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
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/
Extractions: 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
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
Extractions: 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
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
Extractions: 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.
[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
Extractions: Vol. 151, No. 2, pp. 459-515 (2000) Previous Article Next Article Contents of this Issue Other Issues ... EMIS Home 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: Electronic fulltext finalized on: 27 Apr 2001. This page was last modified: 22 Jan 2002. Johns Hopkins University Press
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
Extractions: 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
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
Extractions: PostScript 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
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
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
Extractions: web hosting domain names photo sharing Mathematical Recreations by Ian Stewart 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.
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
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
Extractions: 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
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
Extractions: 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. 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. 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
Extractions: Week of June 16, 2001; Vol. 159, No. 24 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."
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
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
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
Extractions: 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: 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
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
Extractions: 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: 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.
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
Extractions: 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.
