21. Electronic Research Announcements More results from www.ams.org Soap bubbles and isoperimetric problemsFor example, the double bubble conjecture in R^3 was proved only recently. 3 The double bubble conjecture (with J. Hass and R. Schlafly), http://www.ams.org/journal-getitem?pii=S1079-6762-00-00079-2

22. Computer Images Of Double Bubbles By John Sullivan The top row shows a standard double bubble of equal volumes, and a nonstandard case of the double bubble conjecture by Hass and Schlafly in 1995. http://torus.math.uiuc.edu/jms/Images/double/

Standard and Nonstandard Double Bubbles 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

23. Double Bubble Minimizes: Applications To Geometry J. Hass, M. Hutchings, and R. Schlafly, The double bubble conjecture, ElectronicResearch Announcements of the American Mathe. Society, 1995, Vol. 1, pp. http://www.cs.utep.edu/interval-comp/bubble.html

Double Bubble Minimizes: Interval Computations Help in Solving a Long-Standing Geometric Problem It is well known that of all surfaces surrounding an area with a given volume V, the sphere has the smallest area. This result explains, e.g., why a soap bubble tends to become a sphere. More than a hundred years ago, the Belgian physicist J. Plateaux asked a similar question: what is the least area surface enclosing two equal volumes? Physical experiments with bubbles seem to indicate that the desired least area surface is a "double bubble", a surface formed by two spheres (separated by a flat disk) that meet along a circle at an angle of 120 degrees. However, until 1995, it was not clear whether this is really the desired least area surface. Several other surfaces ("torus bubbles") have been proposed whose areas are pretty close to the area of the double bubble. The theorem that double bubble really minimizes was recently proven by Joel Hass from Department of Mathematics, University of California at Davis (email hass@math.ucdavis.edu

24. Science Blog -- Double Bubble Conjecture Proven From Williams College double bubble conjecture proven When two round soapbubbles come together, they form a double bubble (as on the right in Figure http://www.scienceblog.com/community/older/2000/F/200005376.html

From: 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.

25. [math/0406017] Proof Of The Double Bubble Conjecture Proof of the double bubble conjecture We prove that the standard double bubbleprovides the leastarea way to enclose and separate two regions of http://arxiv.org/abs/math.DG/0406017

Mathematics, abstract math.DG/0406017 From: Frank Morgan [ view email ] Date: Tue, 1 Jun 2004 16:10:48 GMT (195kb) Proof of the Double Bubble Conjecture Authors: Michael Hutchings Frank Morgan Antonio Ros Comments: 31 pages, published version Subj-class: Differential Geometry; Metric Geometry Journal-ref: Ann. of Math. (2), Vol. 155 (2002), no. 2, 459489 Full-text: PostScript PDF , or Other formats References and citations for this submission: CiteBase (autonomous citation navigation and analysis) Which authors of this paper are endorsers? Links to: arXiv math find abs

26. Annals Of Mathematics, II. Series, Vol. 151, No. 2, Pp. 459-515, 2000 In the interim, a proof of the double bubble conjecture without the restriction to Keywords double bubble conjecture; equal volumes; spherical cap http://www.emis.de/journals/Annals/151_2/3.html

Annals of Mathematics, II. Series Vol. 151, No. 2, pp. 459-515 (2000) Previous Article Next Article Contents of this Issue Other Issues ... EMIS Home Double bubbles minimize Joel Hass and Roger Schlafly Review from Zentralblatt MATH Zbl 0970.53009 )], see the review below. Reviewed by Robert Finn Keywords: double bubble conjecture; equal volumes; spherical cap Classification (MSC2000): Full text of the article: PDF file (526 kilobytes) Electronic fulltext finalized on: 27 Apr 2001. This page was last modified: 22 Jan 2002. Johns Hopkins University Press ELibM for the EMIS Electronic Edition Metadata extracted from Zentralblatt MATH with kind permission

