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Steven Strogatz (L) and Alan Alda (R) discuss the butterfly effect at an event at the National Museum of Mathematics. (Image courtesy of the National Museum of Mathematics.)

 

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This month's topics:

The math of the lasso

 

flat rope trick

"The Flat Loop trick is described mathematically as static solutions in a rotating frame of an inextensible string or elastic rod held at a distance R from the axis." Here the rope has length $s_1+s_2$, with a loop of length $s_2$ terminating in a much smaller loop (the honda) which can slide along the cord. The cowboy's hand describes a planar circle of radius $R$ with angular frequency $\Omega$. Image adapted from Brun, Ribe and Audoly, Proc. R. Soc. A 470 no. 2171.

"The Math of Whips, Chains and Ropes" was James Gorman's attempt to put a louche spin, so to speak, on An introduction to the mechanics of the lasso, published on the Proceedings of the Royal Society A by Pierre-Thomas Brun, Neil Ribe and Basile Audoly (at EPFL-Lausanne, Orsay and Paris-6, respectively). Writing on the Science webpage of the New York Times, Gorman tells us how this work grew out of Brun's thesis (his advisors were Ribe and Audoly) and how Brun approached the topic: "First, he learned basic trick roping, because, he said, reality is important in applied math. He also read The Lasso: A Rational Guide to Trick Roping, by Carey Bunks. He recruited a professional roper from Disneyland Paris. And he built a robotic arm that could do a simple trick." That trick is the "flat rope trick", illustrated above. (The professional roper is Jesus Garcilazo, who performs that trick and other amazing ones in the video that goes with Gorman's posting). Along with deriving and solving the equations for the movement of the lasso, the article discovers some useful facts for those who would like to learn how the "flat loop" is done. Gorman: "it is important to start with a big enough loop to do the trick. [From the article: $s_2 \approx 3s_1$]. They also discovered, by analyzing video frame by frame, that the hand is actually in a different position in relation to the loop than ropers think it is -- not leading the loop but traveling right along in phase with it." Some additional hints are given in the article. One of them can also be gleaned from careful inspection of Garcilazo's performance: "To avoid accumulating twist in the rope, the cowboy constantly rolls the rope between his thumb and forefinger while spinning it." The other concerns $\Omega$. " [A] minimal ... frequency of ... 1.4 turns per second, or 84 r.p.m., is required to produce Flat Loops. A key to success is therefore to pick a good angular velocity."

Lifesaving statistics

Bayseian statistics helped save a fisherman's life. That's the grabber in a New York Times article by F. D. Flam, published in the "Science" section on September 30, 2014. Here's how the story starts: "Statistics may not sound like the most heroic of pursuits. But if not for statisticians, a Long Island fisherman might have died in the Atlantic Ocean after falling off his boat early one morning last summer. The man owes his life to a once obscure field known as Bayesian statistics--a set of mathematical rules for using new data to continuously update beliefs or existing knowledge." Flam gives a quick survey of the difference between Bayesian statistics and garden-variety, "frequentist" statistics. "The essence of the frequentist technique is to apply probability to data. If ... a coin .. came up heads nine times out of 10, a frequentist would calculate the probability of getting such a result with an unweighted coin. The answer (about 1 percent) is not a direct measure of the probability that the coin is weighted; it's a measure of how improbable the nine-in-10 result is--a piece of information that can be useful" in evaluating the hypothesis that the coin is loaded. "By contrast, Bayesian calculations go straight for the probability of the hypothesis, factoring in not just the data from the coin-toss experiment but any other relevant information--including whether you have previously seen your friend use a weighted coin." We are not told exactly how that other data is "factored in."

Flam refers to Andrew Gelman (Statistics, Columbia) to explain how frequentist evaluation of experimental results almost unavoidably leads to problems ("Even if scientists always did the calculations correctly--and they don't, he argues--accepting everything with a $p$-value of 5 percent means that one in 20 'statistically significant' results are nothing but random noise.") and gives an example of a pubished and appealingly counterintuitive study that Gelman reanalyzed à la Bayes: "the study's statistical significance evaporated." Another example comes from the celebrated "Monty Hall" problem. Flam sets the stage, then gives the Bayesian explanation:

  • "A Bayesian calculation would start with one-third odds that any given door hides the car, then update that knowledge with the new data: Door No. 2 had a goat. The odds that the contestant guessed right--that the car is behind No. 1--remain one in three. Thus, the odds that she guessed wrong are two in three. And if she guessed wrong, the car must be behind Door No. 3. So she should indeed switch."

