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April 2000

 Double Bubbles These images, of a double bubble and its cross-section, were created by John Sullivan of the University of Illinois, Urbana-Champaign, and are used with his permission. Note the curvature in the wall between the two chambers.

In the March 17 2000 Science is a piece by Barry Cipra: "Why Double Bubbles Form the Way They Do," and reporting on the recent solution of the Double Bubble Conjecture. The problem was to give a mathematical proof that the most economical way to enclose two contiguous given volumes is by a combination of three spherical surfaces, just as shown in the John Sullivan's pictures. The solution, by Michael Hutchings of Stanford University, Frank Morgan of Williams College and Manuel Ritoré and Antonio Ros at the University of Granada, proceeds by showing that "any other, supposedly area-minimizing shape can be ever so slightly twisted into a shape with even less area."

More Monkey Math. In the March 2, 2000 Nature, a joint Siberian-Israeli team reports on the investigation of non-verbal serial memory, using the macaque monkey as a test animal. Serial memory refers to the retention of lists. The question is, is a list remembered by chaining (remembering which item follows which other) or is it remembered by associating ordinals (first, second, third, ...) to the various items? It is known that monkeys are very good at chaining. The experiments reported here show that they also have a useful grasp on ordinality. For example, "monkeys were trained on four nonverbal lists, each containing four novel photographs of natural objects ... . The task was to touch the simultaneously presented images in the correct order (A1-A2-A3-A4, B1-B2-B3-B4, C1-C2-C3-C4, D1-D2-D3-D4). When the monkeys had mastered this task, the items were shuffled, taking one item from each list, so that in two derived lists the ordinal number of the items was maintained (for example, A1-D2-C3-B4) whereas in two others it was not (for example, B3-A1-D4-C2). Lists with maintained ordinal position were acquired rapidly and virtually without error, whereas derived lists in which the ordinal position was changed were as difficult to learn as new lists. "

Squeeze in a few more? Kepler conjectured in 1611 that the most efficient way to pack equal-sized spheres (for example, identical oranges) in a box was to use the face-centered cubic configuration. It took a long time to settle this question to everyone's satisfaction. This finally happened two years ago, when Thomas Hales showed that the density of the face-centered cubic arrangement (approximately 74%) could not be improved on. Then the question was considered, suppose the spheres are packed at random, like balls being poured into a container. Was there a maximum density for a random packing? Different experiments led to different estimates of this number, leaving a confusing situation. Charles Seife reports in the March 17 2000 Science on the solution to this problem. There is no such number, and looking for it "makes no more sense than searching for the tallest short guy in the world." Random packings achieved with gentler and gentler pressure on the spheres can get arbitrarily close to Kepler's limit (and as they do so, they become more and more ordered). Seife is reporting on results recently published by S. Torquato, T. M. Truskett and P. G. Debenedetti, of the Complex Materials Theory Group at Princeton University, in Physical Review Letters.

How to win \$1,000,000 - the hard way. An Associated Press story, picked up by the March 26, 2000 Seattle Times, reports that Faber & Faber and Bloomsbury Publishing are offering a million bucks to whoever can prove that every even number is the sum of two primes. Simple? 2=1+1, 4=3+1, 6=3+3, 8=3+5, ... 98=79+19, 100=97+3, ... but the problem has been open since 1742. The stunt is in connection with the upcoming release of "Uncle Petros and Goldbach's Conjecture," by Apostolos Doxiadis. The million dollar assertion is in fact Goldbach's Conjecture. Good luck.

-Tony Phillips
SUNY at Stony Brook