Math II (Несколько текстов для зачёта), страница 2

An artist's cross-section depicts an even larger section of the 3-D "bubble" universe that Margaret Geller and colleagues mapped, containing the stickman (red). Galaxies are arrayed on the surfaces of the bubbles.

Geller is finding faster ways to plot redshifts with help from a fellow CfA scientist, Dan Fabricant. He created a hectospec to permit the MMT to quickly scan the skies. The crowded CfA lab where Fabricant's creation has been realized looks more like a machine shop, however, than a testing ground for delicate astronomical instrumentation. Most of the room, in fact, is taken up by a clumsy-looking circular structure. With 300 thin steel rods radiating evenly from its edge toward a two-foot-diameter stainless steel plate in the middle, it has the appearance of a large eye with a two-foot metal pupil. This ersatz "eye," Fabricant explains, will be placed behind the new MMT mirror, with the mirror's gathered light focused onto prisms at the ends of the rods.

The real stars of Fabricant's creation are two small, innocuous-looking "boxes" suspended from metal tracks. Around this CfA lab, they are better known as Fred and Ginger, though at this moment their progenitor refuses to distinguish one from the other. "Your choice," Fabricant laughs. '"They look alike."

Named by Geller after that famously fluid pair of cinematic dancers, Fred and Ginger must in fact imitate the grace and skill of their namesakes while aiding in the observation of galaxies. Each robot glides along its track suspended above the steel eye. Moving as fast as three feet per second, the robots position the steel rods on the metal surface at points where observers want to get more information. Concealed within each rod is a fiber-optic strand of delicate synthetic quartz. Though each fiber-optic strand is only 250 micrometers in diameter--about the width of two human hairs--it can transmit a galaxy's light to machines that can determine its spectrum and redshift.

What all this ungainly machinery offers is speed. When Geller began mapping the universe, researchers counted themselves lucky to get redshifts for 30 galaxies a night. Fred and Ginger's swift telescopic "dance," however, will allow Geller and her team to get redshifts for several thousand a night.

For someone who dreams of other worlds, Margaret Geller began her journey to celestial cartography with more earthly concerns. Despite immersing herself from an early age in acting, by the time she went to college at the University of California at Berkeley, she says, "I realized that being an actress wasn't really what I thought, because you repeated the same thing over and over. But I was still fascinated by the idea of theater, performing and being some other character. I liked the attention."

Berkeley's big, impersonal classes were a turnoff for Geller. The ones in science, though, tended to be smaller, and Geller gravitated toward physics. Like her father, she planned to pursue graduate studies in solid-state physics, but was advised against it by one of her professors. "He said you want to look for a field that will be exciting ten years after you get your PhD, because that's when you'll be mature as a scientist." He suggested either astronomy or biophysics. Geller grins at the memory. "I couldn't imagine doing biophysics, whereas astronomy seemed sort of exciting."

Dan Fabricant developed this hectospec, which allows optical fibers to be placed on images of individual galaxies so their redshifts and thus distances can be measured.

After becoming only the second woman to get a physics PhD at Princeton, Geller came to the CfA in 1974 to continue work on galaxy clusters. Then, in the 198os, researchers found a large, clark region in the universe where nothing could be seen. Like many, Geller and CfA coworker John Huchra did not believe that such empty regions could be common.

"John and I started to do a survey of nearby clusters of galaxies. It took me a long time to understand that the real issue was the existence of large-scale patterns. People thought that these large-scale patterns didn't exist, so why go look for them?" Geller laughs, her hands as animated as her voice. "One of the things that made us recognize that there might be bigger structures was that study of the dark region. Of course, in the great wisdom of thinking you know the answer, I and many others thought it must be a mistake."

So she set out to prove it so. She and her colleagues decided to think big and map not just a thousand galaxies in a strip across the sky, as was the fashion, but many thousands over a period of years. With few qualms about nights in the cold, remote confines of mountaintop telescopes, her collaborators would obtain much of the data while Geller interpreted it back in Cambridge.

Geller decided that if they were to find any pattern, it was most likely over a wide range. Thus, to get an idea of the shape and size of the universe's "continents and oceans," she figured that neither random sampling nor intensive study of one small patch, which previous surveys had used, would work well. Instead, they would examine thin "slices" running across the sky, each six degrees wide. They hoped the very width of the slices would give a good sampling of the universe's structure.

No one could have predicted what that first slice of the sky would show. "We were lucky that it was such a clean picture," Geller says, "because if it hadn't been, we wouldn't have seen it, as few believed in those kinds of big patterns in the universe." Today, with large mapping surveys under way, studying the structure of the universe has never been more in vogue. "The fact that there were such clear patterns," says Geller, "really captured people's imaginations."

Geller enjoys talking about her own captive imagination during a lecture she gave to a group of Hollywood TV producers shortly before winning a MacArthur Fellowship in 1990. "I sat at the head table," she remembers with a smile, "and was introduced to a number of people, one of whom was Michael Eisner. It did not register with me that it was the Michael Eisner. I explained how we did a strip of the universe to figure out if it had 'continents' and 'oceans,' so I did a demo by cutting up a map. I needed somebody to hold it, so I turned to Eisner and asked him. As soon as he stood up, flashbulbs went off, and I thought, 'Ooh, I must be doing a great job!'"

Her laughter echoes against the stickman picture and into the adjacent corridor, where pictures of the MMT hang.

"Where are they? homework is due. Must be easy."

