MathTrek

The Noisy Game of Baseball

Thu, 10 Apr 2008 18:24:37 -0500

By Julie J. Rehmeyer

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.

Creeping Up on Riemann

Fri, 04 Apr 2008 13:37:47 -0500

By Julie J. Rehmeyer

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.

Humanitarian Statistics

Fri, 28 Mar 2008 12:36:54 -0500

By Julie J. Rehmeyer

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.

Sacred Geometry

Mon, 24 Mar 2008 12:10:54 -0500

By Julie J. Rehmeyer

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.

Spoil-Proofing Elections

Fri, 14 Mar 2008 13:04:53 -0500

By Julie J. Rehmeyer

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 Geometry of Music

Fri, 07 Mar 2008 16:10:33 -0500

By Julie J. Rehmeyer

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.

A Mathematical Tragedy

Fri, 29 Feb 2008 12:36:46 -0500

By Julie J. Rehmeyer

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.

An Attack on Fermat

Thu, 21 Feb 2008 17:57:18 -0500

By Julie J. Rehmeyer

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.

Math on Display

Thu, 14 Feb 2008 14:58:13 -0500

By Julie J. Rehmeyer

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.

The Grammy in Mathematics

Fri, 08 Feb 2008 21:23:27 -0500

By Julie J. Rehmeyer

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.

Getting Old, Faster and Faster

Fri, 01 Feb 2008 11:26:02 -0500

By Julie J. Rehmeyer

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.

Benjamin Franklin Plays Sudoku

Thu, 24 Jan 2008 17:53:34 -0500

By Julie J. Rehmeyer

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.

Checking It Twice

Thu, 17 Jan 2008 18:40:34 -0500

By Julie J. Rehmeyer

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.

Small Infinity, Big Infinity

Thu, 10 Jan 2008 14:42:22 -0500

By Julie J. Rehmeyer

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 Power of Being Influenced

Thu, 03 Jan 2008 13:00:29 -0500

By Julie J. Rehmeyer

Sometimes an idea spreads through society like a newly-mutated cold virus zooming through a class of first-graders. Other times, a good idea never seems to take hold. What makes the difference? Scientists want to know, and marketers want to know even more, since they make their living spreading ideas about their products.

A key reason some ideas are so successful, conventional wisdom has held, is that a few highly influential people espouse them. In his book The Tipping Point, Malcolm Gladwell wrote that what he calls "social epidemics" are "driven by the efforts of a handful of exceptional people." Those exceptional people tend to be experts on a subject who love to talk. Such people can convince dozens of others of their opinions. An excellent sales strategy, then, would be to find those few critical people, persuade them of the value of your product, and leave it to them to convince others.

It's a compelling idea, but does it really work? Social network theorists Duncan J. Watts of Columbia University and Peter Sheridan Dodds of the University of Vermont in Burlington decided to put the notion to a test. What they found is a disappointment for "viral marketers" who specialize in selling products by influencing influential people.