Halfway through the 2005 baseball season, John Olerud was having a great year with the Boston Red Sox. His batting average was .405, far better than that of most players. If someone had offered to wager with you on what his batting average would be for the rest of the season, what would you have bet?

It might seem like .405 would make sense, the same as the first half of the season. But if that had been your choice, you wouldn't have done so well. In the second half of the season, Olerud's batting average was down to .257. And if you'd used that method to bet on Henry Blanco of the Chicago Cubs, who had a first-half-season batting average of .151, you would have lost money too: in the second half of the season, his average was .305. Aaron Hill of the Toronto Blue Jays went from .359 to .226. And the list goes on. ]]>

iStockphoto

In fact, according to a new analysis by Lawrence Brown of the University of Pennsylvania in Philadelphia, predicting that a player's batting average will be the same in the second half of the season as the first half is about the worst plausible method out there. You'd do much better, in fact, by ignoring a batter's individual average and simply predicting .248, the average across all players in Major League Baseball.

Lady Luck is the one messing up your bet. "If somebody does well for the first half of the season, they're probably doing better than their native ability," Brown says. In other words, chances are they're getting lucky. The nasty thing about luck is that it tends not to hold, so the players' averages often go tumbling down.

The hitters who've done well may be bummed, but the hitters who started out badly get a break. Brown's analysis shows that for the same reasons, their averages are likely to pick up.

This phenomenon happens throughout life. Did you pay more for your house than most apparently similar houses? Bad news: It probably wasn't just the lovely roses out front that drove the price up. Instead, you most likely paid too much, and when you go to sell it, odds are you won't do as well.

Similarly, a busy period at a bank's call center will probably be followed by a not-quite-so-busy period. Chances are that a very tall couple's children will be tall but not quite as tall as they are. Statisticians call this phenomenon "regression to the mean."

The trick is to separate out the effects of chance from the real differences in performance. The traditional method is to shrink everyone's scores toward the average. So, for example, you might choose the point halfway between the average for all hitters and an individual's first-half average.

Brown has developed new, more sophisticated statistical techniques for this purpose. The methods have wide applicability, but baseball made a handy test case since baseball fans are rabid keepers of statistics. His best method gave a 40 percent improvement over simply choosing the average across all players.

This method takes into account that some players batted many more times than others during the first half of the season. If a player has batted a lot, his batting average is more likely to reflect his real ability. So Brown's method predicts that the player's performance in the second half will be closer to his performance in the first half.

Which method is best depends, however, on the amount of initial data. After a single month, there's still so little information that the best method is to just take the average across all players. After five months, though, Brown's method does a lot better.

Even Brown's best method couldn't nail individual players' future performances very precisely. Mostly, he says, that's because baseball is a "noisy game," with chance playing a large role. An oracle that knows each player's precise ability could help some, he says. "But I can estimate that such an oracle, if she existed, could do no more than about 10 percent better than my best method—and perhaps not even that well."

References:

Brown, L.D. 2008. In-season prediction of batting averages: A field test of empirical Bayes and Bayes methodologies. Annals of Applied Statistics 2(March):113-152. Abstract available at http://dx.doi.org/10.1214/07-AOAS138.

Prime numbers are maddeningly capricious. They clump together like buddies on some regions of the number line, but in other areas, nary a prime can be found. So number theorists can't even roughly predict where the next prime will occur. The distribution of primes is the great motivating question of number theory.

Prime numbers are like the atoms of mathematics: the simple, indivisible building blocks upon which all the other numbers are built. By definition, a prime number isn't divisible by any number except itself and 1; so, for example, 5 is prime but 4 is not, since 4 = 2 × 2. But while the atoms of chemistry are neatly arranged in a periodic table, the search for a pattern in primes keeps number theorists pondering as they lie in bed at night.

Vexingly, the answer to their questions lies encoded within a single function—one that happens to be enormously difficult to fully understand. The "Riemann zeta function" contains within it the key to the distribution of the prime numbers. But mathematicians have been working on uncovering the function's mysteries since 1859, when Bernhard Riemann formulated a much-celebrated hypothesis about it, and so far, they haven't cracked it. With the recent solutions to Fermat's Last Theorem and the Poincaré conjecture, the Riemann hypothesis could now be considered the biggest puzzle in mathematics—and the Clay Mathematics Institute in Cambridge, Mass., will award the person who solves it a million dollars.]]>

In this visualization of the Riemann zeta function, the different colors show the values of the function on different areas of the complex plane. Jean-François Colonna

Two mathematicians, Ce Bian and Andrew Booker of the University of Bristol in England, now have the first glimpse of an elusive mathematical object that may one day help crack the problem. They have found the first example of a third-degree transcendental L-function.

"There is hardly a problem in number theory that doesn't seem to be connected to L-functions," says Michael Rubinstein of the University of Waterloo in Ontario, Canada. But these functions, though incredibly numerous, have also been incredibly hard to find. "It's like what biologists must feel when finding a new species they'd only seen tracks from before," he says. "You know they're out there and you're trying to find them. Now we've got one."

Mathematicians attack really hard problems like the Riemann hypothesis with a strategy that might initially seem odd: they try to prove a claim that is even bigger and bolder than the original one. By embedding the problem in a larger context, they can build bigger tools to attack it.

To see why that might be useful, imagine that a mosquito is pestering you. If you can't manage to swat it, you might instead try a bug bomb, killing every insect in the room—and being sure to get that darn mosquito in the process. Thus killing all the bugs might be easier than simply killing the one wily mosquito. This technique of generalization is the same one that brought down both Fermat's Last Theorem and the Poincaré conjecture.

In the case of the Riemann hypothesis, mathematicians are considering the whole family of L-functions, of which the Riemann zeta function is just one. They've generalized the Riemann hypothesis to all the L-functions, and they want to use this bigger, badder version to kill the "mosquito" of the original function along with all the others.

The Riemann zeta function can also be visualized using three dimensions. Bernd Thaller

Unfortunately, mathematicians haven't had a single example of an L-function that is reasonably complex to work with. The simplest examples, like the Riemann zeta function, have been known for a long time, and somewhat more complex versions were featured prominently in Andrew Wiles' proof of Fermat's Last Theorem. But the functions get vastly more sophisticated than that, and an understanding of these more complex versions will likely be necessary to prove the Riemann hypothesis.

Until just over a decade ago, mathematicians hadn't even proven that these very complex functions existed. And when Stephen D. Miller of Rutgers University in New Jersey did prove it, he did so indirectly, without providing a single example. So, many number theorists have put a great deal of effort into understanding a type of function they had never even seen. Now, with the aid of 10,000 hours of computing time on a PC, Bian and Booker have finally tracked one of these functions down.

Mathematicians are especially interested in the places where an L-function has the value 0 because the generalized Riemann hypothesis states that all those places will lie on a single line in the plane (with a few well-understood exceptions). This graph shows that special line for Bian and Booker's function. As expected, it has the value 0 in many places. Michael Rubinstein

"It's an amazing computation," says Don Blasius of the University of California, Los Angeles. "It solves a computationally extremely challenging problem that would have been literally undoable until now."

Finding the function required a combination of computational cleverness and theoretical advances. Bian and Booker couldn't find it with absolute exactness because that would involve finding infinitely many irrational numbers that occur in the function. But the researchers deduced the first few hundred of these numbers, to within about 6 decimal places.

Oddly, Booker points out, even though mathematicians had never seen one of these functions, they knew a lot about what they had to be like. The researchers could check their result by making sure their function had all the properties it was expected to have. One of the checks on the computation involved checking the generalized Riemann hypothesis for this particular case. "We're totally confident in our result," Booker says.

Mathematicians will be thrilled if they do someday succeed in proving the Riemann hypothesis, but odds are that they have a long way to go. Booker freely says that his result is just one small step along the way. "Is this thing going to solve the Riemann hypothesis? Well, no," he says. But it may contribute to a solution eventually, and in the meantime, it certainly has some mathematicians excited.

References:

For more information on Bian and Booker's result, go to www.aimath.org/news/gl3.

For more information on the problems the Clay Mathematics Institute has issued a prize for, go to www.claymath.org/millennium.

For more information about prime numbers, go to primes.utm.edu/.

