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"Dr. Mandelbrot coined the term "fractal" to refer to a new class of mathematical shapes whose uneven contours could mimic the irregularities found in nature. "Applied mathematics had been concentrating for a century on phenomena which were smooth, but many things were not like that: the more you blew them up with a microscope the more complexity you found," said David Mumford, a professor of mathematics at Brown University. "He was one of the primary people who realized these were legitimate objects of study." "

"In the beauty and geometric complexity of tile mosaics on walls of medieval Islamic buildings, scientists have recognized patterns suggesting that the designers had made a conceptual breakthrough in mathematics beginning as early as the 13th century. A new study shows that the Islamic pattern-making process, far more intricate than the laying of one's bathroom floor, appears to have involved an advanced math of quasi crystals, which was not understood by modern scientists until three decades ago. The findings, reported in the current issue of the journal Science, are a reminder of the sophistication of art, architecture and science long ago in the Islamic culture. They also challenge the assumption that the designers somehow created these elaborate patterns with only a ruler and a compass. Instead, experts say, they may have had other tools and concepts."

"'The most beautiful experience we can have is the mysterious - the fundamental emotion which stands at the cradle of true art and true science.' "We live in a society absolutely dependent on science and technology," Carl Sagan famously quipped in 1994, "and yet have cleverly arranged things so that almost no one understands science and technology. That's a clear prescription for disaster." Little seems to have changed in the nearly two decades since, and although the government is now actively encouraging "citizen science," for many "citizens" the understanding of - let alone any agreement about - what science is and does remains meager. So, what exactly is science, what does it aspire to do, and why should we the people care? It seems like a simple question, but it's an infinitely complex one, the answer to which is ever elusive and contentious. Gathered here are several eloquent definitions that focus on science as process rather than product, whose conduit is curiosity rather than certainty."

Behind a simple imperative lurks a complex web of masculinity.

"For years, fans of the Batman comics have puzzled over a mystery at the heart of the series: why doesn't Batman just kill his arch-nemesis, the murderous Joker? The two have engaged in a prolonged game of cat-and-mouse. The Joker commits a crime, Batman catches him, the Joker is locked up, and then invariably escapes. Wouldn't all this be much simpler if Batman just killed the Joker? What's stopping him? Enter philosopher Immanuel Kant and the deontological theory of ethics Now, philosophy professors are finding superheroes and comic books to be exceptionally useful tools in helping students think about the complex moral and ethical debates that have occupied philosophers for centuries."

"Neuroscientists understand the structural bases of some of these qualities. Take language, that uniquely human behavior. Underlining it are structures unique to the human brain - regions like "Broca's area," which specializes in language production. Then there's the brain's "extrapyramidal system," which is involved in fine motor control. The complexity of the human version allows us to do something that, say, a polar bear, could never accomplish - sufficiently independent movement of digits to play a trill on the piano, for instance. Particularly striking is the human frontal cortex. While occurring in all mammals, the human version is proportionately bigger and denser in its wiring. And what is the frontal cortex good for? Emotional regulation, gratification postponement, executive decision-making, long-term planning. We study hard in high school to get admitted to a top college to get into grad school to get a good job to get into the nursing home of our choice. Gophers don't do that."

December 2005 Over the millennia, many mathematicians have hoped that mathematics would one day produce a Theory of Everything (TOE); a finite set of axioms and rules from which every mathematical truth could be derived. But in 1931 this hope received a serious blow: Kurt Gödel published his famous Incompleteness Theorem, which states that in every mathematical theory, no matter how extensive, there will always be statements which can't be proven to be true or false. Gregory Chaitin has been fascinated by this theorem ever since he was a child, and now, in time for the centenary of Gödel's birth in 2006, he has published his own book, called Meta Math! on the subject (you can read a review in this issue of Plus). It describes his journey, which, from the work of Gödel via that of Leibniz and Turing, led him to the number Omega, which is so complex that no mathematical theory can ever describe it. In this article he explains what Omega is all about, why maths can have no Theory of Everything, and what this means for mathematicians."

"The helpless, seemingly clueless infant staring up at you from his crib, limbs flailing, drool oozing, has a lot more going on inside his head than you ever imagined. A wealth of new research is leading pediatricians and child psychologists to rethink their long-held beliefs about the emotional and intellectual abilities of even very young babies. In 1890, psychologist William James famously described an infant's view of the world as "one great blooming, buzzing confusion." It was a notion that held for nearly a century: infants were simple-minded creatures who merely mimicked those around them and grasped only the most basic emotions-happy, sad, angry. Science is now giving us a much different picture of what goes on inside their hearts and heads. Long before they form their first words or attempt the feat of sitting up, they are already mastering complex emotions-jealousy, empathy, frustration-that were once thought to be learned much later in toddlerhood. They are also far more sophisticated intellectually than we once believed. Babies as young as 4 months have advanced powers of deduction and an ability to decipher intricate patterns. They have a strikingly nuanced visual palette, which enables them to notice small differences, especially in faces, that adults and older children lose the ability to see. Until a baby is 3 months old, he can recognize a scrambled photograph of his mother just as quickly as a photo in which everything is in the right place. And big brothers and sisters beware: your sib has a long memory-and she can hold a grudge."