27. Science -- Sign In results to analogs of the double bubble conjecture in higher dimensions.(The twodimensional double bubble conjecture was proved by an earlier group of http://www.sciencemag.org/cgi/content/full/287/5460/1910a

You do not have access to this item: Full Text : Cipra, MATHEMATICS:Why Double Bubbles Form the Way They Do, Science You are on the site via Free Public Access. What content can I view with Free Public Access If you have a personal user name and password, please login below. SCIENCE Online Sign In Options For Viewing This Content User Name Password this computer. Help with Sign In If you don't use cookies, sign in here Join AAAS and subscribe to Science for free full access. Sign Up More Info Register for Free Partial Access including abstracts, summaries and special registered free full text content. Register More Info Pay per Article 24 hours for US $10.00 from your current computer Regain Access to a recent Pay per Article purchase Need More Help? Can't get past this page? Forgotten your user name or password? AAAS Members activate your FREE Subscription

28. Proof Of The Double Bubble Conjecture Proof of the double bubble conjecture. Michael Hutchings, Frank Morgan, ManuelRitoré, and Antonio Ros. Abstract. We prove that the standard double bubble http://www.maths.tcd.ie/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

29. Electronic Research Announcements Of The AMS Proof of the double bubble conjecture. Kamber, Franz W. Franz W. Kamber; Peter W.Michor The flow completion of a manifold with vector field http://www.maths.tcd.ie/EMIS/journals/ERA-AMS/era-auth-2000.html

This journal is archived by the American Mathematical Society. The master copy is available at http://www.ams.org/era/ 2000 Table of Contents - Indexed by Authors Adem, Alejandro Alejandro Adem; Jeff H. Smith On spaces with periodic cohomology Afraimovich, Valentin Valentin Afraimovich; Jean-René Chazottes; Benoît Saussol Local dimensions for Poincaré recurrences Bullett, S. R. S. R. Bullett; W. J. Harvey Mating quadratic maps with Kleinian groups via quasiconformal surgery Calegari, Danny Danny Calegari Chazottes, Jean-René Valentin Afraimovich; Jean-René Chazottes; Benoît Saussol Local dimensions for Poincaré recurrences Giambruno, A. A. Giambruno; M. Zaicev Minimal varieties of algebras of exponential growth Grishin, A. V. A. V. Grishin Harvey, W. J. S. R. Bullett; W. J. Harvey Mating quadratic maps with Kleinian groups via quasiconformal surgery Hutchings, Michael Michael Hutchings; Frank Morgan; Manuel Ritoré; Antonio Ros Proof of the double bubble conjecture Kamber, Franz W. Franz W. Kamber; Peter W. Michor The flow completion of a manifold with vector field Kirillov, A. A. A. A. Kirillov

30. Mathematical Recreations A notorious case is the double bubble conjecture, which states that the shape This is the double bubble conjecture. If it is true, the radii of the http://www.fortunecity.com/emachines/e11/86/bubble.html

web hosting domain names photo sharing Mathematical Recreations by Ian Stewart Double Bubble,Toil and Trouble The dodecahedron has 20 vertices, 30 edges and 12 faces- each with five sides. But what solid has 22.9 vertices, 34.14 edges and 13.39 faces -each with 5.103 sides? Some kind of elaborate fractal , perhaps? No, this solid is an ordinary, familiar shape, one that you can probably find in your own home. Look out for it when you drink a glass of cola or beer, take a shower or wash the dishes. I've cheated, of course. My bizarre solid can be found in the typical home in much the same manner that, say, 2.3 children can be found in the typical family. It exists only as an average. And it's not a solid; it's a bubble. Foam contains thousands of bubbles, crowded together like tiny, irregular polyhedra-and the average number of vertices, edges and faces in these polyhedra is 22.9, 34.14 and 13.39, respectively. If the average bubble did exist, it would be like a dodecahedron , only slightly more so. Coalescing Bubbles that enclose unequal volumes have shapes that remain a mathematical challenge.