In late 2006, a statistical study of deaths that occurred after the invasion of Iraq ignited a storm of controversy. This Lancet study estimated that more than 650,000 additional Iraqis died during the invasion than would have at pre-invasion death rates, a vastly higher estimate than any previous. But in January, a World Health Organization study placed the number at about 150,000.

The conflicting findings highlight just how difficult it is to gather reliable information in a war zone. But they also show the increasing involvement of statisticians in informing responses to humanitarian crises. In addition to the work in Iraq, statisticians have gathered evidence that has aided in the prosecution of Slobodan Milosevic, guided reparations for the civil war in Sierra Leone, and helped determine the needs of Katrina survivors, among many other projects. ]]> "You can go to a congressional hearing or an international war crimes tribunal and you can hear the stories," says Lynn Lawry of the International Medical Corps. "But how many are we talking about? How many people are at risk? How many people are affected?"

Jana Asher, a statistician at Carnegie Mellon University (right), discusses field operations with staff in Sierra Leone. R. Conibere

Statisticians are well-suited to answer these questions because they have the tools to put together partial information into a global picture. For example, even if complete records can't be gathered, a statistician can survey a small number of randomly chosen people affected by a crisis and infer from their experiences the likely impact on the population as a whole. For example, Jana Asher of Carnegie Mellon University in Pittsburgh, Pa., developed an estimate of the rates of rape across Sierra Leone by determining how many women from a national sample had been raped.

But humanitarian crises pose huge challenges. Little information may be available—even from before a crisis—about how many people live where. Even if a previous census was taken, the high birth and death rates in developing countries tend to quickly make censuses outdated. Areas within continuing war zones can be unsafe for survey workers.

"When you have a displaced population that has been forced to flee their homes, all the traditional census methods really break down very badly," says David Banks, a statistician at Duke University in Durham, N.C. "The refugees don't have addresses. They're wandering from one camp to another. Communication is poor."

These challenges have to be met with very carefully designed protocols. For example, the Lancet study of Iraq, with the shockingly high mortality rates, was initially criticized for not surveying people who lived in back alleys because the areas were too dangerous for surveyors. Les Roberts, who was at Johns Hopkins University in Baltimore at the time but is now at Columbia University, and his collaborators on the study argued that the critics had misunderstood their randomization technique.

Random surveys are not the only useful statistical method. To tally the number of deaths related to the conflict in Timor-Leste, Romesh Silva and Patrick Ball of the Human Rights Data Analysis Group combined incomplete datasets to generate a broader picture of events. The Indonesian military claimed that its occupation of Timor-Leste had caused no deaths. Many stories had been told of killings and famine, but Silva and Ball wanted solid evidence.

Along with gathering about 8,000 personal accounts conveyed to the Commission for Reception, Truth and Reconciliation, Silva and Ball conducted a census of public graveyards including 319,000 gravestones and a survey of a random sample of 1,400 households about displacements and deaths. The researchers found that the different lines of evidence corroborated one another strongly, adding to the strength of each approach. In addition, Silva and Ball could observe how often names recurred across the different databases and get a much better estimate of the total number of deaths across the country.

They found that Indonesian occupation of Timor-Leste from 1974 to 1999 led to more than 100,000 deaths beyond what would have been expected in peacetime, through a combination of direct killings, famine, and illness.

The conflicting studies in Iraq show just how tricky it is to apply these methods in messy real-life situations. About the Lancet study, Asher says, "I don't think there was anything obvious in what they did that someone can point to and say this method is flawed. But the WHO study used appropriate methodology too."

The most suspect part of the Lancet study, Asher says, is that the researchers didn't supervise the survey workers closely. On the other hand, the World Health Organization relied on government workers to administer the questionnaires. People can be intimidated by government workers and be less inclined to say much, a phenomenon that is particularly common in unstable countries. The only way to resolve the conflict, Asher says, is to do yet another study, with an even more careful design.

References:

Asher, J., D. Banks, and F.J. Scheuren, eds. 2008. Statistical Methods for Human Rights. New York: Springer. See www.springer.com/statistics/social/book/
978-0-387-72836-0.

Iraq Family Health Survey Study Group. 2008. Violence-related mortality in Iraq from 2002 to 2006. New England Journal of Medicine 358(Jan. 31):484-493. Available at http://content.nejm.org/cgi/content/full/358/5/484.

Burnham, G. . . . and L. Roberts. 2006. Mortality after the 2003 invasion of Iraq: A cross-sectional cluster sample survey. Lancet 368(Oct. 21):1421-1428. Abstract available at http://dx.doi.org/10.1016/S0140-6736(06)69491-9.

Silva, R., and P. Ball. 2006. The Profile of Human Rights Violations in Timor-Leste, 1974–1999. A report by the Benetech Human Rights Data Analysis Group to the Commission on Reception, Truth and Reconciliation of Timor-Leste. Available at www.hrdag.org/resources/timor_chapter_graphs/
timor_chapter_page_01.shtml.

Hundreds of years ago in Japan, people offered thanks to the gods by sacrificing a horse or a pig. Horses and pigs, however, were valuable and expensive, so poor folks had a hard time expressing their gratitude. So they came up with a solution: Rather than sacrificing a horse, they would simply draw a painting of a horse on a wooden tablet and hang it in the temple.

Then someone, most likely an impoverished samurai, realized that horses and pigs were hardly the only thing that could be drawn on a tablet. He had the idea of painting something original, something beautiful, something creative. He offered mathematics.]]>

This tablet was hung in the Kinshouzan shrine in the Gifu Prefecture in 1865. It shows 12 different geometric problems. The third problem from the right was presented by a 16-year-old girl. Fukagawa

Hundreds of beautifully painted, multi-colored wooden tablets showing problems and theorems of geometry have adorned Japanese temples. They are called "sangakus," which simply means mathematical tablets. The text on the tablets is written in an ancient form of Chinese, which was the language of scholars, much like Latin in the West. Only in the past couple of decades have these tablets been translated into modern languages in significant numbers.

A Japanese mathematics teacher, Hidetoshi Fukagawa, has been finding, translating, and researching the tablets. This spring, Fukagawa and Tony Rothman of Princeton University will publish a complete history of sangaku, including photographs of many sangakus that have never before been seen outside of Japan.

"Sangakus are exceptional," Rothman says. "They're not only exceptionally beautiful, but the problems are often exceptionally difficult. And the solutions can be very clever. Some of the things they do to solve these problems would never have occurred to me."

The sangakus were made during a period when Japan was mostly isolated from the outside world. The shogun leaders expelled all the foreign missionaries and forbade Japanese from leaving the country on pain of death in the early 1600s. The result was a kind of renaissance in Japan, with a flowering of unique cultural traditions like tea ceremonies, puppet theater, and woodblock prints.

At the same time, the shoguns persuaded the samurai warriors to lay down their weapons and become government functionaries. The pay, however, was low, so the samurai often moonlighted with other jobs. One of these outside jobs was to teach mathematics in the schools.

This tablet was created in 1814, but it was only discovered in 1994 when the temple it was in was about to be destroyed. Fukagawa

Isolated from the development of calculus taking place in the West, these mathematicians and their students created a kind of home-grown geometry with a uniquely Japanese character. Many of the problems were based on origami or folding fans, for example.

Here is an example of a sangaku problem. Take a circle and draw a polygon inside it, with each corner of the polygon on the circle. Choose one of the vertices of the polygon and connect lines from it to all the other vertices, dividing the polygon up into triangles. Within each one of those triangles, draw a circle that just touches each side of the triangle. The sum of the radii of those circles will be constant, no matter which vertex you chose.

One sangaku shows that the sum of the radii of the small circles in the top two drawings, but not the bottom one, will be the same. Ennis

Most sangakus simply state the theorem and provide a diagram, but they don't provide a proof, and this one is no exception. The most straightforward way to prove it relies on Carnot's Theorem, which wasn't proven in the West until 100 years after the sangaku was created.

Rothman believes that sangakus were not just religious offerings, but "acts of bravado and challenges to other people to solve the problem." For example, one sangaku proclaims, "'This answer is correct to 15 decimal places,'" Rothman says. "It's kind of like, 'top that if you can!'"

Starting around 1800, several collections of sangaku problems were made into books, including the solutions, so researchers know the original methods for many of the problems. But a couple of sangakus are unsolved to this day. "One of them results in an equation of the 1024th degree," Rothman says. "A mathematician later got very famous for reducing it to a problem of the 10th degree, but that's still way too big to solve. We have no idea how they did it."

The Kaizu Tenma Shrine in the Shiga prefecture has a sangaku under the right eave which contains 30 problems. It's 10 inches high and 17 feet long. Fukagawa

References:

Fukagawa, H., and T. Rothman. 2008. Sacred Mathematics: Japanese temple geometry. Princeton, N.J.: Princeton University Press. See www.press.princeton.edu/titles/8646.html.

Peterson, I. 2001. Temple circles. Science News Online (April 21). Available at www.sciencenews.org/articles/20010421/mathtrek.asp.

Many more pictures of sangakus are available at www.wasan.jp/english by clicking on the map.

More information on sangaku is available at www.loyola.edu/maru/sangaku.html.

When Ralph Nader recently announced he was entering the 2008 presidential race, many Democrats groaned. It was his fault, they say, that George Bush defeated Al Gore in 2000. But Nader retorted that the Democratic Party has only itself to blame for the loss in 2000.

Mathematicians offer a different perspective. The problem, they say, doesn't lie with Nader or with the Democrats. It lies with our voting system.

Complaints about the obscure Electoral College system are common, but the mathematicians' objection is even more basic. Presidential elections in the United States are decided using a variation of a method known as plurality voting: each person votes for one candidate, and the candidate with the most votes wins.

Seems like the obvious approach—but obvious doesn't always mean effective. "The plurality vote is pretty much the worst voting system there is," says Donald Saari, a mathematician at the University of California, Irvine. ]]> The 2000 election gave a vivid demonstration of plurality voting's limitations. Polls indicated that most people who voted for Nader would have preferred Gore to Bush. The votes for Nader and Gore combined in Florida would have beat Bush. But with the votes divided between them, Bush emerged the winner.

Though this example is especially dramatic, Saari has found that determining voters' preferences from their ballots is often tricky. For example, suppose three candidates, A, B, and C, are competing. The preferences of the voters are as follows:

• 3 people rank A first, B second, and C third;
• 2 people rank A first, C second, and B third;
• 2 people rank B first, C second, and A third; and
• 4 people rank C first, B second, and A third.

Plurality voting would name A the winner, with 5 votes.

On the other hand, suppose one wanted the candidate that was least disliked. Six people rank A last, two people rank B last, and three people rank C last, so in that case, B should win.

Yet another method would be to assign 2 points for a first place vote, 1 point for second place and none for third. In this method, known as the Borda count, C walks away the winner with 12 points, beating out B's 11 points and A's 10.

So who should win the election?

Examples like this lead Saari to conclude that "election outcomes can more accurately reflect the choice of an election rule rather than the voters' wishes." He even jokes to colleagues that for a price, he could rig the election of their next department chair to guarantee that their preferred candidate would win. He would interview everyone in the department to determine their preferences and then choose an election method—one that could be argued to be fair—that would produce the desired outcome.

Indeed, he has found that 69 percent of the time, an election result can be changed by changing the voting rules.

But that doesn't mean there's no basis for choosing the best rule. Although mathematicians haven't settled on a single choice, they've done a lot of work to explore the consequences of choosing one method over another.

Saari's preferred method is the Borda count, because he believes it reflects the voters' wishes most accurately. Suppose, he argues, that voters prefer candidate A to candidate B to candidate C and candidate B then drops out. The voters should still prefer A to C, right? Saari found that for three-candidate elections, the Borda count is the method most likely to ensure that.

Plurality voting, on the other hand, often does terribly at this test. For example, in the contest above, suppose candidate B withdraws. Although A should be the winner just like before, in fact, five people would prefer A to C and six would prefer C to A, making C the winner. Similarly, if C withdraws, A should still win, but in fact, B would.

Saari says this property explains much of the horse-race jockeying between candidates during the presidential election. The media, for example, often speculate on the impact it will have on other candidates if one drops out of the race. In a system like the Borda count, a candidate dropping out wouldn't change the rankings of the other candidates.

But Steven Brams of New York University, another researcher in the area, prefers a different voting method, one known as approval voting. In that method, voters vote yes for each candidate they find acceptable.

"The major problem with the Borda count is its manipulability," Brams says. "If you have a favorite candidate, and your second choice is his or her fiercest competitor, you have no reason to vote sincerely." Ranking the competitor second, after all, will give away a point, weakening your effective support for your favorite. As long as you're confident that your third pick isn't likely to win, you're better off putting your least-favorite candidate second.

Brams acknowledges that approval voting is subject to paradoxes that the Borda count isn't. But, he says, "I think these paradoxes in many cases are artificial, constructed, contrived. They're rare events."

The two are in agreement, however, that the problems with the current system are not rare. "The real examples are everywhere," Saari says. "Just look at Nader."

References:

Brams, S. 2008. Mathematics and Democracy: Designing Better Voting and Fair-Division Procedures. Princeton, N.J.: Princeton University Press. See www.press.princeton.edu/titles/8566.html.

Saari, D. 2001. Decisions and Elections: Explaining the Unexpected. Cambridge, England: Cambridge University Press. See books.cambridge.org/0521004047.htm.

______. 2000. Chaotic Elections! A Mathematician Looks at Voting. Providence, R.I.: American Mathematical Society. See www.ams.org/bookstore.

Further Readings:

Klarreich, E. 2002. Election selection. Science News 162(Nov. 2):280–282. Available at www.sciencenews.org/articles/20021102/bob8.asp.

Peterson, I. 2003. Election reversals. Science News Online (Oct. 18). Available at www.sciencenews.org/articles/20031018/mathtrek.asp.

______. 1998. How to fix an election. Science News Online (Oct. 31). Available at www.sciencenews.org/pages/sn_arc98/10_31_98/mathland.htm.

Donald Saari has a website at www.math.uci.edu/~dsaari/. It includes videos on the Mathematics of Voting, Creating Voting Paradoxes, and an explanation of What Causes Voting Paradoxes.

The connection between mathematics and music is often touted in awed, mysterious tones, but it is grounded in hard-headed science. For example, mathematical principles underlie the organization of Western music into 12-note scales. And even a beginning piano student encounters geometry in the "circle of fifths" when learning the fundamentals of music theory.

But according to Dmitri Tymoczko, a composer and music theorist at Princeton University, these well-known connections reveal only a few threads of the hefty rope that binds music and math. To grasp the true structure of music, he says, we need to understand the geometry of hyperdimensional objects. Doing so has given him new ways of understanding pieces of music that have long baffled theorists and even led him to new insights into the history of music. ]]>

This shows a three-dimensional representation of a four-dimensional space that captures many essential features of musical structure. Dimitri Tymoczko

Tymoczko compares the structure of music to the shape of a rock face that a rock-climber is scrambling up. "If you know the conditions of the rock face, you can predict the motions of the climber," he says. "The structure of the space makes certain choices overwhelmingly natural or convenient. There's something similar that goes on with music. When you think about things abstractly, you can come to understand that the directions that music went aren't completely arbitrary. Composers are exploring the possibilities that musical space presents them with."

Tymoczko built on familiar geometrical analogs for music. For example, musical pitch is often imagined as lying on a line with low notes to the left and high notes to the right. Furthermore, as pitches go higher and higher, the notes repeat in different octaves, such that a low C, a middle C, and a high C all sound very similar. Often, the exact octave of a particular note doesn't matter very much in music. Instead, musicians commonly visualize a "pitch class circle," which comes from the original line by gluing together each point of the line that represents the same note in different octaves. So low C, middle C, and high C, for example, would all be glued together.

Applying the same kind of reasoning to complete pieces of music, Tymoczko created a geometric space in which he could analyze a piece of music with two notes being played simultaneously. He started with a piece of paper and made the horizontal direction represent the pitch of one note and the vertical direction represent the pitch of the other. A piece of music with two voices would correspond to dots moving around in this space.

Then he modified the space to embed musical structure within it. First, Tymoczko used the same method musicians used to create the pitch circle. He glued the left edge of the page to the right edge, turning the horizontal lines into circles and creating a cylinder from the whole page. Then he glued the bottom end of the cylinder to the top, turning the vertical lines into circles as well and creating a donut shape from the entire page.

Next, he noted that the order of the notes in a chord doesn't much matter. That means that the point on his page that has C in the horizontal direction and E in the vertical direction is really the same as the point that has E in the horizontal direction and C in the vertical direction. So he took his space and glued all those points together. It takes a bit of effort to visualize it, but for two simultaneous notes, this turns the donut shape into a Möbius strip.

Two-note chords correspond to points on a Möbius strip.

Tymoczko used the same method to create geometrical spaces to model pieces with any number of simultaneous notes. A piece with three notes, for example, would correspond to points in three-dimensional space. When he wrapped the space around to form circles and identified chords with the same pitches in a different order, he created a twisted prismatic donut. More notes require more than three dimensions, which gets hard to picture but not so hard to describe mathematically.

Having constructed his spaces, he began translating musical principles into their geometric equivalents. He noted that if he plotted a major chord in his geometrical space and reflected it, as if across a mirror down the middle of the space, it turned the chord into a minor one. Rotating a chord to a different spot in the space corresponded to transposing the chord into a different key. Composers need to choose sequences of chords that the ear can make sense of harmonically, and Tymoczko noted that this tends to be accomplished by transitioning between chords using combinations of these geometric rotations and reflections, or approximations to them.

Composers also need to write music in such a way that our minds can link the sounds into simultaneous, overlapping melodies. This is this easiest when each individual melody moves only in fairly small steps.

The trick for composers is to accomplish both those goals, harmonic and melodic consistency, at the same time. "We've just translated that into a math problem," Tymoczko says. "The solution is to use sequences of points close together that are related by rotation, or nearly so."

In the video above, Chopin's E-minor prelude can be visualized as multiple points moving around on a circle. Note that the notes tend to move short distances. (Click here or on the image above to play the video in a new window.) Video courtesy of Dimitri Tymoczko

Music theorists have long found Chopin's E minor prelude puzzling. Although the chord progressions sound smooth to the ear, they don't quite follow the traditional rules of harmony. When Tymoczko looked at the piece and watched the composition's motion through his geometrical space, he saw that Chopin was moving in a systematic way among the different layers of the four-dimensional cubes. "It's almost as if he's an improviser with a set of rules and set of constraints," Tymoczko says.

Another way of visualizing Chopin's composition is through a three-dimensional projection of a four-dimensional space, as in the video above. The chords primarily cluster in the center of the space, usually moving through small distances. (Click here or on the image above to play the video in a new window.) Video courtesy of Dimitri Tymoczko

What's particularly amazing, Tymoczko says, is that the mathematics needed to describe these spaces wasn't even developed in Chopin's time. Nevertheless, he says, "it is unquestionable that he had some cognitive representation of the space. So there was this period of history where the only way Chopin could express this abstract knowledge was through music. His knowledge of four-dimensional geometry was most efficiently expressed through piano pieces."

References:

Tymoczko, D. 2006. The geometry of musical chords. Science 313(July 7):72-74. Available at www.sciencemag.org/cgi/content/full/313/5783/72.

Tymoczko has much more information, including his original compositions, software and additional videos, at www.music.princeton.edu/~dmitri/.

This is part two of a two-part series. Part one appeared last week.

Nearly two centuries ago, Sophie Germain, the first woman known to have discovered significant mathematical theorems, developed a bold plan to prove Fermat's Last Theorem. But this entire plan was nearly lost to history, until David Pengelley of New Mexico State University in Las Cruces and Reinhard Laubenbacher of Virginia Tech in Blacksburg, dug through her notes, long archived in a French library.

Fermat made his conjecture in 1630, but it took more than 350 years for mathematicians to finally come up with a proof of it. Andrew Wiles of Princeton University cracked the problem in 1995. In Germain's day, almost all mathematicians working on the problem tackled only small bits of it at a time. But Germain's approach, had it been successful, would have proven the entire conjecture at one go. Because her work was almost entirely unknown, mathematics ended up reproving some of her results 80 years later.]]> Before Pengelley and Laubenbacher's recent discoveries, mathematicians knew only of a small partial result of Germain's in number theory. But in her manuscripts, they found a simple, direct plan of attack on Fermat's entire theorem. She exploited techniques developed by Carl Friedrich Gauss and laid out her method in a letter to him in 1819, looking for feedback and, perhaps, collaboration.

Sophie Germain (1776–1831) was the first woman known to have discovered significant mathematical theorems.

She had initially written to the master mathematician in 1804, using her male pseudonym of Antoine-August LeBlanc. She shared with Gauss some proofs that grew from her reading of his great work Disquisitiones Arithmeticae. He had responded with enthusiasm, saying "it pleases me that arithmetic has acquired in you so able a friend." Their correspondence continued for 4 years.

Eventually, Gauss discovered her secret. In 1806, Napoleon's armies were marching into Prussia, and Germain became concerned that Gauss might be in danger. She asked a friend who was a commander in the French artillery to find Gauss and ensure his safety. Her friend followed her request—but revealed her identity in the process.

Gauss initially responded with delight, writing to Germain: "The taste for the abstract sciences in general and, above all, for the mysteries of numbers, is very rare.… But when a woman, because of her sex, our customs and prejudices, encounters infinitely more obstacles than men in familiarizing herself with their knotty problems, yet overcomes these fetters and penetrates that which is most hidden, she doubtless has the most noble courage, extraordinary talent, and superior genius."

Yet, in his next letter, Gauss closed their correspondence, saying he had a new job in astronomy and would no longer have time for her mathematical investigations. She heard from him only once more, when his assistant wrote asking for her help in selecting a clock as a gift from Gauss to his wife.

Sophie Germain corresponded for four years with Carl Friedrich Gauss, one of the greatest mathematicians of all time.

Germain seems to have accepted Gauss's closure, though it isn't known whether she helped with the clock. But she didn't stop her work on number theory. And when she developed her method to solve Fermat's Last Theorem, she wrote again.

Fermat's Last Theorem states that there are no nonzero whole numbers x, y, and z such that xⁿ + yⁿ = zⁿ for any n greater than 2. Germain's approach to proving it used Gauss's new technique of modular arithmetic, which divides different numbers by some fixed number and only considers the remainder. So, for example, 4 modulo 3 is 1, and 7 modulo 3 is also 1.

Germain's idea was that it is often easier to prove that Fermat's equation can't be true when taken modulo some "auxiliary prime" P. It was a promising approach, but one aspect of it was challenging. A step in her method was to divide both sides of the equation by x. But if it happened that P divided x evenly with no remainder, then x would be equal to 0 modulo P. And of course, in mathematics, it isn't legitimate to divide by zero. For related reasons, neither y nor z could be evenly divisible by P either.

So if she proved that Fermat's equation didn't hold modulo P, it only showed that the regular, non-modular version didn't hold for x, y, and z that aren't divisible by P. But that wasn't good enough—she needed to prove it with no restrictions.

So Germain devised a way around the problem. She noted that for any particular values of x, y, and z, only finitely many numbers could possibly divide any of them evenly. So she just had to prove that she could carry her method above through for infinitely many values of P. Then, for any values of x, y and z, she would know that for some P, she'd have proven Fermat's equation couldn't be true modulo that number. That would mean that Fermat's equation wouldn't hold for any particular choice of x, y and z, and Fermat's Last Theorem would be proven.

When Germain wrote to Gauss, she knew she had not yet completed the project. "I have never been able to arrive at the infinity, although I have pushed back the limits quite far by a method of trials too long to describe here," she wrote. "You can easily imagine, Monsieur, that I have been able to succeed at proving that this equation is not possible except with numbers whose size frightens the imagination.… But all that is still not enough; it takes the infinite and not merely the very large."

Gauss didn't respond to her letter.

Germain never managed to arrive at infinity, though she pushed her approach a long way. But the effects of her isolation show in her work. Scattered throughout are little mistakes. "We all make mistakes, and colleagues or referees catch them," Laubenbacher says. "She didn't get that."

Pengelley agrees. "I think what the mistakes show more than anything else is that she didn't have other people who read her work and gave her feedback. It's conceivable to me—unbelievably—that some of her most important manuscripts might not have been read by anybody." Pengelley and Laubenbacher presented their findings at the Joint Mathematics Meetings in San Diego in January.

Even without the mistakes, though, her program could not have succeeded. The problem was simply too hard, and nearly 200 years of mathematical development would be needed before it could be cracked. Eventually, she herself proved that her approach couldn't work.

Germain might have been able to accept the overwhelming difficulty of the problem. "I have never ceased thinking about the theory of numbers," she wrote in her letter to Gauss explaining her program. "I will give you a sense of my absorption with this area of research by admitting to you that even without any hope of success, I still prefer it to other work which might interest me while I think about it, and which is sure to yield results."

This is part two of a two-part series. Part one appeared last week.

References:

Laubenbacher, R., and D. Pengelley. 2007. Voici ce que j'ai trouvé: Sophie Germain's grand plan to prove Fermat's last theorem. Preprint available at www.math.nmsu.edu/%7Edavidp/germain.pdf.

Bucciarelli, L., and N. Dworsky. 1980. Sophie Germain: An essay in the history of the theory of elasticity. Hingham, N.Y.: D. Reidel Publishing.

For more information on the aspect of Sophie Germain's number theory work that has long been known, see www.agnesscott.edu/Lriddle/women/germain-FLT/SGandFLT.htm.

For more information on Sophie Germain's life, go to www.pbs.org/wgbh/nova/proof/germain.html.

First of two parts

Around 1630, Pierre de Fermat scribbled his famous note in the margin of a book stating what is now known as "Fermat's Last Theorem." "I have discovered a truly remarkable proof which this margin is too small to contain," he added. His proof has never been found and was almost certainly wrong, but Fermat's conjecture bedeviled mathematicians for centuries to come.

Mathematicians soon realized that the problem was far harder than it first appeared. Number theorists labored endlessly to nibble off small parts of it, but in the early 1800s, one mathematician finally developed a bold strategy that had the potential to solve the whole problem at once. But the entire approach was very nearly lost to history, because until recently, all the notes and manuscripts were moldering unread in a French library. ]]>

Sophie Germain was the first person to develop a realistic plan to prove Fermat's Last Theorem.

The mathematician who developed the approach was respected by luminaries like Carl Friedrich Gauss, Adrien-Marie Legendre, and Joseph-Louis Lagrange, but was marginal in the mathematical community, with no formal training or university position. That's because the mathematician was a woman—indeed, the first woman to do significant research in mathematics.

Sophie Germain has been known for her work in the theory of elasticity and the curvature of surfaces, but until now, her only known work in number theory was a single result that Legendre attributed to her in a footnote.

"What he credited to her in this footnote is in some sense really a misrepresentation of what she did," says Reinhard Laubenbacher of Virginia Polytechnic and State University in Blacksburg. He and David Pengelley of New Mexico State University in Las Cruces searched through her notes in the Bibliothèque Nationale in Paris. Of the over 2,000 pages in the archive, hundreds and hundreds concerned number theory.

Some pages contained mere doodles that degenerated into chicken scratches, but many were filled with remarkable results. Included was a 20-page manuscript Germain had written so meticulously that not a single word was scratched out. "I personally believe," Pengelley says, "that she intended to submit it to the French academy for the prize for Fermat's Last Theorem."

Fermat's Last Theorem states that there are no nonzero whole numbers x, y, and z such that xⁿ + yⁿ = zⁿ for any n greater than 2. (For n = 2, there are lots of solutions, for example, 3² + 4² = 5².) No complete solution to the problem was found until 1995, when Andrew Wiles of Princeton University cracked it using very sophisticated modern techniques from algebraic geometry.

During Germain's time, the main approach to the problem was to tackle it for particular exponents n, and it was known that it would suffice to prove the theorem for prime exponents. And Germain herself used the proof she has been known for, called Sophie Germain's Theorem, to show that the theorem is true for any prime n less than 100, if none of x, y, or z is divisible by n.

This result alone was remarkable, given the challenges Germain faced. As a woman, Germain couldn't enroll in the universities in France. So she took on the identity of a male student who had recently left the school, Antoine-August LeBlanc, reading lecture notes and sending in her homework assignments.

Auguste Eugene Leray painted this portrait of Germain at 14. She had started studying mathematics a year earlier, despite her family's efforts to discourage her. A friend noted in her obituary that she studied "by getting up at night in a room so cold that the ink often froze in its well, working enveloped with covers by the light of a lamp even when, in order to force her to rest, her parents had put out the fire and removed her clothes and a candle from the room."

But somehow, her instructor, the great mathematician Lagrange, discovered her secret. According to a commentator at the time, Lagrange "went to her to express his astonishment in the most flattering of terms," and the commentator goes on to say that "the appearance of this young 'geomètre' made quite a stir." Nevertheless, the barriers against Germain's inclusion in the mathematical community didn't come tumbling down. She still couldn't enroll in the university, and her education was haphazard. And even after she had produced many impressive results, she had difficulty getting access to the French Academy of Sciences.

Indeed, Pengelley and Laubenbacher believe that she must have worked in far greater isolation than previously thought, judging from her notes. Legendre had been thought to have been a mentor for Germain. They both lived in Paris and both worked on Fermat's Last Theorem. But her notes show that they independently proved many of the same theorems and seemed to know little of one another's methods.

"Legendre's techniques were much more ad hoc," Pengelley says. "Germain would develop a theoretical approach or algorithm, not a computational one. She focused on methods of general applicability. She was more the theoretical mathematician." Pengelley and Laubenbacher presented their findings at the Joint Mathematics Meetings in San Diego in January.

Furthermore, Pengelley and Laubenbacher were astonished to find that Germain had something no other mathematician had at that time: a plausible, realistic plan for cracking Fermat's Last Theorem in its entirety, not just one number at a time, and not just if none of x, y, or z is divisible by n. This newly discovered grand plan used a completely different approach than what mathematicians have known from Sophie Germain's Theorem all this time.

"She was out to prove it all in one fell swoop," Laubenbacher says. "She was going to use this new math Gauss had developed. She read Gauss's book and said, 'That's the ticket to proving Fermat!' That was very bold. Nobody thought like that. Her role in 19th-century number theory was revolutionary."

Look for part II next week.

References:

Laubenbacher, R., and D. Pengelley. 2007. Voici ce que j'ai trouvé: Sophie Germain's grand plan to prove Fermat's last theorem. Preprint available at www.math.nmsu.edu/%7Edavidp/germain.pdf.

Bucciarelli, L., and N. Dworsky. 1980. Sophie Germain: An essay in the history of the theory of elasticity. Hingham, N.Y.: D. Reidel Publishing.

For more information on the aspect of Sophie Germain's number theory work that has long been known, see www.agnesscott.edu/Lriddle/women/germain-FLT/SGandFLT.htm.

For more information on Sophie Germain's life, go to www.pbs.org/wgbh/nova/proof/germain.html.

Mathematicians often rhapsodize about the austere elegance of a well-wrought proof. But math also has a simpler sort of beauty that is perhaps easier to appreciate: It can be used to create objects that are just plain pretty—and fascinating to boot.

That beauty was richly on display at an exhibition of mathematical art at the Joint Mathematics Meetings in San Diego in January, where more than 40 artists showed their creations.]]> Michael Field, a mathematics professor at the University of Houston, finds artistic inspiration in his work on dynamical systems. A mathematical dynamical system is just any rule that determines how a point moves around a plane. Field uses an equation that takes any point on a piece of paper and moves it to a different spot. Field repeats this process over and over again—around 5 billion times—and keeps track of how often each pixel-sized spot in the plane gets landed on. The more often a pixel gets hit, the deeper the shade Field colors it.

"Coral Star" shows the motion brought about by one particular dynamical system. Michael Field

The reason mathematicians are so fascinated by dynamical systems is that very simple equations can produce very complicated behavior. Field has found that such complex behavior can create some beautiful images. For example, the dynamical system he depicts in "Coral Star" does some peculiar things as it gets closer to the center (technically, the equation is discontinuous at the origin). So as you get closer and closer to the center, the image gets more and more complex.

"Even apart from the center, the image has quite a lot of depth to it," Field says. "It's a feature of the way it's colored. I'm not so keen on bright primary colors. The shading makes it more interesting."

This image has an unusual 35-fold symmetry, and Field created it as a present for his wife on their 35th anniversary.

Robert Bosch, a mathematics professor at Oberlin College in Ohio, took his inspiration from an old, seemingly trivial problem that hides some deep mathematics. Take a loop of string and throw it down on a piece of paper. It can form any shape you like as long as the string never touches or crosses itself. A theorem states that the loop will divide the page into two regions, one inside the loop and one outside.

The white line above forms a single loop, dividing the page into two regions. Looked at from afar, the image forms a Celtic knot. Robert Bosch

It is hard to imagine how it could do anything else, and if the loop makes a smoothly curving line, a mathematician would think that is obvious too. But if a line is very, very crinkly, it may not be obvious whether a particular point lies inside or outside the loop. Topologists, the type of mathematicians who study such things have managed to construct many strange, "pathological" mathematical objects with very surprising properties, so they know from experience that you shouldn't assume a proof is unnecessary in cases like this one. And this problem did turn out to be very difficult to solve: It took about 20 years after mathematicians began working on the problem to find a correct proof.

Bosch created a simple string-loop on a page and colored the resulting region inside the loop red and the region outside black. From afar, the image looks like two interlaced loops—one red and one black—that form a Celtic knot. For more information about his method for creating the image, see "Artful Routes".

Robert Fathauer, an artist with a mathematical puzzle business in Phoenix, Ariz., found that it doesn't require fancy mathematics to stumble upon remarkable mathematical patterns. He was playing around with various ways of arranging squares in repeating patterns. He started with a red cube and placed five half-sized orange cubes on its exposed faces. Then he put five smaller yellow cubes on the faces of each of those, and five even smaller greenish cubes on the faces of those, and so on.

This "Fathauer crystal" is built from 13 iterations of a fractal pattern, placing cubes on cubes on cubes. Robert Fathauer

"After a few iterations, I noticed that something special was happening with that arrangement," Fathauer says. The shape was approximating a pyramid, with triangular holes punched out. Even more remarkably, he found that the faces of the pyramid formed the Sierpinski Triangle, one of the earliest fractals ever studied.

The ordinary method to form Sierpinski Triangle is to take an equilateral triangle, connect the midpoints of the sides, and punch out the middle triangle. Then do the same thing to each remaining triangle, ad infinitum, to create an object in which any tiny piece looks the same as the whole. This self-similarity makes it a fractal.

Andrew Pike took inspiration from a similar Sierpinski fractal in creating his art. The senior at Oberlin College started with a photograph of the Polish mathematician Waclaw Sierpinski and recreated a version of it with tiles made from the "Sierpinski carpet." To create a Sierpinski carpet, take a square, divide it in a tic-tac-toe pattern, and take out the middle square. Then draw a tic-tac-toe pattern on each remaining square and knock out the middle squares of those. Continuing forever will create the Sierpinski carpet.

Pike stopped short of continuing forever and instead created tiles with different numbers of iterations of the process. Some of the tiles started white, with the knocked-out squares black, and some of them started black, with the knocked-out squares white. This gave him squares that approximated many gradations of gray.

This portrait of Waclaw Sierpinski is made from Sierpinski tiles. Andrew Pike

Then he created a computer program that divided the photograph of Sierpinski into tiny squares, averaged the shades of gray in the picture across each individual square, and selected the Sierpinski tile that was closest in shading. "But it didn't look good," Pike says. "The transitions were really rough."

He couldn't simply make the tiles smaller, because printers can produce dots that are only so tiny. So he used a technique called "dithering." He calculated the error—the difference between the shading of the photograph and the shading of the most similar Sierpinski tile—and spread it between the other nearby tiles. This effectively softened the image, removing the awkward transitions between tiles.

"We chose the image of Sierpinski because it was self-referential," Pike says. Seems appropriate for a technique using self-similar fractals.

Dominique Ribault's "Elephants and Mouse" was inspired by the mathematical art of M.C. Escher and Victor Vasarely. Dominique Ribault

References:

The website for the exhibition is at www.bridgesmathart.org/art-exhibits/jmm08/.

Copies of the catalogue for the exhibit, complete with high-quality reproductions of all the pieces of art, are for sale at www.mathartfun.com.

Peterson, I. 2005. Artful routes. Science News Online (Jan. 1). Available at www.sciencenews.org/articles/20050101/mathtrek.asp.

______. 2003. Constructing domino portraits. Science News Online (April 12). Available at www.sciencenews.org/articles/20030412/mathtrek.asp.

Shortly after September 11, 2001, a small, heavy package wrapped in brown paper arrived in the mail at the Woody Guthrie Archives in New York City. Inside was a mess of wires.

Guthrie's daughter Nora eventually figured out that the suspicious package wasn't a bomb, but rather a recording of her father on a device that predated magnetic tape. After a year of searching, she managed to track down someone with the equipment to play it.

What she finally heard was a bootleg recording of her father singing a live performance in 1949. It was the first time she had ever heard him perform in front of a live audience. He had developed Huntington's chorea and stopped performing when she was a child, and she thought he had never been recorded live. ]]>

Woody Guthrie was an American folk singer and songwriter, known for songs like "This Land is Your Land," "Pretty Boy Floyd," and "Hard, Ain't It Hard." He died in 1967.

So she was determined to preserve the recording. For the first step, she and a team of engineers transferred it into digital format. It was a hair-raising experience. "The wire was really flimsy," says Jamie Howarth, a sound engineer on the job. "It was frustratingly, maddeningly fragile." It snapped over and over, and with every snap, a moment of the recording was lost. And when it didn't snap, it kinked and snarled.

After a 36-hour session, Guthrie and the engineers listened to the recording they produced. The pitch rose and fell independent of Guthrie's singing. They could hear him telling long stories, but only every few words were intelligible. The wire had stretched in places, slowing the recording down. The kinks produced moments of silence.

Howarth is the head of a company that specializes in restoring old analog recordings. If a tape slows down for any reason either during recording or playback, it lowers the pitch and stretches the sound out longer. If it speeds up, the pitch goes higher and the sound goes faster. Howarth had found that slight speed variations occur even in modern recording equipment, creating slight distortions that sound like "wow-wow" or a flutter.

Sound File: Vivaldi Clip 1

This 50-second recording of the NBC Symphony Orchestra playing "Vivaldi Concerto for Orchestra and Two Violins" in 1955 is extremely distorted with a fast flutter and "wow" dips that are painful to listen to.
J. Howarth / Plangent Processes

Fortunately, math can help. Howarth had developed algorithms to correct these recordings. He looks for extraneous sounds, like an air conditioner or fan in the background that creates a rhythmic sound. Instead of simply removing these sounds, he uses them as a clock, a kind of built-in foot-beat in the recording that tells him what the true timing should be. When a recording is made, this background rhythm is even. But when it's played back, it speeds up and slows down in perfect timing with the errors in the recording. That allows Howarth to adjust the timing of the recording to make it much more similar to the original sound.

Sound File: Vivaldi Clip 2

This is the same recording after Howarth's corrections.
J. Howarth / Plangent Processes

When Howarth isn't lucky enough to find a rhythmic background noise, he has another technique. He has found that all analog recordings contain a sort of rhythmic buzz at a specific frequency way above human hearing. This buzz can substitute for a background fan.

Howarth had successfully used these techniques to restore other old recordings, like the film soundtracks for Oklahoma! and Close Encounters of the Third Kind. But the Guthrie recording was such a mess that it forced him to develop new techniques. He turned to Kevin Short, a mathematician at the University of New Hampshire who had done work on signal processing for sound compression.

To test his algorithms, Howarth whapped a pencil eraser against the transport of a tape player during the recording of a single, steady tone, disrupting its even motion. The top sonogram shows the recorded pitch varying wildly with the disruption. He then corrected the recording using his algorithms and removed the distortion entirely, producing the bottom sonogram. J. Howarth / Plangent Processes

The team discovered the many ways that wire makes a lousy material for sound recording. One problem is that wire's round. When the wire kinked, it would twist out of position and the head would no longer be reading the proper side of the wire. The machine still read the low and medium frequencies, but the very high frequency sounds dropped out—including the signal Howarth used as his foot-beat.

Short developed techniques to interpolate the missing information. "We could actually pick up a hum from the Con Edison power supplies," Short says. "It's a pretty nasty noise." Because that hum was lower frequency, it remained even in the twisted sections. Short also brought in more sophisticated techniques to shift the pitches once the algorithm had identified what needed to be done.

