Bubble Trouble

Bubble Trouble

It’s not magic, it’s math!

Originally shared by Richard Green

Double Bubble: Toil and Trouble

For about 100 years, an open problem in mathematics was the Double Bubble Conjecture. This asked the following question: do two bubbles that meet in the usual way enclose and separate two equal volumes of air in such a way as to use the least possible amount of surface area? 

This easily stated question had been assumed to be true without proof as early as 1896. However, it resisted all efforts to prove it until 1995, when Hass, Hutchings and Schlafly showed conclusively, with the help of a computer, that the answer to the question is yes. Assuming that the bubbles are of equal volume, the authors showed that the separating boundary between the bubbles is a flat disc, meeting each bubble at an angle of 120 degrees, and that each of the bubbles stuck to this central disc is a piece of a perfect sphere. Furthermore, the authors showed that this is the most efficient way to enclose two equal volumes by using the least possible amount of surface area.

In 2000, Hutchings, Morgan, Ritoré and Ros announced that the answer to the question is still “yes”, even when the two bubbles involved are of different sizes. In this case, the separating boundary between the bubbles will not be flat, and the entire double bubble configuration will consist of three spherical caps, in such a way that the separating membrane meets each bubble at a 120 degree angle. As in the previously established case where the bubble volumes were equal, this is the most efficient way to enclose two volumes of air so as to minimize the surface area. The proof of the more general result was much more work than the equal volumes case, but the paper still only occupies 30 pages in the Annals of Mathematics (2002), and significantly, the argument does not rely on the aid of a computer.

Even more surprisingly, the double bubble conjecture was soon extended to four-dimensional bubbles by a team of four undergraduates (Reichardt, Heilmann, Lai and Spielman) under the direction of Frank Morgan, who was one of the authors of the second paper mentioned above. The conjecture is also known to be true in certain cases in dimensions even higher than four.

You can hear a four-part podcast series about the double bubble conjecture here (http://www.ams.org/samplings/mathmoments/mm103-bubbles-podcast). Frank Morgan is one of the people featured, and he gives very nice talks. Another of the speakers is James Sethian, who, together with Robert Saye, created this picture.

(Seen via the American Mathematical Society.)

#mathematics #scienceeveryday

Join the Conversation

39 Comments


  1. There only needs to be a balance between the internal pressure, the external pressure, and the cohesion of the bubble film, though, right? Thus, would it not be possible to create a bubble in a vacuum whose internal pressure is just high enough to cancel out film tension without requiring nonzero external pressure?

    Like


  2. The bubble has to have surface tension – that’s what the calculations model. Yin Huang is right – it’s the balance of the positive (outward) pressure from the gas with the negative (inward) pressure from any external gas plus the tension of the bubble that determines the shape. In a vacuum it would take much less gas inside to produce the same shape of bubble.


    Think of a balloon – you could blow one up in a vacuum; it would just inflate much faster than when you’re fighting ambient air pressure.

    Like


  3. I realize it would take less gas but there is no gas. I now think a bubble can’t be made in a vacuum because if you waived the soap bubble wand in a vacuum there’s nothing besides inertia pushing the film, unlike in air where the gas pushes on it.

    Like


  4. Ben Robinson you must provide some gas for the inside from somewhere outside the vacuum.  But it could be stable if you did the right amount.  No wands assumed here, I think.  Inflate implies an external source of gas.

    Like


  5. Anything that disrupts the surface tension of the bubble will make it pop. In a vacuum lets humor the idea that you can get one to form. Know let’s turn back on reality….the bubble would instantly burst. Why? Because in a vacuum, the vapor pressure of the molecules in the bubble will change…so much so that they will boil away. This would mean that no such bubbles can form in the vacuum of space

    Like


  6. Ben Robinson You didn’t specify that the inside of the bubble must be a vacuum.


    Christopher Rucinski Not necessarily. At least in the imperfect vacuum of space, you can create a temperature condition in which certain liquids will not boil away.

