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1. Frank Morgan's Math Chat - DOUBLE BUBBLE CONJECTURE PROVED
Frank Morgan s Math Chat. MATHCHAT. March 18, 2000. double bubble conjecturePROVED. Four mathematicians have announced a mathematical
http://www.maa.org/features/mathchat/mathchat_3_18_00.html
March 18, 2000
##### DOUBLE BUBBLE CONJECTURE PROVED
Four mathematicians have announced a mathematical proof of the Double Bubble Conjecture: that the familiar double soap bubble on the right in Figure 1 is the optimal shape for enclosing and separating two chambers of air. In an address to the Undergraduate Mathematics Conference at the Rose-Hulman Institute of Technology in Indiana on Saturday, March 18, Frank Morgan of Williams College announced that he, Michael Hutchings of Stanford, and Manuel RitorÃ© and Antonio Ros of Granada had finally proved that the double soap bubble had it right all along.
The familiar double soap bubble on the right is now known to be the optimal shape for a double chamber. Wild competing bubbles with components wrapped around each other as on the left are shown to be unstable by a novel argument. Computer graphics by John M. Sullivan, University of Illinois, www.math.uiuc.edu/~jms/Images. When two round soap bubbles come together, they form a double bubble as on the right in Figure 1. Unless the two bubbles are the same size, the surface between them bows a bit into the larger bubble. The separating surface meets each of the two bubbles at 120 degrees. This precise shape is now known to have less area than any other way to enclose and separate the same two volumes of air, even wild possibilities as on the left in Figure 1, in which the second bubble wraps around the first, and a tiny separate part of the first wraps around the second. Such wild possibilities are shown to be unstable by a new argument which involves rotating different portions of the bubble around a carefully chosen axis at different rates. The breakthrough came while Morgan was visiting RitorÃ© and Ros at the University of Granada last spring. Their work is supported by the National Science Foundation and the Spanish scientific research foundation DGICYT.

2. DOUBLE BUBBLES
progress on this problem, proving that the Double Bubble gives the best way of enclosing two the result, titled "The double bubble conjecture", joint with Michael Hutchings and
http://www.math.ucdavis.edu/~hass/bubbles.html
##### SOME RESULTS ON BUBBLES
Bubbles are nature's way of finding optimal shapes to enclose certain volumes. Bubbles are studied in the fields of mathematics called Differential Geometry and Calculus of Variations. While it is possible to produce many bubbles through physical experiments, many of the mathematical properties of bubbles remain elusive. One question that has been asked by physicists and mathematicians is whether bubbles form the optimal (meaning smallest surface area) surfaces for enclosing given volumes. In work with Roger Schlafly, we made progress on this problem, proving that the Double Bubble gives the best way of enclosing two equal volumes.
A double bubble and a competitor, called a torus bubble. Thanks to John Sullivan of the University of Minnesota, for generating these images. Here are more images of bubbles, generated with help from Jim Hoffman at MSRI. A symmetric torus bubble.
A non-symmetric torus bubble.
In this one the inner Delaunay surface is generated by a curve that includes a full period, though only one local maximum and one local minimum.
In tiff format
Another non-symmetric torus bubble.

3. Double Bubble Conjecture -- From MathWorld
D. double bubble conjecture. Double Bubble. search. Eric W. Weisstein. DoubleBubble Conjecture. From MathWorldA Wolfram Web Resource.
http://mathworld.wolfram.com/DoubleBubbleConjecture.html
 INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index ABOUT THIS SITE About MathWorld About the Author DESTINATIONS What's New MathWorld Headline News Random Entry ... Live 3D Graphics CONTACT Email Comments Contribute! Sign the Guestbook MATHWORLD - IN PRINT Order book from Amazon D Double Bubble Conjecture Double Bubble search Eric W. Weisstein. "Double Bubble Conjecture." From MathWorld A Wolfram Web Resource. http://mathworld.wolfram.com/DoubleBubbleConjecture.html Wolfram Research, Inc.

4. Frank Morgan's Math Chat - \$200 DOUBLE BUBBLE NEW CHALLENGE
Geometry Group report, Williams College, 1999). It bears on provingthe double bubble conjecture (see Math Chat of October 25, 1996).
http://www.maa.org/features/mathchat/mathchat_10_7_99.html
October 7, 1999
##### \$200 DOUBLE BUBBLE NEW CHALLENGE
OLD CHALLENGE (Colin Adams). A web comment claims that, "If the population of China walked past you in single file, the line would never end because of the rate of reproduction." Is this true? ANSWER. Probably not, as best explained by Richard Ritter. The current population of China is about 1.25 billion, with about 20 million births per year. We'll assume that the birthrate stays about the same, as the population grows a bit but the births per 1000 drops a bit, under the current one child per family policy. The Chinese walk say 3 feet apart at 3 miles per hour, for a rate of 46 million Chinese per year. So even if no one died in line, the line would shorten by 26 million per year and run out in about 1250/26 = 48 years. (Different assumptions could lead to a different conclusion.) Incidentally, the UN Population Fund projects that the world population will hit 6 billion next week (around October 12). NEW CHALLENGE with \$200 PRIZE for best complete solution (otherwise usual book award for best attempt). A double bubble is three circular arcs meeting at 120 degrees, as in the third figure.

