Pi: A Window into the World of Mathematics

Mathematicians have always been fascinated by Pi, the famous never-ending never-repeating decimal that rounds to 3.14. But why? What makes Pi such an interesting number? Every mathematician has their own answer to that question. For me, Pi’s allure is that it illustrates perfectly the arc of mathematics. Let me explain what I mean by taking you on a short mathematical adventure.

Picture yourself in a kitchen, rummaging the pantry for two cans of food. Let’s say you’ve found two that have circular bases of different diameters d1 and d2. Associated with each circle is a circumference value, the distance you’d measure if you walked all the way around the circle.

Were you to perfectly measure each circle’s circumference and diameter you would discover an intriguing relationship:

In other words, the ratio of each circle’s circumference to its diameter doesn’t change, even though one circle is bigger than the other. (This circumference-to-diameter number is  (“Pi”), the familiar 3.14-ish number.) This is the first stop along the arc of mathematics: the discovery of a relationship between two quantities.

Where this story gets very interesting is when, after grabbing even more cans and measuring the ratio of their circumferences to their diameters—you seem to have lots of free time on your hands—you keep finding the same ratio. Every. Time. This is the second stop along the arc of mathematics: the discovery of a pattern. Shortly after that, you begin to wonder: does every circle, no matter its size, have the same circumference-to-diameter ratio? You have reached the third stop along the arc of mathematics: conjecture. (Let’s call our circumference-to-diameter conjecture The Circle Conjecture.)

At first you consider proving The Circle Conjecture by measuring the ratio C/d for every circle. But you soon realize that this is impossible. And that’s the moment when you start truly thinking like a mathematician and begin to wonder: Can I prove The Circle Conjecture true using mathematics? You have now reached the most important stop along the arc of mathematics: the search for universal truth.

One of the first thinkers to make progress on The Circle Conjecture was the Greek mathematician Euclid of Alexandria. Euclid published a mammoth 13-book treatise text called Elements circa 300 BC in which he, among other accomplishments, derived all the geometry you learned in high school from just five postulates. One of Euclid’s results was that the ratio of a circle’s area A to the square of its diameter d2 is the same for all circles:

This is close to what we are trying to prove in The Circle Conjecture, but not the same. It would take another giant of mathematics—the Greek mathematician Archimedes of Syracuse—to move us onto what is often the last stop on the arc of mathematics: thinking outside the box.

Archimedes went back to Euclid’s five postulates, all but one of which dealt with lines, and extended some of Euclid’s postulates to handle curves. With these new postulates Archimedes was able to prove in his treatise Measurement of a Circle (circa 250 BC) that the area, circumference, and radius r of a circle are related by the equation:

(You may recognize this as the area of a triangle with base C and height r. Indeed, Archimedes’ proof of the formula effectively “unrolls” a circle to produce a triangle and then calculates its area.) Combining Archimedes’ formula with Euclid’s result, and using the fact that r = d/2, yields:

Et Voilà! The Circle Conjecture is proved! (To read more about the mathematical details involved in proving The Circle Conjecture, I recommend this excellent article.)

This little Pi adventure illustrated the core arc of mathematics: discovery of a relationship between to quantities; discovery of a more general pattern; statement of a conjecture; search for a proof of that conjecture; and thinking outside the box to help generate a proof. Let me end our mathematical adventure by encouraging you to embark on your own. Find things you experience in your life that are quantifiable and seem to be related (e.g., how much sleep you get and how awake you feel) and follow the stops along the arc of mathematics. You may soon afterward discover another universal truth: anyone can do mathematics! All it takes is curiosity, persistence, and creative thinking. Happy Pi Day!

 

Oscar E. Fernandez is associate professor of mathematics at Wellesley College. He is the author of Calculus Simplified, Everyday Calculus, and The Calculus of Happiness (all Princeton).

Marcia Bjornerud: Grandmothers of Geoscience

A sheepish admission:  I intermittently check the reviews of my books posted by readers on the website of an online retail behemoth.  I smile at benevolent judgments, cringe at misspellings and misreadings, wonder whether some of the more generic entries were written by bots, and occasionally obsess about comments that get under my skin.  A few weeks ago, in a generally positive review of my PUP book Timefulness: How Thinking Like a Geologist Can Help Save the World, a reader commented that the tone of the text was “grandmotherly”.    

