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.

Math Drives Careers: Author Ignacio Palacios-Huerta

Logical thinking, analytical skills, and the ability to recognize patterns are crucial in an array of fields that overlap with mathematics, including economics. But what does math (or economics, for that matter) have to do with the world’s most popular sport? Economist Ignacio Palacios-Huerta’s recent book, Beautiful Game Theory: How Soccer Can Help Economics  made a splash during the last World Cup, showing how universal economic principles can be understood through soccer. Read on for his thoughts on why the language of modern economics, including behavioral economics, is mathematics.

The Role of Mathematics in my Life as an Economist

To describe the role of mathematics in my life as an economist, I first need to explain what, to me, Economics is all about. So let me take you to one of my favorite books, A Treatise of Human Nature, written almost 300 years ago by David Hume.

Beautiful Game TheoryIn the introduction Hume writes, “‘Tis evident that all the sciences have a relation, more or less, to human nature … Even Mathematics, Natural Philosophy, and Natural Religion, are in some measure dependent on the science of Man, [which is] the only solid foundation for the other sciences”. By the science of man Hume means the understanding of all facets of human nature, including preferences, senses, passions, imagination, morality, justice, and society. This science applies wherever men are making decisions, be it running public institutions or countries, as employees in firms, or as individuals investing in education, taking risks in financial markets, or making family decisions. This science of man is thus what one may initially be tempted to call Economics for, as George Bernard Shaw puts it in my favorite definition, “Economy is the art of making the most of life”.

But of course this definition is incomplete because other social sciences (e.g., sociology, history, psychology, political science) are also concerned with human behavior. So what makes Economics “different”? Here is the difference: the difference is not the subject matter but the approach. The approach is totally different, and a very mathematical one. As such, mathematics plays a critical role in the life of any economist.

Let me elaborate. Continuing with Hume, it turns out that he also anticipated our methodological approach in modern Economics: observation and logical arguments. Which can be translated as: data and data analysis (what we call econometrics), and mathematics, for mathematics is, after all, the language of logic. So in Economics, as in physics, we write down our ideas and theories in mathematical terms to make logical arguments, and then we use more math (statistical, econometrics, etc) to check whether the data appear to be consistent with the theoretical arguments. If they are, the evidence can be said to support the theory; if they aren’t, the theory needs to be refined or discarded. Yes, lots of math and related techniques provide what is our distinct “economics approach to human behavior.” It is not the subject matter but the approach that is different, and it heavily relies on mathematics.

To economists and other social scientists, mathematics has many methodological virtues: it can lend precision to theories, can uncover inconsistencies, can generate hypothesis, can enable concision and promote intelligibility, and can sort out complex interactions, while statistical and econometric analysis can organize and carefully interpret voluminous data.

None of this is obvious when you begin studying Economics (“Why should I take all this math, statistics and econometrics? Why all this pain?”). But I think most of us soon learn to appreciate that the language of modern economics is mathematics, and that it is rightly so. And this is not math for the sake of math (as in pure mathematics), but math with a purpose: modeling human behavior.

Let me conclude by saying that since the economic approach is applicable to all human behavior, any type of data about human activity can be useful to evaluate economic theories. This includes, why not, sports data, which in many ways can be just perfect for testing economic theories: the data are abundant, the goals of the participants are clear, the outcomes are easy to observe, the stakes are high, and the subjects are professionals with experience. If a theory is “correct”, sport is a good setting to check it.

So just as data involving falling stones and apples were useful to Galileo Galilei and  Isaac Newton to test for the first time theories that were important in physics, data from sports can be useful in Economics to do exactly the same. As such in some of my contributions to Economics I have used math to develop theoretical models, and further mathematical tools applied to this type of data to test them.

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:



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

Pi Day Recipe: Apple Pie from Jim Henle’s The Proof and the Pudding

Tomorrow (March 14, 2015) is a very important Pi Day. This year’s local Princeton Pi Day Party and other global celebrations of Albert Einstein’s birthday look to be truly stellar, which is apt given this is arguably the closest we will get to 3.1415 in our lifetimes.

