Welcome to the Universe microsite nominated for a Webby

We’re thrilled to announce that the microsite for Welcome to the Universe by Neil DeGrasse Tyson, Michael A. Strauss, and J. Richard Gott, designed by Eastern Standard, has been nominated for a Webby in the Best Use of Animation or Motion Graphics category. Be sure to check it out and vote for the best of the internet!

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Anurag Agrawal: Monarch overwintering

by Anurag Agrawal

The estimates of the monarch butterfly overwintering population were announced February 9th by WWF Mexico. The butterflies are so dense at their dozen or so mountain-top clustering sites that overwintering butterflies cannot be individually counted. Instead, the area of forest that is densely coated with butterflies (at about 5,000 butterflies per square meter looking up into the canopy) is estimated as a measure of monarch abundance. Butterflies arrive in Mexico around early November and stay until March.

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This winter season (2016-2017), there were approximately 2.9 hectares of forest occupied with dense monarchs (somewhere in the neighborhood of 300,000 million overwintering butterflies). This estimate is down 27% compared to last year. Nonetheless, the previous two years were a 600% increase over the all-time low recorded in the winter of 2013-2014.

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Where does this leave us? This year’s population was higher than predicted by many. The season started with a late spring storm that killed an estimated 5-10% of monarchs in March 2016, and many reported low numbers of adults last summer. Nonetheless, the lower numbers this season compared to last are within the range of year-to-year variation, and overall, the population seems to be relatively stable over the past decade. With these 24 years of data, there are various ways to plot and assess the trends. Below I have plotted the four year averages for six periods beginning in 1992. Any way you slice it, the trend has been negative, and the population is not nearly what it once was. Nonetheless, the downward trend seems to have lessened this last period. Is this the new norm? How dangerously low are these numbers? And what can we do to reverse the trend?

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AgrawalAnurag Agrawal is a professor in the Department of Ecology and Evolutionary Biology and the Department of Entomology at Cornell University. He is the author of Monarchs and Milkweed: A Migrating Butterfly, a Poisonous Plant, and Their Remarkable Story of Coevolution.

 

 

 

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Anurag Agrawal on Monarchs and Milkweed

AgrawalMonarch butterflies are one of nature’s most recognizable creatures, known for their bright colors and epic annual migration from the United States and Canada to Mexico. Yet there is much more to the monarch than its distinctive presence and mythic journeying. In Monarchs and Milkweed, Anurag Agrawal presents a vivid investigation into how the monarch butterfly has evolved closely alongside the milkweed—a toxic plant named for the sticky white substance emitted when its leaves are damaged—and how this inextricable and intimate relationship has been like an arms race over the millennia, a battle of exploitation and defense between two fascinating species. Check the PUP blog each Monday for new installments in our “Monarch Monday” blog series by Anurag Agrawal.

What makes monarchs and milkweeds so special?

AA: Monarchs and milkweed are remarkable creatures, they’re on a wild ride! From the monarch’s perspective, its only food as a caterpillar is the milkweed plant. This makes them highly specialized, highly evolved, and very picky eaters indeed. They’re actually not that unique among butterflies, but they are extreme. Milkweed does everything in its power to defend itself against being eaten by monarchs. They make sandpapery leaves, toxins that can stop a human heart, and a thick poisonous goo that can glue an insect’s mouth shut. Again, although milkweed is not unique among plants, it is extreme. In what is called a coevolutionary arms race, monarchs and milkweed have been continually evolving over the eons to keep up with each other. As such, they have a lot to teach us about the way nature works, the way plants and animals interact, and about the various paths that evolution can take different species. And this is all to say nothing of the monarch’s spectacular annual migration, often over 3,000 miles flown by individual butterflies, using the sun to navigate, and having stored away milkweed’s poisons to protect themselves from being eaten by birds. Monarchs and milkweeds are royal representatives of all interacting species.

Why did you write this book?

AA: After studying monarchs and milkweed myself for over 15 years, I felt like I had a lot I wanted to share, especially with non-scientists and nature lovers. Monarchs and milkweed are such fascinating organisms, and yet so much of their beautiful biology is not widely known. I also wrote the book because there are areas of my own knowledge about monarchs and milkweed that I wanted to immerse myself in, but that I had not yet done any research on. So as an author, getting to visit the overwintering sites in Mexico, to study the population decline of monarch butterflies, and to understand their mating rituals were all fascinating detours from my everyday research life at Cornell University. The book was incredibly fun to write, and getting to work with artists and historians made it all the more rich. I hope that anybody that has an appreciation for nature, an interest in science, or just a curiosity about the ecology of plants and butterflies will enjoy this book. Working on this project has surely altered the course of my own research, the classes I teach, and how I see the natural world.

Why have you highlighted some of the personalities of the scientists studying monarchs and milkweeds in this book?

AA: One of the most amazing things about monarchs and milkweeds is the scientists who have studied them. They were such remarkable characters, especially those pioneering studies back in the 1950s: tremendously creative, sometimes competitive, and with some of their discoveries worthy of a Nobel prize. Getting to know them, both from their discoveries and their personalities, and how they interacted, has enriched my appreciation for how science is done. It also highlights the meandering and sometimes serendipitous nature of discoveries. I wanted to share the thrill of science, its ups and downs, and the process by which it is done with the curious reader.