"When it was done, we were all just awed by this recording," Howarth says. "It was miraculous." Despite all the difficulties in the process, the wire recording was in many ways surprisingly good. "It sounds really, really, really good for its time," he says.

Sound File: Guthrie Clip 1—Before processing

Sound File: Guthrie Clip 2—After processing

To hear a sample of the recording, before and after processing, click on the sound file links above.
(Audio clips courtesy Woody Guthrie Publications, Inc. Copyright © 2007, Woody Guthrie Publications, Inc. Used by permission.)

The restored recording was released last September and was almost immediately nominated for a Grammy. The award ceremony will be broadcast Feb. 10.

2/10/08: It won!

References:

Guthrie, W. 2007. The Live Wire: Woody Guthrie in Performance—1949. Woody Guthrie Foundation. Available for purchase at http://woodyguthrie.org/mm5/merchant.mvc.

Kevin Short's website is at www.math.unh.edu/~kmshort/.

Plangent Processes has a website with many more examples of their work at www.plangentprocesses.com/.

The full list of Grammy award nominations is at www.grammy.com/GRAMMY_Awards/.

It's not so obvious how old a 60-year-old is. Ask most 60-year-olds these days and they'll say they still feel pretty young, since they're healthy and expect many active years to come. In 1900, though, a 60-year-old was, well, old.

This simple fact has big ramifications for demographers. Demographers have long known that on average people are getting older all around the world, and they have worked to assess the likely social impacts of that aging. For example, relatively few young people are around to support old people's pensions. But increased longevity counteracts those impacts by making people of any age in effect younger than they used to be, for example increasing the number of years they are capable of working. So it has been hard to assess how big the impact of an aging population is likely to be. ]]> A trio of demographers has now developed new measures of age to fill this gap. Instead of analyzing people's chronological age, they've measured the number of additional years a person can expect to live, subtracting each person's age from the life expectancy at that point in time in the region they're living in. Even with this new measure, the demographers found that the world's population has grown older and will almost certainly continue to do so. In fact, the pace of aging will accelerate over the next 30 years.

"That's important because fastest aging is associated with the greatest adjustment costs," says Warren Sanderson of the International Institute for Applied Systems Analysis in Austria and the State University of New York at Stony Brook. "When things happen more slowly, the political processes can move slowly and more thoughtfully. When things happen fast, it's hard to adjust."

The researchers hope that the findings will encourage lawmakers to plan for an aging population now. In the United States, the minimum age for a full Social Security payment will soon rise to 67, for example. In the context of an aging society, further changes like this will almost certainly be necessary, the researchers say. "If it goes up gradually, the burden will not be very big," Sanderson says. Since aging is accelerating, problems that seem like mild irritants now will turn into major conundrums quickly, so a slow response may lead to abrupt—and difficult—adjustments.

The researchers found that the mean age of the world population is 30, with 44 years left to live on average. By the end of the century, the mean age is expected to be 45, with 41 years left. So both measures, mean age and average years of life remaining, show aging.

North America is a happy anomaly, though. The average age will go up through the end of the century, (from 37 to nearly 50), but the average years remaining also will go up (from 43 to 48). North America is the only region to show that pattern. Sanderson and his collaborators Wolfgang Lutz and Sergei Scherbov of the International Institute for Applied Systems Analysis in Laxenburg, Austria published their findings online Jan. 20 in Nature.

The increase in average age is caused by changes at both ends of life: People are living longer, and fewer babies are being born. In the United States, the population as a whole is getting older because of the aging of the baby boom generation and the decline in fertility after the baby boomers were born. In China, the one-child policy that was instituted in the 1979 lowered fertility rates, and, throughout the developing world, family planning efforts did the same. "One wave is piling on top of another worldwide to cause this," Sanderson says.

This new measure of the age of the population is especially informative because people make many decisions based on how many more years they expect to live, the researchers argue. For example, as longevity has increased, people have invested more in education since they'll have more time to reap the benefits.

These changes may soften the societal impact of the aging of the population. "If the younger generation is smaller, but they are better educated and can be more productive, that might be a good thing," Lutz says, since the young people, though fewer, would still be able to produce enough to support those too old to work. "Certainly the decline of population is desirable from an environmental perspective," he says. The team next plans to try to quantify these effects.

The other good news environmentally is that world population is likely to stop growing within the century. The researchers made this prediction two years ago, and they have now incorporated newer data that has corroborated it. Their model shows an 88 percent probability that world population growth will end within the century.

References:

Lutz, W., W. Sanderson, and S. Scherbov. In press. The coming acceleration of global population ageing. Nature. Abstract available at http://dx.doi.org/10.1038/nature06516.

Further Readings:

Christensen, D. 2001. Making sense of centenarians. Science News 159(March 10):156-157. Available at www.sciencenews.org/articles/20010310/bob14.asp.

Harder, B. 2006. U.S. population to surpass 300 million. Science News (Oct. 7):238. Available at www.sciencenews.org/articles/20061007/note16.asp.

Christensen, D. 2001. Making sense of centenarians. Science News 159(March 10):156-157. Available at www.sciencenews.org/articles/20010310/bob14.asp.

Harder, B. 2006. U.S. population to surpass 300 million. Science News (Oct. 7):238. Available at www.sciencenews.org/articles/20061007/note16.asp.

Only in the last five years has sudoku been capturing people's recreational time. But 250 years ago, Benjamin Franklin was developing fascinating puzzles with principles quite similar to sudoku, keeping himself occupied while taking a break from his electrical investigations. Now, a mathematician has discovered two Franklin puzzles even more fantastic than those previously known and written a book describing all of Franklin's mathematical endeavors.

In Benjamin Franklin's Numbers: An Unsung Mathematical Odyssey (Princeton University Press, 2007), Paul C. Pasles of Villanova University in Pennsylvania argues that Franklin's mathematical achievements have long been overlooked. Franklin applied common-sense quantitative reasoning in many areas where it had never been used—for example, calculating the economic costs of war and slavery, and making population forecasts before the field of population demographics had been developed.

But his mathematical inclinations come out most dramatically in his "most devious magic squares, odd little amusements that must have required considerable facility with number relationships," Pasles writes.]]> A 3-by-3 magic square is a grid of the numbers 1 through 9 arranged so that every row, column, and diagonal adds up to the same number.

The rows, columns and diagonals of this magic square all add up to 15.

Magic squares can be made larger as well. A 4-by-4 magic square, for example, uses the numbers 1 through 16. Unlike sudoku, magic squares don't require any special properties of subgrids—though Franklin created some squares that have them as well.

A myth dates the original magic square to 2200 B.C., when a turtle is said to have emerged from the Lo River in China with a pattern of markings on its back corresponding to a 3-by-3 magic square. Just as sudoku would spread around the world thousands of years later, so magic squares spread to Tibet, Japan, Thailand, Europe, Africa, and finally America. Franklin probably first encountered magic squares in a 1708 book on recreational mathematics.

As with other aspects of his life, Franklin developed clever innovations in magic squares. "Not being content with these [regular properties of magic squares], which I looked on as common and easy things, I had imposed on myself more difficult tasks, and succeeded in making other magic squares, with a variety of properties, and much more curious," Franklin wrote in a letter. For example, in this 8-by-8 magic square Franklin created, each color-coded pattern adds up to 260.

Each color-coded set of squares sums to 260. Paul C. Pasles, Copyright 2008.

This magic square is filled with many other similar hidden patterns as well.

When a friend of Franklin's mentioned having seen an impressive 16-by-16 magic square, Franklin refused to be outdone.

Franklin called this "the most magically magical of any magic square ever made by any magician." Paul C. Pasles, Copyright 2008.

Each row, column, and chevron-shaped "bent row" in Franklin's 16-by-16 magic square adds up to 2056. In addition, if you choose any 4-by-4 subgrid within the magic square and add up the numbers, you'll get 2056 as well!

Delighted by his own efforts, Franklin dashed off another letter: "This I sent to our friend the next morning, who after some days, sent it back in a letter, with these words:— 'I return to thee thy astonishing or most stupendous piece of the magical square, in which'—but the compliment is too extravagant, and therefore, for his sake, as well as my own, I ought not to repeat it. Nor is it necessary; for I make no question you will readily allow this square of 16 to be the most magically magical of any magic square ever made by any magician."

