1. Double Bubble Page Proof of the double bubble conjecture, F. Morgan, American Mathematical Monthly,108 (March 2001), 193205. Back to Homepage 04/10/23. http://www.ugr.es/~ritore/bubble/bubble.htm

Gzip Postscript file PDF file 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

2. Frank Morgan's Math Chat - DOUBLE BUBBLE CONJECTURE PROVED double bubble conjecture PROVED. Four mathematicians have announced a mathematical proof of the double bubble conjecture that the familiar double http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

3. Frank Morgan's Math Chat - DOUBLE BUBBLE CONJECTURE PROVED double bubble conjecture PROVED. Four mathematicians have announced a proof ofthe double bubble conjecture that the familiar double soap bubble on the 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.

4. Double Bubble Page You can download here the paper ''Proof of the double bubble conjecture'', by Michael Hutchings, Frank Morgan, Manuel Ritor and Antonio Ros. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

5. Frank Morgan's Math Chat - $200 DOUBLE BUBBLE NEW CHALLENGE research 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. GEOMETRIC MEASURE THEORY AND THE PROOF OF THE DOUBLE BUBBLE GEOMETRIC MEASURE THEORY AND THE PROOF OF THE double bubble conjecture FRANK MORGAN AND MANUEL RITOR E Abstract. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

7. DOUBLE BUBBLES An announcement of the result, titled The double bubble conjecture , joint withMichael Hutchings and Roger Schlafly, appeared in Issue 3 of Electronic 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.

8. Furman Mathematics Clanton Visiting Mathematician Is the universe curved? How can one visualize curved space? http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

9. Double Bubble Conjecture -- From MathWorld double bubble conjecture. COMMENT On this Page. SEE Double Bubble. Pages LinkingHere http://mathworld.wolfram.com/DoubleBubbleConjecture.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index DESTINATIONS About MathWorld About the Author Headline News ... Random Entry CONTACT Contribute an Entry Send a Message to the Team MATHWORLD - IN PRINT Order book from Amazon Double Bubble Conjecture SEE: Double Bubble [Pages Linking Here]

10. Proof Of The Double Bubble Conjecture, Michael Hutchings, Frank Proof of the double bubble conjecture Source Ann. of Math. 155 (2002), no. 2, 459489 Abstract http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

11. Double Bubble -- From MathWorld Haas, J. General double bubble conjecture in R^3 Solved. Focus The Newsletterof the 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 DESTINATIONS About MathWorld About the Author Headline News ... Random Entry CONTACT Contribute an Entry Send a Message to the Team MATHWORLD - IN PRINT Order book from Amazon Geometry Surfaces Miscellaneous Surfaces ... 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). In the plane, the analog of the double bubble consists of three circular arcs meeting in two points. It has been proved that the configuration of arcs meeting at equal 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

12. Cwikel The double bubble conjecture says that the familiar double soap bubble is the leastarea way to enclose and separate two given volumes of air. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

13. Proof Of The Double Bubble Conjecture Proof of the double bubble conjecture. Michael Hutchings, Frank Morgan, ManuelRitor?, and Antonio Ros. Abstract. We prove that the standard double bubble 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 Ritor�, 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 Ritor� Departamento de Geometr�a y Topolog�a, Universidad de Granada, E-18071 Granada, Espa�a E-mail address: ritore@ugr.es

14. Frank Morgan At MSRI - Geometric Measure Theory And The Proof Of Frank Morgan Geometric Measure Theory and the Proof of the double bubble conjecture - Lecture 2 http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

15. The Double Bubble Conjecture The double bubble conjecture. Joel Hass, Michael Hutchings, and Roger Schlafly Double bubble; isoperimetric; Received by the editors September 11, 1995 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

16. Double Bubble Conjecture - Science Technology Services double bubble conjecture loop quantum gravity double bubble conjecture what is it and how was it prooved? Web Physics Forums http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

17. About "Double Bubble Conjecture Proved (Math Chat)" double bubble conjecture Proved (Math Chat) have announced a mathematicalproof of the double bubble conjecture that the familiar double soap bubble is 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 Home The Math Library Quick Reference ... Help http://mathforum.org/

18. Double Bubble Conjecture double bubble conjecture http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

19. The Math Forum - Math Library - Higher-Dimensional Geom. double bubble conjecture Proved (Math Chat) Frank Morgan, MAA Online Fourmathematicians have announced a mathematical proof of the Double Bubble http://mathforum.org/library/topics/higherd_g/?keyid=14176664&start_at=51&num_to

20. Historia Matematica Mailing List Archive [HM] Double Bubble Co HM double bubble conjecture Proved Subject HM double bubble conjecture Proved From Antreas P. Hatzipolakis (xpolakis@otenet.gr http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

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