The Double Bubble Conjecture

In honor of Frank Morgan’s visit to OU this week, we thought we’d say a little about his area of research: minimal surfaces.

What is a minimal surface?  The fancy answer is that it is a surface with a mean curvature of zero.  The less fancy answer is that it is the surfaces of minimal area which satisfy whatever constraints you put on them.  The classic example is soap bubbles.  If you dip a wire frame into soapy water, the resulting soap bubble is the one of smallest area (because of the pull of surface tension) which fits the space they are in.  For example, if you dip a wire cube into soapy water you get:

If you look closely you can even see the sheets of soap between the wire frame and the interior cubical bubble.

As you might imagine, the mathematics of such things (even in the 3 dimensions, nevermind higher dimensions!) is quite complicated.  As an easy example, it goes way back to Archimedes and Zenodorus in ancient Greece that a sphere is the shape of minimal area which encloses a given volume.  Even though it seems pretty obvious that should be correct, it wasn’t until the 1800s that it was proven mathematically.

A more difficult example is to consider what happens if you use two bubbles to enclose two adjacent regions. Doing so you obtain:

Computer graphics by John M. Sullivan, University of Illinois

Now imagine you want to prove mathematically that this is the shape of smallest area which encloses two neighboring regions of these volumes?  This is the Double Bubble Conjecture.  In 1995 it was proven by Hass, Hutchings, and Schlafy in the special case that both bubbles have the same volume.  What they did was simplify the problem down to where you simply had to compute 200,260 integrals which they then did by computer.  Not very satisfying, but it was progress!

Then, Frank Morgan, Michael Hutchings, Manuel Ritoré, and Antonio Ros finally proved the conjecture for arbitrary double bubbles in early 2000.  And they did it without using computers!  You can look at their paper here.  It is amazing that such an easy to understand question was only solved in the last 10 years!  One of the authors of that paper, Michael Hutchings, is a professor at the University of California, Berkeley.  You can check out his website to learn more about the state of the art in research into minimal surfaces.  For example, he says

The triple bubble problem in R^3 currently seems hopeless without some brilliant new idea, although again there is a natural candidate surface.

— Michael Hutchings

Of course a good bet would be the triple bubble:

While you’re thinking about that, you can sit back and watch bubbles in slow motion:

P.S.  All the great minds like to think about bubbles from time to time:

Macbeth

A dark Cave. In the middle, a Caldron boiling. Thunder.

Enter the three Witches.

1 WITCH. Thrice the brinded cat hath mew’d.

2 WITCH. Thrice and once, the hedge-pig whin’d.

3 WITCH. Harpier cries:—’tis time! ’tis time!

1 WITCH. Round about the caldron go;
In the poison’d entrails throw.—
Toad, that under cold stone,
Days and nights has thirty-one;
Swelter’d venom sleeping got,
Boil thou first i’ the charmed pot!

ALL. Double, double toil and trouble;
Fire burn, and caldron bubble.

2 WITCH. Fillet of a fenny snake,
In the caldron boil and bake;
Eye of newt, and toe of frog,
Wool of bat, and tongue of dog,
Adder’s fork, and blind-worm’s sting,
Lizard’s leg, and owlet’s wing,—
For a charm of powerful trouble,
Like a hell-broth boil and bubble.

ALL. Double, double toil and trouble;
Fire burn, and caldron bubble.

3 WITCH. Scale of dragon; tooth of wolf;
Witches’ mummy; maw and gulf
Of the ravin’d salt-sea shark;
Root of hemlock digg’d i the dark;
Liver of blaspheming Jew;
Gall of goat, and slips of yew
Sliver’d in the moon’s eclipse;
Nose of Turk, and Tartar’s lips;
Finger of birth-strangled babe
Ditch-deliver’d by a drab,—
Make the gruel thick and slab:
Add thereto a tiger’s chaudron,
For the ingrediants of our caldron.

ALL. Double, double toil and trouble;
Fire burn, and caldron bubble.

2 WITCH. Cool it with a baboon’s blood,
Then the charm is firm and good.

William Shakespeare (1564-1616)

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