Can you share one of your ah-ha! moments from studying monarchs and milkweeds?

AA: One of my favorites was from when I was an assistant professor at the University of Toronto. One day I was eating lunch by myself in a small downtown garden. Just by chance, I happened to sit on a bench beneath a very tall milkweed plant that had a very large monarch caterpillar feeding away. Without giving away all the details, that one hour encounter, in the middle of a city with 3 million people, changed my perspective on monarchs and milkweed forever. It was so unlikely an event, perhaps 1 in a 1,000 that a butterfly had been flying by and happened to lay an egg on this Toronto milkweed, and then a further 1 in 100 chance of that egg hatching and surviving to be that large caterpillar that I could watch it. And probably a 1 in a million event that I would happen to be eating lunch there, that day, to observe the events. In biology one has to work hard, be patient, and occasionally get very lucky! Throughout my studies on monarchs and milkweed, I have had tremendous luck in encountering wonderful biology that has had profound consequences.

Is the monarch butterfly going extinct?

AA: The answer to this very important and timely question is both simple and complex. On the simple side, there is no way the monarch butterfly is going extinct anytime soon. Having said that, the butterfly, and especially the long-distance migration that occurs every fall from Southern Canada and the USA, all the way to Mexico’s highlands in Michoacán, is indeed declining at a rapid pace, and we should all be worried about the sustainability of the annual migration. There’s so much information and misinformation floating around in the news these days about the causes of the monarchs decline. What I’ve tried to do in the book is outline the best knowledge that we have to date and to examine the facts critically, so we can really understand what might be going on. Unfortunately, we don’t have all the answers, but we can reject some of the most prominent explanations for the population decline of the monarch butterfly. As I argue in the book, planting milkweed certainly won’t hurt, but it is unlikely to save the monarchs annual migratory cycle. It is perhaps ironic that I spend eight chapters of the book discussing and detailing the importance of milkweed for monarchs, and nothing could be more true than their intertwined and intense evolutionary battle, but at this stage, and thinking about their conservation, it does not appear that milkweed is what is limiting the monarch’s population. Monarchs will persist for a very long time, but given that they are migratory butterflies that taste their way across North America, their declining population is something we must try to understand. Much more than the monarch is at stake, these butterflies are sentinels for the health of our continent!

Anurag Agrawal is a professor in the Department of Ecology and Evolutionary Biology and the Department of Entomology at Cornell University. He lives in Ithaca, New York. He is the author of Monarchs and Milkweed: A Migrating Butterfly, a Poisonous Plant, and Their Remarkable Story of Coevolution.

Keith Devlin: Fibonacci introduced modern arithmetic —then disappeared

More than a decade ago, Keith Devlin, a math expositor, set out to research the life and legacy of the medieval mathematician Leonardo of Pisa, popularly known as Fibonacci, whose book Liber abbaci has quite literally affected the lives of everyone alive today. Although he is most famous for the Fibonacci numbers—which, it so happens, he didn’t invent—Fibonacci’s greatest contribution was as an expositor of mathematical ideas at a level ordinary people could understand. In 1202, Liber abbaci—the “Book of Calculation”—introduced modern arithmetic to the Western world. Yet Fibonacci was long forgotten after his death. Finding Fibonacci is a compelling firsthand account of his ten-year quest to tell Fibonacci’s story. Devlin recently answered some questions about his new book for the PUP blog:

You’ve written 33 math books, including many for general readers. What is different about this one?

KD: This is my third book about the history of mathematics, which already makes it different from most of my books where the focus was on abstract concepts and ideas, not how they were discovered. What makes it truly unique is that it’s the first book I have written that I have been in! It is a first-person account, based on a diary I kept during a research project spread over a decade.

If you had to convey the book’s flavor in a few sentences, what would you say?

KD: Finding Fibonacci is a first-person account of a ten-year quest to uncover and tell the story of one of the most influential figures in human history. It started out as a diary, a simple record of events. It turned into a story when it became clear that it was far more than a record of dates, sources consulted, places visited, and facts checked. Like any good story, it has false starts and disappointments, tragedies and unexpected turns, more than a few hilarious episodes, and several lucky breaks. Along the way, I encountered some amazing individuals who, each for their own reasons, became fascinated by Fibonacci: a Yale professor who traced modern finance back to Fibonacci, an Italian historian who made the crucial archival discovery that brought together all the threads of Fibonacci’s astonishing story, an American math professor who fought against cancer to complete the world’s first (and only) modern language translation of Liber abbaci, and the widow who took over and brought his efforts to fruition after he lost that battle. And behind it all, the man who was the focus of my quest. Fibonacci played a major role in creating the modern commercial world. Yet he vanished from the pages of history for five hundred years, made “obsolete,” and in consequence all but forgotten forever, by a new technology.

What made you decide to write this book?

KD: There were really two key decisions that led to this book. One was deciding, back in the year 2000, to keep a diary of my experiences writing The Man of Numbers. My first history book was The Unfinished Game. For that, all I had to do was consult a number of reference works. It was not intended to be original research. Basic Books asked me to write a short, readable account of a single mathematical document that changed the course of human history, to form part of a series they were bringing out. I chose the letter Pierre De Fermat wrote to his colleague Blaise Pascal in 1654, which most experts agree established modern probability theory, in particular how it can be used to predict the future.