Leading up to the publication of the forthcoming The Proof and the Pudding: What Mathematicians, Cooks, and You Have in Common by Jim Henle, we’re celebrating the holiday with a recipe for a classic Apple Pie (an integral part of any Pi Day spread). Publicist Casey LaVela recreates and photographs the recipe below. Full text of the recipe follows. Happy Pi Day everyone!

Notes on Jim Henle’s Apple Pie recipe from Publicist Casey LaVela

The Proof and the Pudding includes several recipes for pies or tarts that would fit the bill for Pi Day, but the story behind Henle’s Apple Pie recipe is especially charming, the recipe itself is straightforward, and the results are delicious. At the author’s suggestion, I used a mixture of baking apples (and delightfully indulgent amounts of butter and sugar).


All of the crust ingredients (flour, butter, salt) ready to go:


After a few minutes of blending everything together with a pastry cutter, the crust begins to come together. A glorious marriage of flour and butter.


Once the butter and flour were better incorporated, I dribbled in the ice water and then turned the whole wonderful mess out between two sheets of plastic wrap in preparation for folding. The crust will look like it won’t come together, but somehow it always does in the end. Magical.


Now you need to roll out and fold over the dough a few times. This is an important step and makes for a light and flaky crust. (You use a similar process to make croissants or other viennoiserie from scratch.)


I cut the crust into two (for the top crust and bottom crust) using my handy bench scraper:



The apples cored, peeled, and ready to be cut into slices. I broke out my mandolin slicer (not pictured) to make more even slices, but if you don’t own a slicer or prefer to practice your knife skills you can just as easily use your favorite sharp knife.


Beautiful (even) apple slices:


Action shot of me mixing the apple slices, sugar, and cinnamon together. I prefer to prepare my apple pie filling in a bowl rather than sprinkling the dry ingredients over the apple slices once they have been arranged in the bottom crust. I’m not sure if it has much impact on the flavor and it is much, much messier, but I find it more fun.



The bottom crust in the pie plate:


Arrange the apple slices in the bottom crust:


Top with the second crust, seal the top crust to the bottom with your fingers, and (using your sharp knife) make incisions in the top crust to allow steam to escape:


The apple pie before going into the oven (don’t forget to put a little extra sugar on top):


The finished product:


There was a little crust left over after cutting, so I shaped it into another pi symbol, covered it in cinnamon and sugar, and baked it until golden brown. I ate the baked pi symbol as soon as it had cooled (before thinking to take a picture), but it was delicious!


Apple Pie

The story of why I started cooking is not inspiring. My motives weren’t pure. Indeed, they involved several important sins.

I really am a glutton. I love to eat. As a child, I ate well; my mother was a wonderful cook. But I always wanted more than I got, especially dessert. And of all desserts, it was apple pie I craved most. Not diner pies, not restaurant pies, and not bakery pies, but real, homemade apple pies.

When I was six, I had my first homemade apple pie. It was at my grandmother’s house. I don’t remember how it tasted, but I can still recall the gleam in my mother’s eye when she explained the secret of the pie. “I watched her make it. Before she put on the top crust, she dotted the whole thing with big pats of butter!”

Several times as I was growing up, my mother made apple pie. Each one was a gem. But they were too few—only three or four before I went off to college. They were amazing pies. The apples were tart and sweet. Fresh fall apples, so flavorful no cinnamon was needed. The crust was golden, light and crisp, dry when it first hit the tongue, then dissolving into butter.

I grew up. I got married. I started a family. All the while, I longed for that pie. Eventually I set out to make one.

Success came pretty quickly, and it’s not hard to see why. The fact is, despite apple pie’s storied place in American culture, most apple pies sold in this country are abysmal. A pie of fresh, tart apples and a crust homemade with butter or lard, no matter how badly it’s made, is guaranteed to surpass a commercial product.

That means that even if you’ve never made a pie before, you can’t go seriously wrong. The chief difficulty is the crust, but I’ve developed a reliable method. Except for this method, the recipe below is standard.