Margaret Geller sounds uncharacteristically anxious. In this hour before her class, she sits in the Harvard Science Center's car6 cum study hall, waiting for students to show up for help. Now, with only ten minutes to go, the first student finally comes around. A teaching assistant comes to his aid as Geller rushes off to make sure everything is in place for her class--her performance.

Today, most of the props from her previous class have returned, including a large, clear blue inflatable ball that Geller uses to show how parallel lines behave in a dosed universe. (They intersect.) But as she spends the first ten minutes explaining how students should prepare for the midterm, just a week away, their tension becomes palpable. This is not an easy course by anyone's description, and what is needed, Geller explains now with carefully staged gesture, tone and smile, is not just correct answers but understanding. "If you show you don't understand," she warns, "we'll deduct points. We're looking to see that you really understand what you've been learning."

For a scientist engaged in work that often takes decades before coming to fruition, Geller has achieved moderate fame not only as an astronomer, lecturer, TV show guest and member of the National Academy of Sciences but also as a teacher and adviser. Today, in fact, one of her former graduate students, a new assistant professor at Brown, meets up with her at CfA after class.

"Ian has taken the most beautiful images of clusters," Geller says by way of introduction. Ian Dell'Antonio himself is far more modest. He is studying where dark matter resides in galaxies and why. But for now, he is concerned about the introductory astronomy course he teaches, where a student of Egyptology keeps him on his toes about what the ancient Egyptians did, and did not, know about the practice of astronomy.

As her former student talks, Geller looks happy, dreamy. In a short film she cowrote on her work as an astronomical mapmaker, Geller included a photograph of herself as a young child. In a light summer dress and blowing a dandelion gone to seed, she appears to be not of the moment but of worlds beyond. Geller wears the exact same look now, as if she is indeed among galaxies, gazing out only occasionally to those who remain earthbound.

"There's something really beautiful about science, that human beings can ask these questions and can answer them," Margaret Geller says about her life as a mapmaker of the universe. "You can make models of nature and understand how it works. There's something exquisite and beautiful about that."

CONSTRUCTING ESCHER

Source: Science News, 12/23/2000&12/30/2000, Vol. 158 Issue 26&27, p410, 1/3p, 1bw Author(s): Peterson, Ivars

Nowadays, mathematicians, computer scientists, and others have a variety of speedy computer-based methods for generating hyperbolic patterns and tilings. M.C. Escher didn't have such technology at his disposal. Neither did Henri Poincar and other 19th-century mathematicians who drew various pictures of the hyperbolic plane. They relied on the traditional tools of geometry-compass and straight- edge-to create their diagrams.

In an article to appear in the January 2001 American Mathematical Monthly, however, Chaim Goodman-Strauss of the University of Arkansas in Fayetteville suggests what these procedural details might have been. He offers techniques and instructions for drawing by hand some tilings of the Poincar model of the hyperbolic plane.

"Remarkably, this may be the first detailed, explicit synthetic construction of triangle tilings of the hyperbolic plane to appear," Goodman-Strauss notes.

His directions are built out of basic tasks familiar to a student of Euclidean geometry: bisecting a line segment, drawing a parallel through a given point, drawing a perpendicular through a given point, constructing a circle through three given points, and a handful of other operations.

A suitable combination of those activities enables one to construct, for example, the hyperbolic line that passes through two given points. To achieve this, one must create a geometric scaffolding of lines and points outside a Poincar disk to guide the drawing of arcs and points within the circle's boundary.

Goodman-Strauss worked out the method by extending his expertise in Euclidean geometry to encompass the types of curves and angles necessary to represent hyperbolic structures. He admits that there's probably nothing original in his contribution. "Surely, this was all well-known at the end of the 19th century, just as it has long been forgotten at the dawn of the 21st," he remarks.

Nonetheless, reviving long-lost construction techniques has value. Such exercises offer an illuminating window on not only Escher's art but also the remarkable work of earlier mathematicians who explored non-Euclidean geometries.

"It is wonderfully satisfying to make these pictures by hand, patiently, with pencil and paper, compass and straightedge," Goodman-Strauss adds. "I encourage you to test this theorem for yourself!"

COST ALLOCATION: AN APPLICATION OF FAIR DIVISION

Source: Mathematics Teacher, Oct2000, Vol. 93 Issue 7, p600, 4p, 2 diagrams

Author(s): Goetz, Albert

Although the subject of cost allocation has been extensively discussed in the literature of political economics, it has been generally neglected in mathematical literature. However, cost allocation affords a practical extension of fair-division techniques-one that is readily accessible to secondary school students and that gives them a simple yet powerful application of mathematics to real-world problem solving. A study of the concepts and the mathematics involved in cost allocation is most appropriate in a discrete mathematics course or a modeling course, but a case can be made for including this topic in other courses, as well. This article presents a typical cost-allocation problem with possible solutions and includes suggestions for presenting similar problems in the classroom. The basics of the problem follow closely from Young (1994).

THE SEWAGE-TREATMENT-PLANT PROBLEM, PART 1

Let us consider two towns, Amity and Bender, each of which needs to build a new sewage-treatment plant. Let us further suppose that the cost for Amity to build the sewage-treatment plant is $15 million and that the cost for Bender to construct the plant is $9 million. Were the two towns to pool their resources, the cost of one sewage-treatment plant, built to service both towns, would be $19 million. Should the two towns decide to build only one plant, and if so, how should the cost be divided?

I find that having small groups work on this problem is both productive and enjoyable for students. Each group is first given one of the two towns to represent and asked to plan a negotiating strategy for the town. Each group is then paired with a group that represents the other town so that the groups can work out a solution.