In The Man of Numbers, in contrast, I set out to tell a story that no one had told before; indeed, the consensus among the historians was that it could not be told—there simply was not enough information available. So writing that book would require engaging in a lot of original historical research. I had never done that. I would be stepping well outside my comfort zone. That was in part why I decided to keep a diary. The other reason for keeping a record was to ensure I had enough anecdotes to use when the time came to promote the book—assuming I was able to complete it, that is. (I had written enough popular mathematics books to appreciate the need for author promotional activities!)

The second decision, to turn my diary into a book (which only at the end found the title, Finding Fibonacci), came after The Man of Numbers was published in 2011. The ten-year process of researching and writing that book had turned out to be so rich, and so full of unexpected twists and turns, including several strokes of immense luck, that it was clear there was a good story to be told. What was not clear was whether I would be able to write such a book. All my other books are third-person accounts, where I am simply the messenger. In Finding Fibonacci, I would of necessity be a central character. Once again, I would be stepping outside my comfort zone. In particular, I would be laying out on the printed page, part of my inner self. It took five years and a lot of help from my agent Ted Weinstein and then my Princeton University Press editor Vickie Kearn to find the right voice and make it work.

Who do you expect will enjoy reading this book?

KD: I have a solid readership around the world. I am sure they will all read it. In particular, everyone who read The Man of Numbers will likely end up taking a look. Not least because, in addition to providing a window into the process of writing that earlier book, I also put in some details of that story that I did not fully appreciate until after the book had been published. But I hope, and in fact expect, that Finding Fibonacci will appeal to a whole new group of readers. Whereas the star of all my previous books was a discipline, mathematics, this is a book about people, for the most part people alive today. It’s a human story. It has a number of stars, all people, connected by having embarked on a quest to try to tell parts of the story of one of the most influential figures in human history: Leonardo of Pisa, popularly known as Fibonacci.

Now that the book is out, in one sentence if you can, how would you summarize writing it?

KD: Leaving my author’s comfort zone. Without a doubt. I’ve never been less certain how a book would be received.

DevlinKeith Devlin is a mathematician at Stanford University and cofounder and president of BrainQuake, an educational technology company that creates mathematics learning video games. His many books include The Unfinished Game: Pascal, Fermat, and the Seventeenth-Century Letter That Made the World Modern and The Man of Numbers: Fibonacci’s Arithmetic Revolution. He is the author of Finding Fibonacci: The Quest to Rediscover the Forgotten Mathematical Genius Who Changed the World.

Oscar E. Fernandez on The Calculus of Happiness

FernandezIf you think math has little to do with finding a soulmate or any other “real world” preoccupations, Oscar Fernandez says guess again. According to his new book, The Calculus of Happiness, math offers powerful insights into health, wealth, and love, from choosing the best diet, to finding simple “all weather” investment portfolios with great returns. Using only high-school-level math (precalculus with a dash of calculus), Fernandez guides readers through the surprising results. He recently took the time to answer a few questions about the book and how empowering mathematics can be.

The title is intriguing. Can you tell us what calculus has to do with happiness?

Sure. The title is actually a play on words. While there is a sprinkling of calculus in the book (the vast majority of the math is precalculus-level), the title was more meant to convey the main idea of the book: happiness can be calculated, and therefore optimized.

How do you optimize happiness?

Good question. First you have to quantify happiness. We know from a variety of research that good health, healthy finances, and meaningful social relationships are the top contributors to happiness. So, a simplistic “happiness equation” is: health + wealth + love = happiness. This book then does what any good applied mathematician would do (I’m an applied mathematician): quantify each of the “happiness components” on the left-hand side of the equation (health, wealth, and love), and then use math to extract valuable insights and results, like how to optimize each component.

This process sounds very much like the subtitle, how a mathematical approach to life adds up to health, wealth, and love. But just to be sure, can you elaborate on the subtitle?

That’s exactly right. Often we feel like various aspects of our lives are beyond our control. But in fact, many aspects of our lives, including some of the most important ones (like health, wealth, and love), follow mathematical rules. And by studying the equations that emerge from these rules you can quickly learn how to manipulate those equations in your favor. That’s what I do in the book for health, wealth, and love.

Can you give us some examples/applications?

I can actually give you about 30 of them, roughly the number discussed in the book. But let me focus on my three favorite ones. The first is what I called the “rational food choice” function (Chapter 2). It’s a simple formula: divide 100 calories by the weight (say, in grams) of a particular food. This yields a number whose units are calories per gram, the units of “energy density.” Something remarkable then happens when you plot the energy densities of various foods on a graph: the energy densities of nearly all the healthy foods (like fruits and vegetables) are at most about 2 calories per gram. This simple mathematical insight, therefore, helps you instantly make healthier food choices. And following its advice, as I discuss at length in the book, eventually translates to lower risk for developing heart disease and diabetes, weight loss, and even an increase in your life span! The second example comes from Chapter 3; it’s a formula for calculating how many more years you have to work for before you can retire. Among the formula’s many insights is that, in the simplest case, this magic number depends entirely on the ratio of how much you save each year to how much you spend. And the formula, being a formula, tells you exactly how changing that ratio affects your time until retirement. The last example is based on astronomer Frank Drake’s equation for estimating the number of intelligent civilizations in our galaxy (Chapter 5). It turns out that this alien-searching equation can also be used to estimate the number of possible compatible partners that live near you! That sort of equates a good date with an intelligent alien, and I suppose I can see some similarities (like how rare they are to find).