31. Proof Of The Double Bubble Conjecture, Michael Hutchings, Frank Morgan, Manuel R Proof of the double bubble conjecture. Source Ann. of Math. 155, no. 2 (2002),459–489 Abstract. We prove that the standard double bubble provides the http://projecteuclid.org/Dienst/UI/1.0/Display/euclid.annm/1032210020

Current Issue Past Issues Search this Journal Editorial Board ... Subscriptions Michael Hutchings Frank Morgan and Antonio Ros Proof of the double bubble conjecture Source: Ann. of Math. Abstract: Primary Subjects: Full-text: Open access Download the full-text in the following format: PDF (278 KB) Euclid Identifier: euclid.annm/1032210020 Mathmatical Reviews number (MathSciNet): Zentralblatt Math Identifier: To Table of Contents for this Issue journals search login ... home

32. Page014 March 2001 The double bubble conjecture is a true statement in 1 Proof ofthe double bubble conjecture in R4 and certain higher dimensional cases, http://www.math.utoledo.edu/~jevard/Page014.htm

Mathematical events (In reverse chronological order) Page maintained by Jean-Claude Evard. Last update: June 9, 2003. May 13, 2003: The Strong Perfect Graph Conjecture: Update by Gérard Cornuéjols: Link April 23, 2003: The breakthrough in the direction of the twin prime conjecture announced on March 13, 2003, does not hold. Andrew Granville of the Universite de Montreal and K. Soundararajan of the University of Michigan have found an error in the preprint Small gaps between primes of Daniel Goldston and Cem Yildirim. Additional information: 1. See announcement of the error on the Web site of the American Institute of Mathematics Link 2. See description of the error on the Web site of the American Institute of Mathematics Link 3. Article by Glennda Chui published in the Mercury News of May 7, 2003 Link 4. See Current Research on the Web site of Daniel Goldston Link April 16, 2003: The Poincare Conjecture seems to be on the edge to be solved: In an article published in the Moscow Time, Sara Robinson reports that Grigory Perelman, from the Steklov Institute of Mathematics in St. Petersburg

33. Research Supervision Generalized the recent proof of the double bubble conjecture from R3 to R4 andcertain higher dimensional cases. Papers, Component bounds for http://www.williams.edu/Mathematics/fmorgan/student2.html

Research Supervision Ph.D. Students Julian Lander , MIT, 1984. Gave the first positive results in general codimension on when a minimizing surface inherits the symmetries of the boundary. Thesis: Julian Lander, Area-minimizing integral currents with boundaries invariant under polar actions, Trans. Amer. Math. Soc. 307 (1988) 419-429. Benny Cheng Benny Cheng, Area minimizing cone type surfaces and coflat calibrations, Indiana U. Math. J. 37 (1988) 505-535. Gary Lawlor , Stanford, 1988. Proved the five-year-old angle conjecture on which pairs of m-planes are area-minimizing. Developed a curvature criterion for area minimization and classified all area-minimizing cones over products of spheres. Gave the first example of nonorientable area-minimizing cones. Thesis: Gary Lawlor, A sufficient criterion for a cone to be area-minimizing, Memoirs of the AMS 91, No. 464 (1991) 1-111. Mohamed Messaoudene , MIT, 1988. Analyzed the mass norm in the first nonclassical case, L R . Thesis: Mohamed Messaoudene, The unit mass ball of three-vectors in

34. SMALL Research With Hutchings, Ritore, and Ros, Proof of the double bubble conjecture, ERAAMS 6 (2000) 4549; Ann. Math. 155 (2002) 459-489. http://www.williams.edu/Mathematics/SMALL2005Research.html