 5. The Double Bubble Conjecture Comments on article. The double bubble conjecture. Joel Hass, Michael Hutchings, and Roger Schlafly. Abstract. The classical isoperimetric inequality states that the surface of smallest area enclosing a given volume in \$R^3\$ is a sphere. area surface enclosing two equal volumes is a double bubble, a surface made of two pieces of roundhttp://www.univie.ac.at/EMIS/journals/ERA-AMS/1995-03-001/1995-03-001.html

6. Double Bubble Page
You can download here the preprint Proof of the double bubble conjecture ,by Michael Hutchings, Frank Morgan, Manuel RitorÃ© and Antonio Ros, 2000.
http://www.ugr.es/~ritore/bubble/bubble.htm
 Gzip PostScript file (426 K, includes figures). DVI file (116 K, without figures). PDF file (340 K, includes figures). This paper generalizes previous work by Joel Hass and Roger Schlafly, who proved the conjecture for the equal volumes case. The interested visitor can find more information in Frank Morgan's homepage , and pictures in John Sullivan's and James Hoffman's For more information, the following papers are quite interesting Why double bubbles form the way they do, B. Cipra, Science, 287 (17 March 2000), 1910-1911. Proof of the double bubble conjecture, F. Morgan, American Mathematical Monthly, 108 (March 2001), 193-205. Back to Homepage

7. Double Bubble Conjecture Proven
Four mathematicians have announced a mathematical proof of the Double Bubble Conjexture that the familiar double soap bubble is the optimal shape for enclosing and separating two chambers of air .
Public release date: 27-Mar-2000
Contact: Frank Morgan
Jo.Procter@williams.edu

Williams College
##### Double bubble conjecture proven
Above, a cluster of three bubbles. The central vertical bubble has volume approximately 6.10736, while the thick waist bubble around it has volume 2.85446, and the tiny outer belt bubble has volume 0.0382928. The total surface area is 29.3233. Considering the central and belt bubbles to be two components of a single region, we can think of this as a double bubble with volumes 6.146 and 2.854.
Click here for more bubble images related to this release. WILLIAMSTOWN, Mass., March 28, 2000 Four mathematicians have announced a mathematical proof of the Double Bubble Conjecture: that the familiar double soap bubble (see Figure 1 http://www.math.uiuc.edu/ ~jms/Images/double/ ) is the optimal shape for enclosing and separating two chambers of air. In an address to the Undergraduate Mathematics Conference at the Rose-Hulman Institute of Technology in Indiana on Saturday (March 18), Frank Morgan of Williams College announced that he, Michael Hutchings of Stanford, and Manuel Ritori and Antonio Ros of Granada had finally proved that the double soap bubble had it right all along. When two round soap bubbles come together, they form a double bubble (as on the right in Figure 1). Unless the two bubbles are the same size, the surface between them bows a bit into the larger bubble. The separating surface meets each of the two bubbles at 120 degrees.

8. Double Bubble -- From MathWorld
Haas, J.; Hutchings, M.; and Schlafy, R. The double bubble conjecture. Electron.Res. Morgan, F. The double bubble conjecture. FOCUS 15, 67, 1995.
http://mathworld.wolfram.com/DoubleBubble.html
 INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index ABOUT THIS SITE About MathWorld About the Author DESTINATIONS What's New MathWorld Headline News Random Entry ... Live 3D Graphics CONTACT Email Comments Contribute! Sign the Guestbook MATHWORLD - IN PRINT Order book from Amazon Foundations of Mathematics Mathematical Problems Solved Problems ... Sullivan Double Bubble A double bubble is pair of bubbles which intersect and are separated by a membrane bounded by the intersection. The usual double bubble is illustrated in the left figure above. A more exotic configuration in which one bubble is torus-shaped and the other is shaped like a dumbbell is illustrated at right (illustrations courtesy of J. M. Sullivan). angles ) has the minimum perimeter for enclosing two equal areas (Alfaro et al. 1993, Morgan 1995). It had been conjectured that two equal partial spheres sharing a boundary of a flat disk separate two volumes of air using a total surface area that is less than any other boundary. This equal-volume case was proved by Hass et al.

