Q&A with Frank Farris, Author of Creating Symmetry: The Artful Mathematics of Wallpaper Patterns

Frank A. Farris teaches mathematics at Santa Clara University and is a former editor of Mathematics Magazine, a publication of the Mathematical Association of America. He is also the author of the new Princeton University Press book Creating Symmetry: The Artful Mathematics of Wallpaper Patterns. The book provides a hands-on, step-by-step introduction to the intriguing mathematics of symmetry.

Frank Farris gave Princeton University Press a look at why he wrote Creating Symmetry, where he feels this book will have major contributions, and what comes next.

Before and After: A Peach and a Sierra Stream Become a Pattern, by Frank A Farris

Before and After: A Peach and a Sierra Stream Become a Pattern, by Frank A Farris

What inspired you to get into mathematical writing?
FF: After editing Mathematics Magazine for many years, I developed a passion for communicating mathematics: I didn’t want dry accounts written by anonymous authors; I wanted stories told by people. I wasn’t so interested in problems and puzzles, but in the stories that bring us face to face with the grand structures of mathematics.

Why did you write this book?
FF: Many years ago, I asked the innocent question: What is a wallpaper pattern, really? Creating Symmetry is the story of my dissatisfaction with standard answers and how it led me on a curious journey to a new kind of mathematical art. I took some risks and let my personality show through, while maintaining an honest, mathematically responsible approach. I hope readers enjoy the balance: real math told by a person.

What do you think is the book’s most important contribution?
FF: Most people who see my artwork say they’ve never seen anything like these images and that pleases me immensely. Of course, people have seen wallpaper patterns before, but the unusual construction method I use—wallpaper waves and photographs—gives my patterns an intricacy and rhythm that people wouldn’t create through the usual potato-stamp construction method, where the patterns is made from discrete blocks.

What is your next project?
FF: I am working on a “wallpaper lookbook,” a book for the simple joy of looking at patterns. Creating Symmetry tells people how to make the patterns, and there’s quite a lot of mathematical detail to process. Not everyone who likes my work wants to know all the details, but can still appreciate the “before and after” nature of the images.

Who do you see as the audience for this book?
FF: There are three audiences and they will read the book in different ways. The general reader, who knows some calculus but may be a little rusty, should find a refreshing and challenging way to reconnect with mathematics. Undergraduate mathematics majors will enjoy the book as a summer project or enrichment reading, as it makes surprising connections among topics they may have studied. The professional mathematician will find this light reading—a chance to enjoy the amazing interconnectedness of our field.


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: How many guards are enough?

Today’s Tipping Point Tuesday gives us a behind the scenes look at how mathematics can be used in unique ways in the workplace.

Here’s the scenario: In busy museums, guards keep an eye on the priceless works of art. Suppose a museum wants to schedule the fewest number of guards per museum shift without leaving any art display unmonitored. Marc Chamberland explains how a museum manager could use mathematics to calculate the ideal number of guards per shift.

Continue exploring numbers with Chamberland in his book, Single Digits: In Praise of Small Numbers. Start by reading the first chapter here.

Q&A with Marc Chamberland, author of Single Digits: In Praise of Small Numbers

Marc Chamberland is the Myra Steele Professor of Natural Science and Mathematics at Grinnell College. He is also the creator of the popular YouTube channel Tipping Point Math, which strives to make mathematics accessible to everyone. Continuing on his mathematics mission, Marc Chamberland has authored Single Digits: In Praise of Small Numbers, a book that looks at the vast numerical possibilities that can come from the single digits. j10437Over the course of the coming weeks, we will be exploring the single digits in real life math situations with the author himself by featuring a series of original videos from Tipping Point Math.

Recently Chamberland gave the press a look at the inspiration behind the book, along with some personal insights on being a mathematician, and more:

What was the motivation behind your Tipping Point Math website?

MC: I have long felt that many people are sour on math because they think it is all technical stuff that leads to nowhere. I felt that if they could be exposed to the rich ideas and beauty of mathematics presented in an interesting way, their negative opinion could change.

I had wondered for a while how YouTube could be used since it is such a popular medium. In 2013, I reconnected with Henry Reich, a former student of mine, who created the highly successful channels MinutePhysics and MinuteEarth. With his inspiration and advice, I was convinced that a similar channel for mathematics was possible. Thus the concept of Tipping Point Math was born.

What is the biggest misunderstanding people have about your mathematics profession?

MC: Besides my remarks about people thinking that math is only about technical stuff, there is also the misconception that all of mathematics is known. This is not the case at all. New mathematics is being developed every day. This ranges from very abstract ideas to applications such as signal processing, medical imaging, population modeling, and computer algorithms.