SMALL 2005 Research Descriptions Number Theory: Irrationality and Diophantine Approximation Advisor: Edward Burger Project Description: While almost all numbers are known to be irrational, a fundamental question remains: Given an irrational, how irrational is it? Here we will consider questions from diophantine approximation-an area in which we develop insights into the structure of certain irrational quantities by approximating them by rational numbers. Thus we learn about the mysterious irrational numbers by examining how close they come to the mundane world of rationals. Commutative Algebra Advisor: Susan Loepp Project Description: Consider the set of polynomials in one variable over the complex numbers. We can define a distance between these polynomials that turns out to be a metric. The cauchy sequences with respect to this metric, however, do not all converge. So, we can complete this metric space to get a new metric space in which all cauchy sequences converge. What is this new space algebraically? Surprisingly, it turns out to be the set of formal power series in one variable over the complex numbers. The idea of completing a set of polynomials generalizes to rings. The relationship between a ring and its completion is important and mysterious. We will work to unravel this mystery by constructing examples of rings whose relationship to their completion is truly bizarre. Geometry Advisor: Frank Morgan Project Description: It is well known (Schwarz 1884) that a round soap bubble is the most efficient, least-area way to enclose a given volume of air. The

35. Math Trek: Bubbles And Math Olympiads, Science News Online, June 16, 2001 12)101. Reichardt, BW, et al. In press. Proof of the double bubble conjecturein R4 and certain higher dimensions. Pacific Journal of Mathematics. http://www.sciencenews.org/articles/20010616/mathtrek.asp

Science News Books. Subscribe to Science News ... Science News for Kids Math Trek Bubbles and Math Olympiads Food for Thought Allergic to computing? Science Safari Geo Name Game TimeLine 70 Years Ago in Science News Science News e-LETTER. ... Week of June 16, 2001; Vol. 159, No. 24 Bubbles and Math Olympiads Ivars Peterson Predicting the geometric shapes of soap bubble clusters can lead to surprisingly difficult mathematical problems. Which one of these two configurations of five planar bubbles of equal area has the smaller total perimeter? The more symmetric candidate isn't always the winner. Frank Morgan Frank Morgan of Williams College in Williamstown, Mass., recently illustrated such difficulties when he invited an audience of mathematicians, students, and others to vote on which one of a given pair of different representations of the same number of clustered planar bubbles would have a smaller total perimeter. Assembled for a ceremony at the National Academy of Sciences in Washington, D.C., to honor the 12 winners of the 2001 U.S.A. Mathematical Olympiad (USAMO), audience members were wrong as often as they were right. "These are very tricky questions," Morgan says. "You often can't even come up with reasonable conjectures."

36. Mathenomicon.net : News : Archive A proof of the double bubble conjecture has been announced at the RoseHulmanInstitute of Technology in Indiana, United States. 12th July 2000 http://www.cenius.net/news/archive/default.php

Feedback Site map Archive Older mathematical and site news. Home About this Site News Archive ... Search Return to News Index 21th March 2000 Double bubbles A proof of the Double Bubble conjecture has been announced at the Rose-Hulman Institute of Technology in Indiana, United States. 12th July 2000 Archimedes' lost writing revealed A text written by Archimedes is being restored by scientists in New York. 23rd August 2000 Cambridge raising funds for Newton writings The University of Cambridge is attempting to raise funds to buy an archive of Sir Isaac Newton's papers. 26th February 2001 Death of Claude Shannon It is with regret that Mathenomicon.net reports the death of Claude Elwood Shannon, the noted mathematician who essentially founded information theory. 13th March 2001 Mathenomicon.net moves to new servers Mathenomicon.net is moved to new, commercial servers, and now utilises active server pages. 24th August 2001 Norway launches Abel Prize The government of Norway has established a new mathematical prize similar to the Nobel prizes. 27th October 2001 Mathenomicon.net site traffic up 900%

37. Double Bubble Conjecture - Information Technology Services loop quantum gravity double bubble conjecture. what is it and how was it prooved? Discuss double bubble conjecture Here, Free! Become A Member, Free! http://www.physicsforums.com/archive/t-1086_Double_Bubble_Conjecture.html

Technology Services Mathematics General Math Archives Double Bubble Conjecture loop quantum gravity - Double Bubble Conjecture what is it and how was it prooved? Discuss Double Bubble Conjecture Here, Free! Zefram - Double Bubble Conjecture Well, you can start off with Mathworld (http://mathworld.wolfram.com/DoubleBubble.html). Discuss Double Bubble Conjecture Here, Free!

