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Math Digest

On Media Coverage of Math

Edited by Mike Breen and Annette Emerson, AMS Public Awareness Officers
Mike Breen (AMS), Claudia Clark (freelance science writer), Lisa DeKeukelaere (2004 AMS Media Fellow), Annette Emerson (AMS), Brie Finegold (University of Arizona), Anna Haensch (2014 AMS Media Fellow), Allyn Jackson (Deputy Editor, Notices of the AMS), and Ben Polletta (Boston University)

March 2014

Celebrating Freeman Dyson on his 90th Birthday, by Ben Polletta

Freeman Dyson

It's not that I didn't like Freeman Dyson before -- to tell the truth, I barely knew the guy -- but what's certain is that I like him a whole lot more having read Thomas Lin's recent interview on the occasion of his 90th birthday. Dyson is the quintessential mathematician's mathematician -- a skeptic, a humanist, an iconoclast, a hater of faculty meetings and a problem solver without pretensions of grandeur -- and his musings on science and his ongoing and rich career are full of charm and wisdom. Like many people of a certain age, Dyson was attracted to mathematics by E.T. Bell's tragically gendered classic "Men of Mathematics," whose pull, says Dyson, was in the way it "showed the mathematicians as being mostly crooks ... and not very clever ... it told a kid that 'if they can do it, why can’t you?'" So inspired, Dyson pursued mathematics and science partly as "a subversive activity," and partly as an exercise in creative self-expression. "I had this skill with mathematical tools, and I played these tools as well as I could just because it was beautiful," he says. "I'm not a person for big questions. I look for puzzles. ... I don't care whether they’re important or not." Nonetheless, his quest for interesting puzzles and his singular genius led him Dyson to make influential and often seminal contributions to a huge variety of disciplines -- including number theory, quantum field theory, random matrix theory, medical research (by helping to develop the low-power nuclear reactors that produce isotopes for research hospitals), and, as recently as 2012, evolutionary game theory. But Dyson remains thoroughly humble. "That's really my skill, just doing calculations and applying mathematics to all kinds of problems," he says. "Mathematics applies to all kinds of things. That's one of the joys of being a mathematician." Indeed. A similar humility comes through in Dyson's reminiscences about his groundbreaking work alongside Richard Feynman on quantum electrodynamics, which are worth viewing the accompanying mini documentary to hear.

Dyson's deep concern for social issues led him to a second career as a public intellectual, writing regularly for The New York Review of Books and penning a number of titles for popular audiences. He's fought for peace while advising the military on logistics to help them avoid unnecessary casualties, and he's been in the news recently for his controversial views on climate change. "What I'm convinced of is that we don't understand climate," he says here. "I'm not saying the majority is necessarily wrong. I'm saying that they don't understand what they're seeing." An article in the New York Times Magazine on Dyson and the climate change controversy surrounding him exposes the nuances of both Dyson's personality and his climate change skepticism. Not only does Dyson question the damage that carbon dioxide does to the planet's ecosystems, he believes coal is indispensable in fueling the "the move of the populations of China and India from poverty to middle-class prosperity," which "should be the great historic achievement of the century ... To me that's very precious." Dyson's views on the Ph.D., which are similarly controversial, and have similarly humane and sensible motivations, will resonate with many academics. "I'm very proud of not having a Ph.D.... It forces people to waste years and years of their lives sort of pretending to do research for which they're not at all well-suited," he says. "The Ph.D. takes far too long and discourages women from becoming scientists, which I consider a great tragedy. So I have opposed it all my life without any success at all." You win some, you lose some. Happy birthday, Mr. Dyson. (Photo: Freeman Dyson, Professor Emeritus, Institute for Advanced Study.)

See: "At 90, Freeman Dyson Ponders His Next Challenge," by Thomas Lin. Quanta Magazine, 31 March 2014.

--- Ben Polletta

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How Do I Love Thee? Let Me Extrapolate The Ways, by Anna Haensch

Hannah Fry

Should you take dating advice from a mathematician? According to a recent post on the Binghamton University blog bupipdream: If that mathematician is Hannah Fry (pictured above), then yes.   A lecturer at the Centre for Spacial Analysis at University College London--and self-proclaimed "all round bad-ass"--Fry applied her research in fluid dynamics and complexity theory to tease out some of the less amorous aspects of love in a TEDx talk at Binghamton earlier this week. 

