2D Foams The conjecture is only true if the cluster has no boundaries, ie the bubble clusteris this result, only the case N=2 ( the doublebubble problem) has been http://cnls.lanl.gov/~yi/foam.html
Extractions: With Pieter Swart, Francois Graner, Cyrille Flament, James Glazier A ``fluid foam'' or ``cellular fluid'' is a material that consists of a collection of cells surrounded by a continuous phase of edges tending to minimize its surface energy. This definition covers a class of systems as diverse as soap foams, emulsions, magnetic garnets, and even grain boundaries in crystals. The cellular structure of 2D fluid foams is similar to biological tissues. When cells migrate in aggregates, they move from one closely packed configuration to another. When subject to shear, bubbles in a foam rearrange from one metastable configuration to another. One question we'd like to address is will study of these cellular materials help understand biological cells? For mathematicians, foams may provide an insight to the classic ``isoperimetric problem'': how to determine the minimal perimeter enclosing a cluster of $N$ bubbles with known areas. This problem has attracted much attention recently when Hales proved a two-thousand years old honeycomb conjecture a cluster of 2D bubbles of same area reaches its minimum perimeter when all bubbles are regular hexagons. The conjecture is only true if the cluster has no boundaries, i.e. the bubble cluster is either infinite or has periodic boundary conditions. Besides this result, only the case N=2 ( the double-bubble problem ) has been well studied, N=3 (
Extractions: Washington, DC 20036-2888 USA The Second Interdisciplinary Conference of the International Society for the Arts, Mathematics, and Architecture (ISAMA) , June 24-28, 2000, University at Albany-State University of New York, Albany, New York. W Kicked off with an illustrated presentation by Ivars Peterson of Science News magazine ISAMA 2000 featured a spirited intermingling of art and math, with stimulating doses of poetry, painting, sculpture, model-building, computation, puzzle, theater, dance, and much more. About 75 people, including mathematicians, computer scientists, artists, architects, teachers, and assorted others, gathered in Albany for this meeting, the latest in a series on art and mathematics that began in 1992. The indefatigable Nat Friedman mathematician, sculptor, and ISAMA director organized and hosted the lively event.
Extractions: Last year Hutchings, Morgan, Ritore and Ros announced a proof of the Double Bubble Conjecture, which says that the familiar standard double soap bubble provides the least-area way to enclose and separate two given volumes of air. It was only with the advent of geometric measure theory in the 1960s that mathematicians were ready to deal with such problems involving surfaces meeting along singularities in unpredictable ways. The lectures will discuss modern, measure-theoretic definitions of "surface," compactness of spaces of surfaces, and finally the proof of the double bubble conjecture. Homework will vary from basic exercises to open problems. The text Geometric Measure Theory: A Beginner's Guide (3rd edition) by Frank Morgan will be made available, as well as additional notes and materials. (Students nominated by MSRI sponsors will receive a copy of the book on arrival. Several copies will be available for use by other participants.) There will be sessions on exercises and on open problems.
Extractions: Previous Story ... Related Stories Next Story Source: Williams College Date: WILLIAMSTOWN, Mass., March 18, 2000 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), 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. 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 Prove Double Soap Bubble Had It Right Four mathematicians have announced a mathematical proof of the double bubbleConjecture that the familiar double soap bubble is the optimal shape for http://www.globaltechnoscan.com/19april-25april/soap_bubble.htm
Extractions: Mathematicians Prove Double Soap Bubble Had It Right For Business Opportunities in Engineering Industry please click here 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), 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. 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, 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.
American Scientist: Foams And Honeycombs Keywords Mathematics, Kelvin problem, ideal foam, soapbubble problems, dry foams,wet dimensional space, a problem dating back to a conjecture by Johannes http://www.americanscientist.org/articles/00articles/Klarreich.html
Limsup | The Knot Genus Problem Is NP Complete Geometry Agol, Hass (who recently solved the double bubble problem), and Thurstonhave a preprint today on the computational complexity of the genus of a knot http://limsup.org/articles/02/05/07/2145256.shtml
Extractions: from the dept. Agol , Hass (who recently solved the double bubble problem ), and Thurston have a preprint today on the computational complexity of the genus of a knot in a three-manifold. They show that the determination of the knot genus is NP-hard. (The genus of a knot is the minimal genus of a surface that spans the knot. The paper contains a nice elementary introduction to the relevant definitions and the statement of the theorem.) Featured Reviews Mihailescu Proves Catalan's Conjecture Limsup Login Nickname: Related Links Agol double bubble problem Thurston computational complexity of the genus of a knot ... Also by dave This discussion has been archived. No new comments can be posted. A long-forgotten loved one will appear soon. Buy the negatives at any price. home contribute story older articles past polls ... preferences
CSU- Fresno Math Department Colloquia Dr. Hass, with his collaborator Roger Schlafly, proved last year the double BubbleConjecture which answered a question that had been first asked 2000 years http://zimmer.csufresno.edu/~cleary/colloq/oldcolloq.html
Extractions: The Math Department Colloquia are a series of talks intended for a general audience. Everyone is encouraged to attend and the talks are directed at people who have a reasonable comprehension of the topics in undergraduate mathematics. Come meet our undergraduates, graduate students and faculty as well as our distinguished guest speakers. Colloquia from 1996-1997 academic year: Rescheduled: Monday, May 5, 3pm Dr. Joel Hass from the UC Davis Department of Mathematics at 4:10-5pm in Science 145. There will be refreshments beforehand. Abstract: Dr. Hass will be speaking about the mathematical models of soap films and bubbles, which are modeled by minimal surfaces and constant mean curvature surfaces respectively. Dr. Hass, with his collaborator Roger Schlafly, proved last year the "Double Bubble Conjecture" which answered a question that had been first asked 2000 years ago and had been studied by many great mathematicians throughout history. He will be speaking about some of the innovative techniques used in this important work and will have some computer graphics as well as soap bubbles to illustrate the ideas. Monday April 21 3:10pm in Science 145: Sean Cleary , from the CSU -Fresno Mathematics Department will show the 20 minute video Not Knot , produced by the Geometry Center at the University of Minnesota. There will be a short explanatory talk in conjuntion with the video presentation. "Not Knot" is a computer-generated video which illustrates some important ideas from knot theory and hyperbolic geometry.
