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

Frank Morgan's Math Chat 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 Ritoré 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 Ritoré 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 Bubble Conjecture -- From MathWorld double bubble conjecture Double Bubble Author Eric W. Weisstein © 1999 CRC Press LLC, © 19992003 Wolfram Research, Inc. http://mathworld.wolfram.com/DoubleBubbleConjecture.html

D Double Bubble Conjecture Double Bubble Author: Eric W. Weisstein Wolfram Research, Inc.

3. DOUBLE BUBBLES An announcement of the result, titled "The double bubble conjecture", joint with Michael Hutchings and Roger Schlafly, 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.

4. The Double Bubble Conjecture The double bubble conjecture. Joel Hass, Michael Hutchings, and Roger Schlafly http://www.univie.ac.at/EMIS/journals/ERA-AMS/1995-03-001/1995-03-001.html

This journal is archived by the American Mathematical Society. The master copy is available at http://www.ams.org/era/ Comments on article The double bubble conjecture Joel Hass, Michael Hutchings, and Roger Schlafly Abstract. Retrieve entire article TeX source DVI PostScript 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 Electronic Research Announcements of the AMS Home page

5. 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

Frank Morgan's Math Chat 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.

6. 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/HTML/Conf/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.

7. [HM] Double Bubble Conjecture Proved By Antreas P. Hatzipolakis Subject HM double bubble conjecture Proved Author Antreas P. Hatzipolakis xpolakis@otenet.gr Date Sat, 18 Mar http://mathforum.com/epigone/historia_matematica/jangplexswel

[HM] Double Bubble Conjecture Proved by Antreas P. Hatzipolakis reply to this message post a message on a new topic Back to Historia-Matematica Discussion Group Subject: [HM] Double Bubble Conjecture Proved Author: xpolakis@otenet.gr Date: http://www.williams.edu/Mathematics/fmorgan/ann.html See also: Frank Morgan: Double Bubble Conjecture Proved http://www.maa.org/features/mathchat/mathchat_3_18_00.html APH The Math Forum

8. 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/HTML/Conf/2000/invited.htm

Invited Speakers (Titles (will) have hyper-text links to abstracts). Professor Frank Morgan - Williams College Soap Bubble Geometry Contest Double Bubble Conjecture 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 Cryptography and the Benefits of Ignorance 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

9. Double Bubble Conjecture double bubble conjecture. Two partial Spheres with a separating boundary (which is planar for equal volumes) separate http://hades.ph.tn.tudelft.nl/Internal/PHServices/Documentation/MathWorld/math/m

Double Bubble Conjecture Two partial Spheres with a separating boundary (which is planar for equal volumes) separate two volumes of air with less Area than any other boundary. The planar case was proved true for equal volumes by J. Hass and R. Schlafy in 1995 by reducing the problem to a set of 200,260 integrals which they carried out on an ordinary PC. See also Double Bubble References Haas, J.; Hutchings, M.; and Schlafy, R. ``The Double Bubble Conjecture.'' Electron. Res. Announc. Amer. Math. Soc. Eric W. Weisstein

10. 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 The double bubble conjecture Joel Hass, Michael Hutchings, and Roger Schlafly Abstract. Retrieve entire article TeX source DVI PostScript 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 Electronic Research Announcements of the AMS Home page

11. The Double Bubble Conjecture Calendar Geometric measure theory and the proof of the double bubble conjecture, Lecture 1 Frank Morgan (Scheduled Workshop Talk) Monday, Jun 25, 2001 930 am to 1030 am at the MSRI Lecture Hall, Mathematical Sciences Research Institute, Berkeley http://www.hms.gr/EMIS/journals/ERA-AMS/1995-03-001/1995-03-001.html.old

The Double Bubble Conjecture by Joel Hass, Michael Hutchings and Roger Schlafly Abstract: Article Information: ERA-AMS Volume 1, Issue 03, 1995, pages 98-102 MSC: Primary: 53A10 Secondary: 49Q10 49Q25 TeX Type: AMS-LaTeX View Article: TeX DVI PostScript

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 Ritoré, 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/ Proof of the double bubble conjecture Michael Hutchings, Frank Morgan, Manuel Ritoré, and Antonio Ros Abstract. Retrieve entire article TeX source DVI PostScript 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) Michael Hutchings Department of Mathematics, Stanford University, Stanford, CA 94305 E-mail address: hutching@math.stanford.edu Frank Morgan Department of Mathematics, Williams College, Williamstown, MA 01267 E-mail address: Frank.Morgan@williams.edu Manuel Ritoré Departamento de Geometría y Topología, Universidad de Granada, E-18071 Granada, España E-mail address: ritore@ugr.es Antonio Ros Departamento de Geometría y Topología, Universidad de Granada, E-18071 Granada, España

13. Proof Of The Double Bubble Conjecture Proof of the double bubble conjecture. Michael Hutchings, Frank Morgan, Manuel Ritoré, and Antonio Ros http://www.ii.uj.edu.pl/EMIS/journals/ERA-AMS/2000-01-006/2000-01-006.html