In the talk she shares her top three tips for finding love: don't be afraid to look a little bit ugly, know when to settle, and if you find yourself in a relationship, speak out about what bothers you.  

Fry supports these bits of wisdom with calculations and mathematical reasoning.  On having perhaps a less-than-perfect visage, Fry says, “it’s [the] spread that counts…If some people think you’re attractive, you’re actually better off having some people think you’re a massive minger." 

A finding supported by the analysis of the nerd-friendly number-crunching dating site OkCupid. 

On knowing when to partner-off and settle down, Fry employed a technique called optimal stopping theory.  "In the wild there are certain types of fish who follow this exact structure," Fry says, “they reject all the fish that come up to them during the first 30 percent of the mating season. Then…they accept the next fish that is bigger and burlier than those that had come before.”

Such advice is mathematically sound for humans too, Fry argues, especially when we consider that a suitable mate is almost as hard to find as a highly evolved civilization somewhere in the Milky Way.   

For more mathematical lifehacks, perhaps less romantic in nature, follow Hannah Fry on Twitter @FryRSquared. (Photo courtesy of Hannah Fry.)

See the article: "TEDx: 'The Mathematics of Love'," by Nicolas Vega. Binghamton University Pipe Dream, 31 March 2014.

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A stand-up mathematician, by Allyn Jackson

This article discusses the unusual career of Matt Parker, described as a "stand-up mathematician"--he does stand-up comedy based on mathematical themes. With a degree in applied mathematics earned in his native country of Australia, Parker moved to England and taught school mathematics before realizing that his true calling was stand-up comedy. Although his comedy skits play on people's school memories of Venn diagrams and algebra, he's not aiming merely to poke fun at mathematics. "I am not just making jokes about maths because I think it is an easy way to do comedy," he told New Scientist. "I honestly want to make people enjoy maths more and realise there is more to maths than what they learned at school." The article highlights Parker's 4-dimensional cube crash video on YouTube as an example of his comedy.

See the article: "Dream Job: Stand-up mathematician," by Jessica Hamzelou. New Scientist, 27 March 2014.

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Math used in the search for the Malaysian Jet, by Allyn Jackson

The first two articles cited below discuss the use of mathematics to locate airplane wreckage. The BBC story discusses the June 2009 Air France flight that went missing while flying from Rio de Janeiro, Brazil, to Paris, France. France's aviation accident investigation authority contacted the American firm Metron Inc, to get help. The Metron statisticians used Bayesian techniques (named after the 18th century mathematician Thomas Bayes who created them) to optimize the search for the aircraft wreckage. A first crack at locating the plane did not pan out, but a few months later the Metron team adpated their model and successfully guided searchers to the right spot. The CNN story says that a British company, Inmarsat, and the UK's Air Accidents Investigation Branch collaborated to produce a novel mathematical process for analyzing information about the most likely flight path of the Malaysian Airlines plane that has been missing since March 8. Calling the process "groundbreaking," an Inmarsat official told CNN that the new calculations "underwent a peer review process with space agency experts and contributions by Boeing."

See the articles "How 'groundbreaking' number crunching found path of Flight 370," by Thom Patterson. CNN, 25 March 2014, and "MH370 Malaysia plane: How maths helped find an earlier crash." BBC News Magazine, 22 March 2014. Other coverage: University of Texas at Dallas prof John Zweck explains to NBC News how great circles and trig could help locate the jet and more on Bayes' Theorem in this NPR segment.

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A profile of math writer Dana Mackenzie, by Mike Breen

Dana Mackenzie doing hulaDana Mackenzie is a math and science writer, and author of the recent The Universe in Zero Words: The Story of Mathematics as Told Through Equations, as well as the three most recent volumes of the AMS series What's Happening in the Mathematical Sciences. Mackenzie's parents inspired him in both writing and math: His mother taught him to read and write and typed and bound his stories, while his father taught him the beauty of equations. Mackenzie got his PhD at Princeton and taught at Duke and Kenyon before becoming a science writer. He has many interests that are mentioned in the profile, including chess, hula, and animals. At the end of the article, Mackenzie tells the author of the article a story of the time he hosted a reading for his book The Big Splat: "He [Mackenzie] invited the Santa Cruz Astronomy Club, which set up telescopes outside. A nearby drugstore thought it was a bomb, so the police came, looked through the telescopes and left." Mackenzie concludes the story with "There's something amusing to me about calling the police to save ourselves from science." (Photo: Courtesy of Dana Mackenzie.)