After some more examples, and a reference to Uri Simonsohn (Psychology, Penn), who "said he had looked into Bayesian statistics and concluded that if people misused or misunderstood one system, they would do just as badly with the other," we get back to the fisherman, John Aldridge. He was alone on watch when he fell off his boat, "sometime from 9 p.m. on July 24 [2013] to 6 the next morning," some 40 miles offshore. The Coast Guard implemented SAROPS, their Search and Rescue Optimal Planning System, organized along Bayesian lines. "Over the next few hours, searchers added new information--on prevailing currents, places the search helicopters had already flown and some additional clues found by the boat's captain. The system could not deduce exactly where Mr. Aldridge was drifting, but with more information, it continued to narrow down the most promising places to search." And they found him. "Even in the jaded 21st century, it was considered something of a miracle."

Tony Phillips
Stony Brook University
tony at math.sunysb.edu

Math Digest Math Digest
On Media Coverage of Math

Math Digest includes posts throughout each month by Anna Haensch (Drexel University) and Ben Pittman-Polletta (Boston University). These early-career mathematicians provide their unique insights (and occasionally videos, interviews and podcasts) on math-related topics recently covered by the media.

Recently posted:

He ought to be in papers, by Ben Pittman-Polletta

The Imitation Game is a new biopic about scientist and mathematician Alan Turing's life and work. With the recent premiere (see coverage of the premiere by Roger Friedman on his blog Showbiz 411) and a cover story in Time magazine, there's no shortage of media coverage on the film, but Dan Rockmore's review in The New Yorker is a welcome deeper look into the intellectual life of the film's protagonist. Turing is most famous for his work in computer science and artifical intelligence, exemplified by the eponymous concepts of the Turing machine and the Turing test. Turing was also a homosexual, and his conviction for "gross indecency" and the subsequent sentence of chemical castration led to his suicide 16 days before his 42nd birthday. These facts provide the "narrative hooks" for the new movie, which, according to Rockmore, focuses on Turing's work cracking the Enigma code during World War II. But Turing was also a pioneer in logic, number theory, and mathematical biology, modeling the brain and development. Rockmore, chair of the Department of Mathematics and professor of computer science at Dartmouth College, uses his review of The Imitation Game as an opportunity to highlight one of Turing's most prescient and impressive intellectual works--his essay "Intelligent Machinery." In this paper, Turing sets down his ideas about how a thinking machine might be built. He puts forward a simple model of a network of neurons, and ideas for how it might be educated from "experience"--and especially from rewarding and aversive stimuli--which are foundational to the influential connectionist paradigm in mind and brain research ("Connectionism," Wikipedia).

This paper, as well as others collected in editor B. Jack Copeland's The Essential Turing (reviewed by Andrew Hodges in the AMS Notices), such as "The Chemical Basis of Morphogenesis," illuminate critical issues in science with a clarity rarely seen in today's literature. For instance, in "Intelligent Machinery, a Heretical Idea," Turing describes how "indexes" might be used to organize information in memory. "New forms of index," he suggests, "might be introduced on account of special features observed in the indexes already used." Here and elsewhere, Turing exemplifies his own rationale for modeling human thought. "The whole thinking process is still rather mysterious to us," he says in "Can Machines Think?", "but I believe that the attempt to make a thinking machine will help us greatly in finding out how we think ourselves." Revisiting Turing's work reminds us how much we have to learn from history--how not only the emotional, but also the intellectual struggles of our forebears, mirror and inform our own. Indeed, this is an idea Turing himself discusses in "Intelligent Machinery." At the end of the essay, he proposes that essentially all problems are search problems, and discusses both evolutionary and intellectual searches in this light. "The remaining form of search is what I should like to call the 'cultural search'," he writes, "... the isolated man does not develop any intellectual power ... the search for new techniques must be carried out by the human community as a whole."

See "What's Missing From 'The Imitation Game'" by Dan Rockmore. The New Yorker, 6 November 2014.

--- Ben Pittman-Polletta

Also now on Math Digest: Bletchley Park's Joan Clarke, Shaw Prize winner George Lusztig, Steven Strogatz and Alan Alda at the Museum of Math, Candy Crush's puzzling math, teaching math to people who think they hate math...

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Books, plays and films about mathematics

Citations for reviews of books, plays, movies and television shows that are related to mathematics (but are not aimed solely at the professional mathematician). The alphabetical list includes links to the sources of reviews posted online, and covers reviews published in magazines, science journals and newspapers since 1996.

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