How physics solves a math problem (and a 3D graphics problem)

Should’ve been titled “accidentally stumbling onto an area of active research way out of my depth”.

The Plateau's problem asks for the minimal surface (with minimum surface area) with a given boundary. Because nature seems to solve this problem easily, as demonstrated by soap films, we turn to physics for inspiration. This is simulated by (hyperbolic) mean curvature flow, which turns out to be a hot topic for research in both maths and physics, and somehow also used in 3D graphics.

Files for download:

**Further reading**:

/Mathematical/ side of things:

1) General overview of Plateau's problem:

3) The relevant Wiki articles:

4) An undergrad/masters project on mean curvature flow:

/Physical/ side of things:

3) Implementation on how to simulate soap films:

4) Research into how instability arises:

5) The YouTube video on @cambridgeuniversity for soap film collapse:

6) Relevant Wiki articles (on Young-Laplace equation):

7) An undergrad/masters thesis on physics of soap films:

/Computer graphics/ side of things:

1) How to smooth 3D contours with mean curvature flow:

2) How to improve on the mean curvature flow procedure to solve Plateau's problem:

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[mathemaniac]

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.

[copywright]

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!

[pengin]

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.

[pmmeurcatpics]

I'm a programmer, and around 2:30 was a lightbulb moment for me – "_that's_ why interfaces are called that!"

[alienbroccoli]

A correction, Pressure is not a vector, although we use the words "pressure points inside or outside" quite freely.

[andrewdemos]

this is amazing 26 year old welder here going to school for bachelors in math

[vslaykovsky]

1:57, gentlemen, I recognize this surface!

[scollyer.tuition]

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.

[donnerflieger]

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

[AfrozYamir]

Lovely video incorporating the intuition and the physical aspect of the maths of surfaces and planes

[ihmejakki]

Very cool stuff! I've encountered minimal surfaces while studying droplets in aerosol physics

[spitsmuis]

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!

[tciddados]

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.

[vi.shyyyy]

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!

[me]

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.

[sofialiguori]

Great video! And I love that you put so much resources in the description

[paraseth]

Amazing video! A physics phd here. I love how the physical concepts make it easier to understand complicated geometrical problems.

[dzuchun]

that thing with first time derivative in aluminium look suspiciously like motion of highly viscous medium

[opensocietyenjoyer]

more like: how we can mathematically calculate a physics phenomenon

[SantosAdducci]

Another great video, this guy doesn't miss.