For the filling:
5 cooking apples (yielding about 5 cups of pieces)
1/4 to 1/3 cup sugar
2 Tb butter
1/2 to 1 tsp cinnamon
lemon juice, if necessary
1 tsp flour, maybe

For the crust:
2 cups flour
1 tsp salt
2/3 cup lard or unsalted butter (1 1/3 sticks)

The crust is crucial. I’ll discuss its preparation last. Assume for now that you’ve rolled out the bottom crust and placed it in the pie pan.

Core, peel, and slice the apples. Place them in the crust. Sprinkle with sugar and cinnamon. Dot with butter. Roll out the top crust and place it on top. Seal the edge however you like. In about six places, jab a knife into the crust and twist to leave a hole for steam to escape. Sprinkle the crust with the teaspoon of sugar.

Bake in a preheated oven for 15 minutes at 450° and then another 35 minutes at 350°. Allow to cool. Serve, if you like, with vanilla ice cream or a good aged cheddar.

Now, the crust:

Mix the flour and salt in a large bowl. Place the lard or butter or lard/butter in the bowl. Cut it in with a pastry cutter.

Next, the water. Turn the cold water on in the kitchen sink so that it dribbles out in a tiny trickle. Hold the bowl with the flour mixture in one hand and a knife in the other. Let the water dribble into the bowl while you stir with the knife. The object is to add just enough water so that the dough is transformed into small dusty lumps. Don’t be vigorous with the knife, but don’t allow the water to pool. If the water is dribbling too fast, take the bowl away from the faucet from time to time. When you’re done, the dough will still look pretty dry.

Recipes usually call for about 5 tablespoons of water. This method probably uses about that much.

Actually, the dough will look so dry that you’ll think it won’t stick together when it’s rolled out. In fact, it probably won’t stick together, but trust me. This is going to work.

Tear off a sheet of plastic wrap and lay it on the counter. Place a bit more than half the dough on the sheet and cover it with a second sheet of plastic.

With a rolling pin, roll the dough out between the two sheets. Roll it roughly in the shape of a rectangle.

It won’t look great and it probably would fall apart if you picked it up.

Don’t pick it up. Remove the top sheet of plastic wrap and fold the bottom third up, and fold the top third down, then do the same horizontally, right and left.

Now replace the top sheet of plastic wrap and roll the dough out gently into a disk.

This time it should look pretty decent. This time the dough will stick together.

You should be able to remove the top sheet of plastic and, using the bottom sheet, turn it over into the pie pan. The crust should settle in nicely without breaking.

Form the top crust the same way.

This method rolls each crust twice—usually not a good idea because working the dough makes it tough. But remarkably, crusts produced this way are tender and light. I’m not sure why but I suspect it’s because the dough is fairly dry.

• Cooking apples are tart apples. The best I know is the Rhode Island Greening, but they’re hard to find. Baldwins and Jonathans are decent, but they’re hard to find too. The British Bramleys are terrific. I’ve made good pies from the French Calville Blanc d’Hiver. But we’re not living in good apple times. Most stores don’t sell apples for cooking. When in doubt, use a mixture.
• The lemon juice and the larger quantity of cinnamon are for when you have tired apples with no oomph. The cheese also serves this purpose. It should be a respectable old cheddar and it should be at room temperature.
• Consumption of too many commercial pies makes me loath to add flour or cornstarch to pie filling. The flour is here in case you fear your apples will be too juicy. I don’t mind juice in a pie, in moderation. If adding flour, mix the apples, sugar, cinnamon, and flour in a bowl before pouring into the crust.
• Lard is best. Its melting point is higher than butter’s. It successfully separates the flour into layers for a light, crispy crust. Butter is more likely to saturate the flour and produce a heavy crust. Some like half butter/half lard, preferring butter for its flavor. But the flavor of lard is nice too, and its porkiness is wonderful with apple.

