Frank Morgan's Math Chat - DOUBLE BUBBLE CONJECTURE PROVED 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 http://www.maa.org/features/mathchat/mathchat_3_18_00.html
Extractions: March 18, 2000 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.
Extractions: 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 et al. (1995), who reduced the problem to a set of 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 of . Furthermore, the
DOUBLE BUBBLES An announcement of the result, titled The Double Bubble Conjecture , joint with Michael Hutchings and Roger Schlafly, appeared in Issue 3 of Electronic Research Announcements of http://www.math.ucdavis.edu/~hass/bubbles.html
Extractions: 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.
Soap Bubbles And Isoperimetric Problems 6 Proof of the double bubble conjecture (with F. Morgan, M. Ritore, and A. Ros), Annals of Mathematics 155 (2002), no. 2, 459489. pdf This paper proves that the standard double http://math.berkeley.edu/~hutching/pub/bubbles.html
Extractions: The typical strategy for trying to prove something like this is to assume that you have an area-minimizing double bubble, and show that if it is nonstandard then you can modify it to decrease area. The first difficulty is proving that an area-minimizing surface exists! This is not at all trivial, and was accomplished by Almgren in the 1970's. Also, to get an existence theorem you generally have to enlarge the space of surfaces under consideration to include some "bad" surfaces that you don't like, and then struggle to rule these out. In particular, we cannot a priori guarantee that the enclosed regions, or the exterior region, will be connected. This difficulty is partially addressed below. The triple bubble problem Anyway, here are some papers on this subject. [1] The shortest enclosure of three connected areas in R^2 (with C. Cox, L. Harrison, S. Kim, J. Light, A. Mauer, and M. Tilton), Given three prescribed areas A_1, A_2, A_3, we determine the shortest union of curves in R^2 whose complement has four connected components, three of which are bounded and have areas A_1,A_2,A_3. This is a generalization to multiple regions of the classical isoperimetric theorem. We introduce the ``overlapping bubbles'' trick, with which we can rule out some pathological competitors by temporarily allowing the curves to cross each other. This paper is based on work done at a Research Experiences for Undergraduates program at Williams College under the guidance of Frank Morgan. I highly recommend a program like this for any undergraduate seriously considering a career in mathematics.
Double Bubble Conjecture New from the Undergraduate Math Conference During the 17th annual RoseHulman Undergraduate Math Conference, Professor Frank Morgan of Williams College announced on Saturday http://www.rose-hulman.edu/mathconf/2000/bubble.html
Extractions: 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.
Extractions: new recent what is this? Authors: Michael Hutchings Frank Morgan Antonio Ros (Submitted on 1 Jun 2004) Abstract: Comments: 31 pages, published version Subjects: Differential Geometry (math.DG) ; Metric Geometry (math.MG) Journal reference: Ann. of Math. (2), Vol. 155 (2002), no. 2, 459489 Cite as: arXiv:math/0406017v1 [math.DG] From: Frank Morgan [ view email
Www.kurims.kyoto-u.ac.jp \MR{88e01010} \bibitemM1{M1} Frank Morgan, {\em The double bubble conjecture}, FOCUS, Math. Assn. Amer., December, 1995. \bibitemM2{M2} Frank Morgan, {\em Geometric measure http://www.kurims.kyoto-u.ac.jp/EMIS/journals/ERA-AMS/2000-01-006/2000-01-006.te
Soap Bubble - Wikipedia, The Free Encyclopedia A soap bubble is a very thin film of soapy water that forms a sphere with an iridescent surface. Soap bubbles usually last for only a few moments before bursting, either on their own http://en.wikipedia.org/wiki/Soap_bubble
About "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 theAuthor Frank Morgan, MAA Online http://mathforum.org/library/view/12694.html
Extractions: 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
Research Supervision Clay Mathematics Institute Summer Course in Geometric Measure Theory and the Proof of the Double Bubble Conjecture, MSRI, summer, 2001. Miguel Carri n lvarez, Joseph Corneli, 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 R , Linear Alg. Appl. 265
Electronic Research Announcements Proof of the double bubble conjecture Author(s) Michael Hutchings; Frank Morgan; Manuel Ritor http://www.ams.org/journal-getitem?pii=S1079-6762-00-00079-2
Tulane Math Colloquium: Fall 2003 Fall 2003. Proof of the Double Bubble Conjecture August 28 Frank Morgan, Williams College. Abstract A single round soap bubble provides the leastarea http://www.math.tulane.edu/activities/colloquium/2003-2004.html
Extractions: A single round soap bubble provides the least-area way to enclose a given volume of air. The Double Bubble Conjecture says that the familiar double soap bubble provides the least-area way to enclose and separate two given volumes of air. I'll talk about the problem, the recent proof (Annals of Math. 2002), the latest results, and open questions. No prerequisites; undergraduates welcome. The Differential Geometry of Real-World Shapes: A Case Study - The Mylar Balloon and Elliptic Functions September 4 When we look at Nature, we see shapes everywhere. But why do things take the shapes they do? In this talk, we will describe the shape of a Mylar balloon in terms of elliptic functions. (A Mylar balloon is often found at kids' birthday parties and is formed by taking two disks of Mylar, sewing them together along their boundaries and inflating.) This topic is a prime example of the interplay among physical principles, geometry, analysis and symbolic computation. Undergraduates are welcome. Language Inclusion for Timed Automata
Extractions: October 7, 1999 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.
Bubbles The double bubble conjecture was finally confirmed by a team of four mathematicians in 2000. Another great bubble mystery that has recently been solved is why the bubbles in a http://www.daviddarling.info/encyclopedia/B/bubbles.html
Extractions: Whether alone or in groups joined together, bubbles get their shape by following one simple rule: soap film always tries to form a minimal surface . As a form of amusement for children the blowing of soap-bubbles goes back to ancient times, and is seen depicted on an Etruscan vase in the Louvre. The scientific study of bubbles and films began in earnest in the 1830s with the experiments of Joseph Plateau , who added glycerine to the soap solution to produce bubbles and films that were remarkably durable. The beautiful play of colors seen on soap bubbles is due to the extreme but variable thinness of their films and is an illustration of the interference phenomenon known as Newton's rings. If at any part the film becomes thin enough a black spot appears. Touching this black spot causes the bubble to break instantly, although in its thicker portions the film can be pierced by a needle without losing continuity.
Extractions: The classification of complete stable area-stationary surfaces in the Heisenberg group H^1 , Adv. Math. no. 2 (2010) 561-600. doi:10.1016/j.aim.2009.12.002 preprint version Mean curvature flow and isoperimetric inequalities Examples of area-minimizing surfaces in the sub-Riemannian Heisenberg group H^1 with low regularity , Cal. Var. Partial Differential Equations no. 2 (2009) 179-192. doi:10.1007/s00526-008-0181-6 preprint version Area-stationary surfaces in the Heisenberg group H^1 , Adv. Math. no. 2 (2008) 633-671. doi:10.1016/j.aim.2008.05.011 preprint version The isoperimetric problem in complete annuli of revolution with increasing Gauss curvature , Proc. Royal Society Edinburgh no. 5 (2008) 989-1003. doi:10.1017/S0308210507000777 ( preprint version The relative isoperimetric inequality in Cartan-Hadamard 3-manifolds , J. Reine Angew. Math. The relative isoperimetric inequality outside a convex domain in R^n , Cal. Var. Partial Differential Equations no. 4 (2007) 421-429. Rotationally invariant hypersurfaces with constant mean curvature in the Heisenberg group H^n , J. Geom. Anal. , no. 4 (2006) 703-720.
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 http://www.eurekalert.org/pub_releases/2000-03/WC-Dbcp-2703100.php
Extractions: Williams College 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.