Read the article "Dana Mackenzie, Santa Cruz County Stories: Chess champion balances science writing, hula dancing," by Bonnie Horgos. Santa Cruz Sentinel, 23 March 2014.

--- Mike Breen

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On statistician Nate Silver and FiveThirtyEight, by Annette Emerson

Time profiles statistician Nate Silver, who runs FiveThirtyEight, the "data-crunching digital publication" that predicts, among other things, political and sports results. Dickey writes that Silver "evangelizes on behalf of data and statistics in a profession where they have historically had little place and then frets about the overuse of numbers." He started out at Baseball Prospectus, where he forecast players' seasons and played online poker. From there, he blogged about politics, and moved to New York City, where "one of his most memorable projects, a ranking of New York City's top 50 neighborhoods," was published in New York magazine. Silver became more well-known after he rightly predicted Obama's victory in 2012. FiveThirtyEight was hosted by the New York Times from 2010-2013, but Silver was restless to "bring a generation of data journalists under his tent" in all forms of media, and FiveThirtyEight migrated to ESPN. The photograph of Silver in the article shows in the background some of the many topics that are the subjects of his analysis: midterm elections, tournament brackets, long-term unemployment, Venezuela oil, climate change, minimum wage, taxis and weather extremes. The article focuses on some of the talent that ESPN has "poached" from the Times, The Wall Street Journal and other places.

See the article: "Hey Sports Fans, It's Time for Math Class," by Jack Dickey. Time, 17 March 2014. [Editor's note: Carl Bialik, previously "The Numbers Guy" at The Wall Street Journal, and Joint Policy Board for Mathematics' 2008 Communications Award recipient, is now at FiveThirtyEight, and shows "How Nate Silver Hires," a chart Silver developed of qualities he seeks in potential candidates.]

--- Annette Emerson

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On ways to get more girls into math, by Annette Emerson

Pi Day has passed, but Ruth Charney’s thoughts on what parents and teachers can do to get girls to excel and pursue mathematics ring true year-round. Charney, president of the Association for Women in Mathematics (AWM), says that she’d like to see more women in mathematics, and a day when words used are "Woman" + "Mathematician" = "Mathematician" and not "Woman mathematician." She says, "I think the way to go when talking to children is to show that math is really about puzzle solving, not just doing some rote equations," and goes on to recommend some ways to engage girls in mathematics for the long-run. She suggests girls join a Math Circle, go to a summer math camp, watch Numberphile and TED ED videos, visit Cut the Knot website, and find a mentor.

See the article: "Calculating women: How to get more girls into math," by Lisa Suhay, Christian Science Monitor, 14 March 2014.

--- Annette Emerson

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On perceptions of women’s ability in math, by Annette Emerson

Women doing MathMedia covered a recent study published in the Proceedings from the National Academy of Sciences ("How stereotypes impair women's careers in science") showing that both men and women believe that men are better at math—even though in fact women do as well in math as men. One factor is that in general women tend to discount their own (and other women's) ability. In one experiment female and male job "applicants" (subjects in this experiment) tested equally well in arithmetic but employers (male and female) were more likely to hire the male, based on both gender stereotyping and how confident the men were about their arithmetic test results as opposed to the women.

Actress/mathematician Danica McKellar, committed to addressing the problem of female insecurities about their abilities in math, has written several books for pre-teen and teen females--to entice them into learning math, reduce their fears, and teach them some math concepts. She says in an interview with Learning first Alliance that "on top of the stereotypes about how difficult and "nerdy" it is to study math, girls are also getting the message that they're not supposed to be good at it," and tells girls that they can do math, that it is cool to be smart, and that the confidence they have by knowing math will make them more confident in future work and life experiences.

In "Debunking Myths about Gender and Mathematics Performance" (Notices of the AMS, January 2012) Jonathan M. Kane and Janet E. Mertz say "boys and girls may be born similar in their innate intellectual potential but end up displaying differences due to a variety of sociocultural factors present in their environment." The piece focuses on school performance, but of course early education and test results impact higher education and career choices. The authors cite Gender Gap in Math Performance vs. Equity Indexes in countries around the world, and one of their conclusions is: "Eliminating gender discrimination in pay and employment opportunities could be part of a win-win formula for producing an adequate supply of future workers with high-level competence in mathematics. Wealthy countries that fail to provide gender equity in employment are at risk of producing too few citizens of either gender with the skills necessary to compete successfully in a knowledge-based economy driven by science and technology." It would seem that early intervention to cultivate self-confidence in girls who study and like mathematics is an important step in changing perceptions--and job prospects--of females.