This recipe is taken from:


The Proof and the Pudding

What Mathematicians, Cooks, and You Have in Common

Jim Henle

“If you’re a fan of Julia Child or Martin Gardner—who respectively proved that anyone can have fun preparing fancy food and doing real mathematics—you’ll enjoy this playful yet passionate romp from Jim Henle. It’s stuffed with tasty treats and ingenious ideas for further explorations, both in the kitchen and with pencil and paper, and draws many thought-providing parallels between two fields not often considered in the same mouthful.”—Colm Mulcahy, author of Mathematical Card Magic: Fifty-Two New Effects

Quick Questions for Ian Roulstone and John Norbury, co-authors of Invisible in the Storm

Ian Roulstone (top) and John Norbury (bottom) are authors of Invisible in the Storm: The Role of Mathematics in Understanding Weather and experts on the application of mathematics in meteorology and weather prediction. As we head into hurricane season along the Eastern coast of the United States, we are still not fully recovered from Hurricane Sandy, empty lots still dot the stretch between Seaside and Point Pleasant and in countless other beach communities. But it could have been worse without the advance warning of meteorologists, so we had a few questions about the accuracy of weather prediction and how it can be further refined in the future.

Now, on to the questions!

Ian RoulstoneNorbury


What inspired you to get into this field?

Every day millions of clouds form, grow, and move above us, blown by the restless winds of our ever-changing atmosphere. Sometimes they bring rain and sometimes they bring snow – nearly always in an erratic, non-recurring way. Why should we ever be able to forecast weather three days or a week ahead? How can we possibly forecast climate ten years or more in the future? The secret behind successful forecasting involves a judicious mix of big weather-satellite data, information technology, and meteorology. What inspired us was that mathematics turns out to be crucial to bringing it all together.

Why did you write this book?

Many books describe various types of weather for a general audience. Other books describe the physical science of forecasting for more specialist audiences. But no-one has explained, for a general readership, the ideas behind the successful algorithms of the latest weather and climate apps running on today’s supercomputers. Our book describes the achievements and the challenges of modern weather and climate prediction.

There’s quite a lot about the history and personalities involved in the development of weather forecasting in your book; why did you consider this aspect important?

When reviewing the historical development of weather science over the past three centuries, we found the role of individuals ploughing their own furrow to be at least as important as that of big government organisations. And those pioneers ranged from essentially self-taught, and often very lonely individuals, to charming and successful prodigies. Is there a lesson here for future research organisation?

“We can use mathematics to warn us of the potential for chaotic behaviour, and this enables us to assess the risks of extreme events.”

Weather forecasts are pretty good for the next day or two, but not infallible: can we hope for significant improvements in forecasting over the next few years? 

The successful forecasts of weather events such as the landfall of Hurricane Sandy in New Jersey in October 2012, and the St Jude Day storm over southern England in October 2013, both giving nearly a week’s warning of the oncoming disaster, give a taste of what is possible. Bigger computers, more satellites and radar observations, and even cleverer algorithms will separate the predictable weather from the unpredictable gust or individual thunderstorm. Further improvements will rely not only on advanced technology, but also, as we explain in our book, on capturing the natural variability of weather using mathematics.

But isn’t weather chaotic?

Wind, warmth and rain are all part of weather. But the very winds are themselves tumbling weather about. This feedback of cause and effect, where the “effects help cause the causes”, has its origins in both the winds and the rain. Clouds are carried by the wind, and rainfall condensing in clouds releases further heat, which changes the wind. So chaotic feedback can result in unexpected consequences, such as the ice-storm or cloudburst that wasn’t mentioned in the forecast. But we can use mathematics to warn us of the potential for chaotic behaviour, and this enables us to assess the risks of extreme events.

Are weather and climate predictions essentially “big data” problems?

We argue no. Weather agencies will continually upgrade their supercomputers, and have a never-ending thirst for weather data, mostly from satellites observing the land and sea. But if all we do is train computer programs by using data, then our forecasting will remain primitive. Scientific ideas formulated with mathematical insight give the edge to intelligent forecasting apps.

So computer prediction relies in various ways on clever mathematics: it gives a language to describe the problem on a machine; it extracts the predictable essence from the weather data; and it selects the predictable future from the surrounding cloud of random uncertainty. This latter point will come to dominate climate prediction, as we untangle the complex interactions of the atmosphere, oceans, ice-caps and life in its many varied forms.

Can climate models produce reliable scenarios for decision-makers?

The models currently used to predict climate change have proved invaluable in attributing trends in global warming to human activity. The physical principles that govern average global temperatures involve the conservation of energy, and these over-arching principles are represented very accurately by the numerical models. But we have to be sure how to validate the predictions: running a model does not, in itself, equate to understanding.