38. International Journal Of Mathematics And Mathematical Sciences 8 J. Hass, M. Hutchings, and R. Schlafly, The double bubble conjecture, Electron.Res. 14 F. Morgan, Proof of the double bubble conjecture, Amer. http://www.hindawi.com/journals/ijmms/volume-32/S0161171202207188.html

Home About this Journal MS Tracking System Author Index ... Contents IJMMS 32:11 (2002) 641-699. DOI: 10.1155/S0161171202207188 THE DOUBLE BUBBLE PROBLEM IN SPHERICAL AND HYPERBOLIC SPACE ANDREW COTTON and DAVID FREEMAN Received 28 July 2002 We prove that the standard double bubble is the least-area way to enclose and separate two regions of equal volume in , and in S when the exterior is at least ten percent of S 2000 Mathematics Subject Classification: 49Q10, 53A10. The following files are available for this article: Full-Text PDF for institutional subscribers REFERENCES [1] A. F. Beardon, The Geometry of Discrete Groups , Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. MathSciNet Double bubbles in the -torus , preprint, 2002, http://www.arXiv.org/math.DG/0208120 [3] P. Castillon, On surfaces of revolution with constant mean curvature in hyperbolic space , Ann. Fac. Sci. Toulouse Math. (6) MathSciNet [4] J. Corneli, P. Holt, N. Leger, and E. Schoenfeld, Partial results on double bubbles in and [5] J. Foisy

39. International Journal Of Mathematics And Mathematical Sciences 3 J. Hass and R. Schlafly, Double bubbles minimize, Ann. of Math. M.Ritoré, and A. Ros, Proof of the double bubble conjecture, Ann. of Math. http://www.hindawi.com/journals/ijmms/volume-2005/S0161171205503776.html

Home About this Journal MS Tracking System Author Index ... Contents IJMMS 2005:8 (2005) 1283-1290. DOI: 10.1155/IJMMS.2005.1283 STABILITY IN A BALL-PARTITION PROBLEM THOMAS I. VOGEL Received 16 August 2004 Stability for a capillary problem with surfaces meeting along a singular curve is analyzed using eigenvalue methods. The following files are available for this article: Full-Text PDF for institutional subscribers REFERENCES [1] M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables , National Bureau of Standards Applied Mathematics Series, vol. 55, Superintendent of Documents, U.S. Government Printing Office, District of Columbia, 1965, 3rd printing, with corrections. MathSciNet [2] R. Courant and D. Hilbert, Methods of Mathematical Physics. Vol. I , Interscience Publishers, New York, 1953. MathSciNet [3] J. Hass and R. Schlafly, Double bubbles minimize , Ann. of Math. (2) MathSciNet Proof of the double bubble conjecture , Ann. of Math. (2) MathSciNet [5] J. H. Maddocks, Restricted quadratic forms and their application to bifurcation and stability in constrained variational principles , SIAM J. Math. Anal.

40. Furman Mathematics: Morgan Abstracts The double bubble conjecture says that the familiar double soap bubble whichforms when two soap bubbles come together provides the most efficient way to http://math.furman.edu/activities/clanton/morgan.html

Furman Mathematics Department People Resources Activities ... Advice Abstracts for Frank Morgan, Clanton Visiting Mathematician, 2002-2003 PROOF OF THE DOUBLE BUBBLE CONJECTURE ABSTRACT: A single round soap bubble provides the most efficient, least-area way to enclose a given volume of air. The Double Bubble Conjecture says that the familiar double soap bubble which forms when two soap bubbles come together provides the most efficient way to enclose and separate two given volumes of air. We'll discuss the problem, the recent proof, important contributions by undergraduates, and remaining open problems. THE SOAP BUBBLE GEOMETRY CONTEST ABSTRACT: A guessing contest with demonstrations, explanations, prizes, and some recent math news, including new results by undergraduates.

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