What would you have been if not a mathematician?

MC: In my last year of high school, I developed an unquenchable thirst to explore two academic areas: mathematics and music. Since I eventually became a mathematics professor, I suppose one could say that mathematics “won”. But music was also consuming. I would ask myself, “Why does that piece of music sound so good? Why does it produce particular emotional states? How can I compose music that affects people in different ways?” To this day I still ask some of these questions, I occasionally compose short pieces, and I play the piano, guitar, and sing. Would I have been a musician? Is it too late to change?

What are you reading right now?

MC: I’m reading “The Alchemist” (by Paulo Coelho) out loud to my wife. The simple language and overflowing spirituality is stunning.

Who do you see as the audience for your book, Single Digits?

MC: My audience: those who love beauty. I did not choose topics for their depth or their technical superiority. I principally chose vignettes that I thought are beautiful.

In Memory of John and Alicia Nash

NashGradThe staff and community of Princeton University Press mourns the tragic loss of John and Alicia Nash. In 2001 we had the great privilege of publishing The Essential John Nash, a collection of Professor Nash’s scholarly articles edited by his biographer, Sylvia Nasar, and his longtime colleague and friend, Princeton mathematician Harold Kuhn, (now deceased). The Essential John Nash received impressive public exposure largely because it was published during the release of the Academy Award-winning movie version of Nash’s biography, A Beautiful Mind. Critics and readers admired The Essential John Nash as a faithful representation of Nash’s most important work, made available for a broadly intellectual audience of mathematicians and social scientists. Gratifying as this recognition was for us, during the course of publication, the staff members at PUP who worked on Professor Nash’s book had the great good fortune to get to know him and Alicia, two gentle and wonderful people. Our thoughts and prayers are with their family.

Peter J. Dougherty

Book Fact Friday – #8 Single Digits

From chapter eight of Marc Chamberland’s Single Digits:

How many times should you shuffle a deck of cards so that they’re well-mixed? Gamblers know that three or four times is not sufficient and take advantage of this fact. In 1992, researchers did computer simulations and estimated that seven rough riffle shuffles is a good amount. They took their research further and figured out that further shuffling does not significantly improve the mixing. If the shuffler does a perfect riffle shuffle (a Faro shuffle), in which s/he perfectly cuts the deck and shuffles so that each card from one side alternates with each card from the other side, then a standard 52-card deck will end in the same order that it started in after it is done 8 times.

Single Digits: In Praise of Small Numbers by Marc Chamberland
Read chapter one or peruse the table of contents.

The numbers one through nine have remarkable mathematical properties and characteristics. For instance, why do eight perfect card shuffles leave a standard deck of cards unchanged? Are there really “six degrees of separation” between all pairs of people? And how can any map need only four colors to ensure that no regions of the same color touch? In Single Digits, Marc Chamberland takes readers on a fascinating exploration of small numbers, from one to nine, looking at their history, applications, and connections to various areas of mathematics, including number theory, geometry, chaos theory, numerical analysis, and mathematical physics.
Each chapter focuses on a single digit, beginning with easy concepts that become more advanced as the chapter progresses. Chamberland covers vast numerical territory, such as illustrating the ways that the number three connects to chaos theory, an unsolved problem involving Egyptian fractions, the number of guards needed to protect an art gallery, and problematic election results. He considers the role of the number seven in matrix multiplication, the Transylvania lottery, synchronizing signals, and hearing the shape of a drum. Throughout, he introduces readers to an array of puzzles, such as perfect squares, the four hats problem, Strassen multiplication, Catalan’s conjecture, and so much more. The book’s short sections can be read independently and digested in bite-sized chunks—especially good for learning about the Ham Sandwich Theorem and the Pizza Theorem.
Appealing to high school and college students, professional mathematicians, and those mesmerized by patterns, this book shows that single digits offer a plethora of possibilities that readers can count on.

#MammothMonday: PUP’s pups sound off on How to Clone a Mammoth

The idea of cloning a mammoth, the science of which is explored in evolutionary biologist and “ancient DNA expert” Beth Shapiro’s new book, How to Clone a Mammoth, is the subject of considerable debate. One can only imagine what the animal kingdom would think of such an undertaking, but wonder no more. PUP staffers were feeling “punny” enough to ask their best friends:


Chester reads shapiro

Chester can’t get past “ice age bones”.


Buddy reads shapiro

Buddy thinks passenger pigeons would be so much more civilized… and fun to chase.


