The Double Bubble Theorem
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- čas přidán 28. 06. 2024
- How does soap make bubbles? Why are bubbles round? What shape do two bubbles make when they connect? Although these might seem like questions with obvious answers, the science and math of soap bubbles has all kinds of unsolved problems. In this video, we'll learn about surface tension and the chemistry and physics of soap, we'll learn a fun proof that bubbles should be round using a technique called Steiner Symmetrization, and we'll learn Plateau's Laws for determining the shape of a bubble cluster.
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0:00 Introduction
1:28 Surface Tension
3:47 Surfactants and Soap
4:53 Why are bubbles round?
7:40 Plateau's Laws
10:38 Conclusion
Thank you to Caleb Birtwistle for captioning!
Best overview: Frank Morgan, Geometric Measure Theory: A beginner's guide.
Bubble Chain World Record: www.guinnessworldrecords.com/..., • Longest bubble chain -...
Surface Tension: michaelberryphysics.files.wor...
More Surface Tension: arxiv.org/abs/1211.3854
Steiner Symmetrization: math.williams.edu/symmetrizat...
More Steiner Symmetrization: www.math.utah.edu/~treiberg//S...
Plateau's Laws proof: www.jstor.org/stable/1970949
2D double bubble proof: projecteuclid.org/journals/pa...
3D double bubble proof: math.berkeley.edu/~hutching/p...
Honeycomb proof: arxiv.org/abs/math/9906042
Triple bubble proof: arxiv.org/abs/2205.09102
More Geometric Measure Theory: maths.anu.edu.au/files/CMAPro...
Bees don't actually make a honeycomb shape; they make them circular and heat transforms them into hexagons.
who cares? nerd
holy shit i never knew that. thanks for sharing
They settle into hexagons, because of course they're the bestagons!
@@faytaliti why do you hate nonagons?
I was under the impression that they got that shape because the bees will push the wax outwards as they work, and so over time, as many bees push the perimeters of the wax outwards, they behave a lot like bubbles being pushed by air pressure.
Kurtsgezart? Never heard of them, this is the superior "cool things + birds" channel.
@@channelknightfadran7901 they are just billionaire propaganda, sorry bro
Kurtzegat is really depressing, hopeless and inhuman.
@@kashu7691 The response to the 'Billionaire Propaganda' allegations was quite interesting, I'm not sure if you've seen it. Kurzgesagt opened up on how they are funded and money from the Gates fundation came down to only 4% I think. Not sure if that changes your mind or anything but I think it's worth a read. Being skeptical is of course not a bad thing no matter who is talking.
@@nive7299 thank you for the info =)
@@EliStettner That's like... the exact opposite of their content though?
All of their space technology content is the definition of 'Here's cool shit we could perhaps do one day' or 'heres a cool way the world will definitely never end'.
All of their videos on real world problems tend to take cautiously positive stances, always ending with the message that although things aren't perfect we can definitely make a difference and that we can come out the other side.
The recent immune system videos are just 'Hey look at how cool our human bodies are'.
Hell, go watch the The Human Era
At first I thought "isn't it obvious that since the circle's perimeter could not be reduced, it must have the smallest perimeter?". But then the example using the reasoning that "1 is the biggest number since every other number could be made bigger by squaring" clicked and instantly made me understand your point. Amazing!
Water absolutely can be blown into bubbles. The issue is that in a gravitational field there is a preferred direction for the surface tension to force the water, so it rapidly flows downward and the film squeezes a hole in itself. However, in an inertial reference from (i.e. in space) pure water bubbles are extremely stable.
Lol "however, because in a gravitational field the preferred direction of......"
But I get what you mean tho, and that's really cool actually
168th like
Thanks. What should I look for with/involving water bubbles?
