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A Video That's Worth a Million Words

Award-winning video reveals the simplicity and beauty of an abstract mathematical tool

Abstraction lies at the heart of mathematics. It makes math powerful, but at the same time, it can make math hard to understand. Abstraction makes math simultaneously beautiful and austere, useful and esoteric.

But a picture can tame the mad monster of abstraction, and sometimes, a video can do so even better. Now, a pair of mathematicians has created a video (see http://www.youtube.com/watch?v=JX3VmDgiFnY) that shows how to visualize and understand Möbius transformations, which are a fundamental and highly abstract mathematical tool. The new video, "Möbius Transformations Revealed," has become an Internet sensation, with 60,000 hits on YouTube so far. It also won honorable mention in the Science 2007 Science and Engineering Visualization challenge.

A Möbius transformation begins with a plane and moves each point to a new location according to certain rules. In their video, Douglas N. Arnold and Jonathan Rogness of the University of Minnesota in Minneapolis transform a multicolored square into new shapes using Möbius transformations.  

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A Möbius transformation can turn the square on the left into the bizarre form on the right. Click here or on the image to see a video that shows how the transformation works.

A Möbius transformation alters an entire plane. To understand the transformation, it helps to focus on a square that lies on the plane. A Möbius transformation can alter the square in any of four ways. The first three ways aren't too hard to picture: the transformation can move a square around on the plane, expand or contract the square, or rotate it.

The fourth alteration is especially intriguing. A Möbius transformation can turn the square inside out. The Arnold-Rogness video illustrates this process beautifully, showing how points that start close to the square's center are sent far outward, while points near the edge of the square move toward the center.  

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When a square is turned inside out through a Möbius transformation, it takes on this flower-like shape.

 Next comes the video's magical step. The mathematicians move into the third dimension to provide a way of visualizing the Möbius transformations. They suspend a sphere above the plane and use it a bit like a slide projector. They put a picture onto the sphere, and a light at the top of the sphere shoots an image of the picture down onto the plane. The picture on the sphere is shaped in such a way that when the light projects the image onto the plane, it forms the original square.  

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A light at the top of the sphere projects an image of the square down onto the plane.

 Now imagine moving the sphere while continuing to shine the light from its top. The "slide projection system" will change the image on the plane, producing a Möbius transformation of the image. Move the sphere a bit to the left, and the projected square will move the left. Move the sphere up, and the square will expand. Rotate the sphere around its vertical axis, and the square will also rotate.

If you turn the sphere upside down but keep the light in the same spot above the plane, the square will turn inside out! This is the puzzling "inside-out" transformation.  

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Moving this sphere around produces a Möbius transformation of the image. Turning the sphere upside down turns the image inside out.

 "You need some pretty heavy mathematical machinery that people usually don't do until their first year of grad school to prove the stuff in the video," Rogness says, "but we've been showing this to high school students and they seem to get it."

Rogness and Arnold had both heard that Möbius transformations could be visualized in this way, but when they began working on the video, they realized that they had never seen a proof that the method works. They hunted through textbooks and could not find a reference to the proof, even though all the mathematicians they talked to knew it to be true. Finally, they sat down and proved it themselves.

"It's a folk theorem," Rogness says. "Everyone seems to know it but I'm still not sure when it was first proven or by whom."

The duo has been astonished by the video's popularity. "I put up the YouTube version just so that we could mention it to friends and fellow mathematicians, expecting a few hundred people might watch it," Rogness says. After the video was mentioned on the technology website Slashdot, about 20,000 people viewed it overnight, and the numbers have continued to increase ever since, Rogness says with amazement. "It's been many orders of magnitude more than I expected." 

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A Genetic Basis for Language Tones?

Scottish scientists uncover a striking link between genes for brain size and tonality in spoken language 

language tone

SPEAKING IN TONES: Scientists have linked newer versions of two genes that control brain size during development to the use of tonal languages within a population.

For the most part, the thousands of languages in the world today fall into one of two categories (notable exceptions being Japanese, some Scandinavian dialects and northern Spain's Basque tongue): tonal or nontonal.

Two linguists believe they know the genetic underpinnings for these differences. During a study of linguistic and genetic data from 49 distinct populations, the authors discovered a striking correlation between two genes involved in brain development and language tonality. Populations that speak nontonal languages (where the pitch of a spoken word does not affect its meaning) have newer versions of the genes, with mutations that began to appear roughly 37 thousand years ago.

