Praeteritio and the quiet importance of Pi

pidayby James D. Stein

Somewhere along my somewhat convoluted educational journey I encountered Latin rhetorical devices. At least one has become part of common usage–oxymoron, the apparent paradox created by juxtaposed words which seem to contradict each other; a classic example being ‘awfully good’. For some reason, one of the devices that has stuck with me over the years is praeteritio, in which emphasis is placed on a topic by saying that one is omitting it. For instance, you could say that when one forgets about 9/11, the Iraq War, Hurricane Katrina, and the Meltdown, George W. Bush’s presidency was smooth sailing.

I’ve always wanted to invent a word, like John Allen Paulos did with ‘innumeracy’, and πraeteritio is my leading candidate–it’s the fact that we call attention to the overwhelming importance of the number π by deliberately excluding it from the conversation. We do that in one of the most important formulas encountered by intermediate algebra and trigonometry students; s = rθ, the formula for the arc length s subtended by a central angle θ in a circle of radius r.

You don’t see π in this formula because π is so important, so natural, that mathematicians use radians as a measure of angle, and π is naturally incorporated into radian measure. Most angle measurement that we see in the real world is described in terms of degrees. A full circle is 360 degrees, a straight angle 180 degrees, a right angle 90 degrees, and so on. But the circumference of a circle of radius 1 is 2π, and so it occurred to Roger Cotes (who is he? I’d never heard of him) that using an angular measure in which there were 2π angle units in a full circle would eliminate the need for a ‘fudge factor’ in the formula for the arc length of a circle subtended by a central angle. For instance, if one measured the angle D in degrees, the formula for the arc length of a circle of radius r subtended by a central angle would be s = (π/180)rD, and who wants to memorize that? The word ‘radian’ first appeared in an examination at Queen’s College in Belfast, Ireland, given by James Thomson, whose better-known brother William would later be known as Lord Kelvin.

The wisdom of this choice can be seen in its far-reaching consequences in the calculus of the trigonometric functions, and undoubtedly elsewhere. First semester calculus students learn that as long as one uses radian measure for angles, the derivative of sin x is cos x, and the derivative of cos x is – sin x. A standard problem in first-semester calculus, here left to the reader, is to compute what the derivative of sin x would be if the angle were measured in degrees rather than radians. Of course, the fudge factor π/180 would raise its ugly head, its square would appear in the formula for the second derivative of sin x, and instead of the elegant repeating pattern of the derivatives of sin x and cos x that are a highlight of the calculus of trigonometric functions, the ensuing formulas would be beyond ugly.

One of the simplest known formulas for the computation of π is via the infinite series 𝜋4=1−13+15−17+⋯

This deliciously elegant formula arises from integrating the geometric series with ratio -x^2 in the equation 1/(1+𝑥^2)=1−𝑥2+𝑥4−𝑥6+⋯

The integral of the left side is the inverse tangent function tan-1 x, but only because we have been fortunate enough to emphasize the importance of π by utilizing an angle measurement system which is the essence of πraeteritio; the recognition of the importance of π by excluding it from the discussion.

So on π Day, let us take a moment to recognize not only the beauty of π when it makes all the memorable appearances which we know and love, but to acknowledge its supreme importance and value in those critical situations where, like a great character in a play, it exerts a profound dramatic influence even when offstage.

LA MathJames D. Stein is emeritus professor in the Department of Mathematics at California State University, Long Beach. His books include Cosmic Numbers (Basic) and How Math Explains the World (Smithsonian). His most recent book is L.A. Math: Romance, Crime, and Mathematics in the City of Angels.

Lynn Gamwell on math and the visual arts’ shared cultural history

GamwellMathematicians and artists have historically shared a common interest: inquiry and comprehension of the intricacies of the world around them, whether through numerical or aesthetic design. Illustrating the relationship between math and art from antiquity to present day, Lynn Gamwells Mathematics and Art highlights the significant impact these two linked worlds have on one another. Gamwell recently took the time to answer some questions about her book. Examining the modern disciplines of art and math, she reveals the profound philosophy of self-reflection that these two cultural and intellectual pursuits share. Don’t forget to check out the stunning slideshow following the Q&A.