Gotta be careful with the definition of "bubble". There are *solid* bubbles (air bubble rising in a glass of water; also a drop of water in air (while falling). There are *hollow* bubbles (a soap bubble drifting in air; also a rare "anti-bubble", a hollow shell of air that sits underwater with water inside it). There are solid bubbles of water that "float" on top of water (tricky to produce; they don't last long; an electrostatic field helps). There are solid bubbles of air that sit just under the surface in your glass of water (they don't last long). It's hard or impossible to "blow" *a hollow bubble of water in air;* it breaks before it forms. You can't even make a film by lifting the bubble wand out. Things might change if the wand is very small (microscopic films and microscopic bubbles?) or in *zero-g.* It all depends what is really happening.
In *zero-g,* released water sticks together as a blob. The blob tends toward the shape of a solid sphere (a *ball* ). At first it will be oscillating, but in time the waves damp out (unless you blow on it). (A spinning blob will tend toward an ellipsoid.) With a straw, you can blow a solid air bubble inside the water bubble, yielding a thick-walled *hollow bubble of water in air.* What happens next? (Assume the air bubble is not centered.) Without forces, the air bubble will stay where it is. Do forces like surface tension exert a directional force on the air bubble? If forces push the air bubble to the center, you got it right. If forces push the air bubble to the edge and then the air bubble exits the water bubble, you got it wrong.
In *zero-g,* an anti-bubble *(a hollow bubble of air in water)* might persist longer, because there is no buoyancy to drive it to the surface.
Physics for the Birds is seriously growing to be one of my favorite science channels
Everything just makes sense, from prerequisite knowledge to the more complex things it all just makes sense and is simple
AND listing all the sources? Seriously, such a good channel
You get a sub. Been watching for a while but what’s pushed me to push the button is that you are tackling real world phenomena in a way that both me, an engineer, and my 6 year old can watch and both be totally engrossed. It’s like 3Blue1Brown but more relevant to non-maths nerds and more approachable for little ones.
Thank you!
I feel like getting compared to 3b1b is every math ed channels dream come true. This channel is truly marvelous.
EVERY TIME YOU MAKE A VIDEO IT ABSOLUTELY MAKES MY ENTIRE DAY!! your style is so friendly that i almost don't notice you're combining advanced maths from MULTIPLE disciplines (still can't stop thinking about the jazz video). Thank you for what you make!!
I didn’t learn much more than the basics of surface tension until I took thermodynamics in grad school, and then I understood why they waited so long to teach it haha. The coolest thing I learned in that class was that the vapor pressure inside of a bubble is proportional to its curvature and the difference in pressure that caused provided some the driving force behind smaller bubbles combining to make larger bubbles in a foam.
This feels like a Numberphile / Matt Parker video and I’m all for it!
Yet another great video - hope to watch you on Nebula some day🤞
One thing that always interested me is when you make a double (or higher) bubble the internal walls aren't actually flat (most of the time) but will curve convexly into the larger bubble. I'm not sure if it is a parabolic curve or if it is the curve of a larger theoretical sphere whose radius is based on setting the larger bubble's radius to 1 and and then dividing that by (1-the radius of the smaller bubble). Either way, I have boned light off them just right on to a wall to get some pretty sharp images of the light source. I kind of thought if gravity could be taken out of the equation the internal dividing walls of bubbles could make excellent optical surfaces (especially for how cheap and easy they are to make and even adjust on the fly.
Here's something about curvature. P is pressure, T is surface tension, R₂ and R₁ are the radii of the larger and smaller bubble respectively, R₃ is the radius of the wall or intermediate film between them.
