Strange but True: Turning a Wobbly Table Will Make It Steady

January 25, 2007 Strange but True: Turning a Wobbly Table Will Make It Steady For every table—turn, turn, turn... there is a proof By JR Minkel

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It's a problem as old as civilization: the wobbly table. You may have thought your only recourse against this scourge is a hastily folded cocktail napkin stuffed under the offending leg. If so, take heart, because mathematicians have recently proved a more elegant solution. Just rotate the table.

The intuitive argument, which dates back at least to a 1973 Scientific American column by Martin Gardner, is straightforward. Consider a square table with four equally long legs. Any three of the legs must be able to rest on the floor simultaneously, as a tripod does. Assume the floor undulates smoothly and the fourth leg hovers above it

Image: BURKARD POLSTER TURN THE TABLES on a wobbly table simply by rotating it. Mathematicians have proven the procedure will work. Click on this link for a video demonstrating the proof

Now imagine turning the table about its center while keeping the first three legs grounded, or balanced. Once the table has rotated by 90 degrees, the wobbly leg must lie below the floor. (If you do not see why, imagine pushing down equally on the wobbly leg and a neighboring leg until the neighbor sinks below the floor and the wobbly leg touches down.) And so, at some point along the wobbly leg's arc, it has to hit a spot on which it can rest. As simple as this argument may sound, however, proof was a long time coming.

The first serious mathematical inroad against table wobbling seems to have occurred in the late 1960s with Roger Fenn, a PhD student at the University of London. One day Fenn and his graduate adviser ended up at a coffee shop faced with—you guessed it—an unsteady table. "The table wouldn't stop wobbling and we fiddled it around until we got it to stop," recalls Fenn, who is now at the University of Sussex.

At his adviser's suggestion, Fenn wrote out a proof that for any smoothly curving floor that bulges upward like a hill, there is at least one way to position the table so that it is balanced and horizontal. But he did not reveal how exactly to find that sweet spot, and he quickly tabled the subject. "I didn't think people were going to take this very seriously," he admits. "You say to somebody you've met, 'Well I'm trying to put a table on the floor so it doesn't wobble'; they'll say, 'Oh yeah?'"

The season for proving the table turning hypothesis would not arrive for another 35 years. By then, the idea had become such a part of mathematical lore that two years ago mathematician Burkard Polster of Monash University in Australia included it in an article on neat math tricks for teachers. He promptly received a letter pointing out that the idea would not work if a floor was too uneven or possessed sheer cliffs, such as between tiles.

Polster rose to the challenge. "It's never been really pinpointed exactly what the ground should be like," he says. So he and some of his colleagues ran through the appropriate trigonometry and satisfied themselves that if a floor has no spots that slope by more than 35 degrees, then turning will indeed balance a square or rectangular table. They detail the proof in a paper accepted for publication by the Mathematical Intelligencer. (In one of those odd cases of co-discovery, a retired CERN physicist named André Martin published a similar result a few months before the Australians did.)

Polster's group even spells out a procedure for balancing the table [see video above]. First lift up the leg of the table diagonal from the wobbly leg. Make sure both legs are roughly equal distances off the ground and then begin rotating. "In practice," the researchers write, "it does not seem to matter how exactly you turn your table on the spot, as long as you turn roughly around the center of the table."

So, next time you feel a table start to tilt, put that napkin down and don't be shy about turning the tables on a wobbly dining experience. Rest assured, mathematics is on your side.

RELATED LINKS:
The proof by Polster and his colleagues Martin's proof "Mathsnacks," a column by Polster and Marty Ross for Vinculum magazine A Quarter-Century of Recreational Mathematics Profile: Martin Gardner Profile: Not Just Fun and Games

Mr. Deity - Allmightiness one day at a time

four short films produced, written, and starring Brian Dalton in which he explores the lighter side of religion through these humorous short films. Launched only last Monday, January 17, within a few days there were over 200,000 downloads of them from YouTube, iTunes, and at his own webpage, mrdeity.com. Everyone who watches them — whether religious, nonreligious, or antireligious — enjoys the wit and humor.

Brian is an ex-Mormon who came to skepticism in the late 1990s through the Skeptics Society, because he liked our nonconfrontational approach that allowed him to explore the many options available to believers who harbor doubts. In time he gave up his religion, but remains utterly fascinated by religion in general, and he has written a pilot for a television series based on his character, Mr. Deity. Enjoy!

reposted from: http://www.skeptic.com/eskeptic/07-01-24.html
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Letter to a Christian Nation by Sam Harris - review

In this week’s eSkeptic Kenneth W. Krause reviews Sam Harris’ book Letter to a Christian Nation (Knopf, 2006, ISBN 0307265773). Krause lives in Wisconsin, along the Mississippi River. He is a former prosecutor and criminal defense attorney with degrees in law, history, literature, and fine art. Books editor for Secular Nation, Kenneth has recently contributed as well to Free Inquiry, Skeptic, and The Humanist.

reposted from: http://www.skeptic.com/eskeptic/07-01-24.html
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Sam Harris, author of Letter to a Christian Nation (photograph by Sara Allan)

The End of Faith Revisited: a Good Start

a book review by Kenneth W. Krause

Reason is to morality what design is to construction, and monotheism is a collective intellectual disaster that necessarily implies an international moral emergency. So says Sam Harris in his predictably candid, derisive and hyper-focused Letter to a Christian Nation, as he lectures an audience of agitated religionists who protested similar and, in some instances, identical scolds dispensed through his first book, The End of Faith.