The examples you’ve mentioned have direct relevance to our lives. Is that a feature of the other examples too?

Absolutely. And it’s more than just relevance—the examples and applications I chose are all meant to highlight how empowering mathematics can be. Indeed, the entire book is designed to empower the reader—via math—with concrete, math-backed and science-backed strategies for improving their health, wealth, and love life. This is a sampling of the broader principle embodied in the subtitle: taking a mathematical approach to life can help you optimize nearly every aspect of your life.

Will I need to know calculus to enjoy the book?

Not at all. Most of the math discussed is precalculus-level. Therefore, I expect that nearly every reader will have studied the math used in the book at some point in their K-12 education. Nonetheless, I guide the reader through the math as each chapter progresses. And once we get to an important equation, you’ll see a little computer icon next to it in the margin. These indicate that there are online interactive demonstrations and calculators I created that go along with the formula. The online calculators make it possible to customize the most important formulas in the book, so even if the math leading up to them gets tough, you can still use the online resources to help you optimize various aspects of health, wealth, and love.

Finally, you mention a few other features of the book in the preface. Can you tell us about some of those?

Sure, I’ll mention two particular important ones. Firstly, at least 1/3 of the book is dedicated to personal finance. I wrote that part of the book to explicitly address the low financial literacy in this country. You’ll find understandable discussions of everything from taxes to investing to retirement (in addition to the various formulas derived that will help you optimize those aspects of your financial life). Finally, I organized the book to follow the sequence of math topics covered in a typical precalculus textbook. So if you’re a precalculus student, or giving this book to someone who is, this book will complement their course well. (I also included the mathematical derivations of the equations presented in the chapter appendixes.) This way the youngest readers among us can read about how empowering and applicable mathematics can be. It’s my hope that this will encourage them to continue studying math beyond high school.

Oscar E. Fernandez is assistant professor of mathematics at Wellesley College and the author of Everyday Calculus: Discovering the Hidden Math All around Us and The Calculus of Happiness: How a Mathematical Approach to Life Adds Up to Health, Wealth, and Love.

Sean W. Fleming on Where the River Flows

Rivers are essential to civilization and even life itself, yet how many of us truly understand how they work? Why do rivers run where they do? Where do their waters actually come from? How can the same river flood one year and then dry up the next? Where the River Flows by Sean W. Fleming is a majestic journey along the planet’s waterways, providing a scientist’s reflections on the vital interconnections that rivers share with the land, the sky, and us. Fleming recently took the time to answer some questions about his new book.

Your book is unique in that it explores the geophysics of rivers: where their waters come from, why their flows vary from day to day and decade to decade, and how math and physics reveal the hidden dynamics of rivers. Why is this important?

SF: Every aspect of our lives ultimately revolves around fresh water. It’s needed to grow food and brew beer, to build cars and computers, to generate hydroelectric power, to go fishing and canoeing, to maintain the ecological web that sustains the world. Floods are the most expensive type of natural disaster in the U.S., and droughts are the most damaging disasters globally. Yet as the margin between water supply and demand grows narrower, and tens of millions more people congregate in megacities often located on floodplains, we become more vulnerable to the geophysical subtleties of the global water cycle. It’s an important part of life that we need to understand if we’re going to make smart choices going forward.

Your book anthropomorphizes a lot. Is this just a way to make the subjects more accessible, or is there a little more to it?

SF: I ask questions like “how do rivers remember?” and “how do clouds talk to fish?” and “can rivers choose where they flow?” It’s a fun way to broach complicated topics about the geophysics of rivers. But posing questions like that also prepares us to open our minds to new ways of thinking about rivers. For instance, modern information theory allows us to quantitatively describe the coupled atmospheric-hydrologic-ecological system as a communications pathway, in which the weather literally transmits data to fish species using the watershed as a communications channel—modulating water levels almost like Morse code. There may be no intent in that communication, but mathematically, we can treat it the same way.

What are the main threats that rivers face? Are these challenges consistent, or do they vary from river to river?

SF: It does vary, but broadly speaking, watersheds face four main threats: pollution, land use change, climate change, and deliberate human modification. Pollution ranges from industrial effluent to fecal contamination to emerging contaminants like pharmaceuticals. Converting natural areas to urban land uses increases flooding and erosion and reduces habitat quantity and quality. Climate change is modifying the timing, volume, and dynamics of streamflows. And civil works like dams, flood control structures, and of course water withdrawals and consumption, alter river flows and ecosystems more profoundly than perhaps anything else. The common thread behind all these concerns is that human populations and economies—and therefore water needs, and our direct and indirect impacts on rivers—are growing much faster than our development of sustainable technologies.

How will climate change affect river flows?

SF: Global warming is expected to accelerate the water cycle, increasing both flooding and drought. Other impacts are more regional. Some areas will enjoy larger annual flow volumes, whereas others may suffer reduced water supplies. More precipitation will fall as rain instead of snow, and snowpack will melt earlier, changing seasonal flow timing. That may interfere with salmon spawning migration, for example, or render existing water supply infrastructure obsolete. In part due to anthropogenic climate change, mountain glaciers are retreating, effectively shrinking the “water towers” of the Himalayas, Andes, Alps, and Rockies—the headwaters of the great rivers that support much of the global human population, from the Columbia to the Yangtze to the Ganges.

