Cocktail Party Physics

thrown for a curve

Over at Wired, David Dobbs has a very nice post on famed pitcher Sandy Koufax -- excuse me, "the greatest pitcher ever" -- and the curve ball, a nasty (if you're the batter) trick whereby the baseball dives downward suddenly just before it reaches the plate, faking out the batter. It's fast, too, traveling at least 75 mph and spinning at an angle of around 1500 rpm. It only takes about 0.6 seconds to get from pitcher's hand to home plate -- not a lot of time to react. David's post focuses on the perceptual illusion created as the ball travels towards the batter, and it's that optical illusion that makes the curve ball so bloody hard to hit. Or, as the decidedly salty Mickey Mantle said after being struck out by one of Koufax's infamous curve balls in the 1963 World Series: "How the fuck is anybody supposed to hit that shit?" How indeed? I'll let David explain what's going on perceptually:

[T]he curveball kills you two ways: first, through actual movement; and second, through an extra perceived movement — illusory — that further complicates the task of getting the tiny strip of sweet spot on your bat onto the ball.

The extra perceived movement rises from a difference between the neural dynamics of central vision and those of peripheral vision. This effect of this difference is that a baseball that is rotating horizontally but falling straight down as it comes toward you will appear to fall vertically if you’re looking straight at it — but appear to move sideways if it’s in your peripheral vision. ... This in turn happens because your eyes simply can’t keep up with a pitch as it approaches you and effectively accelerates its path across your field of vision. The ball goes from moving at you to moving past you. At the crucial moment — the last few feet of the ball’s half-second, 60-foot trip  to the plate — you must of necessity switch from seeing the ball with your central vision to seeing it with your peripheral vision.

To add to your troubles, it is in this tenth of a second or so that the curveball also moves the most in reality. ... So just as the ball’s real downward and sideways motion is greatest, the curve’s apparent break is exaggerated by visual dynamics.

But there's some terrific physics involved here, too, and Jen-Luc Piquant also uncovered a little-known slice of 20th century history where baseball and science came together for one shining moment to prove that the curve ball really does curve. (For links to everything you could possibly want to know about the physics of baseball, check out this site.)

The man who brought those worlds together was Lyman Briggs who served as director of the National Bureau of Standards (today it's known as the National Institute of Standards and Technology) from 1933 to 1945. Born on a farm in Battle Creek, Michigan, Briggs was an esteemed physicist, despite never attending high school; he won admission to Michigan State College by examination, graduating second in his class four years later. He was also a lifelong baseball fan, having played outfield on the college baseball team during the 1890s.

This was a period when the question of whether the curve ball actually curved was hotly debated. Among the true believers was St. Louis Cardinals pitcher Dizzy Dean. "Ball can't curve?" he famously declared during the 1930s. "Shucks, get behind a tree and I'll hit you with an optical illusion." But anecdotes aren't a substitute for scientific data. So once Briggs officially retired, he decided to do the experiments to settle the matter. And he was well-connected enough to enlist the aid of the pitching staff of the Washington Senators and their manager, Cookie Lavagetto, to do so. It wasn't just a question of baseball, either: the question related to NIST's ongoing research into ballistics and projectiles: the rate of spin is related to how much the ball (or projectile) is deflected at different speeds. Apparently the NSB (now NIST) conducted lots of experiments with golf balls and baseballs; one of Briggs' publications was a 1945 paper entitled, "Methods for Measuring the Coefficient of Restitution and the Spin of the Ball."

So really, Briggs was the perfect man to take on the question of the curveball. Based on his earlier research, he already knew that, in 1852, a German enigneer named Heinrich Magnus had accounted for the curved path of a cannon ball by describing a kind of "whirlpool of air" created around the projectile. This is now known as the "Magnus effect." (That said, Magnus wasn't the first to notice this. In 1672, Isaac Newton correctly inferred the cause after observing tennis players in his Cambridge college, while a British artillery engineer named Benjamin Robins used a similar concept to explain deviations in the trajectories of musket balls in 1742. Isn't the Internet an amazing place?) Per Wikipedia:

Generally the Magnus effect describes the laws of physics that make a curveball curve. A fastball travels through the air with backspin, which creates a high-pressure zone in the air ahead of and under the baseball. The baseball's raised seams augment the ball's ability to churn the air and create high pressure zones. The effect of gravity is partially counteracted as the ball rides on and into energized air. Thus the fastball falls less than a ball thrown without spin (neglecting knuckleball effects) during the 60 feet 6 inches it travels to home plate. On the other hand, a curveball, thrown with topspin, creates a high-pressure zone on top of the ball, which deflects the ball downward in flight. Instead of counteracting gravity, the curveball adds additional downward force, thereby gives the ball an exaggerated drop in flight.

Briggs took things a few steps further and set out to explore how much the curve of a baseball depends on its spin and its speed. Ideally, he wanted to study balls thrown by actual pitchers but it proved too difficult to photograph the flight path, even with a strobe camera flashing 20 times per second. So he switched to rotating a baseball on a rubber tree thereby giving it spin, and the ball was then struck by wood projectiles shot from a large mounted airgun. By doing so, he managed to to measure the speed and the curve; he just couldn't figure out how to measure the spin as well. Per his own notes, the images captured were so small "that the marks put on the ball to measure the spin could not be seen." And spin was looking to be the critical variable when it came to determining how much a ball curved.

To measure the spin of a pitched ball, he enlisted the pitching staff of the Washington Senators at Griffith Stadium. (Historical documents give us the names of those who helped: pitchers included Pedro Ramos and Camilio Pascual; Ed FitzGerald was the catcher for the experiments.) He attached one end of a light flat tape to the ball and then laid the rest of the tape -- a very long piece of tape, 60 feet or so -- loosely along the ground between the mound and home plate, making sure there were no twists. After the pitched curveball was caught, he simply counted the number of twists that had appeared in the tape. The number ranged from 15 or 16 down to 7 or 8 twists in the tape. Knowing the distance the ball traveled (60 feet), Briggs could deduce the pitch speed was 100 feet per second, and concluded that the maximum spin would be 1600 rpm.

But Briggs still wasn't satisfied; the dude was thorough. How could he mark the ball in such a way as to determine the spin most precisely? It just so happens that NSB had a wind tunnel -- maybe they still do -- specifically for the purpose of studying aerodynamics. This allowed him to precisely control most of the variables. He tossed baseballs into the wind tunnel, and let them freefall against the horizontal wind streams, which naturally caused the ball to curve. When a baseball finally hit the ground after the requisite 0.6 seconds, it bounced off a sheet of cardboard treated with lampblack, which left a smear on the ball, indicating point of impact. His conclusion: "An increase in the speed of the pitch beyond 100 feet per second reduced the curve only slightly and the important thing was the spin." Spin rather than speed was the critical factor in causing a pitched ball to break. And a curveball can curve up to 17-1/2 inches as it travels from the pitcher's mound to home plate.

NSB announced the results on March 29, 1959, and Briggs subsequently published his results in the American Journal of Physics. What does it all mean? Well, it's probably not going to help star pitchers perfect their curveballs, unless they're really fast at doing calculations in their heads. That kind of skill comes with practice, practice, practice. But knowing the precise relationship between all those variables and the forces acting on the ball is extremely useful in its own right -- if not for baseball, then most definitely for ballistics. And hey -- baseball players got to contribute to the forward march of science.