As we explain, although climate prediction is hugely complicated, mathematics helps us separate the predictable phenomena from the unpredictable. Discriminating between the two is important, and it is frequently overlooked when debating the reliability of climate models. Only when we take such factors into account can we – and that includes elected officials – gauge the risks we face from climate change.

What do you hope people will take away from this book?

From government policy and corporate strategy to personal lifestyle choices, we all need to understand the rational basis of weather and climate prediction. Answers to many urgent and pressing environmental questions are far from clear-cut. Predicting the future of our environment is a hugely challenging problem that will not be solved by number-crunching alone. Chaos and the butterfly effect were the buzzwords of the closing decades of the 20th Century. But incomplete and inaccurate data need not be insurmountable obstacles to scientific progress, and mathematics shows us the way forward.


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



Paying It Forward, Using Math: Oscar Fernandez’s ‘Everyday Calculus’ Donated to Libraries in Franklin County, PA

Everyday Calculus, O. FernandezWhat a week!

It was recently announced that one of our books, Everyday Calculus by Oscar Fernandez, is to be donated by the United Way of Franklin County, in partnership with the Franklin County Library System, to public libraries all throughout Franklin County. The decision recognizes the 2013 Campaign Chair, Jim Zeger, who has demonstrated a dedication to service and a “willingness to teach others” during the course of his four-year tenure on the board of directors.

But the choice of text was far from random; Everyday Calculus was selected “because of the need for materials that support financial and mathematical literacy within our library systems,” says Mr. Zeger. He’s one to know; before coming to United Way, Zeger studied math at Juniata College and taught mathematics at the Maryland Correctional Institute. He also served for a number of years as part of the Tuscarora School District school board, and “is very supportive and understanding of the value of relating and connecting applied math to students.”

Bernice Crouse, executive director of the Franklin County Library System, accepted the books and has found them a place in each County library, including the bookmobile, in order to make them more accessible to readers. According to Crouse, this book fits perfectly with Pennsylvania Library Association’s PA Forward initiative, which “highlights Financial Literacy as a key to economic vitality in Pennsylvania.”

Mr. Fernandez is reportedly “delighted” and “honored” by the decision, and looks forward to further collaborating with United Way.

Concepts in Color: Beautiful Geometry by Eli Maor and Eugen Jost

If you’ve ever thought that mathematics and art don’t mix, this stunning visual history of geometry will change your mind. As much a work of art as a book about mathematics, Beautiful Geometry presents more than sixty exquisite color plates illustrating a wide range of geometric patterns and theorems, accompanied by brief accounts of the fascinating history and people behind each.

With artwork by Swiss artist Eugen Jost and text by acclaimed math historian Eli Maor, this unique celebration of geometry covers numerous subjects, from straightedge-and-compass constructions to intriguing configurations involving infinity. The result is a delightful and informative illustrated tour through the 2,500-year-old history of one of the most important and beautiful branches of mathematics.

We’ve created this slideshow so that you can sample some of the beautiful images in this book, so please enjoy!

Plate 00
Plate 4
Plate 6
Plate 7
Plate 10
Plate 15.1
Plate 16
Plate 17
Plate 18
Plate 19
Plate 20
Plate 21
Plate 22
Plate 23
Plate 24.2
Plate 26.2
Plate 29.1
Plate 29.2
Plate 30
Plate 33
Plate 34.1
Plate 36
Plate 37
Plate 38
Plate 39
Plate 40.2
Plate 44
Plate 45
Plate 47
Plate 48
Plate 49
Plate 50
Plate 51

Beautiful Geometry by Eli Maior and Eugen Jost

"My artistic life revolves around patterns, numbers, and forms. I love to play with them, interpret them, and metamorphose them in endless variations." --Eugen Jost

Figurative Numbers

Plate 4, Figurative Numbers, is a playful meditation on ways of arranging 49 dots in different patterns of color and shape. Some of these arrangements hint at the number relations we mentioned previously, while others are artistic expressions of what a keen eye can discover in an assembly of dots. Note, in particular, the second panel in the top row: it illustrates the fact that the sum of eight identical triangular numbers, plus 1, is always a perfect square.