What’s so important about understanding the science of rivers? What does it add to our view of the world?

SF: Just think about floods. Knowing how urbanization or deforestation may affect flooding, or how some kinds of flood control can backfire, or how the flood forecasting behind an evacuation order works, is important for making informed choices. There’s also a philosophical aspect. A dramatic view of a river meandering across a desert landscape of red sand and sagebrush at twilight is made even richer by being able to look deeper and recognize the layers of causality and complexity that contributed to it, from the rise of mountains in the headwaters as a continental plate split apart over millions of years, to the way the river shifts its channel when a thunderstorm descends from the skies to deliver a flash flood.

A consistent theme across the book is the interconnectedness of ideas. Why this emphasis? What’s the significance of those connections?

SF: A fundamental and amazing fact of nature is that not only can so much be so effectively described by math, but the same math describes so many different phenomena. Consider debris flows, a sort of flood-landslide hybrid posing serious dangers from Japan to California to Italy. It turns out we can understand phenomena like debris flows using cellular automata, a peculiar kind of computer simulation originally created to explore artificial life. What’s more, cellular automata also reveal something about the origins of fractal patterns, which occur in everything from tree branches to galaxies to the stock market. Recognizing that ideas from one field can be so powerful in another is important for pushing science forward.

The book seems to present a conflicted view of global water security. It paints an extraordinarily dark picture, but it is also very optimistic. Can you explain?

SF: Grave challenges often drive great achievements. Consider some United Nations numbers. Over a billion people don’t have sufficient water, and deprivation in adequate clean water claims—just through the associated disease—more lives than any war claims through guns. By 2050, global water demand will further increase by a stunning 55%. Little wonder that a former World Bank vice-president predicted the 21st century will see water wars. Yet there’s compelling evidence we can get serious traction on this existential threat. Advances in policy and technology have enabled America to hold its water demand at 1970s levels despite population and economic growth. A focused science investment will allow us to continue that success and replicate it globally.

FlemingSean W. Fleming has two decades of experience in the private, public, and nonprofit sectors in the United States, Canada, England, and Mexico, ranging from oil exploration to operational river forecasting to glacier science. He holds faculty positions in the geophysical sciences at the University of British Columbia and Oregon State University. He is the author of Where the River Flows: Scientific Reflections on Earth’s Waterways.

PUP math editor Vickie Kearn: How real mathematicians celebrate Pi Day

Who doesn’t love Pi (aka Pie) Day? Residents here in Princeton, NJ love it so much that we spend four days celebrating. Now, to be honest, we’re also celebrating Einstein’s birthday, so we do need the full four days. I know what I will be doing on 3.14159265 but I wondered what some of my friends will be doing. Not surprisingly, a lot will either be making or eating pie. These include Oscar Fernandez (Wellesley), Ron Graham (UCSD), and Art Benjamin (who will be performing his mathemagics show later in the week). Anna Pierrehumbert (who teaches in NYC) will be working with upper school students on a pi recitation and middle school students on making pi-day buttons. Brent Ferguson (The Lawrenceville School) has celebrated at The National Museum of Mathematics in NYC, Ireland, Greece, and this year Princeton. Here he is celebrating in Alaska:

Pi

The Princeton University Math Club will be celebrating with a party in Fine Hall. In addition to eating pie and playing games, they will have a digit reciting contest. Tim Chartier (Davidson College) will be spending his time demonstrating how to estimate pi with chocolate chips while also fielding interview requests for his expert opinion on March Madness (a lot going on this month for mathematicians). Dave Richeson (Dickinson College) goes to the local elementary school each year and talks with the fifth graders about pi and its history and then eats creatively rendered pi themed pie provided by the parents.

You might be wondering why we celebrate a mathematical constant every year. How did it get to be so important? Again I went back to my pi experts and asked them to tell me the most important uses of pi. This question is open to debate by mathematicians but many think that the most important is Euler’s Identity, e(i*pi) + 1 = 0. As Jenny Kaufmann (President of the Princeton University Math Club) puts it, “Besides elegantly encoding the way that multiplication by i results in a rotation in the complex plane, this identity unites what one might consider the five most important numbers in a single equation. That’s pretty impressive!” My most practical friend is Oscar and here is what he told me: “There are so many uses for pi, but given my interest in everyday explanations of math, here’s one I like: If you drive to work every day, you take many, many pi’s with you. That’s because the circumference of your car’s tires is pi multiplied by the tires’ diameter. The most common car tire has a diameter of about 29 inches, so one full revolution covers a distance of about 29 times pi (about 7.5 feet). Many, many revolutions of your tires later you arrive at work, with lots and lots of pi’s!” Anna is also practical in that she will be using pi to calculate the area of the circular pastry she will be eating, but she also likes the infinite series for pi (pi/4 = 1 – 1/3 + 1/5 – 1/7 etc.). Avner Ash (Boston College) sums it up nicely, “ We can’t live without pi—how would we have circles, normal distributions, etc.?”