"HERE IS A mathematician's nightmare I heard in the 1980s when that irritating, unconforming, self-regarding provocateur Benoît Mandelbrot was suddenly famous - fractals, fractals everywhere. The mathematician dreamed that Mandelbrot died, and God spoke: "You know, there really was something to that Mandelbrot." Sure enough. Mandelbrot created nothing less than a new geometry, to stand side by side with Euclid's - a geometry to mirror not the ideal forms of thought but the real complexity of nature. He was a mathematician who was never welcomed into the fraternity ("Fortress Mathematics," he said, where "the highest ambition is to wall off the windows and preserve only one door"), and he pretended that was fine with him. When Yale first hired him to teach, it was in engineering and applied science; for most of his career he was supported at I.B.M.'s Westchester research lab. He called himself a "nomad by choice." He considered himself an experienced refugee: born to a Jewish family in Warsaw in 1924, he immigrated to Paris ahead of the Nazis, then fled farther and farther into the French countryside."

"True art would never be mistaken for a crude, paint-by-the-numbers copy. But a researcher has developed a statistical tool for determining whether a purported masterpiece is only a skilled imitation, suggesting that art may be a numbers game after all. Using high-resolution digital images and complex mathematical formulas, associate professor Hany Farid of Dartmouth College analyzed works by Renaissance artists to determine their authenticity. His computer program was able to accurately separate eight drawings by 16th century Flemish artist Pieter Bruegel the Elder from five drawings by imitators. It also found that portions of a painting by Italian artist Pietro di Cristoforo Vannucci, known as Perugino, were probably done by Perugino's apprentices. Farid described his work, presented Monday in the Proceedings of the National Academy of Sciences, as "simply another tool that is contributing to the dialogue of art authenticating" and said more work is needed before digital analysis of art could be done on a wider scale. Art experts reacted warily to the prospect that a masterpiece could be reduced to the sum of its digital parts. "

"A few months after the World Trade Center attacks, a strange message appeared on a U.S. Army computer: "Your security system is crap," it read. "I am Solo. I will continue to disrupt at the highest levels." Solo scanned thousands of U.S. government machines and discovered glaring security flaws in many of them. Between February 2001 and March 2002, Solo broke into almost a hundred PCs within the Army, Navy, Air Force, NASA, and the Department of Defense. He surfed around for months, copying files and passwords. At one point he brought down the U.S. Army's entire Washington, D.C., network, taking about 2000 computers out of service for three days. U.S. attorney Paul McNulty called his campaign "the biggest military computer hack of all time." But despite his expertise, Solo didn't cover his tracks. He was soon traced to a small apartment in London. In March 2002, the United Kingdom's National Hi-Tech Crime Unit arrested Gary McKinnon, a quiet 36-year-old Scot with elfin features and Spock-like upswept eyebrows. He'd been a systems administrator, but he didn't have a job at the time of his arrest; he spent his days indulging his obsession with UFOs. In fact, McKinnon claimed that UFOs were the reason for his hack. Convinced that the government was hiding alien antigravity devices and advanced energy technologies, he planned to find and release the information for the benefit of humanity. He said his intrusion was detected just as he was downloading a photo from NASA's Johnson Space Center of what he believed to be a UFO. Despite the outlandishness of his claims, McKinnon now faces extradition to the United States under a controversial treaty that could land him in prison for years-and possibly for the rest of his life. The case has transformed McKinnon into a cause célèbre. Supporters have rallied outside Parliament with picket signs. There are "Free Gary" websites, T-shirts, posters. Rock star David Gilmour, the former guitarist for Pink Floyd, even recorded

"It was once a staple of public school cafeterias that blended the indulgent and the nutritious, satisfying parents and children both. But chocolate milk is uncontroversial no more. Dozens of districts have demanded reformulations. Others have banned it outright. At the center of these battles are complex public health calculations: Is it better to remove sugary chocolate flavorings at the risk that many students will skip milk altogether, missing out on crucial calcium and Vitamin D? Or should schools instead make tweaks - less fat, different sweeteners, fewer calories - that might salvage the benefits while minimizing the downside?"

Parenting sites indicate many New York City parents want caregivers to teach their children a language.

Some interesting questions as to whether parents can "know" it's a good thing or a bad thing to have their children learn a second language. There are clearly cognitive and social costs and benefits that must be weighed.

"Now, as the United States and other Western nations struggle to recover from a debt and property bubble of their own, a growing number of economists are pointing to Japan as a dark vision of the future. Even as the Federal Reserve chairman, Ben S. Bernanke, prepares a fresh round of unconventional measures to stimulate the economy, there are growing fears that the United States and many European economies could face a prolonged period of slow growth or even, in the worst case, deflation, something not seen on a sustained basis outside Japan since the Great Depression. "

The TOK question here is can you compare Japan of the 1990s with the US and Western Europe of today? To what degree can economists, or any human scientists, suggest that one complex situation is "like" another one?