Faith, by some accounts, is conviction to a belief despite facts and reason. When overwhelming majorities accept lesser standards of intellectual integrity, faith becomes institutionalized and potentially dangerous. Faith in omnipotence only exacerbates the problem. Christianity and Islam, especially, because they define in- and out-groups in terms of perpetual rewards and punishments, are inherently dangerous. Monotheists define morality according to no objective standard outside the Bible. Instead, religious affiliation necessarily depends upon the foundational texts that attempt in vain to identify both ethical and non-ethical behavior. Hence, for the faithful, any conceivable act may be defined as moral so long as the relevant god’s text can be interpreted to support it. Humans may piously slaughter other humans in any number, in any imaginable way, simply because their faith allows them to trust in an omnipotent and omni-benevolent creator who desires or all too regularly demands that they do so.

The Christian faithful, Harris contends, have inherited some of the most unethical standards imaginable. Consistent with Proverbs, Leviticus, Deuteronomy, Exodus, Mark and Matthew, pious parents are obliged to thrash or kill their disobedient children. The Abrahamic texts insist that followers stone not only the adulterer, but also those who labor on the Sabbath. Although some Christians will argue that New Testament morals have superseded those of the Old, Harris counters that, according to the former, Jesus instructed his followers to remain true to the ancient laws. The Church incinerated heretics for more than 500 years as it cited allegedly validating chapter and verse. Augustine supposed that dissenters should be tortured; Aquinas directed that they be murdered. Both Luther and Calvin encouraged the slaughter of innocent apostates and Jews. In no Christian text was Jesus said to have objected to slavery; in many was the practice condoned.

The first four of the Ten Commandments, Harris observes, had nothing to do with morality. The rest clearly did, but were hardly original. Virtually every culture edified similar principles in its annals, laws, and myths. Regardless, morality predates recorded history and, perhaps, humanity itself. Our closest primate relatives demonstrate some degree of kin altruism and broader social concern. The point, of course, is that religious ethics represent only one phase of our moral evolution, a phase that humanity can and must transcend.

Hopelessly antiquated religions have grown increasingly counterproductive as sources of moral guidance. Christians delight in imagining themselves supremely ethical in their opposition to embryonic stem cell research and abortion. But neither stem cell use nor legal abortions harm anything capable of either experiencing loss or inspiring a reasonable sense of loss in others. Insisting that human “souls” can inhabit the microscopic recesses of a Petri dish is not a moral argument. Rather, it is the imposition of both intellectual and moral primitivism.

Indeed, religious ethics often seem impervious to empathy. Many Christian conservatives oppose vaccination for the human papillomavirus, now the most common sexually transmitted disease in America, largely because they consider HPV an obstacle to premarital sex. The Vatican contests condom use even to thwart the spread of HIV. Christopher Hitchens summarized the crisis well when he pointed out that Mother Teresa “was not a friend of the poor. She was a friend of poverty. She said that suffering was a gift from God.” Teresa might have performed admirable deeds for humanity as an individual, but it should be clear that she brought such goodness despite, rather than because of, her religion. That so much suffering can be directly attributed to religion, Harris concludes, should inform us that honest and thorough criticism of religious faith is both our intellectual and our moral responsibility.

To religious moderates, Harris offers neither sanctuary nor convenient alliance. Temperance and tolerance are not solutions to this deadly predicament. To the contrary, religious liberalism’s demand for respect only lends ostensible though certainly not actual credibility to religious dogma and fanaticism. Moderates simply cannot continue to have it both ways, the author demands. Either human beings created the Bible, or they did not. Either Christ was a man, or he was not. If so, the fundamental and necessary tenets of Christianity are and have always been false. At some point, Harris persists, one side will win and the other will lose.

Speaking truth to both religious and secular power, however redundantly, has become Sam Harris’s claim to fame, perhaps even his raison d’être. Although the underlying problem is a bit more complex than the author appears to recognize, his assessment of faith’s threat to human survival is sound. Arguably, much of the developed world seems well on its way to begetting the end of faith, or, more precisely, faith of the licensed, monotheistic variety. In America, of course, the crisis is more severe; a mere letter to a Christian nation will never suffice. But if such letters are read and rejoined by the right Christians, perhaps history might prove them to have been a very good start indeed.