    Like


  7. Ben Robinson You realize that waving a wand around is hardly the only way to make bubbles, right? Under normal conditions, you can create a bubble by blowing air into a straw that has a soap film stretched across its (opposite) end. No waving needed.

    Like


  8. U need something inside the bubble to keep the surface from attracting its neighbour atoms..


    It might be done in vacuum


    Charge the surface and charge the center of the bubble space with the same polar.


    The charge might thanks to the repellency keep the bubble in shape.

    Like


  9. Ben Robinson 


    Alright, if you’re requiring the inside of the bubble to also be a vacuum, then it doesn’t work. As for the second part, your original question was “can a bubble exist in a vacuum?”, not “can a perfectly spherical bubble exist in a vacuum?”


    Christian Grenfeldt 


    You can’t charge a vacuum, unless you decide that ions “don’t count” towards a non-vacuum.

    Like


  10. Yin Huang If you read it again, the vacuum was an idea of a way to get the “perfect sphere” as in “and that each of the bubbles stuck to this central disc is a piece of a perfect sphere.


    It goes without saying that one needs a bubble to exist before you can ascertain its shape.

    Like


  11. Christopher Rucinski 


    I assume you meant to write “perfect sphere.”


    If that’s the case, then the “vacuum” bit is perfectly irrelevant. Beyond a certain level of precision, a structure that’s a “perfect sphere” is impossible in any real-world scenario.

    Like


  12. Christopher Rucinski Ben Robinson 


    Room temperature itself would introduce constant perturbations that would force the bubble to deviate from the “perfect sphere,” assuming you could even somehow start as a “perfect sphere.”


    Moreover, if you’re concerned about intra-bubble convection, the bubble’s own surface tension will, in zero gravity with a sufficiently controlled and “calm” external environment (e.g. a good enough vacuum with good enough heat control) and a sufficiently homogeneous surface material, dampen any deformation away from a sphere.


    After all, the process of blowing through a bubble wand is an aerodynamic mess, and even in Earth gravity and an outdoor environment, small bubbles created by the wand become rough approximations of a sphere very quickly.


    Edit: Ben Robinson 


    Well, the article said perfect sphere & I couldn’t understand how this could be possible


    Well, it is a mathematical proof, not a physical observation.

    Like


  13. So only proof in concept, the only way to make this happen would be to have 2 gas bubbles in a vessel filled with a quiescent inviscid liquid & having that vessel in zero G & in a vacuum.


    Not 100% sure but I think the bubbles would join without coalescing & be as spherical as possible.

    Like


  14. Ben Robinson 


    So only proof in concept.


    A bit like F = ma, I suppose.


    quiescent inviscid liquid


    Why a liquid in a vacuum? “Very still air” (or simply a “naked” vacuum) would work just as well. Just make two gas bubbles in zero-G in a low-density gas, with sufficient environmental isolation, and wait long enough for things to calm down.


    Again, if you really want “perfection,” it’s not possible under any real-world conditions.

    Like


  15. Ben Robinson Yin Huang Yes perfect is impossible. The only instance of “perfect” is in the shared version of this story from American Mathematical Society. That original version had no such mention of “perfect”. This was just a journalism mistake in writing another version of the story for their audience.


    One would think that a bowling ball is very, very spherical. In fact, possibly one of the closest things to being a perfect sphere; however, if you  were to expand the bowling ball to the size of Earth, then you will see that the Earth IS smoother – not necessarily more of a perfect sphere as the equator bulges out.


    There are also things with gravity that would not allow a perfect sphere to form. If you are on Earth trying to create that perfect bubble, then the top of the bubble has a slightly less gravity effect, then those on the bottom. So a smaller bubble would create less of this difference; however, this is why doing it in space would be more viable to create a “perfect” sphere. I still think having a bubble in an enclosed space, within outer space, with a very favorable environment inside…would be the best way to create a “perfect” sphere.