The Women Doing Mathematics poster shown above highlights women mathematicians working in various fields.

See: "Study: Women Who Can Do Math Still Don't Get Hired," by Shaila Dewan, New York Times, 11 March 2014 and "Can women do math? New study finds both sexes believe men are better - regardless of a person's actual ability," by Mark Prigg. Daily Mail, 11 March 2014.

--- Annette Emerson

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Sloan Fellow Seizes the Surprising Power of Symmetry, by Anna Haensch

Amusement park ride

The beautiful, and for some, terrifying symmetry of the Chair-O-Plane means that understanding the movement of one rider is enough to understand the movement of every rider. Image: Flickr Creative Commons, Eric.Parker .

As reported in news@Northeastern, finding and exploiting the symmetries in complicated mathematical systems is the bread and butter of Northeastern University's Ivan Loseu (below), winner of a 2014 Sloan Research Fellowship.

Ivan Loseu

Ivan Loseu, originally from Belarus, will spend his next two years as a Sloan Research Fellow working on his research in representation theory. Image: Courtesy of Michael Finkelberg.

Math Digest reached Loseu by Skype to ask him about symmetries, and his current research in representation theory. As complex math is often best understood through motivating examples, he points us towards Hamiltonian mechanics.

"Let's say that we have a mechanical system with symmetry," he says. "This symmetry gives rise to conservation laws, and using conservation laws you can reduce the dimension, or the number of variables involved."

But what does a highly symmetric Hamiltonian system actually look like?

"Well…," Loseu looks around on his desk as though to avail himself of the nearest Hamiltonian system, "take this!" He removes one of his headphones and holding the cord in his hand, begins spinning the earbud, "as you rotate this headphone the laws of motion don't change."

"Since you have additional constraints on how this object can move," he explains, you can really think about the whole headphone system as having one less dimension. This process is called reduction. But because of the symmetry, it's also possible to reclaim the original system from the reduced system, something which can often be quite difficult in mathematics.

Of course mankind's fascination with the beautify and structure of symmetry is nothing new. In a TED Talk (embedded below), University of Oxford professor Marcus du Sautoy, explains how sophisticated mathematical concepts are embedded in ancient paintings:

For other ways that symmetries can make you rich and famous--or at least rich--check out this Freakanomics Radio podcast.

Congratulations to all of the 2014 Sloan Research Fellows, we wish you many years of success and symmetry.

See the article: "Magic and symmetry in mathematics," by Angela Herring. news@Northeastern, 11 March 2014.

--- Anna Haensch

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On Tim Chartier's Bracketology, by Mike Breen

Tim ChartierIt's a little hard to believe that a word like bracketology (the process of predicting the games in the NCAA Basketball Tournament), which didn't even exist until a few years ago, is now used commonly, but because of the popularity of the tournament, often billed as March Madness, it's used--and performed--frequently. Tim Chartier (left) of Davidson College and Math Ambassador for the MAA, has been quite busy recently doing interviews because of his successful use of math to predict the outcomes of games in the tournament. Chartier and Davidson students have been looking into predicting outcomes of the NCAA Men's Division I College Basketball Tournament for a few years and now that Warren Buffett has offered $1 billion to anyone who can predict every game in the tournament correctly, linear algebra has acquired some added luster. In Chartier's class's first year of using math to make picks, one Davidson student, Daniel Martin, finished in the top 1% of predictions in ESPN's online contest. That won't get you Buffett's billion (in fact, looks like we'll all have to wait until at least 2015), but it could be enough to win your office pool. In this front-page article, Chartier explains how he and his students use math--and some intution--to make picks in the tournament. Weights are assigned to various factors, such as the quality of a team's opponents--which turns out to be very important--to arrive at predictions. This year, Chartier is trying to incorporate momentum into the algorithm: "Yes, you have momentum down the stretch, but are you doing that against good teams? You’ve been able to sustain winning streaks, but are you doing that against good teams?" Photo: Laura McHugh, MAA.