In an instant, several thoughts collided in my head.  The first was indignation – I’m not a grandmother!  Nanoseconds later, I reminded myself that as a fifty-something mother of three sons I certainly could be (and in fact hope to be in a few years).  Next, I chastised myself for falling into the very trap of vanity-rooted time denial that my book exhorts us all to avoid.  And then, my mind moved to the question of what exactly “grandmotherly” means in our culture, and whether a reader would apply the word “grandfatherly” to a work written by a male scientist in his 50s.  On that count, I felt less sure about the right answer.

So many words for women in our culture are tinged with accusation or insult: “mistress” is freighted in a way that “master” is not; “dame” has been demoted to slang (and has horsy connotations) but “sir” hasn’t; “matronly” is not exactly a compliment.  And I chafe, as a “Fellow” of a couple of professional organizations that there is no obvious female equivalent:  Am I a “Gal of the Geological Society of America”?

But as I turned the word “grandmotherly” over in my mind, viewing it from all sides, I saw mostly respect: acknowledgment of experience, persistence, hard-won wisdom, and the right to a voice that should be heard and heeded. 

The fact is that there are far too few grandmothers in any of the sciences and certainly the geosciences in particular.  There was Mary Anning (1799-1847) of Lyme Regis, discoverer of Jurassic sea monsters and arguably the first professional paleontologist;  geophysicist Inge Lehmann (1888-1993), who showed that the Earth’s inner core is solid, a discovery essential to understanding the planet’s magnetic field;  Marie Tharp (1920-2006) who created the first maps of the deep seafloor – more than half of Earth’s surface; Tanya Atwater (born 1942) who worked out the tectonic evolution of western North America over the past 60 million years. 

But I personally had no senior female mentors in my undergraduate and graduate school years.  And according to the American Geological Institute, even today women represent only 15% of the full professors in the geosciences in US universities[1].  I wasn’t fully aware of it as a student, but I see now that the absence of academic grandmothers was an impediment to my own development as a scientist.  There were no exemplars for how to be taken seriously in an overwhelmingly male, highly competitive work environment; no instructions for how to synchronize biological and tenure clocks; no reassurances that success was even possible.  In graduate school, the small cohort of women in my program supported each other but on our own could not allay the chronic anxieties we all shared.  How different our experiences as young scientists would have been with just one grandmotherly figure to turn to.

So, if I am now being bestowed the mantle of grandmother, honoris causa, I humbly accept.  Perhaps one day, our most esteemed scientists, both male and female, will be recognized with that most coveted of all awards: “Grandmother of the National Academy of Sciences”.

Marcia Bjornerud is professor of geology and environmental studies at Lawrence University. She is the author of Reading the Rocks: The Autobiography of the Earth and a contributing writer for Elements, the New Yorker’s science and technology blog. She lives in Appleton, Wisconsin.

Mark Serreze: Becoming A Scientist

In honor of Earth Day, Princeton University Press will be highlighting the contributions that scientists make to our understanding of the world around us through a series of blog posts written by some of our notable Earth Science authors. Keep a look out for this series all month long.

Mark Serreze, investigating the pressure ridges in the Arctic.

What is it that leads someone to become a scientist? It varies, but from what I’ve seen, it’s often a combination of nature and nurture. Just as some people seem to have an inherent knack for writing making music, or cooking, I think that some of us are wired to become scientists. In turn, there is often someone we can look back to—parents or perhaps a teacher—that encouraged or inspired us to pursue a science career.

I had an interest in science from when I was very young, and I was always full of questions about the natural world. The first book I ever owned is “The Golden Book of Science” 1963 edition—featuring 1-2 page essays on everything from geology to insects to the weather. Each night, at my insistence, my mother would read one of them to me. To this day, I still own the book.

When I wasn’t reading, I could spend hours outside marveling at the organized industriousness of ants as they built their anthills, or looking at colorful rocks with a magnifying glass. I was enthralled with the burgeoning manned space flight program, and, sitting beside my mother and staring at the black TV while she ironed clothes, watched in awe at the Project Gemini rocket launches.   

As for the nurture part, I had an advantage in that both of my parents were chemists with Master’s degrees. This was at a time it was quite unusual for women to hold advanced degrees. They met in the laboratory. Mom was a whiz when it came to thermodynamics, and Dad apparently knew everything there was to know about acrylic plastics. Ours was indeed an odd household. While my siblings and I chafed under a rather strict Catholic upbringing, at the same time we were very much free-range kids, and scientific experimentation of all sorts was quite acceptable.  