Tux reads shapiro

Tux always wanted to be an evolutionary biologist…


Stella reads Shapiro

Stella thinks 240 pages on a glorified elephant is a little excessive. Take her for a walk.


Murphy reads shapiro

A mammoth weighs how much?! Don’t worry, Murphy. The tundra is a long way from New Jersey.


Glad we got that out of our systems. Check out a series of original videos on cloning from How to Clone a Mammoth author Beth Shapiro here.

Win a copy of Relativity: 100th Anniversary Edition by Albert Einstein through Corbis!

We are teaming with Corbis Entertainment to offer this terrific giveaway through their official Albert Einstein Facebook page. Contest details below, but please head over to the “official Facebook page of the world’s favorite genius” to enter!

Enter for a chance to win a FREE COPY of “Relativity: 100th Anniversary Edition” by Albert Einstein!

Math Drives Careers: Paul Nahin on Electrical Engineering and √-1

Paul Nahin is the author of many books we’ve proudly published over the years, including An Imaginary Tale, Dr. Euler’s Fabulous Formula, and Number Crunching. For today’s installment in our Math Awareness Month series, he writes about his first encounter with √-1.

Electrical Engineering and √-1

It won’t come as a surprise to very many to learn that mathematics is central to electrical engineering. Probably more surprising is that the cornerstone of that mathematical foundation is the mysterious (some even think mystical) square-root of minus one. Every electrical engineer almost surely has a story to tell about their first encounter with √-1, and in this essay I’ll tell you mine.

Lots of different kinds of mathematics have been important in my personal career at different times; in particular, Boolean algebra (when I worked as a digital logic designer), and probability theory (when I wore the label of radar system engineer). But it’s the mathematics of √-1 that has been the most important. My introduction to √-1 came when I was still in high school. In my freshman year (1954) my father gave me the gift of a subscription to a new magazine called Popular Electronics. From it I learned how to read electrical schematics from the projects that appeared in each issue, but my most important lesson came when I opened the April 1955 issue.

It had an article in it about something called contra-polar power: a desk lamp plugged into a contra-polar outlet plug would emit not a cone of light, but a cone of darkness! There was even a photograph of this, and my eyes bugged-out when I saw that: What wondrous science was at work here?, I gasped to myself —I really was a naive 14-year old kid! It was, of course, all a huge editorial joke, along with some nifty photo-retouching, but the lead sentence had me hooked: “One of the reasons why atomic energy has not yet become popular among home experimenters is that an understanding of its production requires knowledge of very advanced mathematics.” Just algebra, however, was all that was required to understand contra-polar power.

contra power scan

Contra-polar power ‘worked’ by simply using the negative square root (instead of the positive root) in calculating the resonant frequency in a circuit containing both inductance and capacitance. The idea of negative frequency was intriguing to me (and electrical engineers have actually made sense of it when combined with √-1, but then the editors played a few more clever math tricks and came up with negative resistance. Now, there really is such a thing as negative resistance, and it has long been known by electrical engineers to occur in the operation of electric arcs. Such arcs were used, in the very early, pre-electronic days of radio, to build powerful AM transmitters that could broadcast music and human speech, and not just the on-off telegraph code signals that were all the Marconi transmitters could send. I eventually came to appreciate that the operation of AM/FM radio is impossible to understand, at a deep, theoretical level, without √-1.

When, in my high school algebra classes, I was introduced to complex numbers as the solutions to certain quadratic equations, I knew (unlike my mostly perplexed classmates) that they were not just part of a sterile intellectual game, but that √-1 was important to electrical engineers, and to their ability to construct truly amazing devices. That early, teenage fascination with mathematics in general, and √-1 in particular, was the start of my entire professional life. I wish my dad was still alive, so I could once again thank him for that long-ago subscription.

Math Drives Careers: Author Louis Gross

Gross jacketLouis Gross, distinguished professor in the departments of ecology, evolutionary biology, and mathematics at the University of Tennessee, is the author, along with Erin Bodine and Suzanne Lenhart, of Mathematics for the Life Sciences. For our third installment in the Math Awareness Month series, Gross writes on the role mathematics and rational consideration have played in his career, and in his relationship with his wife, a poet.

Math as a Career-builder and Relationship-broker

My wife is a poet. We approach most any issue with very different perspectives. In an art gallery, she sees a painting from an emotional level, while I focus on the methods the artist used to create the piece. As with any long-term relationship, after many years together we have learned to appreciate the other’s viewpoint and while I would never claim to be a poet, I have helped her on occasion to try out different phrasings of lines to bring out the music. In the reverse situation, the searching questions she asks me about the natural world (do deer really lose their antlers every year – isn’t this horribly wasteful?) force me to consider ways to explain complex scientific ideas in metaphor. As the way I approach science is heavily quantitative, with much of my formal education being in mathematics, this is particularly difficult without resorting to ways of thought that to me are second nature.

