ❤
bla bla bla bla *PURE MATH NOW, NO YUCKY PHYSICS*