 9. PROOF OF THE DOUBLE BUBBLE CONJECTURE 6762(00)000792PROOF OF THE double bubble conjectureMICHAEL HUTCHINGS, FRANK MORGAN, MANUEL RITORÂ´E, AND ANTONIO inR3. Thedouble bubble conjecture, long assumed true (P, pphttp://www.ams.org/era/2000-06-06/S1079-6762-00-00079-2/S1079-6762-00-00079-2.pd

 10. Proof Of The Double Bubble Conjecture In R4 And Certain Higher 2003PROOF OF THE double bubble conjecture IN R4AND CERTAIN HIGHER DIMENSIONAL CASESBen W The double bubble conjecture. Conjecture 1.1 (double bubble conjecture). The leastareahttp://www.math.albany.edu:8000/PacJ/p/2003/208-2-9.pdf

11. The Double Bubble Conjecture
org/era/. Comments on article. The double bubble conjecture. Joel Hass,Michael Hutchings, and Roger Schlafly. Abstract. The classical
http://www.mpim-bonn.mpg.de/external-documentation/era-mirror/1995-03-001/1995-0
This journal is archived by the American Mathematical Society. The master copy is available at http://www.ams.org/era/ Comments on article
##### Joel Hass, Michael Hutchings, and Roger Schlafly
Abstract. Retrieve entire article
##### Article Info
• ERA Amer. Math. Soc. (1995), pp. 98-102 Publisher Identifier: S1079-6762(95)03001-0 Mathematics Subject Classification . Primary 53A10; Secondary 49Q10, 49Q25. Key words and phrases . Double bubble; isoperimetric Received by the editors September 11, 1995 Communicated by Richard Schoen Comments
Joel Hass Department of Mathematics, University of California, Davis, CA 95616 E-mail address: hass@math.ucdavis.edu Michael Hutchings Department of Mathematics, Harvard University, Cambridge, MA 02138 E-mail address: hutching@math.harvard.edu Roger Schlafly Real Software, PO Box 1680, Soquel, CA 95073 E-mail address: rschlafly@attmail.com Hutchings was supported by an NSF Graduate Fellowship
• 12. Proof Of The Double Bubble Conjecture
The master copy is available at http//www.ams.org/era/. Proof of the double bubbleconjecture. Michael Hutchings, Frank Morgan, Manuel RitorÃ©, and Antonio Ros.
http://www.mpim-bonn.mpg.de/external-documentation/era-mirror/2000-01-006/2000-0
This journal is archived by the American Mathematical Society. The master copy is available at http://www.ams.org/era/
##### Michael Hutchings, Frank Morgan, Manuel RitorÃ©, and Antonio Ros
Abstract. Retrieve entire article
##### Article Info
• ERA Amer. Math. Soc. (2000), pp. 45-49 Publisher Identifier: S 1079-6762(00)00079-2 Mathematics Subject Classification . Primary 53A10; Secondary 53C42 Key words and phrases . Double bubble, soap bubbles, isoperimetric problems, stability Received by the editors March 3, 2000 Posted on July 17, 2000 Communicated by Richard Schoen Comments (When Available)

 13. Proof Of The Double Bubble Conjecture Proof of the double bubble conjecture We prove that the standard double bubble provides the leastarea way to enclose and separate two regions of prescribed volume in \$\rr^3\$. euclid.annm/http://rdre1.inktomi.com/click?u=http://ProjectEuclid.org/getRecord?id=euclid.an

14. Double Bubble Conjecture
on Saturday Morning that he, Michael Hutchings of Stanford College, and ManuelRitori and Antonio Ros of Granada have proved the double bubble conjecture.
http://www.rose-hulman.edu/class/ma/web/mathconf/2000/bubble.html
##### New from the Undergraduate Math Conference
During the 17th annual Rose-Hulman Undergraduate Math Conference, Professor Frank Morgan of Williams College announced on Saturday Morning that he, Michael Hutchings of Stanford College, and Manuel Ritori and Antonio Ros of Granada have proved the Double Bubble Conjecture. The Double Bubble Conjecture is that the familiar double bubble (on the right below) is the optimal shape for enclosing and separating two chambers of air, where optimal means minimizes the surface area needed to enclose two chambers of specified volumes. The proof relies on showing the wild competing bubbles with components wrapped around each other (shown on the left) are unstable. This is done by a new argument involving rotating different portions of the bubble around a carefully chosen axis at different rates. [Computer graphics by John M. Sullivan, University of Illinois, www.math.uiuc.edu/~jms/Images/double/ The breakthrough came while Morgan was visiting Ritori and Ros at the University of Granada last spring. Their work is supported by the National Science Foundation and the Spanish scientific research foundation DGICYT. The proof of two equal bubbles was accomplished earlier by Hass, Hutchings, and Schlafly and required the use of a computer to compute the volumes for competing bubbles. The new proof for the general case involves only ideas, pencil and paper.