One of the most important questions one asks on Pi Day is how many digits can you recite? The largest number I got was 300 from the Princeton Math Club. However, there are quite a few impressive numbers from others, as well as some creative answers and ways to remember the digits. For example, Oscar can remember 3/14/15 at 9:26:53 because it was an epic Day and Pi Time for him. Art Benjamin can recite 100 digits from a phonetic code and 5 silly sentences. Ron Graham can recite all of the digits of pi, even thousands, as long as they don’t have to be in order. Dave Richeson also knows all of the digits of pi which are 0,1,2,3,4,5,6,7,8,and 9.

No matter how you celebrate, remember math, especially pi(e) is useful, fun, and delicious.

Vickie Kearn is Executive Editor of Mathematics at Princeton University Press.

J. Richard Gott: What’s the Value of Pi in Your Universe?

Carl Sagan’s sci-fi novel Contact famously introduced wormholes for rapid transit between the stars. Carl had asked his friend Kip Thorne to tell him if the physics of wormholes was tenable and this led Thorne and his colleagues to investigate their properties. They found that traversable wormholes required exotic matter to prop them open and that, by moving the wormhole mouths one could find general relativity solutions allowing time travel to the past. A quantum state called the Casimir vacuum whose effects have been observed experimentally, could provide the exotic matter. To learn whether such time machines could be constructible in principle, we may have to master the laws of quantum gravity, which govern how gravity behaves on microscopic scales. It’s one of the reasons physicists find these solutions so interesting.

But in Contact there is lurking yet another fantastic sci-fi idea, which gets less publicity because it was not included in the movie version. In the book, the protagonist finds out from the extraterrestrials that the system of wormholes throughout the galaxy was not built by them, but by the long gone “old ones” who could manipulate not only the laws of physics but also the laws of mathematics! And they left a secret message in the digits of pi. In his movie Pi, Darren Aronofsky showed a man driven crazy by his search for hidden meanings in the digits of pi.

This opens the question: could pi have been something else? And if so, does pi depend on the laws of physics? Galileo said: “Philosophy is written in this grand book…. I mean the universe … which stands continually open to our gaze…. It is written in the language of mathematics.” The universe is written in the language of mathematics. Nobel laureate Eugene Wigner famously spoke of the “unreasonable effectiveness of mathematics” in explaining physics. Many philosophers take the Platonic view that mathematics would exist even the universe did not. And cosmologist Max Tegmark goes so far as to say that the universe actually is mathematics.

Yet maybe it is the other way around. The laws of physics are just the laws by which matter behaves. They determine the nature of our universe. Maybe humans have simply developed the mathematics appropriate for describing our universe, and so of course it fits with what we see. The mathematician Leopold Kronecker said, “God created the integers, all the rest is the work of man.” Are the laws of mathematics discovered by us in the same way as we discover the laws of physics? And are the laws of mathematics we discover just those which would have occurred to creatures living in a universe with physics like ours? In our universe, physics produces individual identical particles: all electrons are the same for example. We know about integers because there are things that look the same (like apples) for us to count. If you were some strange creature in a fractal universe containing only one object—yourself—and you thought only recursively, you might not ever think of counting anything and would never discover integers.

What about π = 3.14159265.…? Might it have a different value in a different universe? In our universe we have a fundamental physical dimensionless constant, the fine structure constant α which is related to the square of the value of the electric charge of the proton in natural geometrical Planck units (where the speed of light is 1 and the reduced Planck constant is 1 and Newton’s gravitational constant is 1). Now 1/α = 137.035999… Some physicists hope that one day we may have a mathematical formula for 1/α using mathematical constants such as π and e. If a theory for the fine structure constant could be developed giving a value in agreement with observations but allowing it to be calculated uniquely from pure mathematics, and if more and more digits of the constant were discovered experimentally fulfilling its prediction, it would certainly merit a Nobel Prize. But many physicists feel that no such magic formula will ever be discovered. Inflation may produce an infinite number of bubble universes, each with different laws of physics. Different universes bubbling out of an original inflating sea could have different values of 1/α. As Martin Rees has said, the laws of physics we know may be just local bylaws in an infinite multiverse of universes. String theory, if correct, may eventually give us a probability distribution for 1/α and we may find that our universe is just somewhere in the predicted middle 95% of the distribution, for example. Maybe there could be different universes with different values of π.

Let’s consider one possible example: taxicab geometry. This was invented by Hermann Minkowski. Now this brilliant mathematician also invented the geometrical interpretation of time as a fourth dimension based on Einstein’s theory of special relativity, so his taxicab geometry merits a serious look. Imagine a city with a checkerboard pattern of equal-sized square blocks. Suppose you wanted to take a taxicab to a location 3 blocks east, and 1 block north of your location, the shortest total distance you would have to travel to get there is 4 blocks. Your taxi has to travel along the streets, it does not get to travel as the crow flies. You could go 1 block east, then 1 block north then 2 blocks east, and still get to your destination, but the total distance you traveled would also be 4 blocks. The distance to your destination would be ds = |dx| + |dy|, where |dx| is the absolute value of the difference in x coordinates and |dy| is the absolute value of the difference in y coordinates. This is not the Euclidean formula. We are not in Kansas anymore! The set of points equidistant from the origin is a set of dots in a diamond shape. See diagram.

Gott

Image showing an intuitive explanation of why circles in taxicab geometry look like diamonds. Wikipedia.