    But Yin Huang, you have not stated what this possible temperature condition would be to allow a bubble to remain stable in a vacuum – when it should be boiled away!

    Like


  16. M theory suggests that nothing is impossible, but back to the bubbles in vacuum, how do you prevent them from bursting, evaporating or coalescing? 


    Plus a low density gas would have particles hitting the bubble creating vibrations that misshape it.


    A 0 viscosity liquid such as a Bose Einstein Condensate would prevent the bubbles from coalescing because of the wave like particles.


    Stillness is essential to avoid generating heat & heat won’t be acquired because of the vessel being surrounded by the vacuum.

    Like


  17. Ben Robinson 


    back to the bubbles in vacuum, how do you prevent them from bursting, evaporating or coalescing?


    How do you prevent normal soap bubbles from doing the above?


    Plus a low density gas would have particles hitting the bubble creating vibrations that misshape it.


    Well, yes. Thus, the impossibility of a perfectly spherical, perfectly static equilibrium. This caveat doesn’t disappear if you surround the gas bubbles in a zero-viscosity liquid.


    heat won’t be acquired because of the vessel being surrounded by the vacuum.


    Heat can be transferred via radiation. Moreover, simply being above absolute zero at all ensures the presence of thermal kinetic vibrations.

    Like


  18. Heat doesn’t transfer into a vacuum, like a person floating naked in space wouldn’t feel cold, plus what radiation is hitting it? You can shield radiation, quarks & such can’t be stopped as far as I know but that’s another subject.


    I believe a 0 viscosity liquid would be evenly dence excluding the 2 bubbles.

    Like


  19. Christian Grenfeldt Yeah, that was definitely my first idea that I presented above. I have been trying to figure out how the stated…specific temperature condition… could make a difference. Temperature does not change the equilibrium vapor pressure point (the boiling point), so it cannot make the bubble stable. In outer space, a bubble will boil away causing the surface tension of the bubble to collapse. That simple!

    Like


  20. Christopher Rucinski 


    Temperature does not change the equilibrium vapor pressure point (the boiling point)


    No, but if you’d bothered to look at my PDF link, how good the vacuum is affects where the boiling point is. With the right material at a sufficiently low temperature, you can prevent boiling in artificial near-vacuums.


    You’re free to personally limit all bubble experiments to convenience-store liquid soap, but that’s not a limitation inherent to the bubble model.

    Like


  21. Yin Huang I did look at it. You made it seem like modifying the temperature would change the boiling point. Now your explanation is understandable. I have not done any calculations, but I have looked at the pressure that is present in outer space (in between solar systems…I assume the pressure will be close to that within the solar system, but probably slightly here)


    http://en.wikipedia.org/wiki/Orders_of_magnitude_(pressure)


    Actually, I just did the calcualtion. Based off of the Pa for the pressure in between the solar systems. I got a PSIA of 2.175566158e-18…or to be more compatible with the PDF you gave *0.000000000000000002175566158*  


    Just for reference, the lowest PSIA value listed in the PDF was *0.00005* or 5.0e-5. The next one was twice the PSIA and had an temp difference of 20 C.


    I tried to put it in a trendline in Excel, but none of the lines actually fit…with the exception of the Polynomial…and that I doubted with very low PSIA values.


    So the boiling point of that bubble substance would have to be very, very low. Wish I could give you a very accurate number, as it would show just how low of Kelvins we are talking about here.


    Can Liquid Helium produce bubble inside of outer space? Or something similar? 

    Like


  22. See probleem – on võrdväärne LEKAALI konstrueerimisega!


    Mind hämmastas juba “joonestamises”: kuratlikult sujuvad liidesed – erinevate ringjoonte ühitamisel. MIKS kuid, ma ei osanud “teada”.

    Like

Leave a comment

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

%d bloggers like this: