41. College Mathematics Journal, The: Mathematical Recreations: Double Bubble, Toil A famous outstanding conjecture, the double bubble conjecture, states thatwhere two bubbles coalesce, the shape consists of three spherical surfaces. http://www.findarticles.com/p/articles/mi_qa3773/is_199805/ai_n8791409

@import url(/css/us/style1.css); @import url(/css/us/searchResult1.css); @import url(/css/us/articles.css); @import url(/css/us/artHome1.css); Home Advanced Search IN free articles only all articles this publication Automotive Sports FindArticles College Mathematics Journal, The May 1998 Content provided in partnership with 10,000,000 articles Not found on any other search engine. Featured Titles for ASA News ASEE Prism Academe African American Review ... View all titles in this topic Hot New Articles by Topic Automotive Sports Top Articles Ever by Topic Automotive Sports Mathematical Recreations: Double Bubble, Toil and Trouble College Mathematics Journal, The May 1998 by Seppala-Holtzman, David N Save a personal copy of this article and quickly find it again with Furl.net. It's free! Save it. Mathematical Recreations: Double Bubble, Toil and Trouble, Ian Stewart. Scientific American 278:1 (January 1998) 104-107. A famous outstanding conjecture, the "Double Bubble Conjecture," states that where two bubbles coalesce, the shape consists of three spherical surfaces. This was recently proved by Joel Hass and Roger Schlafly for the special case of two coalescing bubbles enclosing equal volumes. The unequal volume case remains unsolved although it is known that, whatever the configuration, the result is a surface of revolution. The first of Plateau's laws, the 120-degree rule, would yield a beautifully simple result, if the Double Bubble Conjecture were true. Let the respective radii of the two coalescing bubbles be r and s, and let t be the radius of the spherical surface (assuming the conjecture) that forms the boundary between them. Then 1/r = 1/s + 1/t. DNS-H

42. Mathematics Magazine: Math Gene: How Mathematical Thinking Evolved And Why Numbe Hass, Joel, General double bubble conjecture in R3 solved, Focus 20 (5) (May/June2000) 45. Stewart, Ian, Bubble trouble It s taken 170 years, http://www.findarticles.com/p/articles/mi_qa3789/is_200012/ai_n8914115

@import url(/css/us/style1.css); @import url(/css/us/searchResult1.css); @import url(/css/us/articles.css); @import url(/css/us/artHome1.css); Home Advanced Search IN free articles only all articles this publication Automotive Sports FindArticles Mathematics Magazine Dec 2000 Content provided in partnership with 10,000,000 articles Not found on any other search engine. Featured Titles for ASA News ASEE Prism Academe African American Review ... View all titles in this topic Hot New Articles by Topic Automotive Sports Top Articles Ever by Topic Automotive Sports Math Gene: How Mathematical Thinking Evolved and Why Numbers are Like Gossip, The Mathematics Magazine Dec 2000 by Campbell, Paul J Save a personal copy of this article and quickly find it again with Furl.net. It's free! Save it. Devlin, Keith, The Math Gene: How Mathematical Thinking Evolved and Why Numbers are Like Gossip, Basic Books, 2000; xvii + 328 pp, $25. ISBN 0-465-01618-9. Hass, Joel, General double bubble conjecture in R3 solved, Focus 20 (5) (May/June 2000) 4-5. Stewart, Ian, Bubble trouble: It's taken 170 years, but now the problem is solved, New Scientist (25 March 2000) 6. Cipra, Barry, Dana Mackenzie, and Charles Seife, Rounding out solutions to three conjectures, Science (17 March 2000) 1910-1912. Continue article Advertisement Michael Hutchins (Stanford University), Frank Williams (Williams College), Manuel Ritore and Antonio Ros (University of Granada) have announced a proof of the general double bubble conjecture in R3. The conjecture says that the surface enclosing two given volumes that has smallest area consists of two bubbles connected by a common surface. The proof proceeds by showing that other candidates for optimality could be deformed into other feasible solutions with smaller surface area, i.e., cannot be even locally optimal. Undergraduate students of Morgan's had already proved the double bubble conjecture in twoand in four-dimensional space.

43. BBC News | SCI/TECH | Double Bubble Is No Trouble Mathematicians reveal why double soap bubbles take the shape they do. socalled double bubble conjecture - that the familiar double soap bubble is the http://news.bbc.co.uk/1/hi/sci/tech/685243.stm

low graphics version feedback help You are in: Sci/Tech Front Page World UK ... AudioVideo Tuesday, 21 March, 2000, 12:24 GMT Double bubble is no trouble Nature will allow bubbles like this... By BBC News Online science editor Dr David Whitehouse Four mathematicians have announced a proof of the so-called 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, Frank Morgan of Williams College, Massachusetts, announced that he, Michael Hutchings of Stanford, and Manuel Ritori and Antonio Ros of Granada, had finally proved what the double soap bubble had known all along. When two round soap bubbles come together, they form a double bubble. 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. Mathematicians have expressed surprise that when two bubbles are joined in this way that the interior surface that separates them is not bowed all that much.

44. Invited Speakers Soap Bubble Geometry Contest double bubble conjecture. Biography DOUBLE BUBBLECONJECTURE Professor Frank Morgan Williams College http://www.rose-hulman.edu/mathconf/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

45. HALES'S FLYSPECK SLIDE PRESENTATION (September 15, 2004) Image of a torus bubble; double bubble conjecture What is the surface minimizingway of enclosing two equal volumes? In 1911, David Boys in his book on Soap http://www.math.pitt.edu/~thales/flyspeck/blog/utah_tphol.html

FLYSPECK September 15, 2004 TPHOL Conference Park City Utah (IMAGE) Image from AMS Notices cover counterexample to Kempe's proof The four color theorem The Four Color Theorem was first proved in 1880s by Kempe. As a result of the proof, Cayley had Kempe elected as a Fellow of the Royal Society. After more than a decade, a counterexample was found to the proof. More on the 4CT Appel-Haken proof 1976 : The proof used 1200 hours of supercomputer time, and the journal The publication included 400 microfiche pages of diagrams and lemmas. Even More on the 4CT There was a second generation proof several years ago by Robertson, Seymour, and Thomas in 1995. There is the formal proof effort by B. Werner and G. Gonthier, Danof Reaction to the 4CT 1990. Paul Halmos. ``We are still far from having a good proof of the Four Color Theorem ... The present proof relies in effect on an Oracle, and I say down with Oracles! They are not mathematics.'' 1982. Frank Bonsall. ``...if the solution involves verification of special cases ... such a solution does not belong to mathematical science at all ... We cannot possibly achieve what I regard as the essential element of a proof our own personal understanding if part of the argument is hidden away in a box ... Perhaps no great harm is done, except for the waste of resources that would be better spent on live mathematicians,...'' (IMAGE) Image of a sphere (IMAGE) Image of a double bubble (IMAGE) Image of a torus bubble Double bubble conjecture What is the surface minimizing way of enclosing two equal volumes?

46. University Of Pittsburgh: Department Of Mathematics The problem of two bubbles, known as the double bubble conjecture, The doublebubble conjecture, Electron. Res. Announc. Amer. Math. Soc. 1 (1995), no. http://www.math.pitt.edu/articles/kelvin.html

Table of Contents Fall 2001 Cannonballs and Honeycomb: Kelvin If we turn to the next page after the Kepler conjecture in Kepler's Six-Cornered Snowflake , we find a discussion of the structure of the bee's honeycomb. The rhombic dodecahedron was discovered by Kepler through close observation of the honeycomb. The honeycomb is a six-sided prism sealed at one end by three rhombi. By sealing the other end with three additional rhombi, the honeycomb cell is transformed into the rhombic dodecahedron. Figure 8 The cannonball packing of balls leads to honeycomb cells. It is also related to more general foam problems. If we tile space with hollow rhombic dodecahedra, and imagine that each has walls made of a flexible soap film, we have an example of a foam. Kelvin The problem of foams, first raised by Lord Kelvin, is easy to state and hard to solve. How can space be divided into cavities of equal volume so as to minimize the surface area of the boundary? The rhombic dodecahedral example is far from optimal. Lord Kelvin proposed the following solution. Truncated octahedra fill space (see Figure 9).

47. CIM Bulletin #9 Double Bubbles. In the March 17, 2000 Science is a piece by Barry Cipra Why Do, and reporting on the recent solution of the double bubble conjecture. http://at.yorku.ca/i/a/a/h/13.htm

Topology Atlas Document # iaah-13 from CIM Bulletin #9 What's New in Mathematics Race to settle Catalan conjecture: it's people vs. computers Ivars Peterson reports in the December 2, 2000 Science on recent progress towards the resolution of this 150-year-old conjecture. Catalan noted that 8 = 2 and 9 = 3 are consecutive integers and conjectured that they were the only set of consecutive whole powers. This translates to the Fermat-like statement that the equation xp - yq = 1 has no whole-number solutions other than 3 Incompressible is incomprehensible New encription algorithm A new Federal encryption algorithm was reported in the October 20, 2000 Chronicle of Higher Education. The article, by Florence Olsen, relates how the Commerce Department, after a 4-year search, has declared the new federal standard for protecting sensitive information to be Rijndael, an algorithm named after its inventors Vincent Rijmen and Joan Daemen. The two Belgians beat out 20 other entries, including teams from IBM and RSA. The new encryption algorithm, of which no mathematical details were given, can be made stronger as more powerful computer processors are developed. This was an entry requirement for the competition. According to Raymond G. Kammer of NIST, which managed the selection process, it should be good for about 30 years, "that is, if quantum computing doesn't manifest itself in five or six years." Updating Ramanujan Double Bubbles Squeeze in a few more?

48. Student Papers At The NES/MAA Spring 2000 Meeting The double bubble conjecture in R3 has recently been proved. In this talk we willdiscuss the double bubble conjecture in the sphere, S3, and in hyperbolic http://www.southernct.edu/organizations/nesmaa/studentpapersspring2000.html

NORTHEASTERN SECTION OF THE MAA Spring 2000 MEETING June 16-17, 2000 St. Paul's School, Concord, New Hampshire Student Papers Schedule and Abstracts You can click on individual abstracts in the schedule or jump to the full set of student paper abstracts Student Papers Session I: . Double Bubble Conjectures Andrew Cotton, Harvard University and Williams College The Isoperimetric Problem on the Cylinder David Freeman, Harvard University and Williams College The Isoperimetric Problem on Singular Surfaces John Spivack, Williams College Student Papers Session II: Diophantine OlympicsWhat Does It Mean to Be a World Champion? Amanda Folsom, The University of Chicago Diophantine OlympicsCan Perfect Squares Score a Perfect 10? Alexander Pekker, Stanford University Diophantine OlympicsHow Do the Primes Factor in the Competition? Julia Snyder, Williams College Diophantine OlympicsIs It Possible for a Bad Team to Earn a Gold Medal? Rungporn Roengpitya, Williams College Student Papers Session III: Mall Time! Irma M.E.T. Servatius, Massachusetts Academy of Mathematics and Science Introduction to Geometric Dissection Nancy Peratto, Keene State College

49. American Scientist Online - A Lucid Interval (The general case of the doublebubble conjecture was proved a few years The double bubble conjecture. Electronic Research Announcements of the AMS http://www.americanscientist.org/template/AssetDetail/assetid/28331/page/6

Home Current Issue Archives Bookshelf ... Subscribe In This Section Search Book Reviews by Issue Issue Index Topical Index ... Classics Site Search Advanced Search Visitor Login Username Password Help with login Forgot your password? Change your username see full issue: November-December 2003 Volume: Number: Page: DOI: COMPUTING SCIENCE Other Formats: PDF Postscript Postscript (gzipped) A Lucid Interval Intervals at Work G , done with two pendulums attracted to large brass weights. The interval analysis assessed various contributions to the uncertainty of the final result, and discovered a few surprises. An elaborate scheme had been devised for measuring the distance between the swinging pendulums, and as a result this source of error was quite small; but uncertainties in the height of the brass weights were found to be an important factor limiting the overall accuracy. Would we be better off if intervals were used for all computations? Maybe, but imagine the plight of the soldier in the field: A missile is to be fired if and only if a target comes within a range of 5 kilometers, and the interval-equipped computer reports that the distance is [4,6] kilometers. This is rather like the weather forecast that promises a 50-percent chance of rain. Such statements may accurately reflect our true state of knowledge, but they're not much help when you have to decide whether to light the fuse or take the umbrella. But this is a psychological problem more than a mathematical one. Perhaps the solution is to compute with intervals, but at the end let the machine report a definite, pointlike answer, chosen at random from within the final interval.

50. Frank Morgan At MSRI - Geometric Measure Theory And The Proof Of The Double Bubb Measure Theory and the Proof of the double bubble conjecture Lecture 1. Measure Theory and the Proof of the double bubble conjecture - Lecture 1. http://www.msri.org/publications/ln/msri/2001/minimal/morgan/1/

MSRI Streaming Video Series Frank Morgan - Geometric Measure Theory and the Proof of the Double Bubble Conjecture - Lecture 1

51. Frank Morgan At MSRI - Geometric Measure Theory And The Proof Of The Double Bubb Measure Theory and the Proof of the double bubble conjecture Lecture 2. Measure Theory and the Proof of the double bubble conjecture - Lecture 2. http://www.msri.org/publications/ln/msri/2001/minimal/morgan/2/

MSRI Streaming Video Series Frank Morgan - Geometric Measure Theory and the Proof of the Double Bubble Conjecture - Lecture 2

52. Cwikel The double bubble conjecture says that the familiar double soap bubble is theleastarea way to enclose and separate two given volumes of air. http://www.math.princeton.edu/~seminar/2002-03-sem/MorganAbstract10-7-2002.html

Frank Morgan Williams College Proof of the Double Bubble Conjecture Abstract The Double Bubble Conjecture says that the familiar double soap bubble is the least-area way to enclose and separate two given volumes of air. I'll discuss the proof, the latest results, and open questions.

53. Seminar Abstract The double bubble conjecture says that the familiar double soap bubbleis the leastarea way to enclose and separate two given volumes of air. http://www.math.princeton.edu/~seminar/2002-03-sem/10-2-2002weekly.html

Current Seminars updated 10/2/ 2002 Week of October 2 - October 4, 2002 Statistical Mechanics Seminar Topic: Stationary Nonequilibrium States and the Continuing Quest for a Proof of Fourier's Law Presenter: Joel Lebowitz , Rutgers University Date: Wednesday, October 2, 2002, Time: 2:00 p.m., Location: Jadwin 343 Abstract: I will present a selective overview of the current state of our knowledge (more precisely of our ignorance) regarding the derivation of Fourier's Law, J( r) = -kappa gradient T( r); J the heat flux, T the temperature and kappa, the heat conductivity. This law is empirically well tested for both fluids and crystals, when the temperature varies slowly on the microscopic scale, with kappa an intrinsic property which depends only on the system's equilibrium parameters, such as the local temperature and density. There is however at present no rigorous mathematical derivation of Fourier's law and ipso facto of Kubo's formula for kappa, involving integrals over equilibrium time correlations, for any system (or model) with a deterministic, e.g. Hamiltonian microscopic evolution. Recent numerical and some analytical developments will also be discussed. Department Colloquium Topic: Compatibility, microgeometry and materials

54. Citebase - Double Bubbles In The 3-torus The double bubble conjecture on the flat 2torus, 2001. Geometric measuretheory and the proof of the double bubble conjecture. http://citebase.eprints.org/cgi-bin/citations?id=oai:arXiv.org:math/0208120

55. Citebase - Double Bubbles Minimize G/A, 20 J. Hass, M. Hutchings, and R. Schlafly, The double bubble conjecture,ERA Amer. Math. Soc. 1 (1995), 98102. G/A, 21 J. Hass and R. Schlafly, http://citebase.eprints.org/cgi-bin/citations?id=oai:arXiv.org:math/0003157

56. Michigan Undergraduate Mathematics Conference 2002 An example of a double bubble appears in the image to the right. Proof ofthe double bubble conjecture in March 2001 American Mathematical Monthly http://webapps.calvin.edu/academic/math/mumc2002/index.shtml.bk

Fourth Annual Michigan Undergraduate Mathematics Conference A conference for Undergraduate Mathematics Students and Faculty February 16, 2002 Calvin College Grand Rapids, Michigan [Featured Speaker] [Keynote Address] [Student Talks] [Poster] ... Who Wants to Be a Mathematician? See You Next Year in Dearborn! The 2003 MUMC will be held at the University of MichiganDearborn on Saturday, February 15, 2003. Visit the conference website for additional information. There are a number of other Undergraduate Mathematics Conferences across the country. Here are links to a few of them: Rose-Hulman Conference on Undergraduate Mathematics Hudson River Undergraduate Mathematics Conference (April 27, 2002) Featured Speaker Frank Morgan from Williams College will be our feature speaker. Frank Morgan is currently the Second Vice-President of the Mathematical Association of America and has long been involved in undergraduate research projects and has advised numerous students and groups of students at both graduate and undergraduate levels. At Williams College, where he currently teaches in the Mathematics and Statistics Department, he was the founding director of the very successful SMALL undergraduate research project Professor Morgan works in minimal surfaces and studies the behavior and structure of minimizers in various dimensions and settings. (If you don't know what that means, think of soap bubbles as 2-dimensional surfaces in a 3-dimensional space.) He has written four books:

57. Archived Front Pages 21/4/2000 Onwards On February 25, 2000 it was announced that the double bubble conjecture had Reference to this conjecture dates back to 1873. When two soap bubbles meet http://www.madras.fife.sch.uk/maths/ArchHomePages9.html

Welcome to our P7 visitors today! Today we see the start of a week of visits from our future S1 students. They will work through a timetable of sample lessons given by the various subject departments. At Mathematics some may even be reading this sentence at this precise moment and are about to be shown other SENTENCES created by our present S1 students. Their homework will be to create their own sentences, send them to us and if they stretch our minds into complicated enough logical knots then we will publish them here. May at the Nrich Site Try some excellent sets of problems for and . They have a deadline of 22nd May for sending in solutions. In this month's Interact Magazine our S3 students: Sam Larg, Dave Stewart, Richard Mason, Joe Neilson, Matthew Broadbent and Ross Craig have had their work published on the problem Never Prime Sue Liu in S5 had an excellent month with solutions published to By the quad - quick solve Shape and territory and Napoleon's Hat Congratulations to all these students for their excellent effort and results.

58. Tulane Math Colloquium Fall 2003 The double bubble conjecture says that the familiar double soap bubble providesthe leastarea way to enclose and separate two given volumes of air. http://www.math.tulane.edu/activities/colloquium/2003-2004.html

59. You Can’t Hear The Shape Of A Drum (Carolyn Gordon Et Al shown true for n up to 4000000 by 1993. http//www.pbs.org/wgbh/nova/proof/wiles.html. double bubble conjecture Proved (Michael Hutchings et al. 2000) http://www.mathsci.appstate.edu/Student/Orgs/Mathclub/math.html

Some Recent Mathematical Results and Open Problems Worth a Million Dollars$$$$$$$!! In 1966 the mathematician Mark Kac asked the question, Can you hear the shape of a drum? That may seem like a strange question at first, but it's no stranger than asking if one can ``see'' the chemistry of a star. In the 1991 solution, mathematicians came up with examples of drums that have different shapes but have exactly the same characteristic vibration frequencies. You wouldn't hear any difference if you listened to these drums with your eyes shut tight. http://www.ams.org/index/new-in-math/hap-drum/hap-drum.html In the margin of a book, next to the statement that x n + y n = z n I have discovered a truly remarkable proof which this margin is too small to contain http://www.pbs.org/wgbh/nova/proof/wiles.html Double Bubble Conjecture Proved (Michael Hutchings et al. 2000) The double soap bubble on the left is the optimal shape for enclosing and separating two chambers of air (a given volume) using the least amount of material (surface area). In 1995 the special case of two equal bubbles was heralded as a major breakthrough on this problem when proved with the help of a computer. The new general case involves more possibilities than computers can now handle. The new proof uses only ideas, pencil, and paper. http://www.maa.org/features/mathchat/mathchat_3_18_00.html

60. Go To Http//www.cs.appstate.edu/~sjg/class/1010/mathematician General double bubble conjecture in R^3 Solved, Focus The Newsletter of theMathematical Association of America, May/June 2000, Volume 20, Number 5, p. http://www.mathsci.appstate.edu/~sjg/class/1010/mathematician/mathematicianrefer

Go to http://www.cs.appstate.edu/~sjg/class/1010/mathematician/mathematicianreferences.html in order to click on the underlined web links. Note that if you see a reference to the Heart of Math book, then you should look there. Thomas Fuller One of Fuller's Calculations A History of Computers the history of computer speed MAD page http://www.math.buffalo.edu/mad/special/fuller_thomas_1710-1790.html African Slave and Calculating Prodigy: Bicentenary of the Death of Thomas Fuller, by Fauvel and Gerdes, Historia Mathematics 17 (1990), 141-151. The Great Mental Calculators: The Psychology, Methods, and Lives of Calculating Prodigies, Past and Present, by Steven B. Smith, 1983, On Mathematics in the History of Sub-Saharan Africa, by Paulus Gerdes, Historia Mathematica 21 (1994), 345-376, p. 345, 361-2, 366, 373. Maria Agnesi Maria Gaetana Agnesi The Living Witch of Agnesi http://www.astr.ua.edu/4000ws/witch-of-agnesi.html Why bother to learn Calculus - http://www.karlscalculus.org/why.html Definition of Calculus Women of mathematics : a biobibliographic sourcebook edited by Louise S. Grinstein p. 1-5. The Witch of Agnesi: A Lasting Contribution from the First Surviving Mathematical Work Written by a Woman - A commemoritive on the 200th anniversary of her death, by S. I. B. Gray and Tagui Malakyan, The College Mathematics Journal, Vol 30, No 4, September 1999, p. 258-268.

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