Pythagorean Metamorphosis

Pythagorean Metamorphosis shows a series of right triangles (in white) whose proportions change from one frame to the next, starting with the extreme case where one side has zero length and then going through several phases until the other side diminishes to zero.

The (3, 4, 5) Triangle and its Four Circles

The (3, 4, 5) Triangle and its Four Circles shows the (3, 4, 5) triangle (in red) with its incircle and three excircles (in blue), for which r = (3+4-5)/2 = 1, r = (5+3-4)/2 = 2, rb = (5+4-3)/2 = 3, and rc = (5+4+3)/2 = 6.

Mean Constructions

Mean Constructions (no pun intended!), is a color-coded guide showing how to construct all three means from two line segments of given lengths (shown in red and blue). The arithmetic, geometric, and harmonic means are colored in green, yellow, and purple, respectively, while all auxiliary elements are in white.

Prime and Prime Again

Plate 15.1, Prime and Prime Again, shows a curious number sequence: start with the top eight-digit number and keep peeling off the last digits one by one, until only 7 is left. For no apparent reason, each number in this sequence is a prime.

0.999... = 1

Celtic Motif 1

Our illustration (Plate 17) shows an intriguing lace pattern winding its way around 11 dots arranged in three rows; it is based on an old Celtic motif.

Seven Circles a Flower Maketh


Plate 19, Parquet, seems at first to show a stack of identical cubes, arranged so that each layer is offset with respect to the one below it, forming the illusion of an infinite, three-dimensional staircase structure. But if you look carefully at the cubes, you will notice that each corner is the center of a regular hexagon.


Plate 20, Girasole, shows a series of squares, each of which, when adjoined to its predecessor, forms a rectangle. Starting with a black square of unit length, adjoin to it its white twin, and you get a 2x1 rectangle. Adjoin to it the green square, and you get a 3x2 rectangle. Continuing in this manner, you get rectangles whose dimensions are exactly the Fibonacci numbers. The word Girasole ("turning to the sun" in Italian) refers to the presence of these numbers in the spiral arrangement of the seeds of a sunflower - a truly remarkable example of mathematics at work in nature.

The Golden Ratio

Plate 21 showcases a sample of the many occurrences of the golden ratio in art and nature.

Pentagons and Pentagrams

Homage to Carl Friedrich Gauss

Gauss's achievement is immortalized in his German hometown of Brunswick, where a large statue of him is decorated with an ornamental 17-pointed star (Plate 23 is an artistic rendition of the actual star on the pedestal, which has deteriorated over the years); reportedly the mason in charge of the job thought that a 17-sided polygon would look too much like a circle, so he opted for the star instead.

Celtic Motif 2

Plate 24.2 shows a laced pattern of 50 dots, based on an ancient Celtic motif. Note that the entire array can be crisscrossed with a single interlacing thread; compare this with the similar pattern of 11 dots (Plate 17), where two separate threads were necessary to cover the entire array. As we said before, every number has its own personality.

Metamorphosis of a Circle

Plate 26.2, Metamorphosis of a Circle, shows four large panels. The panel on the upper left contains nine smaller frames, each with a square (in blue) and a circular disk (in red) centered on it. As the squares decrease in size, the circles expand, yet the sum of their areas remains constant. In the central frame, the square and circle have the same area, thus offering a computer-generated "solution" to the quadrature problem. In the panel on the lower right, the squares and circles reverse their roles, but the sum of their areas ins till constant. The entire sequence is thus a metamorphosis from square to circle and back.

Reflecting Parabola

Ellipses and Hyperbolas

When you throw two stones into a pond, each will create a disturbance that propagates outward from the point of impact in concentric circles. The two systems of circular waves eventually cross each other and form a pattern of ripples, alternating between crests and troughs. Because this interference pattern depends on the phase difference between the two oncoming waves, the ripples invariably form a system of confocal ellipses and hyperbolas, all sharing the same two foci. In this system, no two ellipses ever cross one another, nor do two hyperbolas, but every ellipse crosses every hyperbola at right angles. The two families form an orthogonal system of curves, as we see in plate 29.2.