Now if the blocks were smaller, there would be more dots, still in a diamond shape. In the limit where the size of the blocks had shrunk to zero, one would have a smooth diamond shape as shown in the bottom section of the diagram. The set of points equidistant from the origin has a name—a “circle!” If the circle has a radius of 1 unit, the distance along one side of its diamond shape is 2 units: going from the East vertex of the diamond to the North vertex of the diamond along the diagonal requires you to change the x coordinate by 1 unit and the y coordinate by 1 unit, making the distance along one side of the diagonal equal to 2 units (ds = |dx| + |dy| = 1 + 1 units = 2 units). The diamond shape has 4 sides so the circumference of the diamond is 8 units. The diameter of the circle is twice the radius, and therefore 2 units. In the taxicab universe π = C/d = C/2r = 8/2 = 4. If different laws of physics dictate different laws of geometry, you can change the value of π.

This taxicab geometry applies in the classic etch-a-sketch toy (Look it up on google, if you have never seen one). It has a white screen, and an internal stylus that draws a black line, directed by horizontal and vertical control knobs. If you want to draw a vertical line, you turn the vertical knob. If you want to draw a horizontal line you turn the horizontal knob. If you want to draw a diagonal line, you must simultaneously turn both knobs smoothly. If the distance between two points is defined by the minimal amount of total turning of the two knobs required to get from one point to the other, then that is the “taxicab” distance between the two points. In Euclidean geometry there is one shortest line between two points: a straight line between them. In taxicab geometry there can be many different, equally short, broken lines (taxicab routes) connecting two points. Taxicab geometry does not obey the axioms of Euclidean geometry and therefore does not have the same theorems as Euclidean geometry. And π is 4.

Mathematician and computer scientist John von Neumann invented a cellular automaton universe that obeys taxicab geometry. It starts with an infinite checkerboard of pixels. Pixels can be either black or white. The state of a pixel at time step t = n + 1 depends only on the state of its 4 neighbors (with which it shares a side: north, south, east, west of it) on the previous time step t = n. Causal, physical effects move like a taxicab. If the pixels are microscopic, we get a taxicab geometry. Here is a simple law of physics for this universe: a pixel stays in the same state, unless it is surrounded by an odd number of black pixels, in which case it switches to the opposite state on the next time step. Start with a white universe with only 1 black pixel at the origin. In the next time step it remains black while its 4 neighbors also become black. There is now a black cross of 5 pixels at the center. It has given birth to 4 black pixels like itself. Come back later and there will be 25 black pixels in a cross-shaped pattern of 5 cross-shaped patterns.

Come back still later and you can find 125 black pixels in 5 cross-shaped patterns (of 5 cross-shaped patterns). All these new black pixels lie inside a diamond-shaped region whose radius grows larger by one pixel per time step. In our universe, drop a rock in a pond, and a circular ripple spreads out. In the von Neumann universe, causal effects spread out in a diamond-shaped pattern.

If by “life” you mean a pattern able to reproduce itself, then this universe is luxuriant with life. Draw any pattern (say a drawing of a bicycle) in black pixels and at a later time you will find 5 bicycles, and then 25 bicycles, and 125 bicycles, etc. The laws of physics in this universe cause any object to copy itself. If you object that this is just a video game, I must tell you that some physicists seriously entertain the idea that we are living in an elaborate video game right now with quantum fuzziness at small scales providing the proof of microscopic “pixelization” at small scales.

Mathematicians in the von Neumann universe would know π = 4 (Or, if we had a taxicab universe with triangular pixels filling the plane, causal effects could spread out along three axes instead of two and a circle would look like a hexagon, giving π = 3.). In 1932, Stanislaw Golab showed that if we were clever enough in the way distances were measured in different directions, we could design laws of physics so that π might be anything we wanted from a low of 3 to a high of 4.

Back to the inhabitants of the von Neumann universe who think π = 4. Might they be familiar with number we know and love, 3.14159265…? They might:

3.14159265… = 4 {(1/1) – (1/3) + (1/5) – (1/7) + (1/9) + …} (Leibnitz)

If they were familiar with integers, they might be able to discover 3.14159265… But maybe the only integers they know are 1, 5, 25, 125, … and 4 of course. They would know that 5 = SQRT(25), so they would know what a square root was. In this case they could still find a formula for

3.14159265. . . =
SQRT(4) {SQRT(4)/SQRT(SQRT(4))}{SQRT(4)/SQRT(SQRT(4) + SQRT(SQRT(4)))}{SQRT(4)/ SQRT(SQRT(4) + SQRT(SQRT(4) + SQRT(SQRT(4))))} …

This infinite product involving only the integer 4 derives from one found by Vieta in 1594.

There are indeed many formulas equal to our old friend 3.14159265… including a spectacular one found by the renowned mathematician Ramanujan. Though every real number can be represented by such infinite series, products and continued fractions, these are particularly simple. So 3.14159265… does seem to have a special intimate relationship with integers, independent of geometry. If physics creates individual objects that can be counted, it seems difficult to avoid learning about 3.14159265… eventually—“If God made the integers,” as Kronecker suggested. So 3.14159265… appears not to be a random real number and we are still left with the mystery of the unreasonable effectiveness of mathematics in explaining the physics we see in our universe. We are also left with the mystery of why the universe is as comprehensible as it is. Why should we lowly carbon life forms be capable of finding out as much about how the universe works as we have done? Having the ability as intelligent observers to ask questions about the universe seems to come with the ability to actually answer some of them. That’s remarkable.