 

physicists put on their poker face

Jen-Luc Piquant has donned her poker face today and for good reason. This morning the Spousal Unit was chuffed to find his latest online issue of Discover magazine was available -- because it contains an article by yours truly on poker-playing physicists. (November issue! Pick up a copy!)  They're everywhere these days: not just the Spousal Unit, but also string theorist Jeff Harvey, particle physicists Michael Binger and Marcel Vonk (both of whom have done extremely well on the professional circuit), a former grad student of Harvey's named Eduard Antonyan, Liv Boeree and Michael Piper (both pals of Binger's), and who knows how many others? Discover loved the idea of poker-playing physicists as much as I did when I pitched it over the summer, and editor Bob Keating did a great job chopping my 3500-word monster draft manuscript into the more manageable length ultimately published in the magazine (clearly, writing books has ruined what little brevity I once possessed).

Still, I'd hate for all that effort to go to waste, plus there was all kinds of great material from the sources I interviewed for the piece that never made it past the transcript stage. That's why the Internet invented blogs: as a place to supplement the inevitable space limitations of traditional media for readers who might want a bit of "behind the scenes" bonus material. I mean, who wouldn't want to know that the first time Jeff Harvey played Texas Hold 'Em was for a fundraiser tournament in Chicago, where the role of "dealer" was played by an Albert Einstein action figure? So here goes:

Walk into one of the nicer poker rooms in Vegas, and you might be surprised by the eerie quiet, in sharp contrast to the cacophony in C major emanating from the  slot machines on the main casino floor. There are tables of players studying their cards with hushed intensity, punctuated by the gentle clink of poker chips in the background and the shuffling of the cards. We've sampled most of the Vegas poker rooms over the years, and the MGM Grand is one of The Spousal Unit’s favorites, with elegant marble rims on the tables, and a clear view of the casino’s signature lion habitat. Not that the serious players spend much time watching the lions.  There are odds to be calculated, bets to be made, and if all goes well, money to be won. And that requires laser focus and significant mental stamina.

In general, physicists hate to gamble. "I don't like gambling at all," says Antonyan. "I don't enjoy it and there's nothing in it for me to compensate for the clear negative EV decision of gambling." Harvey's not a fan, either: "Personally I don't like to gamble on games where the house has the odds, but I'm not critical of people who do." And while the Spousal Unit gamely learned craps with me while I was writing The Calculus Diaries (it was research, people!), he hasn't been tempted to play craps since.

Binger doesn't mind gambling, per se, but he learned the pitfalls of blackjack as an undergraduate, when he wrote a computer program to beat the game through card-counting (or, as the casinos like to call it, “cheating”) for his senior project. But then he tried to put his strategy into practice. He lost a pile of cash playing blackjack on an ill-fated trip to Reno, and was barred from six casinos in one day for card-counting in a desperate attempt to recoup his losses. “I realized I wasn’t going to get rich playing blackjack,” he recalls. Poker was different: as he studied the game and pondered the underlying mathematics, Binger realized that poker could be a “beatable game."

That seems to be the strongest lure for poker-playing physicists: it's a game of skill and strategy, not a game of pure chance (although let's face it, luck does play a role, too). Vonk has always loved games, but his love for poker rests on the combination of "math skills" and "people skills," as he puts it. "Good poker requires that you make sound game-theoretic decisions but there  is still plenty of freedom to try and outsmart your opponents," he says. "Other casino games miss that second element. All you can do in blackjack or roulette is make the best possible mathematical decisions, and even then, you will still lose in the long run. I have never been attracted to those games. It's the fact that you play against other people that makes poker so interesting, and that makes it possible to actually be a winner at the game."

And poker favors a similar skill set. "The analytical thinking we develop certainly makes it easier to learn and keep improving at poker, but you also need to be very competitive, a little addictive, and need to have a pretty stable psyche," Antonyan says. "There are a lot of players who could have been really good but can't win because they 'tilt' [lose their cool and start making bad strategic decisions] too much and too often." Binger also cites the mental and emotional discipline required, qualities physicists tend to also develop. "In poker, you often have to sit there for hours, patiently waiting for your spot, and sometimes endure months on end of running bad," he says. "Similarly, in physics research, there are times when you are stuck in the middle of a lengthy calculation or project, and cannot see the end in sight. But you have to persevere and stay focused and positive to get through it."

Is the math really enough to be a poker badass at the professional level? Binger fields this question a lot, and says the probability and equity calculations and statistical analysis he applies give him an edge in the game. Vonk finds that his post-game analysis of how he played specific hands benefits from his mathematical skills. But both Vonk and Binger admit that there are also plenty of other players who really don't know much about the underlying math; they have a good feel, or instinct, for how to play the game. "There are many people who hate math but are great poker players, but there are hardly any players who lack the people reading abilities and still manage to be good poker players," says Vonk. "Mathematical knowledge can to a large extent be replaced by intuition and experience. After a player has played a million hands of poker, even if he does not know the math at all, he will have a decent feeling about when it is profitable to draw to a flush and when it is not."

That said, knowing the math means you can acquire this kind of knowledge much more quickly, and those skills can give an edge in very rare situations that don't often occur in a poker game. "To be a great player, you need both!" Vonk insists. Chris "Jesus" Ferguson is one of the best players in the world, and definitely relies on math and game theory when he plays:

 

Antonyan estimates that the game of poker is "90% simple math/general strategy, and 10% understanding the dynamics of the table and/or the attitudes of one or more players towards you as they develop." The math part rests on basic probability theory, and as the article makes clear, the probabilities of poker are a bit more complicated because there are many more possible combinations of hands -- and you're working with incomplete information. Vonk breaks down the process to a few basic questions: What cards do I have? What range of cards do I think my opponent has? Given these, what is the probability I will win the hand after all cards have been dealt? And most important: given that probability, will I make money in the long run when I pay the bet? The best one can do, most of the time, is "make a very broad guess," he says. Per the Spousal Unit (blogging at Preposterous Universe way back in 2004):

"Texas Hold 'Em is so popular because it manages to accurately hit the mark between 'enough information to devise a consistently winning strategy' and 'not enough information to do much more than guess.' The charm in such games is that there is no perfect strategy, in the sense that there is no algorithm guaranteed to win in the long run against any other algorithm. The best poker players are able to use different algorithms against different opponents as the situation warrants."

To get a sense for how the probabilities can play out in poker, consider the following three possible pairs of hole cards:

Jack-10 suited

Ace-7 unsuited

Pair of sixes

Sean posed this question on his blog, Cosmic Variance, back in 2006: Which hand is most likely to win if you choose to stay in the pot all the way to the showdown, against other pairs of randomly chosen hole cards? The answer took a whole 'nother blog post to delineate. Mathematically, it depends on the number of opponents. The probability that you will win goes down as the number of opponents goes up, because there are more ways for you to be beaten. That said, some hands play well against very few opponents, while others play well against many opponents. It all depends on the circumstances.

