The fifty-micron Möbius strip. This tiny object was grown as a crystal of the Niobium-Selenium compound NiSe3 by a Japanese team at Hokkaido University and NTT Basic Research Labs, Atsugi. The team, led by Satoshi Tanda, "soaked a mixture of Se and Nb powder at 740oC in an evacuated quartz tube" using a furnace with "a large temperature gradient; this creates a crucial non-equilibrium state inside the quartz tube in which selenium circulates through vapour, mist and liquid (droplet) phases," as reported in the May 23 2002 Nature. Their "brief communication" was entitled: "Crystal Topology: A Möbius strip of single crystals." The crystals formed on the selenium droplets. Three different topologies obtained are shown in the figure. More information is available at the Hokkaido Low Temperature Physics Group website
Baby Math on NPR. On Weekend All Things Considered for May 11, 2002 Michelle Trudeau introduced us to Karen Wynn of the Yale Infant Cognition Laboratory. Wynn and her colleagues have shown that infants (5, 6, 7-month-olds) are aware of number. You can't ask them of course, but you can capitalize on babies' low boredom threshold. When presented with too much of "the same" (e.g. puppets that sing 4-note phrases only) they look away. But they snap to attention when the puppet starts a series of 2-note phrases. Similar results with visual patterns. The report is available online.
Solving an Escher Puzzle. "The Print Shop" is one of Maurits Escher's more paradoxical creations. In the lower left-hand corner we see, through a window, a man looking at a print on the wall of a print shop. But the top of the print swells out of the shop and as we follow it clockwise through the picture it leads us back to the outside of the shop where we started, so the shop itself is in the print. This is a continuous version of the "picture within itself" that we see, in the US, on Land O'Lakes Butter boxes and in Holland on packages of Droste chocolate. The center of the print has "a large, circular patch that Escher left blank. His signature is scrawled across it." So Sara Robinson describes it in the July 30 2002 New York Times, where she tells how Hendrik Lenstra, a mathematics professor at Berkeley and at Leiden, solved the riddle of what goes in the center. The key turned out to be the revelation, by a friend who had watched Escher at work, that the artist had kept the distortions conformal (i.e. angle-preserving, like the Mercator projection). Lenstra was able to exploit this feature to give a complete mathematical analysis of the print, and to fill in the patch. The solution has been beautifully presented on the web-page Escher and the Droste effect on the Leiden website. The page shows the original print and an amazing animation of the solution. Do not miss this.
Nitin Kayal, Neeraj Saxena, Manindra Agrawal.
Image courtesy IIT Kanpur.
Post-mortem on the Geometry Center. The analysis is carried out by Jeffrey Mervis in Science for July 26, 2002: "The Geometry Center, 1991-1998. RIP." The Geometry Center was created at the University of Minnesota as one of the first NSF-funded Science and Technology Centers. "From the start, the Geometry Center faced long odds. Even its mission was controversial." The mission was "to attempt to introduce computer graphics and visualization into pure mathematics and geometry," Mervis was told by David Dobkin, who chaired the center's governing board. "It wanted to change the field, but people weren't ready for that." Another problem was the budget: $2 million a year from NSF funds otherwise typically doled out in $25,000 parcels to single investigators. "We were immediately a target for people who said we didn't deserve all that money," said Richard McGehee, who directed the Center during its final years. There is no lack of suspects, and Mervis glances at several others. But he gives the final word to Don Lewis, head of the NSF mathematics division at the time: "I didn't see any progress, so I pulled the plug."
The Geometry Center which, as McGehee remarks "had one of the first 100 Web sites" lives on virtually at the U of M.
143-Year-Old Problem Still Has Mathematicians Guessing - the headline stretches almost across the top of a page in the July 2 2002 Science Times. And right in the middle is a picture of the man himself, with the caption "In 1859, Bernhard Riemann made a hypothesis on prime numbers that hasn't been proved or refuted." The occasion is a meeting at NYU earlier this year, where "more than a hundred of the world's leading mathematicians" gathered to "swap hunches, warn of dead ends and get new ideas that could ultimately lead to a solution" of the Riemann Hypothesis. Bruce Schechter wrote this article, a beautiful piece of mathematical reporting. It blends ancient history (Hardy, Gauss, Riemann) with modern history (Hugh Montgomery, Peter Sarnak, Andrew Wiles) and enough authentic background about prime numbers, complex numbers and the zeta function to keep the exposition honest. Of course after this wonderful buildup the news is disappointing, if not surprising: "Mathematicians at the conference agreed that there was no ... clear evidence of a trail head" from which to set off in pursuit of the still elusive hypothesis. Even more tantalizing, the Riemann Hypothesis now appears as the door to a universe of undiscovered mathematics. As Montgomery puts it: "It should be the first fundamental theorem."
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