15. Invited Speakers
double bubble conjecture Professor Frank Morgan Williams CollegeSaturday, March 18, 2000, 930-1030 Room E104 of Moench Hall.
http://www.rose-hulman.edu/class/ma/web/mathconf/2000/invited.htm
Invited Speakers (Titles (will) have hyper-text links to abstracts).
• Professor Frank Morgan - Williams College

• Biography:
Frank Morgan works in minimal surfaces and studies the behavior and structure of minimizers in various dimensions and settings. His three texts on Geometric Measure Theory: a Beginner's Guide 1995, Calculus Lite 1997, and Riemannian Geometry: a Beginner's Guide 1998, will soon be joined by The Math Chat Book 1999, based on his live call-in Math Chat TV show and Math Chat column, both available at www.maa.org. (contd) Professor Morgan's home page
• Professor Nigel Boston - University of Illinois at Urbana-Champaign

• Biography:
Nigel Boston grew up in England and attended Cambridge and Harvard. After a year in Paris and two in Berkeley, he went to the University of Illinois where he has been ever seince, except for six months at the Newton Institute. His original work was in algebraic number theory and closely related to the work used to prove Fermat's Last Theorem. (contd) Professor Boston's home page
##### Abstracts
C RYPTOGRAPHY AND THE B ENEFITS OF I GNORANCE
Professor Nigel Boston - University of Illinois at Urbana Champaign
Friday, March 17, 2000 1:35 P.M. E104

 16. Double Bubble Conjecture Proved (Math Chat) double bubble conjecture Proved (Math Chat) Four mathematicians have announced a mathematical proof of the double bubble conjecture that the familiar double soap bubble is the optimal shape forhttp://rdre1.inktomi.com/click?u=http://mathforum.org/library/view/12694.html&am

17. Computer Images Of Double Bubbles By John Sullivan
I created these images to illustrate the proof of the equalvolume caseof the double bubble conjecture by Hass and Schlafly in 1995.
http://torus.math.uiuc.edu/jms/Images/double/
##### John M Sullivan
Click on any image for a larger (640x800) version.
Please send me email ( jms@uiuc.edu ) for permission to publish these images,
or to obtain even larger TIFF versions, with different background colors.
These images show bubble clusters near equilibrium. The top row shows a standard double bubble of equal volumes, and a nonstandard cluster in which one bubble is a torus, forming a waist around the other. I created these images to illustrate the proof of the equal-volume case of the Double Bubble Conjecture by Hass and Schlafly in 1995. The bottom row shows a standard double bubble of unequal volumes (consisting of three spherical caps meeting at equal 120-degree angles), and a nonstandard bubble of the same volumes, in which the larger region is broken into two components (one a tiny ring around the other region). I created these images to illustrate the proof of the general Double Bubble Conjecture by Hutchings, Morgan, Ritore and Ros in 2000. In all four cases, the cluster is a surface of revolution. More details about the geometry of the examples with unequal volumes, including pictures of the generating curves, are available

18. Double Bubble
GENERAL double bubble conjecture IN R 3 SOLVED. In March 2000, the proofof the general double bubble conjecture in R 3 was announced
http://www.rit.edu/~rehsma/news993/bubble.html
##### GENERAL DOUBLE BUBBLE CONJECTURE IN R SOLVED
In March 2000, the proof of the general double bubble conjecture in R was announced by four mathematicians: Michael Hutchings of Stanford University, Frank Morgan of Williams College, and Manuel RitorÂ and Antonio Ros of the University of Granada. Their proof completes a long history of work on the problem. Experiments with blowing soap bubbles give rise not only to spheres, but also to more complicated conglomerations of bubbles. These can be foams with complicated geometries, but when only two components are enclosed the shape assumed is known as a "standard double bubble." This is made of pieces of three round spheres, meeting along a common circle at an angle of 120 degrees. The double bubble conjecture asserts that this shape is the most efficient one possible in enclosing two given volumes. The study of optimal shapes has taken us to the point where the techniques have real connections in physical and biological applications. This has begun to be seen in exciting new studies of foams, crystal growth, and other complex structures. From FOCUS, the Newsletter of the MAA, May/June 2000.

19. [HM] Double Bubble Conjecture Proved By Antreas P. Hatzipolakis
HM double bubble conjecture Proved by Antreas P. Hatzipolakis. reply Thedouble bubble conjecture, long assumed true (see P, pp. 300
http://mathforum.org/epigone/historia/jangplexswel