Against one opponent, the sixes will win 62.8% of the time, versus 57.3% for Ace-7 and 56.2% for Jack-10 suited. Against four opponents, those odds are reversed: Jack-10 suited will win 27.3% of the time, versus 20.7% for Ace-7 and 17.9% for the pair of sixes. Why does this happen? “Against only one randomly-chosen pair of hole cards, there is a substantial chance that the sixes won’t need to improve; likewise the ace can often come out on top just by itself, so the Ace-7 is second-best,” Sean explains. “But against four randomly-chosen pairs of hole cards, chances are excellent that someone will improve, and Jack-10 suited has the best chance.”

The probabilistic outcomes change again if we pit these three hands against each other, two at a time. In that case, sixes are slightly more likely to beat Ace-7, and Ace-7 is likely to beat Jack-10 suited, but Jack-10 suited is likely to beat a pair of sixes.

The sixes are the best starting hand all by themselves. For one of the latter two to win, favorable community cards must appear on the flop, turn, or river. The only way for the Ace-7 to beat paired sixes is for either an ace or a seven to turn up -- or, less likely, for just the right combination of four cards to land on the board to make a straight or flush.

Pit those same sixes against Jack-10 suited, and the situation is reversed. In that scenario, there are more ways for Jack-10 suited to improve. The cards are “connectors,” so there are more possible cards that would give low straights (7-8-9) and high straights (Q-K-A), plus the hole cards are suited, making it much easier to make a flush.

So Jack-10 suited will usually beat a pair of sixes. But it won’t usually beat Ace-7 if the ace is of the same suit. For instance, if four more suited cards come up, the Jack-10 suited will have a flush, but the Ace-7 will have a higher flush, and will win the hand.

See? Poker is a very complicated game. And even more complications arise during the betting process: "what happens after the first two cards are dealt and after each card thereafter," the Spousal Unit once wrote.

If determining the edge and the odds were all there were to succeeding at poker, probability theory would suffice, and if it were a purely logical game like chess, it would merely require impressive feats of calculation to determine the winning series of moves.

But remember that poker is a game of limited information, where players must deduce what cards their opponents are likely to have based on their knowledge of the odds and clues from other players’ behavior. There may not be a single answer. As Harvey -- ever the string theorist -- puts it in the Discover article: “Chess is like classical mechanics. Poker is like quantum mechanics. In chess, there is only one right move. In poker, there is a probability distribution of right moves.” Harvey admits that one of his classic errors is "calling wen I think I am beat for other reasons (betting patterns, tells, and so on)," but he calls anyway because "the math says I should. At times like that, I need to pay less attention to the math."

Yet another complicating factor is that human beings aren't always predictably rational, particularly when it comes to poker: if you assume your opponent is skilled and rational, and he isn't, your strategy could backfire and fall victim to "beginner's luck." I found this enlightening analysis over at Cardplayer.com, outlining the different between an optimal strategy and an exploitive strategy (Ferguson's favorite) in No-Limit Texas Hold 'Em (and note the very specific circumstances described throughout: change even one element and it might call for a different strategy):

"Let's say you're playing no-limit hold 'em against a calling station who never folds pre-flop no matter what the bet is, but will sometimes fold after the flop if he misses completely. He just insists on seeing the flop. Now say you're dealt two aces and you each have a few thousand blinds in front of you. The optimal strategy is probably to make a small raise, both building a pot and disguising your hand. But with this player in the game, a much better play is to move all in, knowing he'll call you. To take maximum advantage of this terrible opponent, you need to employ an exploitive strategy. The optimal strategy would still win you money but against bad players, other strategies might win you more money. ... An optimal strategy is designed to protect you against opponents who play well. But when we can find ways to do better than optimal strategy against certain players, we do it."

The article also mentions mathematical/computational great John von Neumann, who with Oskar Morgenstern (an economist) wrote the definitive treatise on game theory and poker -- specifically the art of bluffing, which fascinated von Neumann -- in 1944: Theory of Games and Economic Behavior. It wasn’t a bestseller, but it did yield an intriguing insight into the art of the bluff: you should always bluff with your worst hand, not a mediocre “bubble” hand. If betting is slow, it might be worth calling, or “limping” into the game, with a mediocre hand, because your chances of winning are pretty good against other mediocre hands. A bad hand won’t win unless everyone else folds, so an aggressive raise is the best strategy.

Indeed, there are rare cases where game theory dictates you should fold pocket aces before the flop when playing a tournament. In non-tournament play, the goal is not just to win the hand but to make the most money. In a tournament, you want to outlast your opponents to win it all. That might entail intentionally opting not to maximize your monetary gains on one specific hand to remain competitive in the tournament. You sacrifice short-term gain to achieve the long-term goal.

It doesn’t pay to be too consistent in Texas Hold ‘Em. I once played poker with a group that included Harvey. I consistently bet when I had a strong hand, checked when I had a “bubble” hand, and folded when I had a bad hand. So when I made trip aces after the flop, I pushed all-in, going heads-up with Harvey. He had pocket Queens, an otherwise strong hand had there not been an ace on the board. It’s hard to fold pocket Queens but that’s just what Harvey did. He correctly analyzed his chances, based on my all-too-predictable style of play, and he had the discipline to stick to his strategy. I won the hand, but didn’t win much money because Harvey folded before he’d committed many chips to the pot.

The optimal strategy can also depend on what type of poker is being played: your strategies will be different for No-Limit Texas Hold-Em, for a No-Limit tournament, a Limit Texas Hold 'Em "ring game", and different again for online poker. "The only people playing online are serious," says Harvey. "They also use software to keep track of their opponents' statistics, which is consistent with the rules of the site, even if it seems a bit like cheating." Here's the difference: in a live game, you have to remember/keep track of opponents' style of play yourself, i..e, when they raise and in what position. The online software can analyze thousands of hands being played at the same time, and that larger sample space makes for a more accurate statistical analysis. Antonyan excels at online poker, and has won a tidy sum to date, although he rarely plays more than four hours a day. He's since left physics, but not to pursue poker: he's putting his quantitative abilities to work on Wall Street as a "quant" with a high frequency trading group.

"It's much more about modeling, statistical analysis and game theory at that level," says Harvey. "I'd have to spend as much time learning and playing poker as I do on physics." And to date, he's been unwilling to do that, unlike, say, Binger, who took time off from physics after scoring big in the 2006 Wold Series of Poker. Despite winning around $4 million through his third place finish, and an extra 2 million since, Binger still lives pretty simply: he uses his winnings to bankroll his travels and more poker tournament action, going to Australia, New Zealand, South Africa, Monte Carlo, London, Venice, San Remo, Aruba the Bahamas, and all over the US. That's a lot of frequent flyer miles! (You can follow his exploits on Twitter: @mwbinger.) And check out the stats: six tournament wins, second place in Bluff player of the year in 2008, and currently ranked sixth worldwide in the Bluff Power Rankings.

The mathematicians have had a good run when it comes to analyzing poker, but the Spousal Unit is (rather cheekily) on record predicting that physicists will prove to be the better poker players in the future? His reasoning? No-Limit Texas Hold 'Em is such a complex system that "we cannot derive a dominant strategy in a closed form. Game theorists and mathematicians study simplified systems about which they can actually prove theorems," he wrote. This is a decent strategy for two players going heads-up, but for a full table, pre-flop, "it becomes a question of which approximations to make and which models to choose for your opponents." Physicists, let's face it, are often pretty adept at choosing the best models.