This journal is archived by the American Mathematical Society. The master copy is available at http://www.ams.org/era/ Proof of the double bubble conjecture Michael Hutchings, Frank Morgan, Manuel Ritoré, and Antonio Ros Abstract. Retrieve entire article TeX source DVI PostScript 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) Michael Hutchings Department of Mathematics, Stanford University, Stanford, CA 94305 E-mail address: hutching@math.stanford.edu Frank Morgan Department of Mathematics, Williams College, Williamstown, MA 01267 E-mail address: Frank.Morgan@williams.edu Manuel Ritoré Departamento de Geometría y Topología, Universidad de Granada, E-18071 Granada, España E-mail address: ritore@ugr.es Antonio Ros Departamento de Geometría y Topología, Universidad de Granada, E-18071 Granada, España

14. 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

Foundations of Mathematics Mathematical Problems Solved Problems Geometry ... 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. et al. showed that the separating boundary which minimizes total surface area is a portion of a sphere which meets the outer spherical surfaces at dihedral angles curvature of the partition is simply the difference of the curvatures of the two bubbles, where R is the radius of the interface and and are the radii of the bubbles (Isenberg 1992, pp. 88-95). Furthermore, for three bubbles with radii

15. Double Bubble Page You can download here the preprint ''Proof of the double bubble conjecture'',by Michael Hutchings, Frank Morgan, Manuel Ritoré 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

16. About "Double Bubble Conjecture Proved (Math Chat)" double bubble conjecture Proved (Math Chat). Library Home FullTable of Contents Suggest a Link Library Help Visit this http://mathforum.org/library/view/12694.html

Double Bubble Conjecture Proved (Math Chat) Library Home Full Table of Contents Suggest a Link Library Help Visit this site: http://www.maa.org/features/mathchat/mathchat_3_18_00.html Author: Frank Morgan, MAA Online Description: Four mathematicians have announced a mathematical proof of the Double Bubble Conjecture: that the familiar double soap bubble 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, 2000, Frank Morgan of Williams College announced that he, Michael Hutchings of Stanford, and Manuel Ritoré and Antonio Ros of Granada had finally proved that the double soap bubble had it right all along... Levels: High School (9-12) College Languages: English Resource Types: Problems/Puzzles Articles Math Topics: Higher-Dimensional Geometry Suggestion Box Home The Math Library ... Search http://mathforum.org/ webmaster@mathforum.org

17. About "$200 Double Bubble New Challenge (Math Chat)" SMALL undergraduate research Geometry Group report, Williams College,1999). It bears on proving the double bubble conjecture. http://mathforum.org/library/view/10992.html

$200 Double Bubble New Challenge (Math Chat) Library Home Full Table of Contents Suggest a Link Library Help Visit this site: http://www.maa.org/features/mathchat/mathchat_10_7_99.html Author: Frank Morgan, MAA Online Description: Challenge: This open problem appears as Conjecture 4.10 in "Component bounds for area-minimizing double bubbles," by Cory Heilmann, Yvonne Lai, Ben Reichardt, and Anita Spielman (NSF "SMALL" undergraduate research Geometry Group report, Williams College, 1999). It bears on proving the Double Bubble Conjecture. Levels: High School (9-12) College Languages: English Resource Types: Articles Math Topics: Higher-Dimensional Geometry Suggestion Box Home The Math Library ... Search http://mathforum.org/ webmaster@mathforum.org

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 RitorŽ 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. Newsletter - May, 2000 Curiosity Seminar Series. Alumni News. Math/Stat Web Sites. Proverbs for the Millennium.General double bubble conjecture in R 3 Solved. Winter 1999 Dean's List. http://www.rit.edu/~rehsma/news993/19993head.html

Department of M ATHEMATICS S TATISTICS NEWSLETTER Volume 14 Number 3 May 2000 Articles Math Awareness Month Randolph Scholar Shawn Dwyer's Co-op at Kodak Putnam Competition Math Club Activities Prof. Jack Weiss to Leave RIT Prof. Farnsworth has Leave ... Prof. Wilcox's Retirement Plan s Curiosity Seminar Series Alumni News Math/Stat Web Sites Proverbs for the Millennium General Double Bubble Conjecture in R Solved Winter 1999 Dean's List Mathematics and Statistics Newsletter Articles Written By: Prof. James Runyon Prof. Wanda Szpunar-Lojasiewicz Prof. Rebecca E.Hill Prof. Theodore Wilcox Prof. Jack Weiss Shawn Dwyer Internet Edition Layout by: Rebecca E. Hill Send your News item to: E-mail: rehsma@rit.edu or Rochester Institute of Technology Department of Mathematics and Statistics 85 Lomb Memorial Drive Rochester, NY 14623 Back to R. Hill's Home Page Back to Department of Mathematics and Statistics Home Page

20. 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 Ritor? and Antonio Ros. http://www.kurims.kyoto-u.ac.jp/EMIS/journals/ERA-AMS/2000-01-006/2000-01-006.ht

This journal is archived by the American Mathematical Society. The master copy is available at http://www.ams.org/era/ Proof of the double bubble conjecture Michael Hutchings, Frank Morgan, Manuel Ritoré, and Antonio Ros Abstract. Retrieve entire article TeX source DVI PostScript 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) Michael Hutchings Department of Mathematics, Stanford University, Stanford, CA 94305 E-mail address: hutching@math.stanford.edu Frank Morgan Department of Mathematics, Williams College, Williamstown, MA 01267 E-mail address: Frank.Morgan@williams.edu Manuel Ritoré Departamento de Geometría y Topología, Universidad de Granada, E-18071 Granada, España E-mail address: ritore@ugr.es Antonio Ros Departamento de Geometría y Topología, Universidad de Granada, E-18071 Granada, España

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