"The pressure in a bubble is inversely proportional to its radius since P = 4T/R. The radius of the intermediate film is dictated by the difference in the pressures on either side of it. These pressures are 4T/R₁ and 4T/R₂ respectively. It immediately follows that P₃ = 4TR₃ = 4T/R₂ - 4T/R₁. So finally we have the simple equation 1/R₃ = 1/R₃ - 1/R₁." (From Gems of Geometry by John Barnes)
I always found it interesting that the face between 2 bubbles is more or less flat. Which makes sense if both bubbles have the same internal pressure
The internal pressure of the bubble scales inversely with radius (crude idea: smaller bubble = higher SA/vol ratio = more surface tension per volume = higher pressure; more exactly: P = 4*gamma/R) so a smaller bubble will bulge slightly into a connected larger bubble. But that bulge may be difficult to see because two bubbles of similar size will have a minimal pressure difference (minimal bulge) while a duo of widely different sizes may have more of a bulge but less of a connecting surface to see it (unless one of the bubbles is huge)
@@alberthung6191 Yes, the radius of the face between two bubbles is dictated by the difference in the pressures on either side of it. Let P be pressure, T surface tension, R1 ,R2, and R3 the radii of the smaller bubble, the bigger bubble, and connecting or intermediate surface respectively. The pressures are 4T/R1 and 4T/R2. So P3 = 4T/R3 = 4T/R2 - 4T/R1. So finally 1/R3 = 1/R2 - 1/R1. Must be tricky to verify in some cases though. Photographs?
Never knew how similar bubbles were to the phospholipid bilayer in biology
It’s insane how much I love this channel!
It's been amazing watching your channel grow from just 20k subs not too long ago to 80k now! I think that you'll hit 100k in no time. I think that no matter what, given your content's extraordinarily high standard of quality and interesting and highly researched topics, you're severely underrated.
Thank you for including your sources in the description!!!
I just found your channel and I love it, it's such a great format that makes it easy to learn!
This channel is consistently wonderful, thanks for the great content.
This channel makes me like physics and birds.
This channel is a blessing thank you so much for existing
The perfect balance of entertaining and informative, bravo
I am fascinated and awed, thank you for sharing this!
Do we know how close to a perfect sphere a real soap bubble is? Has anyone actually done measurements on a real bubble to see how close to the math it is?
In theory it should be as close as possible, but there is a finite smallest edge size due to the minimal distance between molecules in the Lennard Jones potential. But they are arguably the closest things to a sphere we have on Earth
@@rafaelalmada723 “in theory” that’s why we should actually measure it, maybe learn some new stuff by how much it is off from a real sphere
@@crsmith6226 there may be some way of measuring it through Mie scattering, but I am not an experimentalist so it's out of my expertise 😔😔
With no wind or external forces, bubbles should be as close to perfect spheres as protons
I’m sure in real life, the bubbles are slightly bottom-heavy due to gravity.
You could confirm this experimentally through high-optic photography from multiple angles
Well you can “see” the thickness of a bubble from the diffracted colors, so it must be aspherical by at least a couple 100nm
You're making an excellent series of videos! 10/10 awesome job!
Excelente video, era justo lo que estaba buscando!! felicitaciones!
Every video just keeps getting better! Truly reminds me of the early days of CZcams when you would discover wonderful channels like Veratasium, Minute Physics, Vsauce etc. Please make more videos!
another fantastic video! you just don't miss
another banger science video. Great job
A great video from you as always!
Fantastic, as always.
Another great video from you! These are the types of things that make people interested in pure maths! Sincerely, a Topology and measure theory student
really enjoyed this video, thankyou
Never expected the jump from soap bubbles to measure theory. Thanks for enlightening me :)
Really loved this video it made me think a lot!
Another great one! Love it
Wow great video, thank you!
Frank Morgan was my college real analysis professor. He’s so unbelievably smart and kind-I never expected him to get a shoutout in a math CZcams video!
This was an awesome explainer!! I just finished a Master's thesis on a related problem :)
Amazing!! math, programming... etc. Whatever you get excited about. So share with us! You are awesome. Keep it up. Can't wait to see 1000s of videos from you
Thank you for introducing me to a new mathematical rabbit-hole that I have not heard of before. Differential geometry + measure theory? Sign me up !
Nice video, love the content!