He also had a corollary: "Phenomenologists and astrophysicists will be better poker players than string theorists." I expect Harvey to respond in kind to the obvious throw down. In fact, I think Discover magazine should sponsor a tournament in Vegas pitting the poker-playing physicists against each other -- or against mathematicians like Ferguson. Just to keep things interesting...

There’s a saying that Texas Hold ‘Em consists of long stretches of boredom punctuated by three minutes of sheer terror. Poker never lacks for suspense: you can play a hand flawlessly from a probabilistic standpoint, but there is still the possibility you’ll lose; statistical anomalies do happen. Even with pocket aces and a flop of 9-9-2, your chance of winning a heads-up showdown against pocket queens is only 92%. Harvey recently faced just that scenario – and a third queen appeared as the very last card. His opponent “sucked out on the river.”

So poker also requires nerves of steel, and an emotional equilibrium that Harvey, for one, admits he does not possess. “You need to be unflappable,” he says. “Bad luck can’t bother you. It’s too easy to get ‘tilted,’ and start playing looser, more erratic, or too passive.” More often than not, he says, “My emotions get the better of me.”

I dug up anecdotal evidence to support that. That same Chicago game where Harvey "read" me and folded his pocket queens also included Peter Sagal, host of NPR's "Wait Wait Don't Tell Me." He, Harvey and another pal, Chris Lackner, once went to a riverboat casino in Indiana to play poker. Sagal recalls harvey "getting a really hot hand and betting on it. I remember his hand was actually shaking as he pushed in the chips." And alas, Harvey lost to a bad beat. But it made me feel better, since I suffer from the same display of nerves whenever I play poker. Some of us just don't have what it takes, in the long run, to be truly world-class players. But I'll be working on perfecting my poker face. You can count on it.

games people play

Confession: I'm not much of a gamer. I know that loses me major "geek chic" points in the trendier online circles, but to really delve into these marvelous virtual gaming worlds requires the kind of large uninterrupted blocks of leisure time I haven't enjoyed for at least a decade. (Oh, how I miss large blocks of leisure time!) It's a bummer because I know I am missing out on some really cool stuff -- like the 2005 "corrupted blood" incident that broke out in World of Warcraft. I heard about it at the AAAS meeting in San Diego from  a Rutgers scientist named Nina Fefferman, who became fascinated by the in-game parallels to real-world epidemics and people's behavioral responses.

Assuming I am not the only person who has never played World of Warcraft -- although Jen-Luc Piquant and the Spousal Unit are now urging me to give it a whirl -- what happened was this. Blizzard Entertainment regularly updates the game, introducing new challenges for advanced players. One such update was a new dungeon called Zul'Gurub controlled by a demon (or "end boss" in game parlance) called Hakkar. Only really high-level players could even find Zul'Gurub. I guess the point is to slay Hakkar, who naturally is quite difficult to kill. One of his secret weapons -- wielded when the demon is about the expire -- is a spell called "Corrupted Blood," which inflicts periodic damage on infected players, gradually draining away their vitality points. The only "cure" is to finally kill Hakkar.

It was supposed to just infect nearby players, most of which by default would be at a high level. So as damaging as the spell could be, it was just supposed to be an added annoyance to make the game space a bit more challenging. But then things went horribly wrong. Because of a glitch in the programming, the virtual pets, or animal companions, of players -- while technically "nonplayable" -- could also become infected and spread the disease, although they didn't show any symptoms. Those pets left the Zul'Grub space and spread "Corrupted Blood" to the lower levels, where it literally wreaked havoc. While advanced players would take the damage, lesser players were "killed" outright. At least three servers were affected, and in the end Blizzard had to reboot the whole thing to fix the glitch. Played out in the game space, wherever the corrupted blood plague broke out, it looked a little something like this:

Those periodic bloody bursts are the spell inflicting damage. You can see players "dying" in the game space, and those still alive bolting around in a panic, with some taking advantage of healing spells and other strategies. But in general chaos reigns. And that's what Fefferman found so fascinating, because it was actual data that seemed to showcase how human beings behave during an epidemic -- and that behavior is not always rational, or courageous. In fact, there were a few players who teleported out of Zul'Gurub and deliberately spread the disease out of malice. Some players tried to help, administering healing spells, but others panicked and fled after being infected and also carried the disease to uninfected lower levels.

All of this mimics real-world behavior in actual epidemics, according to Fefferman, who pointed out that people choose to play certain roles in this type of crisis, much as they did in the game. There were opportunists, conspiracy theorists, and Fefferman found one video showing a player who decided his role would be to stand on the steps while carnage raged around him, narrating the action and foretelling the end of the world -- yes, he became That Guy, the self-appointed Doomsday Prophet. At least he wasn't one of the malicious infectors. Then there was the sheer idiocy of players who ignored the warnings and went to infected areas out of curiosity, thereby becoming infected themselves. And so she became convinced that we could learn a great deal from studying plague outbreaks in virtual gaming worlds.

Fefferman went on to co-author a paper (with Eric Lofgren) in Lancet Infectious Diseases discussing some of the implications of the "Corrupted Blood" outbreak for improving real-world epidemiological models. Such models, while useful, necessarily must make assumptions about human behavior -- specifically, they use mathematical rules to approximate human behavior. And it would be frankly immoral to deliberate introduce a pathogen into a controlled population to study how things played out firsthand. But what if someone could design a new disease specifically for a virtual online community and study the situation that way, using the collected data to further refine epidemiological models? That would be a dream come true for Fefferman.

Blizzard has always maintained that World of Warcraft is just a game and was never designed to mirror anything "real." Hakkar's spell is the stuff of online fantasy. Also, while players "died"  of the plague in the game, a virtual death of an avatar just isn't the same as real, physical death -- plus there are ways to resurrect one's character in the game even after one's demise. Which might be why you had the malicious infectors deliberately spreading the disease, essentially behaving like terrorists. In fact, Charles Blair of the Center of Terrorism and Intelligence Studies thinks World of Warcraft could help scientists study how terrorist cells operate because it "involves real people making real decisions in a world with controllable bounds, which could provide a more realistic model for military intelligence analysts."

Real people making real decisions that affect others around them is why the game holds such fascination for Fefferman as well. That, and the fact that the Corrupted Blood plague was completely unplanned -- unlike the zombie plague that Blizzard deliberately spread to promote World of Warcraft: Wrath of the Lich King. And frankly, players get really invested in their avatars; they don't take their "deaths" lightly. It's more than just an in-game annoyance. These are people who really get into the fantasy world; their avatars are extensions of themselves. So there is real emotional stress that comes into play when a deadly plague breaks out in the game space.

Fefferman is not not alone in her enthusiasm for the potential of online gaming platforms to improve models for real-world systems, especially those heavily reliant on predicting human behavior. Other scientists have suggested the use of role-playing games as a platform on which to model how infectious diseases spread throughout a population, the most obvious being Second Life. (Second Life has also attracted the attention of folks in behavioral economics who build economic models, particularly the collapse of a virtual bank in what turned out to be eerily similar to the real-world economic collapse of 2008.)The Spousal Unit linked recently to a great online interview with game designer Jane McGonigal, who is particularly gung-ho on the potential for video games to save the world.