At one point, after getting a chemistry set for Christmas, I thought I might become a chemist myself. These were not the boring, defanged chemistry sets of today – back then, they included chemicals that, when properly mixed, yielded career-inspiring reactions. I later got heavily into model rocketry, astronomy, and civil engineering, building small dams across the stream running past our house to improve the habitat for the frogs. Included among the more foolish (albeit highly educational) endeavors was a scientifically-based experiment on the feasibility of riding ice floes down the Kennebunk River. Then there was the time when an experiment in pyrotechnics gone wrong ended up with a frantic call to the fire department to douse a five-acre conflagration in the neighbor’s field.

Years before I ever got into college I knew I was going to be a research scientists of some type, for, through nature and nurture, the roots were already there. As I talk about in my book, Brave New Arctic, a number decisions and events came together – mixed with some blind, dumb luck – to eventually steer me towards a career in climate science. What I could never have foreseen is how, through these events and decisions, and then through 35 years of research, I’d find myself in the position to tell the story about the dramatic transformation of the North.

Climate scientists, like myself, have to deal with an added challenge that climate change is a highly polarized subject. There are the frequent questions from the media: Will there be a new record low in Arctic sea ice extent this year? Why does it matter? Why is the Arctic behaving so differently than the Antarctic? It can be overwhelming at times. Then there are the emails, phone calls and tweets from those who simply want to rant. While I get a lot of emails from people fully on board with the reality that humans are changing the climate and want to get straight answers about something they’ve heard or read about, I also have a growing folder in my inbox labeled “Hate Mail”. Some very unflattering things have been said about me on social media and across the web. I’ve had to grow a thick skin.  

Making a career as a research scientist is not for everyone. Science is not the sort of thing that is easy to put aside at the end of the day. It gnaws at you. The hours are long, and seldom lead to monetary riches. It can also be a frustrating occupation, such as when realizing that, after months of research pursuing a lead, you’ve hit a dead end.

We chose to be scientists because it’s what we love to do. We live for those “aha” moments when the hard work pays off, and we discover something new that advances our understanding.

In writing this book I was forced to dig deeply to understand my own evolution as a scientist, and to document insights from other scientists who, like me, were there at the beginning when the Arctic still looked like the Arctic of old. It’s been an adventure, and when I someday retire (which is a very hard thing for scientists to do,) I hope to be able to look back and say that that this book opened some eyes, and inspired others to follow their own path to becoming a scientist.

 

Mark Serreze is director of the National Snow and Ice Data Center, professor of geography, and a fellow of the Cooperative Institute for Research in Environmental Sciences at the University of Colorado at Boulder. He is the coauthor of The Arctic Climate System. He lives in Boulder, Colorado.

Noah Wilson-Rich on city beekeeping

the bee jacketNoah Wilson-Rich is an unconventional beekeeper who spends most of his time building bee hives on hundreds of buildings, including major stadiums, in nine different cities. These urban settings now support live bee populations and the environmentally friendly trend is only growing. As author of The Bee: A Natural History, Wilson-Rich establishes himself as an authority not only on the species but on conservation as well. An article on his beekeeping and speaking tour appeared recently in The Wall Street Journal.

Wilson-Rich emphasizes the urgency of preserving the bees’ population, pointing out that his urban hives are just one step in the right direction. The Wall Street Journal reports:

Mr. Wilson-Rich is researching ways to improve bee health, so he also carries test tubes to collect samples. He believes urban beekeeping is part of the solution. “Anybody who eats fruits and veggies needs bees. We have to protect our pollinators!” he says.

Wilson-Rich goes on to speak about some little known facts about bees, their habits, and what exactly makes them so uniquely necessary to humans. Read the rest of the article here.

Noah Wilson-Rich is founder and chief scientific officer of The Best Bees Company, a Boston-based beekeeping service and research organization. He is author of the book The Bee: A Natural History.