Interesting, thank you. Greetings from Popayan, Colombia.
this channel is gonna take off really fast
Love this, physics made easy
dude i was searching for why bubble are spherical and i got a much more interesting subject than the one i was looking for thanks.
I only ever learned about surface tension superficially as an undergrad and quickly became very confused about how it worked when actually having to deal with it in grad school research. Your explanation made it so much clearer.
Question: How does viscosity affect bubbles? I notice incidentally generated bubbles tend to be smaller and last longer in viscous solutions. I suspect it's just that the viscosity allows non-equilibrium states to persist for longer?
Great video! Where do you get your ideas for some of these videos? I don't know how you find such niche, but interesting topics
I learned about surface tension in chemistry actually, when my teacher talked about minescus and glassware. We then discusses surface tension with intermolecular forces for individual molecules with individual molecules
Just last night I re-rewatched most of the videos, who'd ave thought that that today would give me another
I love this.
Blub Blub is the sound they make
unproven conjecture
@@vespuccini feels right tho
While I'm not a bird, I always enjoy learning these neat little parts of physics.
The point at the end of the video is the center of a lot of philosophical debate in computational complexity theory (e.g. the field that asks questions like P vs. NP). We have the conception of problems which are inherently *hard* to compute (say NP-hard problems), and we think of different computation models being roughly the same power (Church-Turing Thesis). However, we see a lot of examples of those problems in nature being computed all the time. Oftentimes, the hard problems we see being solved in nature are examples of "easy instances" of broader hard problems. This helps us dig down deeper about what the hard part of a problem really is. Most of this is still pretty up in the air and I think our organization of complexity theory will change a lot in the coming decades.
A great theologian named Sir mixalot spoke a lot about the double bubble back in 1992 A.D.
love the content
Gem of a youtube channel! crimminaly under subed
The CONCEPT is simple, the MATH is hard. That's why they're difficult to prove, but very intuitive to understand at a conceptual, high level.
I'm a molecular bio major (and chem minor but whatever), and I genuinely am always awe-struck by the physics behind biological facts. As someone specifically interested in molecular bio, I wish we went more into detail on the physics between molecules in a biological system. My other bio major friend and I were discussing this recently: this is a gross minimization but, chemistry may be the study of molecular interactions and components and biology may be the study of life and lifeforms and how they work/interract systematically, but physics is the mathematics between all living and non living entities in the universe. That will never not fascinate me. My university doesn't offer biophysics classes, but I will find a way to take one for sure.
ur so good human
Watching this video made me realise how much high school physics I have forgotten over the years. I could tell I know the bits of surface tension due to some exposure before but the dots were too sparse in my mind 😂
It would be interesting to hear you talk about Ken Brakke's "Surface Evolver" - a program that solves complicated minimization of surface problems. And how about the packing of spheres? Another problem with a long history and recent progress.
Nice vid
the analogy at 6:50 makes my head turn inside out
I wonder how much of the difficulty in some areas of math comes down to lacking notation or representation of functions, etc. Often, when something is discontinuous, it feels abnormal to treat it mathematically, as if "math didn't like it", and yet nature has no problem with those. Like trying to model someone kicking a ball before learning about Dirac's delta.
great channel! subscribed.
and remember, nature does NOT owe you an explanation. it exists.
it's up to us to understand it
This made me feel I’m chilling with a warm friend.
I think it helps to notice that surface tension can also be understood as free-surface energy. Then minimizing total energy is equal to minimizing the surface area. Treating surface tension as a force does not intuitively lead to the minimization of the surface area (unless you invoke some further arguments :) )
0:04 Wait there's such thing as a bubble stacking competition?! 😱
I'm curious how do you find all the research papers to make the timeline at about 9:00 how are you sure that nothing has been missed?
*me, secretly not a bird:* _"I'm in"_
i know this is irrelevant, but i absolutely love your voice, especially when you say "double bubble"
3:14 hmm but isnt the density of water always constant only the pressure changes?