The popular children's online multiplyer game, Whyville, has also gotten into the epidemiological act with its own family-friendly version of a plague: Why-Pox, similar to chicken pox, in which players' avatars develop red spots on their faces, among other symptoms. According to Fefferman, even though it isn't "real," kids are very attached to their avatars and find this quite upsetting. Whyville was designed from the start to be educational entertainment, so WhyPox is a designed disease, first introduced in 2002. The game now has its own virtual Center for Disease Control, where children can help "investigate" outbreaks via the Infection Simulator and Epidemic Simulator, learning the fundamentals of epidemiology in a fun, hands-on way. These principles can be difficult to teach, says Fefferman: "It's hard to explain why students should do something." With the simulations, students don't just blindly follow orders, they understand the purpose of the actions they take.

So gaming is now the hot trendy approach to epidemiology -- and education. Fefferman was one of several speakers at this particular AAAS session, which focused on the question of whether one could truly use video games to teach. There needs to be a learning goal apparently -- not just the goal of learning to play the game -- and some kind of genuine science content that goes beyond merely zapping, say, chemical symbols a la Space Invaders. And because educators love to measure progress, there must be some way to assess whether the material has actually been learned. This is actually pretty critical when it comes to educational video games, because unless they are carefully designed, kids can learn to place the game without actually absorbing the underlying scientific knowledge.

In addition to the whole "Corrupted Blood" incident in World of Warcraft, and Whyville's outbreak of Whypox, I learned about things like The Evolution Readiness Project, an NSF-funded game designed for fourth graders to teach them about evolutionary principles using plants in a virtual greenhouse. Eric Klopfer of MIT's Media Lab is designing educational games for mobile platforms -- games students can play for fun on their iPhones, cells, or other mobile devices, while still learning something useful. One pilot game was called Weather Links, which Klopfer described as "Pokemon meets weather prediction." Players had to figure out how to predict the weather six hours before a key battle is to take place. Another in development is Beetle Breeders, based on Mendelian genetics. 

"We spend too much time teaching the outcomes of science, as meanings of words," said Diane Ketelhut of Temple University, who has designed her own epidemiological games and is keenly interested in building games that assess student performance through a new program called SAVE Science. "We need to teach the process of scientific inquiry beyond memorizing the steps of the scientific method." Traditional assessment tools are inadequate: standardized tests lack any context -- "nothing we do in science is decontextualized, it always has a purpose" -- and while regular performance assessments might provide that context, it is not cost-effective, highly subjective, and doesn't necessarily assess true student understanding.

That's where a well-designed video game might be able to help, says Ketelhut, who is experimenting with embedding assessment tools into virtual gaming environments, hopefully with such skill that students don't even know they're being tested. A database records all student actions, and the game should allow for nonlinear problem-solving pathways -- students should be able to integrate the spirit of inquiry with the content. That's different from most classrooms, which are usually organized very linearly. But most of us don't really learn that way.

In one of her pilot games, "Sheep Trouble," students got to design their own avatars and travel in time to medieval England, where they had to  help a farmer determine whether an outbreak of illness among a new breed of sheep was science or "bad magic." Ketelhut isn't making grand claims based on one small pilot study, but the student response was overwhelmingly positive. They were engaged, got to play an active role, explore the virtual environment, and solve a practical problem. One student, caught on videotape, put it well: even though she knew it was a test, "They didn't tell you what to do like on a normal test. It seemed like a real-life question, not just a question on a piece of paper."

So, can a video game teach science? Most of the speakers responded with "Yes, but..." I liked Klopfer's analysis in particular: "Can a car teach science? Yes, some kids will learn science from a car, but most won't." He thinks the value of his mobile gaming platforms, and other educational video games, is best realized when it precedes the classroom but then is integrated into it. "Video games are preparation for future learning, and they definitely can help promote interest and excitement in science," he said. Entertainment is still entertainment -- but that doesn't mean it can't be useful in getting kids excited and motivated to learn. Just make sure they don't accidentally hack into a defense computer by mistake.

50 years of getting yer (video) game on

*Raises right hand* I am not now, nor have I ever been, a gamer. A horrible confession for any geek to make, I know, but most video games bore and frustrate me. I tried Myst early on and got totally fed up; it felt too much like taking the logic section of my GREs: I've tried everything I can think of and I can't get anything to work! What do you people want from me? I have a vivid enough imagination, thanks, that I don't need a game to experience being another character. Books will do just fine for that, and writing fanfic takes care of the interaction part without having to fend off other pesky players trying to kill my character and then camping it so it can't respawn (although I would probably do that if I were a gamer. reee reeee reeee).

Well, it's not entirely true that I'm not a gamer. I'm not keen on First Person Shooters (FPS), Massively Multiplayer (MMP) games, Massively Multiplayer Online Role-Playing Games (MMORPG) (like World of Warcraft; or Years of Yarncraft as the webcomic Sluggy Freelance spoofed it.), Multi-User Dungeons (MUD), or Real-Time Strategy (RTS)  (Civilization) games (and yes, I realize a lot of these categories overlap). I'm fascinated by the whole online avatar creation thing, but  MUDs actually make me a little queasy, and Second Life just seems, well, silly, unless it involves something like Sean Carroll's talk on the arrow of time given at Second Life's Galaxy Dome in Spaceport Bravo. The educational potential for those kinds of activities makes the teacher in me salivate. Otherwise, not so much.

I do, however, confess to an unhealthy love of Tetris, Solitaire, online pinball, and Snood. (In fact, is there a twelve-step program for Snood?  I really need that; I've got it open right now, in fact and am Alt-Tabbing between my browser and the game. What a loser.) The attraction of these games for me, especially Tetris, is the spatial element. I've always liked the kind of puzzles were you fit things together, or which take the calculation of angles to score points. I loved geometry in high school (at least the constructing part) and liked playing pool and air hockey for the same reason.

And once upon a time, I was totally hooked on Pong, the arcade version of Atari's popular electronic tennis game that first came out in 1972. We had one pizza parlor in my town and whenever my friends and I headed there after our high school's Friday night football or basketball game, we'd play the arcade version while waiting for our pie. I was killah. Developing all that excellent hand-eye coordination from playing air hockey and old fashioned table pinpall totally paid off.

If you're a regular, long-time gamer, you may or may not know that one of the first video games—a forerunner of Pong called "Tennis for Two"—was developed not at Atari, but at Brookhaven National Laboratory. On an oscilloscope screen. Brookhaven's then-Chief of the Instrumentation Division, William A. Higinbotham, and Robert Dvorak, Sr., built the game as an exhibit for an open house, which proved to be so wildly popular that it threatened to overshadow Brookhaven's real mission and its six Nobel Prizes. Brookhaven has their own little synopsis of how the game came to be, and why it was never patented, but there's another video of a couple of people playing the recreation of the game, called "The Second Ever Video Game."

As with most firsts, there's a little disagreement about which game actually was the first videogame. And whether Tennis for Two was really a video game or not depends in part on how picky you are about nomenclature. The image produced on the oscilloscope's CRT screen did not use video's graphics rastering, in which a shape is converted to pixels for display, a process that gives contemporary video games their 3-D quality. Although the images in Tennis for Two are projected onto a CRT screen with an electron gun like any other video image, the CRT oscilloscope (unlike most contemporary oscilloscopes which use LED displays) only displays fluctuations in voltage, and the image reflects the vector or geometric shape of those fluctuations rather than a relative sampling of the shapes mapped onto a grid of pixels. Rasterization allows faster display time and changes in those displays than vector graphics do, hence their use in contemporary videogames. That's why the capacity and speed of graphics cards is so important in gaming; the more powerful your graphics card is (i.e., the more capacity it has for processing position and shape algorithms), the quicker your display changes. And that matters when you're gaming in real time. Display time is part of what made Tennis for Two so innovative:

“The real innovation in this game is the use of those ‘new-fangled’ germanium transistors that were just becoming commercially available in the late 1950s,” said Peter Takacs of Brookhaven Lab’s Instrumentation Division, who is currently working to rebuild a playable Tennis for Two. “Higinbotham used the transistors to build a fast-switching circuit that would take the three outputs from the computer and display them alternately on the oscilloscope screen at a ‘blazing’ fast speed of 36 Hertz. At that display rate, the eye sees the ball, the net, and the court as one image, rather than as three separate images.”

One unintentional innovation in this game was the satisfying little click players heard when they "hit" the "ball" back to the other player across the screen. This was the result of the physical movement of the switches mentioned above, but it may have had a role in the game's popularity by actually making it easier to play. Sound plays a significant role in our perception of speed and motion, with both inputs processed more or less together in the brain. Different areas of the brain process visual, auditory, and touch input, but new studies show that the boundaries between those areas are pretty blurry, with neurons from one area encroaching on the territory of the others. This means that perception of one stimulus may affect the perception of another without involving higher brain functions—one of the reasons that gamers' reflexes can be so quick. Sight and sound work in tandem with each other and with touch in a feedback loop that grows smoother with practice.

This same feedback loop is part of what helps people learn to type. Typewriter keyboards, especially the electric or early electronic ones had a satisfying clicking sound and "touch" with some resistance to reinforce what typing felt and sounded like. People who learned to type on these keyboards often still have strong preferences for how their computer keyboards sound and feel. We like a little resistance and noise, rather than silence and a perfectly smooth touch. (Blackberry's new Storm has actually added a tactile touch screen that depresses when you touch the buttons for this reason.) But the rhythm and sound of your typing can actually be extremely revealing. "Researchers at the University of California, Berkeley, have found a way to turn the clicks and clacks of typing on a computer keyboard into a startlingly accurate transcript of what exactly is being typed." So much of the text can be recovered through sound alone that it may actually be possible to "recover" someone's password just from a sound recording of their typing.

Hmm, I guess someone playing Tetris with the arrow keys would kinda foil that.  And of course playing Snood with my trackball. And if it weren't for Pong, I'd would never have gotten hooked on either of them in the first place. Pong, the Gateway Drug. Who knew? And to think it all started with a harmless desire to make science exciting!

this is pong on drugs

Jen-Luc Piquant has been urging me to indulge in a bit of meditation to calm my nerves as we head into the home stretch for the Big Nuptial Event this coming weekend. I suspect she was thinking of something a bit more New Age-y, such as communicating with my molecules of consciousness, but personally, I find playing endless games of Scrabble or Solitaire to be oddly soothing in times of stress. Jen-Luc has her way of meditating, and I have mine. (Some people find exercise meditative, but I tend to draft articles or blog posts in my head in those situations.) And in an earlier era, I could have turned to Pong. You remember Pong, don't you?  At least, you should if you were a kid in the 1970s. It was one of the first video arcade games ever developed, courtesy of Atari, and definitely the first to achieve widespread popularity, moving from the arcades to people's homes soon after it was introduced to the marketplace.

Pong was little more than a computer version of table tennis (or, as we called it growing up, "ping-pong"). It had simple monochromatic visuals and made a "pong" type sound whenever the "ball" (a little white dot) was hit by the "paddles" (two white rectangular bars at either end of the screen). It wasn't the first time someone thought of playing an electronic version of ping-pong: in 1958, a scientist at Brookhaven National Lab named William Higinbotham invented "Tennis for Two," played on an oscilloscope, but it failed to penetrate much further than the lab, perhaps because very few American homes had their very own oscilloscope.

Eight years later, a man named Ralph Baer working for Sanders Associates wrote a short paper describing a system for playing simple video games on a TV set, leading to his development of a computer version of ping-pong -- which he duly patented and licensed to Magnavox. In 1972, the company launched the Magnavox Odyssey, the first home video game console, offering a dozen different games, including table tennis. Nolan Bushnell saw the system demonstrated at a trade show that year, and one month later co-founded Atari with Ted Dabney, each contributing $250 of starting capital. The original idea was to create a video car driving game for arcades, but Bushnell realized this was a pretty tall order, given the limited graphical capabilities available at the time, so he asked his electronic engineer, Al Alcorn, to create a ping-pong game. And thus, Pong was born.

The original arcade version had a coin-operated switch, and the instructions were simplicity itself: "Avoid missing ball for high score." Finally, an arcade game that even those in the most drunken of stupors could play! The system was field-tested at Andy Capp's tavern in Sunnyvale, California (not to be confused with the fictitious town of Sunnydale central to the Buffyverse), and within a day of its introduction, people were lined up outside the bar waiting for their turn. In fact, the machine broke down rather quickly: there was a milk carton placed inside to catch the inserted coins, and when it was overflowing with quarters, they jammed the switch. By March 1973, Atari had sold a whopping 10,000 systems. The home console version proved just as popular: it was the must-have Christmas gift for 1975.   Pong was even featured in several Saturday Night Live segments during the show's first season -- you know, back when it was truly ground-breaking and actually funny.

Pong seems to be riding the nostalgia wave of late, evidenced by last year's hugely popular American Express commercial featuring tennis player Andy Roddick (video clip is here). It's pretty funny. Warned by his trainer in advance that his opponent "returns everything," Roddick walks onto toe court to find himself facing the telltale rectangular white band -- rendered in 3D this time, thanks to our vastly improved graphical capabilities these days. Pong proves to be a formidable opponent, returning shot after shot until Roddick has an epiphany: he realizes Pong can only move side to side, not forward or back. So he jettisons his usual killer power serve and tips the ball neatly over the net in a simple drop shot so that it dribbles into his opponent's side of the court, winning the game -- because Pong can't move forward to return the ball. The commercial even spawned an interactive Web game called Stop Pong. (Try it! S'fun!)

The Roddick commercial serves to illustrate the chief weakness of Pong: as video games go, it was pretty mindless, and quickly got boring. I have dim memories of a Pong console in our home when I was growing up. If memory serves, our favorite thing to do was align the paddles in such a way that the little white "ball" was passed back and forth in a perfectly straight line. Neither side ever missed. We'd leave it like that, and go play a few real games of ping-pong, because we had a bona fide table in the garage, mounted on a couple of sawhorses. (When your grandfather, father, and brother are all in construction, there's always a sawhorse laying about the place.) Pong was an interesting novelty, but it just couldn't beat the real live game in terms of pulse-pounding excitement and unpredictability.