What’s the basic idea of your book?

LG: I started with the assumption that how people understand reality relates directly to the concepts of mathematics that develop in their culture. Mathematics is a search for patterns, and artists, in turn, create visualizations of the patterns discovered in their time. So I describe a general history of mathematics and the related artwork.

Since you begin in Stone Age times, your book covers over 5000 years. Is there a historical focus to the book?

LG: Yes, there are 13 chapters, and the first gives the background up to around 1800 AD. The other 12 chapters are on the modern and contemporary eras, although I occasionally dip back into pre-modern times to give the background of a topic. A central question that drove my exploration of the modern era was: where did abstract, non-objective art come from? Between around 1890 and 1915, many artists stopped depicting people and landscapes and start using pure color and form as the vocabulary of their art. Why? I argue that modern art is an expression of the scientific worldview. Beginning in the late nineteenth century and continuing today, researchers describe bacteria, cells, radiation, and pulsars that are invisible to the unaided eye, as well as mathematical patterns in nature.

Can you give a few examples of the relation of math and art?

LG: Italian Renaissance artists, such as Leonardo da Vinci, constructed the space in paintings such as The Last Supper using linear perspective, which is a geometric projection invented in the 1430s by the architect Filippo Brunelleschi. In the twentieth century, Swiss Constructivists such as Karl Gerstner created symmetrical patterns based on the mathematics of group theory, which measures the amount of symmetry in a system, such as atoms and sub-atomic particles. The contemporary America artist Jim Sanborn uses topology, which is the projection of geometric shapes onto surfaces that are stretched and distorted. For example in photographs of cliffs in Ireland, Jim first projected concentric circles onto the rocks and then took the photograph with a long exposure at moonrise. These artists are, of course, interested in many other things besides mathematics; aesthetic issues are their primary focus.

The examples you give are artists who are inspired by math; are mathematicians ever influenced by art?

LG: Mathematics are rarely inspired by a particular piece of art (since most artists use elementary arithmetic and geometry), but rather they aspire to include in their proofs general aesthetic qualities, such as purity, simplicity, and elegance.

You mention Leonardo da Vinci; didn’t he use the Golden Ration?

LG: No. It is a common misconception that a ratio described by Euclid as “mean and extreme ratio” has been used by artists throughout history because it holds the key to beautiful proportions. This myth was begun in the early nineteenth century by a German scholar who called Euclid’s ratio “golden.” The myth took a tenacious hold on Western intellectuals because, as science was beginning to take them off their privileged pedestal, it assured them that all beauty is based on a ratio embodied in human anatomy. There is no science supporting this claim.

Your book is a global history; did you find that there is a difference between math in the East and West?

LG: Yes, because a culture’s understanding of mathematics is based in its understanding of reality. In antiquity, Eastern mathematics in based in Taoism, the view that nature is composed of myriad parts that came together by self-assembly into a harmonious whole. Thus Chinese mathematicians discerned patterns in numbers, such as the Luoshu (magic square), in which numbers in the rows, columns, and diagonals have the same sum (the harmonious whole). On the other hand, Western cultures believed that a divine person (The Egyptian sun-god Ra, the God of Abraham, Plato’s carpenter) had imposed order on formless chaos. Thus Westerners went looking for this order, and they found it in the movement of the stars (the Babylonian zodiac), and the planets (Kepler’s Laws of Planetary Motion). Although there was a difference between Eastern and Western math when there was little contact, in today’s culture there is one global math.

The book includes the diverse fields of art, philosophy, mathematics, and physics; what is your educational background?

LG: I have a BA in philosophy and a PhD in art history. I’m self-taught in the history of science and math.

At 576 pages, this is a long book with extensive endnotes and 500+ illustrations; how long did it take you?

LG: 12 years of research and writing, plus one year in production.

Did you make any discoveries about art that especially surprised you?

LG: Yes. When I started my research I thought that artists during the modern era (the twentieth- and twenty-first centuries) would have only a vague knowledge of the math of their times, because of the famed “two cultures” divide. But I found specific historical evidence (an artist’s essay, manifesto, interview, or letter), which demonstrated that the artist had direct knowledge of a particular piece of mathematics and had embodied it in his or her art. Examples include: Aleksandr Rodchenko, Henry Moore, Piet Mondrian, Max Bill, Dorothea Rockburne, as well as musicians, such as Arnold Schoenberg, and poets, such as T. S. Eliot and James Joyce. Again, I would stress that for such artists mathematics is a secondary interest at best, and they are concerned with materials, expressive content, and purely aesthetic issues.

Any surprising discoveries about math and science?

LG: Yes, here are two. Much of what is taught as physics is really philosophy (interpretation) of physical data. An example is the Copenhagen interpretation of quantum physics, which was taught as THE gospel truth from its announcement in 1927 to around 1960. In fact, there are other ways to interpret the same laboratory data, which were largely ignored. I’m used to such dogmatism in the art world, where artists and critics are known to proclaim what art IS, but I expected to find a more cool-headed rationalism in the laboratory. Alas, we’re all human beings, driven by our passions. Another example is the strong resistance to Platonism (the view that abstract objects exist outside time and space) in modern culture, even though Platonism is the view held by most working mathematicians (i.e., they believe they are discovering patterns not creating them). While doing research, I found myself viewed with suspicion of being a religious missionary (disguised as a scholar) because I gave a sympathetic reading of historical religious documents (in other words, I tried to describe reality from their point of view). In fact, my outlook is completely secular. I came to realize that many secularists are unable to separate Platonism from its long association with religious doctrine, which touches a nerve in certain otherwise dispassionate academics.

Are you planning another project? What are you going to do next?

LG: I’m going to take some time off and regroup. I’ve started to think about writing something for children.

Check out the slideshow highlighting just a few of the book’s stunning images:

Lynn Gamwell is lecturer in the history of art, science, and mathematics at the School of Visual Arts in New York. She is the author of Exploring the Invisible: Art, Science, and the Spiritual (Princeton).

Romance, Crime, and… Mathematics? Presenting the new trailer for LA Math

LA Math by James D. Stein, emeritus professor in the Department of Mathematics at California State University, is full of A-listers and wannabes, lovers and lawyers, heroes and villains. And it’s also full of math—practical mathematics knowledge, ranging from percentages and probability to set theory, statistics, and the mathematics of elections. Check out the new trailer for this unconventional and highly readable book of mathematical short stories here:

#ThanksEinstein: Hanoch Gutfreund on the revelation of relativity

Einstein meme 2The Revelation of Relativity

By Hanoch Gutfreund

Hanoch Gutfreund is professor emeritus of theoretical physics at the Hebrew University of Jerusalem, where he is also the academic director of the Albert Einstein Archives. This is the story about how Einstein’s General Theory of Relativity revolutionized his teaching, understanding, and career.

My present day interest in Einstein evolved late in my academic life. It started when as Rector and then President of the Hebrew University, in the 1990’s, I became aware of the unique cultural asset possessed by the university – the Albert Einstein Archives. When I stepped down from the presidency, with einstein lightthe encouragement of my successor, I began to devote more and more time to promote the Einstein – H.U. connection, through public lectures on various Einstein topics and by organizing and helping to organize Einstein exhibitions in different places in the world.

As professor of theoretical physics, for many years I taught everything that Einstein did in his miraculous year – 1905. However, only in the late nineties did I read the original papers with commentaries by John Stachel. For me this was a revelation. Einstein’s way of thinking, his motivations, his introductions and conclusions – all this was very different from the way these topics were treated in ordinary textbooks. I believe that if I had known and understood what I know and understand today, my students would have appreciated and benefited from my lectures even more. Motivated by this revelation, I decided to fill a gap in my own physics education. As a student, I never had a course in general relativity. In the learning process, the historical context and Einstein’s intellectual struggle were for me at least as important as the scientific results.

Teinstein speed of lighto mark the 50th anniversary of the Israeli Academy of Science, we displayed the most important manuscript in the Einstein Archives, the manuscript of Einstein’s seminal paper on general relativity. Each one of the 46 pages of this manuscript was enclosed in a dimly illuminated box. People visited this exhibit as if they were entering a shrine.

Following this experience, I met with Jurgen Renn, director of the Max Planck Institute for the History of Science. We discussed an option to publish this manuscript as part of a comprehensive account of Einstein’s intellectual odyssey to general relativity.

Gutfreund_RoadtoRelativityThis meeting led to a fruitful collaboration, which has now produced The Road to Relativity: The History and Meaning of Einstein’s The Foundation of General Relativity. It attempts to make the essence of general relativity accessible to broader audiences. We have also initiated the recently published, 100th anniversary edition of Einstein’s popular booklet on the special and general theory of relativity, with extensive background material and a reading companion, intended to resent Einstein’s text in a historical and modern context. We are already considering other Einsteinian projects in the future. This year, as the world marks the 100th anniversary of general relativity, there are many requests addressed to the Albert Einstein Archives and to myself for assistance in organizing special exhibitions, for participation in scientific conferences and in public events, for interviews in the media and for help and advice in various other initiatives. It’s an exciting time, and I remain very grateful for this inspiring phase in my life.

Hanoch Gutfreund is professor emeritus of theoretical physics at the Hebrew University of Jerusalem, where he is also the academic director of the Albert Einstein Archives.

Check out the earlier post in this series by Jimena Canales.

#ThanksEinstein image courtesy of the official Albert Einstein Facebook page.

An exclusive trailer for Alice’s Adventures in Wonderland, featuring illustrations by Salvador Dalí

ALICE WAS BEGINNING TO get very tired of sitting by her sister on the bank, and of having nothing to do: once or twice she had peeped into the book her sister was reading, but it had no pictures or conversations in it, “and what is the use of a book,” thought Alice, “without pictures or conversations?”

Thus begins Alice’s Adventures in Wonderland, one of the most beloved classics of children’s literature. Commemorating the 150th anniversary of its publication, this illustrated edition of Alice’s Adventures in Wonderland, edited by Lewis Carroll expert Mark Burstein, features rarely seen illustrations by Salvador Dalí. In the introduction, Burstein discusses Dalí’s connections with Carroll, the nature of wonderland, and his treatment of the towering (though sometimes shrinking) figure of Alice.

Take an exclusive peek inside the curiously mathematical world into which Alice famously falls, here:

Five places you didn’t expect to encounter applied math

You don’t need to step into a classroom to have a run-in with mathematics. Professionals from a range of backgrounds — engineering, economics, physics, biology, computer science — use mathematics every day. To celebrate the publication of the much-anticipated Princeton Companion to Applied Mathematics, edited by Nicholas J. Higham, we’re thinking about all of the unique places and situations where applied mathematics is at work. Here is a list of just a few, compiled with a little help from our numerically inclined friends.

On the Golf Course

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Golf involves mathematics, and not just when keeping score. The flight of your golf ball is affected by how air interacts with the surface of the ball. Did you know that the dimples in golf balls have a purpose, one with a mathematical explanation? Douglas N. Arnold, professor of Mathematics at the University of Minnesota, tells us more:

In the middle of the nineteenth century, when rubber golf balls were introduced, golfers noticed that old scuffed golf balls traveled farther than new smooth balls, although no one could explain this unintuitive behavior. This eventually gave rise to the modern dimpled golf ball. Along the way a great deal was learned about aerodynamics and its mathematical modeling. Hundreds of different dimple patterns have been devised, marketed, and patented. However, even today the optimal dimple pattern lies beyond our reach, and its discovery remains a tough challenge for applied mathematics and computational science.

Check out Dr. Arnold’s entry, “The Flight of a Golf Ball,” where he explains why golf ball dimples are an important part of your Saturday morning tee time.

On Wall Street

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Wall Street is all about the numbers. Whether modeling the risk of a single stock or mapping the complex interactions that make up the world’s financial structure, mathematics helps the financial sector to study and evaluate systemic risk.