"You can consider this as the first of the many possible studies that we could do to try to find a genetic basis for language and language typology and the different populations that speak a language," says Patrick Wong, an assistant professor of communication sciences and disorders at Northwestern University, who was not involved in this study.

In English, the pitch at which a word is spoken conveys emotion but usually does not affect its meaning. But in many sub-Saharan Africa, Southeast Asian and Latin American languages tone changes the meaning of words. For instance, the Chinese word huar said in a high pitch means flower, but in a dipping pitch means picture.

The new research, published this week inProceedings of the National Academy of Sciences USA ties this difference to two genes, ASPM and Microcephalin. The exact functions of both genes are still open to debate, but they are known to affect brain size during embryonic development. "They presumably have something to do with brain structure, because there are deleterious mutations of the genes that lead to microcephaly" (a condition in which a person's brain is much smaller than the average size for his or her age), says senior study author, Robert Ladd, a professor of linguistics at the University of Edinburgh in Scotland.

Ladd and colleague Dan Dediu, a fellow linguist at the university, focused on one particular variation of each of these two genes. "They're versions of these genes that are not only newer, but also show signatures of strong natural selection in modern humans," Ladd says. In their report, the authors note that previous studies indicate that these popular new mutations do not appear to affect intelligence, brain size or social ability. But based on their strong correlation with language tone, they surmise that they may contribute to slight differences in the cerebral cortex, the outermost layer of the brain, which, among many other functions, plays a role in our ability to understand language.

Ladd and Dediu compared 24 linguistic features—such as subject-verb word order, passive tense, and rounded vowels—with 981 versions of the two genes found in the 49 populations studied. Most of the language contrasts could be explained by geographic or historical differences. But tone seemed to be inextricably tied to the variations of ASPM and Microcephalin observed by the authors. The mutations were absent in populations that speak tonal languages, but abundant in nontonal speakers.

Northwestern's Wong says that in a field in which researchers struggle to determine whether differences arise from experience or genetics, the new study "gives us an idea that there is a genetic side to things." He says the research indicates that small differences in brain organization determined by genetic makeup may be amplified by cultural factors and contact with other languages through war or migration, creating today's dichotomy in language tonality.

"Even remarkable correlations can arise by coincidence—or, in this case, possibly by prehistoric migration factors that are currently unknown to anthropology and archaeology—so we can't rule that out," Ladd says. "The next step is to attempt to correlate individual genotypes with measurably different behaviors on experimental tasks that are plausibly related to language and speech." 

Our Enemy Hands
 

IT’S hard to see Americans as under-washed. Sales of antibacterial soap, tooth whiteners and “intimate hygiene” products (wipes and sprays) are skyrocketing. Scientists actually connect the rising rates of asthma and allergies in the West to our overzealous cleanliness. And yet, in a compulsively sanitized culture, cleaning one part of the body — the hands — seems to be more honored in the breach than the observance. Studies show that hospital doctors resist washing their hands, and gimlet-eyed researchers report that only about 15 percent of people in public restrooms wash their hands properly.

Our ancestors would have been bewildered by this discrepancy between relentlessly scrubbed bodies and neglected hands. Depending on their era and culture, they defined “clean” in a wide variety of ways. A first-century Roman spent a few hours each day in the bathhouse, steaming, parboiling and chilling himself in waters of different temperatures, exfoliating with a miniature rake — and avoiding soap. Elizabeth I boasted that she bathed once a month, “whether I need it or not.” Louis XIV is reported to have bathed twice in his long, athletic life, but was considered fastidious because he changed his shirt three times a day.

But through all these swings of the hygiene pendulum, one practice never went out of style — humble, ordinary hand-washing. Which was fortunate, because hand-washing is the one cleansing practice canonized by modern science, a low-tech but effective way to prevent getting and passing on the common cold and infections from Clostridium difficile to MRSA, SARS and bird flu.

Hand-washing made sense in the ancient world, when food was eaten in the hands. Theophrastus’s “Characters,” written in the fourth century B.C., paints a portrait of a hairy, scabby sloven named Nastiness, who doesn’t wash his hands after dinner. But hand-washing was more than pragmatic: it was also a sign of honor and civility, something you offered your guests, via a basin and towel, as soon as they arrived. Since the Greeks believed that any respectful relationship, with gods as well as humans, demanded cleanliness, washing was a necessary prelude to prayer, and sanctuaries usually had fonts of water at their entrances.