If only we'd had access to today's advanced home computers (never mind graphic-heavy, interactive game consoles like PlayStation and X-Box). Then we could have indulged in the groovy psychedelic permutations of Plasma Pong, described by PC World as "Pong on acid," and recently named one of the top five indie games by Wired.com. The Website (launched earlier this year) has already surpassed one million hits. Plasma Pong has even warranted mention in the hallowed pages of the Washington Post. It's not the first time someone's re-invented Pong; apparently there was a version that featured former president Bill Clinton's head as the ball, and a demo prototype of a system that used body sensors to control the paddles. But Plasma Pong is just whacked-out enough to have the same broad popular appeal as the original. The WaPo reporter described is as being "like playing ping-pong while floating in melted lollipops.... It's over the top, freaky, high-speed, and mellow all at once." What could be better than that? (You can see an actual game of Plasma Pong being played here.)

Plasma Pong is the brainchild of George Mason University student Stephen Taylor, who got bored during a winter break spent at his parents' house (we hear ya, buddy!), and started writing code for a revamped version of Pong. He didn't bother tweaking the basic rules: he re-invented the visuals and added some interesting 2D physics simulations in the form of liquid plasmas. A simple mouse click sends a jet of liquid across the "court" -- or, alternatively, creates a suction effect to draw the ball toward you. The bright colors constantly pulsate and change, occasionally sending particles flying around the screen.

There's also a "sandbox function," to alter the viscosity of the "liquid," for example, to indulge those who can't resist doing their own tweaking. "You can toggle all the controls, turn it into a giant bowl of Jell-O if you want to," Taylor told the Washington Post. The game was an instant success: when Taylor first posted it online, the GMU serve slowed to a crawl because the game had been downloaded 50,000 times.

Kids today. Sheesh. I don't know what Taylor's majoring in, but it's pretty clear he's well-versed in math and science, not to mention being extraordinarily creative: his game required him to create algorithms for such advanced calculations as viscosity, gravity, vorticity, and other physical forces that affect the movement of a liquid. And trust me, that's no small feat. That's because plasmas -- ionized gases that technically make up a fourth state of matter because the ionization gives what would otherwise be a gas distinct fluid properties -- are extremely difficult physical systems to model. (I wrote about lab-based simulations of space plasmas in a post last year.)

There's an entire field of physics (fluid dynamics) devoted to gaining better understanding, and developing better predictive models, for complex fluids (plasmas, smoke, fire, or typical liquids). There are so many variables, and the systems can change so rapidly in response to the tiniest variation in just a single variable, that it's extremely difficult to predict a fluid's behavior beyond the near-term, or effectively manipulate it to one's advantage. Taylor, it turns out, relied on a research paper written by Jos Stam, a well-known scientist in the gaming industry, to develop Plasma Pong.

The complexity of fluid behavior is what makes Plasma Pong so much fun: not only do you get bright pretty psychedelic colors, but the addition of fluid dynamics means the system creates sudden, unpredictable movements: the ball can easily get caught in eddies and currents. Sometimes you can use this to your advantage. For example, firing a plasma into your opponent's playfield can create an eddy, enabling you to score. But it's not 100% controllable -- some players might find that frustrating. I think it's a great introduction to the unpredictability of Nature.

Still, Taylor's already moved on to the next version of the game. First, he'll graduate from college. Then, he'll launch his own company to market a new version of Plasma Pong under a different name (to avoid trademark infringement concerns). He's working on making a multi-player version of the game to enable a GMU student to play someone in Tokyo, for example. And he'd love to upgrade the graphics to 3D, rather than 2D, plasma simulations -- an even more difficult feat than what he's already accomplished. If he succeeds, it'll be like playing Pong inside a lava lamp. Jen-Luc can't wait...

juggle me this

Over more years than he'd probably care to remember, Master William Lee has become a weekend fixture in New York City's Washington Square Park. Most people know him as "that juggling Chinaman," which might strike readers as appallingly racist were it not how Lee refers to himself. "And how many of you want to see the Chinaman burn?" he routinely asks gathered onlookers as he teases them with the promise of juggling flaming torches, among other feats. It's all in fun, of course: Lee is a consummate entertainer, combining his crack juggling skills with a few illusionist tricks and a running commentary of jokes, punctuated by occasional pleas for donations. And he's the latest in a long line of juggling jokesters that dates back to ancient Egypt.

That's right, the earliest historical record of juggling (estimated date is between 1994 and 1781 BC) can be found in hieroglyphic depictions of female dancers and acrobats throwing balls that adorn the tomb of an unknown prince (see figures in third row from top). Master Lee's Chinese ancestors weren't far behind: by 770-476 BC, Chinese historians were writing about juggling warriors, such as Lan Zi, who supposedly juggled seven swords; and Xiong Yiliao who juggled nine balls at the same time during a battle between two warring states. One wonders why such warriors were not simply killed outright, but purportedly their skilled pyrotechnics were so impressive they could end conflicts before they began. Hmmm.

By 400 BC, juggling had spread to Greece. In Rome, an officer in the Roman legion named Sidonius Apollinaris supposedly entertained his troops by performing juggling tricks. Even the Irish got into the act: the legendary hero Cuchulainn was able to juggle nine apples, while a few centuries later, historical records show that a royal court buffoon diverted King Conaire by juggling nine swords, nine silver shields and nine balls of gold. And in 1066 -- yes, that pivotal year -- the warrior-bard of William of Normandy (named Taillefer), juggled before enemy lines and reportedly made the first kill at the Battle of Hastings. So it was a very effective diversionary tactic.

Nonetheless, by the mid 1800s juggling was pretty much comprised of street and court performers and "filler" acts at theaters and music halls throughout Europe and North America. And why not? It's fun, and good, clean (mostly) entertainment. But is there anything truly useful to be learned from it? The experts say sure! Peter Beek and Arthur Lewbel wrote a November 1995 article in Scientific American on the subject (which you can find on Lewbel's science of juggling Website), asserting the following:

Juggling definitely has uses beyond hobby and enterntainment. It is complex enough to have interesting properties and simple enough to allow the modeling of these properties. Thus, it provides a context in which to examine other, more complex fields.... One is the study of human movement and the coordination of the limbs. Another is robotics and the construction of juggling machines. The third is mathematics: juggling patterns have surprising numerical properties.

According to Beek (a movement scientist at the Free University in Amsterdam) and Lewbel (a professor of economics at Boston College), the first scientific paper on the subject appeared in 1903 in the American Journal of Psychology. The author, Edgar James Swift, documented the rate at which some students learned to toss two balls in one hand. Within 40 years, the advent of computers meant that scientists could calculate the trajectories of thrown objects. (This is also about the same time International Jugglers Association was founded.) Mostly, the focus through the 1960s was building on Swift's work: using juggling as a task to compare general methods of learning sensorimotor skills.

Juggling finally started getting more respect in the 1970s, when MIT scientist Claude E. Shannon created the first juggling machines out of an Erector set, and also formulated his "juggling theorem" correlating the position of the balls and the action of the hands while juggling. By the 1980s, mathematicians had gotten into the act. And they're still coming up with newer and better computer models for this simple human activity.