Euler's e

Plate 33, Euler's e, gives the first 203 decimal places of this famous number - accurate enough for most practical applications, but still short of the exact value, which would require an infinite string of nonrepeating digits. In the margins there are several allusions to events that played a role in the history of e and the person most associated with it, Leonhard Euler: an owl ("Eule" in German); the Episcopal crosier on the flag of Euler's birthplace, the city of Basel; the latitude and longitude of Königsberg (now Kaliningrad in Russia), whose seven bridges inspired Euler to solve a famous problem that marked the birth of graph theory; and an assortment of formulas associated with e

Spira Mirabilis


Plate 36 shows a five-looped epicycloid (in blue) and a prolate epicycloid (in red) similar to Ptolemy's planetary epicycles. In fact, this latter curve closely resembles the path of Venus against the backdrop of the fixed stars, as seen from Earth. This is due to an 8-year cycle during which Earth, Venus, and the Sun will be aligned almost perfectly five times. Surprisingly, 8 Earth years also coincide with 13 Venusian years, locking the two planets in an 8:13 celestial resonance and giving Fibonacci aficionados one more reason to celebrate!

Nine Points and Ten Lines

Our illustration Nine Points and Ten Lines (plate 37) shows the point-by-point construction of Euler's line, beginning with the three points of defining the triangle (marked in blue). The circumference O, the centroid G, and the orthocenter H are marked in green, red, and orange, respectively, and the Euler line, in yellow. We call this a construction without words, where the points and lines speak for themselves.

Inverted Circles

Steiner's Prism

Plate 39 illustrates several Steiner chains, each comprising five circles that touch an outer circle (alternately colored in blue and orange) and an inner black circle. The central panel shows this chain in its inverted, symmetric "ball-bearing" configuration.

Line Design

Plate 40.2 shows a Star of David-like design made of 21 line parabolas.

Gothic Rose

Plate 44, Gothic Rose, shows a rosette, a common motif on stained glass windows like those one can find at numerous places of worship. The circle at the center illustrates a fourfold rotation and reflection symmetry, while five of the remaining circles exhibit threefold rotation symmetries with or without reflection (if you disregard the inner details in some of them). The circle in the 10-o'clock position has the twofold rotation symmetry of the yin-yang icon.


Pick's Theorem

Plate 47 shows a lattice polygon with 28 grid points (in red) and 185 interior points (in yellow). Pick's formula gives us the area of this polygon as A = 185 + 28/2 - 1 = 198 square units.

Morley's Theorem

Variations on a Snowflake Curve

Plate 49 is an artistic interpretation of Koch's curve, starting at the center with an equilateral triangle and a hexagram (Star of David) design but approaching the actual curve as we move toward the periphery.

Sierpinski's Triangle

The Rationals Are Countable!

In a way, [Cantor] accomplished the vision of William Blake's famous verse in Auguries of Innocence:

To see the world in a grain of sand,
And heaven in a wild flower.
Hold infinitely in the palm of your hand,
And eternity in an hour.

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Click here to sample selections from the book.

Place Your Bets: Tim Chartier Develops FIFA Foe Fun to Predict World Cup Outcomes

Tim ChartierTim Chartier, author of Math Bytes: Google Bombs, Chocolate-Covered Pi, and Other Cool Bits in Computing has turned some mathematical tricks to help better predict the outcome of this year’s World Cup in Brazil.

Along with the help of fellow Davidson professor Michael Mossinghoff and Whittier professor Mark Kozek, Chartier developed FIFA Foe Fun, a program that enables us ordinary, algorithmically untalented folk to generate a slew of possible match outcomes. The tool weighs factors like penalty shoot-outs and the number of years of matches considered, all with the click of a couple buttons. Chartier used a similar strategy in his March Mathness project, which allowed students and basketball fans alike to create mathematically-produced brackets – many of which were overwhelmingly successful in their predictions.

Although the system usually places the most highly considered teams, like Brazil, Germany, and Argentina at the top, the gadget is still worth a look. Tinker around a bit, and let us know in the comments section how your results pan out over the course of the competition.

In the meantime, check out the video below to hear Chartier briefly spell out the logic of the formula.

Happy calculating!