Read the article: "Math whiz unlocks NCAA bracket formula," by David K. Li. New York Post, 10 March 2014. Chartier's work to help predict outcomes in the tournament was covered by many in the media, including The Atlantic, The New York Times, Bloomberg TV, and the CBS Evening News.

--- Mike Breen

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To Understand the Stock Market, Start by Learning Physics, by Anna Haensch

Wolf of Wall Street

The opening bell rings, stockbrokers hit the floor, and with frantic shouts and flying scraps of paper, millions of orders to buy and sell are tabulated. As each millisecond passes, the price of a single stock jumps and falls at dizzying speeds, and according to a recent article in New Scientist, traces out a totally random pattern.

But of course a mathematician doesn't call anything random unless she really means it. According to a recent paper in Physics Review Letters, the price of a stock can be precisely modeled as a random walk. In a physical setting, this behavior is better described by Brownian motion, and usually involves microscopic particles moving about in liquid. Whatever its name, as time passes the price of a stock moves up and down subject to the push and pull of asking prices and selling prices.

Brownian motion

This yellow particle exhibits Brownian motion as the liquid particles bump up against it with varying amounts of force from every direction. Now imagine the yellow dot is a stock price, and the black dots are thousands of stockbrokers shouting into telephones. Image: courtesy of Wikimedia Commons.

For a mathematician who wants to understand the seemingly random fluctuations of the stock market, this is great news. Math Digest called Didier Sornette, a professor at ETH Zurich, who was an investigator on the project. "Notwithstanding the complex strategies of buying and selling"-- of course--"the resulting behavior is that of a normal fluid obeying Einstein relations," he says.

So, although the price changes with each millisecond, and it's being perpetually bombarded by millions of forces--in this case "buy" and "sell" orders--we can still make sense of its behavior through the Einstein relations. And better yet, our analogy of stock prices as a particle in water is complete with quantifiable notions of velocity and drag.

Unfortunately, it's not just as simple as using the models that Sornette and his team have assembled to forecast the normal market. But in the event of a calamity, the models will be helpful. "Suppose a big surprise hits the market," he says, "it's like the brownian particle in fluid that is deflected with an external push. The order book will resist, the different layers will react."

In this way, we can use the models developed by Sornette and his team to help predict the outcome of a highly volatile jolt to the stock market. But even with such a reassuring finding, this mathematician might still keep her assets in that highly stable shoebox on top of her fridge.

See the article: "Stock prices fluctuate like particles doing a Brownian motion dance," by Lisa Grossman. New Scientist, 5 March 2014.

--- Anna Haensch

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Preserving the Internet in a disaster, by Lisa DeKeukelaere

A Japanese researcher's "rules of thumb" for scientists designing the location of physical infrastructure for internet connectivity may help ensure that future disasters, such as earthquakes, don’t intersect and damage connections between network nodes. The researcher, Hitoshi Saito, developed the rules using the principles of integral geometry, under the assumptions that a disaster would occur within a finite area, and that nodes inside this area would fail with a certain probability. He then proved the validity of these rules by testing various network configurations against data from major earthquakes Japan has experienced. If implemented, his rules would represent a change from network design focused on protection from disaster to design aimed at avoiding disaster altogether through strategic node placement and interconnection. The research is posted on the ArXiv.

See the article: "Mathematical Proof Reveals How To Make The Internet More Earthquake-Proof." MIT Technology Review, 3 March 2014.

--- Lisa DeKeukeleare

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Obituary for Lee Lorch, by Allyn Jackson

This obituary describes the life of mathematician and civil rights activist Lee Lorch. The article puts particular emphasis on Lorch's leadership in the campaign to desegregate Stuyvesant Town in Manhattan, an effort that helped make housing discrimination illegal nationwide. After this campaign, Lorch was fired from his job at City College, despite an excellent record as a scholar and teacher (years later, in 1990, City College awarded him an honorary doctorate). Positions at Penn State and Fisk University likewise foundered due to his reputation as a "troublemaker." Lorch finally got a position at Philander Smith College, an all-black institution in Little Rock, Arkansas. There he and his wife were among those assisting a group of African American schoolchildren who attended a white school. This group, known as the "Little Rock Nine," became emblematic of the civil rights struggle. Around this time Lorch was called before the House Un-American Activities Committee, where he refused to answer questions about his supposed ties to Communists. Philander Smith College did not renew his contract, and Lorch was blacklisted from U.S. colleges and universities. In 1959, he and his family moved to Canada, where he first joined the faculty at the University of Alberta. He went to York University in Toronto in 1968 and remained there for the rest of his career.