Elizabeth Shakman Hurd speaks out against religious-citizenship test

Hurd_BeyondReligious_F15Elizabeth Shakman Hurd, author of Beyond Religious Freedom, calls the requirement by an advanced democratic country of a mandatory religious test for citizenship outright pernicious. In her recent Al Jazeera op ed, Hurd condemns the Republican suggestion and promotion of an amendment that would ban Muslim Syrian refugees from entering the country, in response to the tragic terror attacks in Paris. Explaining that, “the grown-ups in the room need to take this poisonous talk seriously and stop it now,” Hurd also adds:

To subject prospective refugees to a religious test would also do violence to the complex realities of the Syrian war and the millions of Syrian men, women and children who are suffering so tragically as a result of it. The goal of the Syrian opposition in 2011 was to put an end to the state’s brutal treatment and exploitation of the Syrian people. The Syrian war has complex roots in economic deprivation, social injustice and everyday oppression. To reduce this deeply complex regional and international conflict to a problem of “Islamic terrorism” simply misreads reality.

While Hurd recognizes that religion plays a significant role in the Syrian war, she notes that the war itself, “cannot be reduced to religion or religious dynamics.” Syrian refugees, she says, should  not be solely defined by specific and unreliable religious parameters that a U.S. government department created.

Read the full piece in Al Jazeera here.

Elizabeth Shakman Hurd is associate professor of political science at Northwestern University. She is the author of The Politics of Secularism in International Relations (Princeton) and the coeditor of Comparative Secularisms in a Global Age and Politics of Religious Freedom.

Jonathan Zimmerman: How consensual is casual sex on campus?

zimmerman jacketIn a recent op ed in Washington Post on the question of consensual sex on college campuses, Jonathan Zimmerman, author of Too Hot to Handle: A Global History of Sex Education, writes, “… if we want to protect our students, not just their colleges, we will have to begin a deeper dialogue about the meaning of sex itself.” In an approach that departs from debates that have focused on what constitutes ‘legal’ sex, Zimmerman questions the ability of students to emotionally connect in such an intimate setting in extremely limited periods of time:

We might succeed in cajoling more students into some kind of verbal consent. But that’s a script, a bedroom contract between sexual vendors. Yes, it will make the whole transaction legal. But consensual? Really? If you met somebody an hour ago, how can you tell what they want? And since you know so little about them, aren’t you more likely to do something that they don’t want, no matter what kind of “consent” they have given?

According to Zimmerman, university online courses, workshops and informational resources about consensual sex on campus fail to emphasize the vital notions of emotional connection and communication. Due to this lack of communication, he suggests that although female students may verbally give consent, they are still pressured to do things they would normally avert.

Read Zimmerman’s full piece in the Washington Post here.

Jonathan Zimmerman is professor of Education and History at New York University. He has also authored Small Wonder: The Little Red Schoolhouse in History and Memory and Innocents Abroad: American Teachers in the American.

Anat Admati on the stark reality of post-2008 banking

Admati-BankersNewClothes_pbkThere are a few lessons still unlearned from the 2008 financial recession, according to Anat Admati, co-author of The Banker’s New Clothes: What’s Wrong with Banking and What to Do about it. “After such a major trauma, we want to believe all is well again,” Admati wrote in her Bloomberg piece on Monday. “But the reality in banking is different and stark.”

Admati turns her attention to former chair of the Federal Reserve, Ben Bernanke’s new book, The Courage to Act. While she applauds Bernanke for appreciating the significance of “equity capital in protecting the economy from financial shocks”, she is skeptical of the supposed progress resulting from regulations implemented by the Federal Reserve post-2008. Admati writes in Bloomberg:

A clear lesson is that banks need much more capital, specifically in the form of equity. In this area, the reforms engendered by the crisis have fallen far short. Regulators focus on “risk-weighted” and accounting-based capital ratios that, among their many flaws, rely on banks to assess the riskiness of their assets. Using off-balance-sheet accounting, derivatives and other tools, banks have become adept at manipulating these ratios. Annual stress tests aren’t much better: They employ the same flawed measures and cannot reliably predict how an actual crisis, which may come from an unexpected direction, would play out in an opaque and interconnected financial system.

Admati argues that a larger amount of equity given to banks would offer substantial benefits to society with minimal costs, halting the precarious practice of creditors allowing the largest banks in the world to borrow money under the assumption of government intervention in dire situations.

Read the rest of Admati’s analysis here .

Anat Admati is the George G. C. Parker Professor of Finance and Economics at Stanford’s Graduate School of Business.