At least there are well-defined parameters for a computer modeling simulation. I won't go into excessive detail, because these have already been elaborated at great length by people far more knowledgeable than we are: namely, Beek and Lewbel, but also former mechanical engineer-turned entertainer Jack Kalvan, who once worked on his own juggling robot while employed with IBM Research, and has since analyzed spatially optimal patterns for juggling using computer modeling.

We'll reiterate this much information: The three most common objects used by jugglers are balls, rings, and clubs. For a single juggler, there are three basic patterns. The "cascade" is the most common, in which an odd number of balls are tossed from one hand to the other. Then there is the "fountain," in which balls are thrown and caught with the same hand, usually used for an even number of objects. Finally, there is the "shower," in which all the objects are tossed in a circle. A juggler might also choose to "multiplex": throwing more than one object from a single hand simultaneously.

Sounds simple enough in theory, but as anyone who's tried to juggle could could tell you, it's harder than it looks. That's even more true when it comes to figuring out how to mathematically model the process. True, the fundamental motion behind juggling is essentially standard projectile motion, involving multiple projectiles with interweaving paths. And the patterns of those paths are periodic cycles: they repeat, rather than change continuously.

Furthermore, the number of possible patterns is relatively small. But no two throws or catches will ever be exactly the same because there are so many variables associated with throwing: angle of release, release velocity, height of the throws, etc. Skilled jugglers are able to tightly control such variables to throw the objects as consistently as possible. (Add a second person to the mix, and it becomes even more complicated, even though the patterns between multiple jugglers are generally based on the single person patterns.)

Any mathematical model for juggling must incorporate both ball motion and hand motion. Hand motion modeling is pretty difficult, since the dynamics of the human arm are incredibly complex. It helps, though, that the hand motions in juggling closely correspond, by necessity, to the positions and velocities of the objects during throws and catches. Objects other than balls, like rings or clubs, have markedly different physical characteristics and must be treated differently in any computer model. Balls can be modeled with standard particle system dynamics, while the less-uniformly shaped clubs and rings work better when modeled as a rigid body system.

According to Ron Grahan, a mathematician at the University of California, San Diego, mathematical models of juggling might give performers a better understanding of the science behind their tricks, and possibly help them to develop new juggling routines. (He should know: he's a past president of the International Jugglers Association, as well as the American Mathematical Society and the Mathematical Association of America.) Using one of the many computer programs currently available that identify "legitimate" patterns and animate them -- many now available on the Web for easy download, such as Kalvan's Optimal Juggler -- jugglers can see what a particular pattern looks like before trying it out in real life. They can even check out juggling feats that are (to the best of our knowledge) humanly impossible, just for fun. In fact, mathematical theory has already suggested a few novel juggling patterns beginning to gain in popularity. (To check out a bit of virtual juggling, go here. Or watch this movie here.)

One would think that anyone with a background in math or physics would have a bit of an "edge" over the rest of us mere mortals when attempting to learn juggle. But such might not be the case; it is, ultimately, a learned motor skill involving excellent hand/eye coordination. While there is certainly a physics aspect to the act of juggling, having a physics background doesn't necessarily make one a better juggler, according to David Ehrenstein, a physicist and editor of Physical Review Focus. He learned to juggle as a child, long before he began studying physics, taught by a couple of fellow performers in a theater production ("Channukah '78!") at the Jewish Community Center in Rockville, Maryland. "I doubt it would have helped," he says now. "The learning is so subconscious, like riding a bike."

Not surprisingly, his skills improved the more he practiced -- and he practiced a lot in those early years, juggling every day for more than two months to master the basics: juggling three lacrosse balls, which he says are easier to juggle than tennis balls. By 14, he'd learned to juggle five balls, a much more difficult feat than performing tricks with three objects (even if, in Master Lee's case, one or more of those objects is on fire).  "The difficulty goes up exponentially with each additional object," explains Ehrenstein, a fact most audiences don't appreciate. "People assume the dependence is more linear, so there's not a good audience-appreciation ratio" to the amount of effort required to learn that particular skill.

Numerous studies on how people master the art of juggling support Ehrenstein's assertion about the difficulty of juggling more than three objects. They show that learning the simple shower pattern can take a few hours or days, using three balls, but learning times increase to weeks and months for four balls, extending to years for five balls. If you're curious, the world record for the greatest number of objects juggled is 12 rings, 11 balls, or eight clubs. An object's shape, once again, is an important variable. But so is basic Newtonian mechanics. Beek and Lewbel's article points out that the need for either speed or height increases rapidly with the number of objects being juggled.

Okay, so maybe a math or physics background doesn't necessarily give you an edge in learning to juggle, but Ehrenstein concedes that there does seem to be quite a few examples of physics, math or computer nerds who also love to juggle. We can only speculate as to the reasons for this, but Ehrenstein thinks it might partly be a social thing: "Who else would have time -- at least during high school -- to practice juggling alone rather than going out with friends?" Jen-Luc Piquant is skeptical about this hypothesis, given the burgeoning number of high-tech gamers who rarely see the light of day. But it's more convincing if you consider Ehrenstein's second point: that there's a natural allure to juggling because it's "a mechanical, logical process that has the kind of certainty we all like in math and physics, as opposed to, say, the humanities."

As for the ongoing proliferation of mathematical models, juggling robots, and anaylses of the underlying physics, he notes, "Once you have a bunch of physicists doing something, it can't be long until they start to use physics principles to analyze it. (This is true. Geeks will scientifically analyze just about anything, even which Applebee's restaurant is the most expensive. Jen-Luc stumbled upon this classic exchange at Overheard in New York:  First Geek: "Literally, it is the most expensive Applebee's in the universe."  Second Geek: "Ah, not so. In a constantly expanding universe the probability approaches 100% that somewhere out there exists a more expensive Applebee's." First Geek: "... Let's just eat at KFC." Jen-Luc, who is quite the food snob, is appalled at the very thought of Applebee's and KFC, and suggests our two Geeks check out the cheaper places in Chinatown or Little India instead.)

For Ehrenstein, at least, all this combined with his early love of theater and performing: for him it was a chance to be noticed. And while adult responsibilities have cut down his juggling time drastically, he still indulges from time to time, performing for small, informal audiences: things like World Year of Physics nights at the New Deal Cafe in Maryland. Once he attended a nuclear physics conference to give a talk on science communication. At the end, he brought out some bean bags to perform a few simple tricks, describing them as protons and neutrons and cracking jokes about fission and fusion. "They just ate it up," he recalls. "I'm always amazed at how easy it is to get audiences to laugh at even incredibly stupid jokes if you're juggling at the same time."

Which might explain the enduring popularity of Master Lee. True, he's a bona fide comedian, and a damn good one at that, but we doubt very much he presents his "A" list material for free on Saturday afternoons in Washington Square Park. Not that his audience minds: they love his act, love being teased and invited to participate in feats like slicing through a cucumber balanced on the volunteer's bared belly with a very large sword... while blindfolded. No mathematical model can prepare someone for that. We wonder what happened to his very first volunteer. And whether he has liability insurance.