The challenges in explaining how quantitative approaches are critical to science, and that science advances in part through better and better ways to apply mathematics to the responses of systems we observe around us, arise throughout education, but are particularly difficult for those without a strong quantitative bent. An example may be helpful. One of the central approaches in science is building and using models – these can be physical ones such as model airplanes, they can be model systems such as an aquarium or they can be phrased in mathematics or computer code. The process of building models and the theories that ultimately arise from collections of models, is painstaking and iterative. Yet each of us build and apply models all the time. Think of the last time you entered a supermarket or a large store with multiple checkout-lines. How did you decide what line to choose? Was it based on how many customers were in each line, how many items they had to purchase, or whether they were paying with a check or credit card? Did you take account of your previous experience with that check-out clerk if you had it, or your experience with using self-checkout at that store? Was the criterion you used some aspect of ease of use, or how quickly you would get through the line? Or was it something else such as how cute the clerk was?

As the check-out line example illustrates, your decision about what is “best” for you depends on many factors, some of which might be quite personal. Yet somehow, store managers need to decide how many clerks are needed at each time and how to allocate their effort between check-out lines and their other possible responsibilities such as stocking shelves. Managers who are better able to meet the needs of customers, so they don’t get disgusted with long lines and decide not to return to that store, while restraining the costs of operation, will likely be rewarded. There is an entire field, heavily mathematical, that has been developed to better manage this situation. The jargon term is “queuing models” after the more typically British term for a waiting line. There is even a formal mathematical way of thinking about “bad luck” in this situation, e.g. choosing a line that results in a much longer time to be checked out than a different line would have.

While knowing that the math exists to help decide on optimal allocation of employee effort in a store will not help you in your decision, the approach of considering options, deciding upon your criteria and taking data (e.g. observations of the length of each line) to guide your decision is one that might serve you well independent of your career. This is one reason why many “self-help” methods involve making lists. Such lists assist you in deciding what factors (in math we call these variables) matter to you, how to weight the importance of each factor (we call these parameters in modeling) and what your objective might be (costs and benefits in an economic sense). This process of rational consideration of alternative options may assist you in many aspects of everyday life, including not just minor decisions of what check-out line to go into, but major ones such as what kind of car or home to purchase, what field to major in and even who to marry! While I can’t claim to have followed a formal mathematical approach in deciding on the latter, I have found it helpful throughout my marriage to use an informal approach to decision making. I encourage you to do so as well.

Check out Chapter 1 of Mathematics for the Life Sciences here.

Alan Turing’s handwritten notebook brings $1 million at auction

turing jacket

Alan Turing: The Enigma

Old journals can be fascinating no matter who they belong to, but imagine looking over the old notebook of the mathematician credited with breaking German codes during WWII.

The Associated Press and other venues reported that a handwritten notebook by British code-breaker Alan Turing, subject of the 2014 Oscar-winning film “The Imitation Game,” a movie based on our book, Alan Turing: The Enigma, brought more than $1 million at auction from an anonymous buyer on Monday. Originally given to Turing’s mathematician-friend Robin Gandy, the notebooks are thought to be the only ones of their kind, and contain Turing’s early attempts to chart a universal language, a precursor to computer code. (In an interesting personal wrinkle, Gandy had used the blank pages for notes on his dreams, noting that, “It seems a suitable disguise to write in between these notes of Alan’s on notation, but possibly a little sinister; a dead father figure, some of whose thoughts I most completely inherited.”)

Andrew Hodges, author of Alan Turing: The Enigma, commented that “the notebook sheds more light on how Turing ‘remained committed to free-thinking work in pure mathematics.'” To learn more about the life of Turing, check out the book here.

Math Drives Careers: Author Oscar Fernandez

We know that mathematics can solve problems in the classroom, but what can it do for your business? Oscar Fernandez, author of Everyday Calculus, takes a look at how knowledge of numbers can help your bottom line.

Why You Should Be Learning Math Even If You Don’t Need It for Your Job

I want to tell you a short story about epic triumph in the midst of adversity. Okay, I’m exaggerating a bit, but hear me out.

A couple of years ago, I approached Boston Scientific—an S&P 500 component—with a crazy idea: let me and a team of students from Wellesley College (a liberal arts college for women) and Babson College (a business school) do consulting work for you. It was a crazy idea because what could I—a mathematician who knew nothing about their business—and some students—who hadn’t even graduated yet—possibly offer the company? Plenty, it turns out, all thanks to our common expertise: mathematics.