“The complexity, unpredictability, and evolving nature of financial markets continues to provide an enormous challenge to mathematicians, engineers, and economists in identifying, analyzing, and quantifying the issues and risks they pose,” write Dr. René A. Carmona and Dr. Ronnie Sircar of Princeton University.

In their entry, “Financial Mathematics,” Dr. Carmona and Dr. Sircar discuss how the finance industry uses mathematics. They also examine the role of mathematics in understanding and regulating financial markets in light of the financial crisis of 2008.

On Your Phone’s Weather App

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Do you check the 10-day forecast during the weekend before a big outdoor event, fingers crossed for clear skies and no rain? There’s math behind that “chance of thunderstorms” prediction. NWP [numerical weather prediction] helps meteorologists to predict weather patterns for more than a week ahead. Better numerical schemes are partially responsible for moving us forward from the weather prediction methods of fifty years ago.

In his article “Numerical Weather Prediction,” Peter Lynch presents the mathematical principles of NWP and illustrates the process by considering some specific models and their application to practical forecasting. Dr. Lynch describes the many conditions that can be better predicted using NWP:

NWP models are used to generate special guidance for the marine community. Predicted winds are used to drive wave models, which predict sea and swell heights and periods. Prediction of road ice is performed by specially designed models that use forecasts of temperature, humidity, precipitation, cloudiness, and other parameters to estimate the conditions on the road surface. Trajectories are easily derived from limited-area models. These are vital for modeling pollution drift, for nuclear fallout, smoke from forest fires, and so on. Aviation benefits significantly from NWP guidance, which provides warnings of hazards such as lightning, icing, and clear-air turbulence.

In the Airport Security Line

baggage claim

On your next trip through airport security, take a look at the x-ray machine. Once an object, like your suitcase, is scanned, the image can be viewed from multiple angles by a security officer. Threat detection software can also be used to locate problematic items. There is math at work here too.

W. R. B. Lionheart, professor of Applied Mathematics at the University of Manchester, explains this technology in his entry “Airport Baggage Screening with X-Ray Tomography.”

While Researching Your Next Vacation

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Getting ready for your first vacation of the fall? Buying tickets, making dinner reservations, researching tourist attractions — what did we do without the internet? Or rather, what did we do before the organized internet?

When the internet was still in its early stages, search engines were not as advanced as they are today, and webpage results were ranked by simple rules. Searching for “New York sightseeing” may have led you to the page where the search term appears the most, instead of a page with the most useful information. Today, search engines use a more advanced method for ranking web pages: grouping pages into authority pages, which have many links to them, and hub pages, which point to many authorities. The catch is that these terms depend on one another. How does this work? In the Princeton Companion to Applied Mathematics, editor Nicholas Higham explains the mathematics behind webpage ranking.

Looking for more examples of math in the world? Check out this video from SIAM, where SIAM conference attendees are asked how they use math in their work. Math really is all around us.

Tipping Point Math Tuesdays With Marc Chamberland: What’s the Best Paper Size?

Tipping Point Tuesday takes on a global debate!

The United States and Canada use paper that is 8.5 inches by 11 inches, called US letter. However, the rest of the world officially uses A4 paper, which has a different aspect ratio. Which paper size is better, US letter or A4? Find the mathematical answer with the help of Marc Chamberland in a video from his YouTube channel Tipping Point Math.

 

Marc Chamberland takes on more mathematical scenarios in his book Single Digits: In Praise of Small Numbers. Read the first chapter here.

Tipping Point Math Tuesdays with Marc Chamberland: What’s the Pizza Theorem?

For the inaugural Tipping Point Math Tuesday, let’s find the math in something everyone loves, Pizza!

Have you ever opened your takeout pizza box and found that the slices were not cut evenly? What’s a party host to do? Marc Chamberland, author of Single Digits: In Praise of Small Numbers shows how mathematics can help you use the pizza theorem to evenly divide a pizza among yourself and your hungry friends:

 

Craving more math? Preview the first chapter of Chamberland’s book here.