Born in New York City in 1915, Lorch received his PhD in 1941 from the University of Cincinnati and has 85 publications in MathSciNet. Active in the AMS, Lorch often brought before the Council cases concerning human rights and discrimination. When he received the Gung and Hu Award for Distinguished Service from the MAA in 2007, the crowd at the prize ceremony gave him a long standing ovation. The ceremony took place at the Joint Mathematics Meetings in New Orleans a year and a half after the devastation of Hurricane Katrina, and Lorch used his acceptance speech to remind the audience to reach out to their mathematical colleagues in the New Orleans area. Known for his humanity, warmth, and humor, Lorch served as a conscience for the mathematical community and will be greatly missed.

See the article: "Lee Lorch, Desegregation Activist Who Led Stuyvesant Town Effort, Dies at 98," by David Margolick. The New York Times, 2 March 2014. [Also see "Mathematician Lee Lorch fought tirelessly against racism," in the Toronto Star.]

--- Allyn Jackson

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Ed Frenkel on an outdated curriculum, by Mike Breen

Rubiks cube Edward Frenkel, who has been frequently covered by the media recently, wrote this Op-Ed for the Los Angeles Times in which he compares current math education with teaching art by teaching only whitewashing and never allowing students to see masterpieces. He acknowledges that the math taught now to almost everyone is important but thinks that students should be offered glimpses of modern mathematics by being exposed to subjects such as symmetry groups, modular arithmetic, and Riemannian geometry. Those subjects can't be mastered by young students but Frenkel writes of his visit with 5th and 6th graders in New York:

I used a Rubik's Cube to explain symmetry groups: Every rotation of the cube is a "symmetry," and these combine into what mathematicians call a group. I saw students' eyes light up when they realized that when they were solving the puzzle, they were simply discerning the structure of this group.

Abstraction is an important skill in today's world, and math can help people acquire that skill. Frenkel thinks that allotting about 20% of class time to the "power and exquisite harmony" of modern mathematics would break the vicious circle of people who hate math and aren't good at it, as well as doing away with the question "Why study math?"

See the article: "How our 1,000-year-old math curriculum cheats America's kids," by Edward Frenkel. Los Angeles Times, 2 March 2014. [Also see "5-Year-Olds Can Learn Calculus," by Luba Vangelova in The Atlantic. In this article, Vangelova writes about educators who are letting children experiment with simple aspects of complex subjects (such as using mirrors to introduce transformations), as opposed to teaching complex parts of simple subjects (e.g. multiplication table drills).]

--- Mike Breen

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On some interesting curves, by Claudia Clark


Several different types of cycloid (dotted red, blue, and orange lines) take shape as a circular wheel rolls. Top: the point is outside the wheel’s rim; Middle: on the rim; and Bottom: inside the rim. Image: "Reflections on a Rolling Wheel," by Eugen Jost from Beautiful Geometry by Eli Maor and Eugen Jost, used courtesy of Princeton University Press.


In this article, Maor and Jost discuss the beauty, history, and mathematical properties of four families of equations. Take the logarithmic spiral, for instance, which can be represented by the polar equation r = e . Jakob Bernoulli, who discovered most of the logarithmic spiral’s properties in the 17th century, dubbed it "spira mirabilis." It is apparent that if θ is increased by equal amounts, r will increase by a constant ratio. One resulting characteristic of the logarithmic spiral is that "any sector with given angular width Δθ is similar to any other sector with the same angular width." Or consider the cycloid, "the curve traced by a point on the rim of a circle that rolls along a straight line without slipping." Among the cycloid’s claims to fame: in 1673, Dutch physicist Christiaan Huygens determined that the arc of an inverted cycloid is "the curve down which a particle, moving under the force of gravity, will take the same amount of time to reach a given final point, regardless of the initial position of the particle." Some 23 years later, Newton, Leibniz, L’Hospital, and the Bernoulli brothers found that this same curve is "the curve along which a particle, again subject only to the force of gravity, will slide down in the least amount of time."

See the article: "Twisted Math and Beautiful Geometry," by Eli Maor and Eugen Jost. American Scientist, March-April 2014, pages 140-145.

--- Claudia Clark

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