Mathematics, often depicted in movies as something pocket-protector-carrying people with less than stellar social skills do, is actually quite ubiquitous. I’d even say that mathematicians are the unsung heroes of the world. Alright, that’s a bit of hyperbole. But think about it. Deep in the catacombs of just about every company, there are mathematicians. They work in low light conditions, hunched over pages of calculations stained with days-old coffee, and think up ways to save the company money, optimize their revenue streams, and make their products more desired. You may never notice their efforts, but you’ll surely notice their effects. That recent change in the cost of your flight? Yep, it was one of us trying to maximize revenue. The reason that UPS truck is now waking you up at 6 a.m.? One of us figured out that the minimum cost route passes through your street.

But we’re do-good people too. We help optimize bus routes to get children to school faster and safer. We’ve spent centuries modelling the spread of disease. More recently, we’ve even reduced crime by understanding how it spreads. That’s why I was confident that my team and I could do something useful for Boston Scientific. Simply put, we knew math.

We spent several weeks pouring over data the company gave us. We tried everything we could think of to raise their revenues from certain products. Collectively, we were trained in mathematics, economics, computer science, and psychology. But nothing worked. It seemed that we—and math—had failed.

Then, with about three weeks left, I chanced upon an article from the MIT Technology Review titled “Turning Math Into Cash.” It describes how IBM’s 200 mathematicians reconfigured their 40,000 salespeople over a period of two years and generated $1 billion in additional revenue. Wow. The mathematicians analyzed the company’s price-sales data using “high-quantile modeling” to predict the maximum amount each customer was willing to spend, and then compared that to the actual revenue generated by the sales teams. IBM then let these mathematicians shuffle around salespeople to help smaller teams reach the theoretical maximum budget of each customer. Genius, really.

I had never heard of quantile regression before, and neither had my students, but one thing math does well is to train you to make sense of things. So we did some digging. We ran across a common example of quantile modelling: food expenditure vs. household income. There’s clearly a relationship, and in 1857 researchers quantified the relationship for Belgian households. They produced this graph:

fernandez 1

That red line is the linear regression line—the “best fit to the data.” It’s useful because the slope of the line predicts a 50 cent increase in food expenditure for a $1 increase in household income. But what if you want information about the food expenditure of the top 5% of households, or the bottom 20%? Linear regression can’t give you that information, but quantile regression can. Here’s what you get with quantile regression:

fernandez 2

The red line is the linear regression line, but now we also have various quantile regression lines. To understand what they mean let’s focus on the top-most dashed line, which is the 95th percentile line. Households above this line are in the 95th percentile (or 0.95 quantile) of food expenditure. Similarly, households below the bottom-most line are in the 5th percentile (or 0.05 quantile) of food expenditure. Now, if we graph the slopes of the lines as a function of the percentile (also called “quantile”), we get:

fernandez 3

(The red line is the slope of the linear regression line; it doesn’t depend on the quantile, which is why it’s a straight line.) Notice that the 0.95 quantile (95th percentile) slope is about 0.7, whereas the 0.05 quantile (5th percentile) slope is about 0.35. This means that for every $1 increase in household income, this analysis predicts that households in the 95th percentile of food expenditure will spend 70 cents more, whereas households in the 5th percentile will spend only 35 cents more.

Clearly quantile regression is powerful stuff. So, my team and I went back and used quantile regression on the Boston Scientific data. We came up with theoretical maximum prices that customers could pay based on the region the product was sold in. As with IBM, we identified lots of potential areas for improvement. When my students presented their findings to Boston Scientific, the company took the work seriously and was very impressed with what a few students and one professor could do. I can’t say we generated $1 billion in new revenue for Boston Scientific, but what I can say is that we were able to make serious, credible recommendations, all because we understood mathematics. (And we were just a team of 5 working over a period of 12 weeks!)

April is Mathematics Awareness Month, and this year’s theme is “math drives careers.” After my Boston Scientific experience and after reading about IBM’s success, I now have a greater appreciation of this theme. Not only can mathematics be found in just about any career, but if you happen to be the one to find it (and use it), you could quickly be on the fast track to success. So in between celebrating March Madness, Easter, Earth Day, and April 15th (I guess you’d only celebrate if you’re due a tax refund), make some time for math. It just might change your career.

Photo by Richard Howard.

Photo by Richard Howard.

Oscar Fernandez is the author of Everyday Calculus. He is assistant professor of mathematics at Wellesley College.