Mathematics Awareness Month 2015: Math Drives Careers

Internet search, pharmaceuticals, insurance, finance, national security, medicine, ecology. What is the link between these diverse career fields? Students graduating with a mathematical sciences degree can find a professional future in all of these fields, and a wide range of others as well. This year’s Mathematics Awareness Month takes a step out of the classroom to show just where mathematics can lead after graduation.

Mathematics Awareness Month is an annual celebration dedicated to increasing public understanding of and appreciation for mathematics. The event, which started in 1986 as Mathematics Awareness Week, adopts a different theme each year. This year’s theme is “Math Drives Careers,” and PUP is excited to bring you a series of guest posts from our authors. Check back all this month for posts about using math to raise revenues, to understand sports and economics, and to solve complex problems.

The organizers of Mathematics Awareness Month explain the importance of mathematics in today’s workforce:

“Innovation is an increasingly important factor in the growth of world economies. It is especially important in key economic sectors like manufacturing, materials, energy, biotechnology, healthcare, networks, and professional and business services. The advances in and applications of the mathematical sciences have become drivers of innovation as new systems and methodologies have become more complex. As mathematics drives innovation, it also drives careers.”
Check out this official Mathematics Awareness Month poster, which includes career descriptions for 10 individuals who used their love for math to find rewarding careers:

 

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Follow along with @MathAware and take a look at Math Awareness Month on Facebook.

Tim Chartier and the Mega Menger

math bytesHow many business cards are needed to complete a level 3 Menger Sponge? What is a level 3 Menger Sponge? Tim Chartier, author of Math Bytes: Google Bombs, Chocolate Covered Pi, and Other Cool Bits in Computing, explains.

In a Huffington Post article, A Million Business Cards Presents a Math Challenge, Chartier asks readers to go in their wallet and check for  business cards. If there aren’t any, “they may be part of a worldwide math challenge. Over the past month, people around the world have been building a mathematical structure out of more than a million business cards.” That mathematical structure is a Mega Menger, but before we get there, let’s discuss what a level 1 Menger Sponge is.

A level 1 Menger Sponge is a fractal consisting of twenty cubes. Each cube is made up of 6 business cards, so a level 1 Menger needs 120 business cards. A level 2 Menger is created using 20 level 1 Mengers, a level 3 Menger is made with 20 level 2 Mengers, so on and so on.

Here at the Press, we don’t have a million business cards to complete a Mega Menger, but we do have a lot of books. Using Chartier’s code (which you can play with here: http://lifeislinear.davidson.edu/sierpinski.html), we were able to create 2-D versions of a Menger Sponge called a Sierpinski Carpet using our jacket images. Below, see if you can figure out which books are featured in the fractal images below! Click the images to see the related title.

1.
zombies

2.
penguins

3.
amazons

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weeden

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Invisible in the Storm wins the 2015 Louis J. Battan Author’s Award, American Meteorological Society

Congratulations to Ian Roulstone & John Norbury, co-authors of Invisible in the Storm: The Role of Mathematics in Understanding Weather, on winning the 2015 Louis J. Battan Author’s Award given by the American Meteorological Society.

The prize is “presented to the author(s) of an outstanding, newly published book on the atmospheric and related sciences of a technical or non-technical nature, with consideration to those books that foster public understanding of meteorology in adult audiences.” In the announcement of the prize, the committee said Invisible in the Storm “illuminates the mathematical foundation of weather prediction with lucid prose that provides a bridge between meteorologists and the public.”

For more information about the 2015 AMS awards: http://www.ametsoc.org/awards/2015awardrecipients.pdf


bookjacket

Invisible in the Storm
The Role of Mathematics in Understanding Weather
Ian Roulstone & John Norbury

This is how you survive the zombie apocalypse

Williams College math professor Colin Adams risks life and limb to record these survival guide videos. Ready your gear–armor, baseball bat, calculus textbook–and prepare for the onslaught.

Part 1: Why we can’t quite finish the zombies off.

Part 2: Escaping zombies in hot pursuit.

Credit: PBS’s NOVA and director Ari Daniel.
 


bookjacket Zombies and Calculus
Colin Adams