This channel is the cure for my summer brain rot
Here's another cool thing you can do with soap bubbles:
The minimum Steiner tree of some points is the graph that connects the points using the minimum possible distance.
Multiple soap bubbles together can be used to create the Steiner tree of a set of points, since it shares similar properties to that of joined soap bubbles (like having only 120 degree angles).
In fact, you can try this by taking two glass plates arranged one on top of the other, connecting them at some points by sticking some pegs between the plates and finally dunking the whole thing in some soapy water. When you take it out you will see that between the plates bubbles will have formed attached to the pegs and in the shape of the Steiner tree of those pegs.
Here's the catch: the minumum Steiner tree problem is NP-hard and we can simulate classical physics in polynomial time. This proves that P = NP!
Well, not really. While (as far as I know) it's not been proved, the final arrangement is almost certainly a local optima. And even if it isn't, it will take a long time for the bubbles to settle when many pegs are used.
There are other ways of doing this "physics prank" but this one is probably the most amusing to me.
incredible video as always - this has become one of my fav channels on youtube. Are you a berkeley physics student? just noticed one of the birbs in your header has a cal hat. if so, go bears
Yep, I graduated from Cal's physics department. Go Bears!
lesgoo birdphysics video
Interesting 👀
i actually have an irrational phobia of clusters of bubbles like these ever since i was a kid and they still throw chills down my spine when i see them
Triptophobia?
@@douggaudiosi14 no, only specifically for bubble clusters, but other so called 'tryptophobia' images or settings still don't scare me. If i'm in the shower and see such bubble clusters i wil literally scream my lungs out and i have to carefully wash myself to avoid making these soap bubbles. Especially if they're big, uneven and there are a lot of them. Like for me, going into a bubble bath is like the purest form of torture and i'll probably pass out from fear drown and die.
Do you have any idea what started this phobia?
What do you use to make your illustrations/animations?
Bubbles have so much in common with cell's plasmatic membrane, their estructure is the inverse (the tails to the outside and heads to the inside and in the plasmatic membrane, heads to the outside and the hydrophobic tails to the inside)
At 8:20 WHY donmutliple esges meet in fours because in 3d soace yiu csnt fit mlre than four or is something else going on?
Fantastic Work, but how does this fit into the “4th Phase of Water” and the science behind it?
I don't understand the comparison at @3:26. The vast majority of attraction in water is from hydrogen bonding. Both this and the attractive dispersion/LJ potential are interactions only with nearest-neighbor molecules, with hydrogen bonding being more directional. What is the attractive force at 3:26?
How can we apply this to the multiverse bubble theory?
Good asmr artist
Fucking fascinating, keep making these man
Hexagon is the bestagon. Mr. Grey sends his regards.
"I study bubbles for a living" strikes me as someone that had a question at age 1 and has simply refused to give up on answering it. Talk about persistence!🤣
I know it’s gonna be a good day when I get to be one of the birds that learns physics
thank you for your's gorgeous videos, huge hugs from russia
How much matj background do you have im curious?
Soft&Wet! Go beyond!
the gravity surface tension analogy doesn't work for an upside down item, like a spout with a mesh on top of it, or a straw with your thumb on it. it's sometimes upside down.
oh, the guy that pronounces "sh" weird is back.
Good vid!
It's interesting to me that the shape of a bubble is not differentiable. Not sure I understood correctly what you were saying there
1:05i see so the stacking guy was before that cuz everyone knows that physics don't apply until someone discoveres them
Try beatboxing bro, it fixes the tsch sound - speaking from experience. NICE VIDEO.
Now that I think about it, the 120° rule even explains why bubbles that form at the edge of the water surface in a water bottle look like "----o----"
How would the rules change, if the ruble is inside a liquid?
I am a stable configuration of bubbles
Bubbles are just good analogs for electron fields in atoms and molecular configurations