![]() While calibration techniques have proven very successful in fixed boundary problems, their applicability to fixed volume constraints has previously been limited. ![]() Metacalibration is a generalization of previously used calibration techniques and was developed at BYU by my adviser, Dr. ![]() In my research I further developed a new method of proof called metacalibration in order to tackle the as-yet-unproven triple bubble conjecture. With no special symmetry, the triple bubble conjecture appears to be exceedingly difficult, if not impossible, to prove using this method. The triple bubble conjecture in space remains unproven. ![]() The final proof relied largely on the fact that the area-minimizing figure must be a surface of revolution. Schwarz first proved in 1884 that the sphere was surface area minimizing for figures that enclose a fixed volume, but was not until 2002 that Frank Morgan and others proved the double bubble conjecture in R3. The three-dimensional analogs of these multiple bubble problems have proven even more difficult to solve. This complexity proves to be a significant barrier to further results. For example, Wichiramala’s dissertation had to consider fifty-four possible configurations in order to prove minimization of the standard triple bubble. Unfortunately, this approach is marred by an ever-increasing combinatorial complexity. This method was also employed by Wacharin Wichiramala, whose doctoral dissertation proves the corresponding result for three separated areas. Using this result, students of the 1990 SMALL group under Frank Morgan proved that the so-called “standard double bubble” was perimeter-minimizing among all figures separately enclosing two fixed areas. This reduced the argument to listing all combinatorial types meeting these requirements. Some advancements in planar multiple bubble problems were made by Frank Morgan, who showed that perimeter-minimizing figures consist of circular arcs meeting at vertices of degree three, forming 120° angles. The traditional approach has been to use calculus of variations to isolate properties of the area-minimizing figure and compare all possible figures of this type. Unfortunately, this has been proven only in some few cases. It is conjectured that the minimizer is the standard shape that soap bubbles form when clumped together. Over the past few decades, mathematicians have become increasingly interested in “multiple bubble problems.” These problems ask which figure among all those that separately contain a given number of volumes has the least surface area. My project was to further develop the technique of metacalibration, a new method of minimization proof, in order to prove the triple bubble conjecture: that the standard triple bubble is the least surface area way to separately enclose three given volumes. Gary Lawlor, Mathematics Education Goal/Purpose Besides, the researchers show how to design the experiments, constraining the soap films between two surfaces in such a way as to obtain the appropriate curves.Donald Sampson and Dr. The study shows that these calculations may be related to Plateau's problem, that is, to find the shape adopted by a soap film under certain boundary restrictions. That was the origin of the calculus of variations, which was also used in other classic problems, like that of the catenary (the shape of a chain suspended by its endpoints) and the isoperimetric curve (a curve which maximises the area it encloses). The mathematician Johann Bernoulli found the answer centuries ago when he realised that it was a cycloid (the curve described by a point on a circle rolling along a line). What shape must a wire be in order that a ball travels down it from one end to the other (at a different height) as rapidly as possible? The answer is the brachistochrone (from the Greek brachistos, the shortest, and cronos, time), the curve of fastest descent. The professor offers the example of the famous problem of the brachistochrone curve. "Of course there are other ways to solve variational problems, but it turns out to be surprising, fun and educative to obtain soap films in the shape of brachistochrones, catenaries and semicircles", Criado emphasises. Soap films always adopt the shape which minimises their elastic energy, and therefore their area, so that they turn out to be ideal in the calculus of variations, "where we look for a function that minimises a certain quantity (depending on the function)", adds the researcher. Together with his colleague Nieves Álamo, he has just published his work in the American Journal of Physics. "With the aid of soap films we have solved variational mathematical problems, which appear in the formulation of many physical problems", explains Carlos Criado, professor at the University of Málaga, speaking to SINC.
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