For the Romans and Greeks, well-washed hands were a natural accompaniment to fairly clean bodies. The medieval and Renaissance focus on clean hands is more surprising, because those ages had little interest in washing beyond the wrist. It’s true that the Crusaders imported the idea of the Turkish bath into Europe, but even if your town had a bathhouse, it merited a visit only once every week or two.

Clean hands were a different story. Monastery cloisters featured a stone trough for hand-washing, and medieval paintings of interiors often show a ewer, a basin and a cloth for drying hands in a corner of the room. Etiquette books ordered hand-washing before and after meals, and people who neglected it inspired scorn: Sone de Nansay, the wandering hero of a 13th-century French poem, noted with dismay that Norwegians did not wash at the end of a meal.

Among the most fervent hand-washing advocates were medieval poets, who found it difficult to describe a banquet without affirming that everyone present washed his hands before eating, then once again afterward. Unless you washed your hands, you had no claim to gentility.

That belief persisted through the 17th century, even as bodily griminess reached new heights. Doctors assured people that they were more susceptible to the plague if they opened their pores in warm water, and terrified Europeans shunned water and washing, except for their hands. Since forks were not in general use until the 18th century, hand-washing still had a practical function as well as a symbolic one: the Dutch in the age of Rembrandt scandalized French visitors by eating without first washing their hands.

By the mid-19th century, people were timidly experimenting with bathing, but scientists still believed that disease spread through decaying matter and bad smells. When Ignaz Semmelweis insisted that Viennese doctors wash their hands in between performing autopsies and delivering babies, he was ridiculed, even though the practice greatly reduced death from puerperal fever. Semmelweis’s simple but radical idea gained currency only in the 20th century. The germ theory slowly triumphed — but until the development of sulfa and antibiotics, almost the only way to fight microbes was by washing them off.

Even with antibiotics, washing off microbes remains an excellent idea. This ancient mark of courtesy is now celebrated in public health campaigns, and the Centers for Disease Control and Prevention has anointed it as “the single most important means of preventing the spread of infection.” So, learn from science as well as the wisdom of our ancestors, and wash your hands.

 

Newfound Sea Anemones Really Get Around

Sea anemones normally anchor themselves to the seafloor. But new species found lurking in the waters surrounding the windswept Aleutian Islands near Alaska swim and walk across the sea floor.

Scientists discovered the anemones, which could represent two species, as well as a new species of kelp as part of a two-year scientific survey of the waters around the Aleutians.

"Since the underwater world of the Aleutian Islands has been studied so little, new species are being discovered, even today," said Stephen Jewett, a marine biologist at the University of Alaska, Fairbanks, and the dive expedition leader.

Overall, scientists say only about 10 percent of the species of life on this planet have been seen or catalogued.

The researchers are consulting experts to verify that the Aleutian anemones are in fact new species, but the consensus so far is that they are. Sea anemones are animals that typically use a foot to anchor to rocks. Some are known to detach when attacked or if their environment changes and food becomes scarce. The new species likely belong to a class of anemones that can detach and drift with ocean currents.

The kelp, dubbed Aureophycus aleuticus, is a type of brown algae that might represent a new genus, or even family (a larger biological classification that can include more than one genus), of the seaweed. Up to 10 feet long, the kelp was discovered near thermal vents in the region of the Islands of the Four Mountains.

Jewett and his team are studying the Aleutian waters to gauge the overall health of the islands and life there. Already, the team found evidence that the rugged and remote islands are not immune to human activity.

"Pollutants traveling through air and water pathways from temperate latitudes have been showing up in the area," Jewett said. "Debris and oil spills from World War II in the Aleutians have left their mark behind in unexploded ordinance and local sources of pollutants."

The team is analyzing water samples collected during dives for nutrient and oxygen levels, acidity, temperature and radioactive chemicals left over from underwater nuclear tests conducted at Amchitka Island between 1965 and 1971.

"Climate change, with changes in water temperature, wind patterns and currents, may impact the region's biological life," Jewett said. "It is important that we collect this information before any major changes occur."  

Analysis: Hurdles remain for stem cells

For all the excitement, big questions remain about how to turn this week's stem cell breakthrough into new treatments for the sick. And it's not clear when they'll be answered.

