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# 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

1. “Perfect sphere” maybe in 0 gravity in a vacuum but can a bubble exist in a vacuum?

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2. No a bubble cannot form in a vacuum. The force from the outside of the bubble is needed, and there is no force in space acting on the bubble surface

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3. 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?

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4. I’m not sure, how can it hold out the wall of the bubble with no pressure? Instant implosion I would think.

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5. Ben Robinson

Internal pressure is (slightly) positive. External pressure is effectively zero. Internal pressure is in equilibrium with film cohesion force.

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6. 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.

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7. 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.

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8. 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.

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9. How can it be a vacuum if it has even a little gas in it. It can’t happen, end of story.

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10. 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

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11. No it won’t work in vacuum.

The force tension in the water will pull it together.

I dont remeber the word. Surface tension?

Water attract water and u will end upp with a single drop of water.

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12. 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.

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13. Yin Huang Of course, where would you get the air in a vacuum?

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14. 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.

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15. Yin Huang  The second gas enters the vacuum it ceases to be a vacuum. It cannot have air because convection currents & evoporation will warp the bubble rendering it an imperfect sphere.

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16. 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.

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17. 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.

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18. Yin Huang it does seem like his “perfect vacuum” idea carried over to his second part of the question

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19. 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.

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20. 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.

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21. Yin Huang yes I meant sphere, my bad. Also, what kind of temp condition is needed for – lets say – a regular bubble substance that you can buy in the store.

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22. Yin Huang Well, the article said perfect sphere & I couldn’t understand how this could be possible, but nothing is impossible even if it’s a relative infinity to one shot.

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23. 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.

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24. 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.

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25. 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.

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26. 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!

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27. As far as the proof in concept idea….yeah. All most everything math does is a model. You simplify the environment to best suit your needs without sacrificing what really happens.

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28. Christopher Rucinski

The temperature condition depends on the bubble substance used, and how good the vacuum is: http://www.jbind.com/pdf/Cross-Reference-of-Boiling-Temps.pdf

You could, of course, use something quite different from standard bubble soap, like molten lead. Then you would have the opposite challenge: keeping it from solidifying as opposed to keeping it from boiling away.

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29. 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.

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30. 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.

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31. 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.

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32. Then I’ll use a thick lead shield.

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33. Very interesting 🙂

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34. Any heat energy in the liquid will boil and leave some frozen ice. .

so the bubble will never get a chance…no matter what

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35. 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!

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36. 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.

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37. 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?

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38. See probleem – on võrdväärne LEKAALI konstrueerimisega!

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

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