But still, the magician strove to out-magic himself. He devised a new game, a "magic circle of circles."

Others had played with magic circles as well, but Franklin's are far more elaborate. Paul C. Pasles, Copyright 2008.

The numbers along each radius of the circle add up to 360, and each of the concentric circles of numbers also adds up to 360. In addition, the numbers along each colored circle also add up to 360. The following diagram shows how Franklin drew the colored "excentric" circles.

The numbers along each of these circles add up to 360 as well. Paul C. Pasles, Copyright 2008.

Pasles found a hidden connection between Franklin's magic squares and magic circles. Imagine taking one of Franklin's squares and moving the rows around into a circle, spreading it out like a fan. Then the rows would turn into the radii of a magic circle, while the columns would turn into the circumferences drawn in black. But what of Franklin's idiosyncratic bent rows? They, Pasles deduces, become the excentric circles!

Despite Franklin's wizardry, his magic squares seemed to all lack one key feature: their diagonals don't sum to the right number. Critics declared them inferior, their magic tarnished. Franklin claimed in a letter to have created a magic square with both diagonals and bent rows summing correctly, but no record of such a square was known.

Now Pasles has found two magic squares Franklin created that each contains all of the various properties. A 16-by-16 square turned up in a facsimile of a letter that had previously been overlooked, and an 8-by-8 magic square appeared in an early translation of a letter. Pasles shows both magic squares in his new book.

Pasles says that while Franklin's mathematics stayed in the realm of simple arithmetic arguments, its importance shouldn't be dismissed. Franklin was involved in the foundations of decision science, statistics, demographics, and other areas, and his magic squares and circles show a remarkable ingenuity. "Those inspired moments are as much a part of mathematics as the really heavy stuff," Pasles says.

A portrait of Benjamin Franklin by Jean-Baptiste Greuze, painted in 1777.

References:

Pasles, P.C. 2007. Benjamin Franklin's Numbers: An Unsung Mathematical Odyssey. Princeton, N.J.: Princeton University Press. See http://press.princeton.edu/titles/8478.html.

______. 2006. A bent for magic. Mathematics Magazine 79(February):3-13. Abstract available at http://www.maa.org/pubs/mag_feb06_toc.html.

Peterson, I. 2006. Magic square physics. Science News Online (July 1). Available at www.sciencenews.org/articles/20060701/mathtrek.asp.

______. 2006. Counting Franklin's magic squares. Science News Online (June 24). Available at www.sciencenews.org/articles/20060624/mathtrek.asp.

______. 2002. More than magic squares. In Mathematical Treks: From Surreal Numbers to Magic Circles. Washington, D.C.: Mathematical Association of America. See http://www.maa.org/pubs/books/mtr.html.

______. 1996. More than magic squares. Science News Online (Oct. 12). Available at www.sciencenews.org/pages/sn_arch/10_12_96/mathland.htm.

Pickover, C.A. 2002. The Zen of Magic Squares, Circles, and Stars: An Exhibition of Surprising Structures across Dimensions. Princeton, N.J.: Princeton University Press. See http://press.princeton.edu/titles/7131.html.

Paul Pasles has a web page about Franklin's mathematics at www.pasles.com/Franklin/.

For more information about magic squares, go to www.mathworld.wolfram.com/MagicSquare.html.

Franklin's autobiography is freely available online at www.earlyamerica.com/lives/franklin/.

To order any of the books listed above from Amazon.com, please click on the book's title. Sales generated through these links contribute to the Society for Science & the Public's programs to build interest in and understanding of science.

Counting is hard. Neither people nor machines seem to be able to do it reliably. And that's a nightmare for election officials who need an accurate ballot count to decide elections.

Eighteen states require officials to double-check the machine counts by hand for a portion of the ballots. But election officials have had little guidance on what to do with the recount results. If the election is close and the recount finds a few errors, should a registrar call for a larger recount or go ahead and certify the result? Most laws left it to their discretion.

Now Philip Stark, a statistician at the University of California, Berkeley, has developed a recount method that guarantees a 99 percent chance that the result is the same as it would be with a full hand count. Several counties in California plan to try out the method on ballot measures during the presidential primaries this year. If this trial and others go smoothly, California could adopt the method statewide. ]]> The idea behind hand recounts is that even though people are not necessarily more accurate than machines, their mistakes tend to be different. Machine counts have problems because of software errors, because the voters used a kind of ink the optical scanners couldn't read, or because a memory card or stack of ballots never made it to be counted. Hand counts go wrong when someone loses track of the count, a paper jam disrupts the printing of the paper ballots, or a box of paper ballots disappears. Counting the ballots both ways can uncover the errors in each method.

Recounting by hand is expensive, though, and if an election is a landslide, a full recount probably isn't necessary. In that case, a random check of a few precincts to catch large-scale fraud or foul-ups will suffice. But in a close race, a few miscounted ballots could decide the outcome.

As part of a complete review of California's election procedures, election officials turned to Stark to develop a method to determine just how big a recount is necessary.

"No flat percentage, short of 100 percent, gives high confidence in all circumstances," Stark says. The appropriate percentage depends on the number of precincts and ballots, the size of the apparent margin of victory, and the number of mistakes the recount finds. A very close race with only a few ballots and lots of mistakes will require a full recount. For many races, though, a recount of 1 percent of precincts, randomly chosen—the number currently required by California law—will be enough to generate 99 percent confidence.

"There are going to be mistakes," he says. "There are always mistakes." The question is just whether there are few enough mistakes in the 1 percent sample to suggest that the election results are very likely right. He first figures out how many mistakes there would have to be in all the ballots to change the election result. Then he counts the number of mistakes in his sample. If he sees very few errors in his sample, it is possible but unlikely that all the ballots would have had enough mistakes to change the election result. Standard techniques from statistics quantify just how unlikely that is. If the probability that the outcome is correct is less than 99 percent chance, Stark keeps counting, including additional randomly chosen precincts. He stops only when he's generated sufficient confidence or he's counted all the ballots.

Stark's method addresses some of the concerns about the trustworthiness of electronic voting machines. "Regardless of how arcane, esoteric, or just wrong the machine's mistakes might be—whether it is a simple programming bug or deliberate fraud—you can guarantee that you have good chance of finding it, if it's big enough to alter the outcome" with this method, Stark says. "That assumes, though, that the mistake doesn't simultaneously affect the audit trail." He also points out that it also necessitates that the machines reliably produce such a paper trail, which they often don't.

The 2000 presidential race was so close in Florida, Stark says, that a full statewide recount would almost certainly have been necessary to guarantee the correct outcome with high confidence. He points out that that race was extraordinarily close, as shown by a thought experiment. He imagined that every person in Florida who voted in that election had decided how to vote by tossing a coin. For that scenario, he then calculated that there would have been an 82 percent chance that the margin between the two candidates would have been larger than it was in real life.

A federal bill (H.R. 811) is currently under review in the House of Representatives that would require audits of federal elections and mandate that all electronic voting machines produce a paper trail for audits. In a close race, the bill would require a larger percentage of ballots to be checked. But because it doesn't take into account how many errors have been found, it can't generate a 99 percent confidence rate, as Stark's method does.

References:

Stark, P. In press. Conservative statistical post-election audits. Annals of Applied Statistics. Preprint available at http://www.stat.berkeley.edu/~stark/Preprints/
conservativeElectionAudits07.pdf.

Further Readings:

Klarreich, E. 2002. Election selection. Science News 162(Nov. 2):280-282. Available at www.sciencenews.org/articles/20021102/bob8.asp.

Peterson, I. 2003. Election reversals. Science News Online (Oct. 18). Available at www.sciencenews.org/articles/20031018/mathtrek.asp.

Peterson, I. 1998. How to fix an election. Science News Online (Oct. 31). Available at www.sciencenews.org/pages/sn_arc98/10_31_98/mathland.htm.

Infinity is bigger than any number. But saying just how much bigger is not so simple. In fact, infinity comes in infinitely many different sizes—a fact discovered by Georg Cantor in the late 1800s.

Now a mathematician has come up with a new, different proof. Based on a simple game, the proof uses a strategy that might someday shed light on one of the great unsolved questions in mathematics.]]> The smallest infinity is the one you'd get to if you could count forever. The numbers 1, 2, 3, 4… are called the