Scientists have to learn more about the new kind of cell the landmark research produced. They have to find a different way to make it, to avoid a risk of cancer. And even after that, there are plenty of steps needed to harness this laboratory advance for therapy.

So if you ask when doctors and patients will see new treatments, scientists can only hedge.

"I just can't tell you dates," says James Thomson of the University of Wisconsin-Madison, one of the scientists in the U.S. and Japan who announced the breakthrough on Tuesday.

"The short answer is: It's still going to be years," Dr. John Gearhart, a stem cell expert at the Johns Hopkins School of Medicine who was familiar with the work, said Wednesday.

Such a delay isn't unusual. It can often take a long time for medical payoffs to flow from basic scientific findings.

For example, the inspiration for a group of cystic fibrosis drugs now being tested in people or animals goes back 18 years to a genetic discovery. And more generally, gene therapy — the notion of fixing or replacing defective genes — has been studied in people for more than 15 years without much success.

At least, federal money for research into the new kind of cell won't be a problem, said Story Landis, head of the National Institutes of Health's Stem Cell Task Force. The task force is about to invite scientists to apply for new grants for such work, she said.

This week's advance has apparently solved a supply problem for the study of embryonic stem cells. These cells are valued for their ability to morph into any of the cell types of the body. Scientists had long searched for a way to produce embryonic cells that carry the genes of a particular person.

Such cells could be used for at least three purposes. The most highly publicized one is to create transplant tissue for treating disease. In the shorter term, they could be used to create "diseases in a dish," colonies of cells bearing illness-promoting genes that could reveal the vulnerable roots of medical conditions. And finally, scientists could use such cells for rapidly screening potential medicines in the laboratory.

Until this week's announcement, scientists who wanted to make such cells looked to an expensive, cumbersome cloning process that destroyed embryos, making it an ethical lightning rod. And it hadn't yet worked with human embryos.

The new technique is much simpler. It makes human skin cells behave like embryonic stem cells without using embryos at all.

End of problem? Not unless these altered skin cells can truly replace embryonic cells, and that's not clear yet, a prominent scientist says.

Paul Berg, a Stanford University Nobel laureate who helped establish federal guidelines for human research on genetically manipulated cells, said the celebration over this week's announcement is premature.

"I'm amazed at the ethicists" saying the problem of needing embryos has been solved, Berg said. "We're not in the clear — this is a first step."

So what are the next steps?

The first basic question to solve is how similar iPS cells are in behavior and potential to the embryonic cells that scientists have studied for nearly a decade.

"My guess is that we'll find that there are significant differences," said Dr. Robert Lanza of Advanced Cell Technology, which has been trying to produce stem cells from cloned human embryos. "I'd be surprised if these cells can do all the same tricks as well as stem cells derived from embryos."

Another big question is how to make iPS cells in a different way. The breakthrough technique treats skin cells by using viruses to carry in a quartet of genes. Those viruses disrupt the DNA of the skin cells. When that happens, there's a risk of cancer.

That's show-stopper when it comes to creating tissue to transplant into people. So scientists have to figure out a way to make iPS cells without those DNA-disrupting viruses.

Scientists should be able to find other ways to slip the genes into the skin cells, Thomson said. Other scientists suggest that a purely chemical treatment, not inserting genes at all, might be able to get the same result.

The cancer-risk problem should be solved quickly, maybe within a year or so, said Doug Melton, co-director of the Harvard Stem Cell Institute.

Before then, iPS cells could be used in lab studies to study the early roots of genetic disease or to screen drugs. But of course, it's anybody's guess when a useful treatment would result from that.

Even with the cancer problem solved for transplant uses, there's another big hurdle:

The whole idea of using embryonic stem cells or iPS cells for treating people with conditions like diabetes and Parkinson's disease via transplant is itself far from proven. Scientists will need to learn how to turn iPS cells into the right kind of tissue, and how to use that tissue in a way that will treat a person's disease.

Such studies, in the lab, animals and finally people, will take years.

As far as that obstacle goes, Thomson said, the breakthrough announced this week changes nothing.

"We have a lot of work to do." 

In this undated photo released by Kyoto University Prof. Shinya Yamanaka of Department of Stem Cell Biology Tuesday, Nov. 20, 2007, mouse cells are shown. (AP Photo/Shinya Yamanaka) 
In this undated photo released by Kyoto University Prof. Shinya Yamanaka of Department of Stem Cell Biology Tuesday, Nov. 20, 2007, mouse cells are shown. 


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