UniverseGottJ. Richard Gott is professor of astrophysics at Princeton University. His books include The Cosmic Web: Mysterious Architecture of the Universe. He is the coauthor of Welcome to the Universe: An Astrophysical Tour with Neil DeGrasse Tyson and Michael A. Strauss.

Just in time for Pi Day, presenting The Usefulness of Useless Knowledge

In his classic essay “The Usefulness of Useless Knowledge,” Abraham Flexner, the founding director of the Institute for Advanced Study in Princeton and the man who helped bring Albert Einstein to the United States, describes a great paradox of scientific research. The search for answers to deep questions, motivated solely by curiosity and without concern for applications, often leads not only to the greatest scientific discoveries but also to the most revolutionary technological breakthroughs. In short, no quantum mechanics, no computer chips. This brief book includes Flexner’s timeless 1939 essay alongside a new companion essay by Robbert Dijkgraaf, the Institute’s current director, in which he shows that Flexner’s defense of the value of “the unobstructed pursuit of useless knowledge” may be even more relevant today than it was in the early twentieth century. Watch the trailer to learn more:

The Usefulness of Useless Knowledge by Abraham Flexner from Princeton University Press on Vimeo.

Marc Chamberland: Why π is important

On March 14, groups across the country will gather for Pi Day, a nerdy celebration of the number Pi, replete with fun facts about this mathematical constant, copious amounts pie, and of course, recitations of the digits of Pi. But why do we care about so many digits of Pi? How big is the room you want to wallpaper anyway? In 1706, 100 digits of Pi were known, and by 2013 over 12 trillion digits had been computed. I’ll give you five reasons why someone may claim that many digits of Pi is important, but they’re not all good.

Reason 1
It provides accuracy for scientific measurements

Pi1

This argument had merit when only a few digits were known, but today this reason is as empty as space. The radius of the universe is 93 billion light years, and the radius of a hydrogen atom is about 0.1 nanometers. So knowing Pi to 38 places is enough to tell you precisely how many hydrogen atoms you need to encircle the universe. For any mechanical calculations, probably 3.1415 is more than enough precision.

Reason 2
It’s neat to see how far we can go

Pi2

It’s true that great feats and discoveries have been done in the name of exploration. Ingenious techniques have been designed to crank out many digits of Pi and some of these ideas have led to remarkable discoveries in computing. But while this “because it is there” approach is beguiling, just because we can explore some phenomenon doesn’t mean we’ll find something valuable. Curiosity is great, but harnessing that energy with insight will take you farther.

Reason 3
Computer Integrity

Pi3

The digits of Pi help with testing and developing new algorithms. The Japanese mathematician Yasumasa Kanada used two different formulas to generate and check over one trillion digits of Pi. To get agreement after all those arithmetic operations and data transfers is strong evidence that the computers are functioning error-free. A spin-off of the expansive Pi calculations has been the development of the Fast Fourier Transform, a ground-breaking tool used in digital signal processing.

Reason 4
It provides evidence that Pi is normal

Pi4

A number is “normal” if any string of digits appears with the expected frequency. For example, you expect the number 4 to appear 1/10 of the time, or the string 28 to appear 1/100 of the time. It is suspected that Pi is normal, and this was evidenced from the first trillion digits when it was seen that each digit appears about 100 billion times. But proving that Pi is normal has been elusive. Why is the normality of numbers important? A normal number could be used to simulate a random number generator. Computer simulations are a vital tool in modeling any dynamic phenomena that involves randomness. Applications abound, including to climate science, physiological drug testing, computational fluid dynamics, and financial forecasting. If easily calculated numbers such as Pi can be proven to be normal, these precisely defined numbers could be used, paradoxically, in the service of generating randomness.

Reason 5
It helps us understand the prime numbers

Pi5

Pi is intimately connected to the prime numbers. There are formulas involving the product of infinitely numbers that connect the primes and Pi. The knowledge flows both ways: knowing many primes helps one calculate Pi and knowing many digits of Pi allows one to generate many primes. The Riemann Hypothesis—an unsolved 150-year-old mathematical problem whose solution would earn the solver one million dollars—is intimately connected to both the primes and the number Pi.

And you thought that Pi was only good for circles.

SingleMarc Chamberland is the Myra Steele Professor of Mathematics and Natural Science at Grinnell College. His research in several areas of mathematics, including studying Pi, has led to many publications and speaking engagements in various countries. His interest in popularizing mathematics resulted in the recent book Single Digits: In Praise of Small Numbers with Princeton University Press. He also maintains his YouTube channel Tipping Point Math that tries to make mathematics accessible to a general audience. He is currently working on a book about the number Pi.

Michael Strauss: Our universe is too vast for even the most imaginative sci-fi

As an astrophysicist, I am always struck by the fact that even the wildest science-fiction stories tend to be distinctly human in character. No matter how exotic the locale or how unusual the scientific concepts, most science fiction ends up being about quintessentially human (or human-like) interactions, problems, foibles and challenges. This is what we respond to; it is what we can best understand. In practice, this means that most science fiction takes place in relatively relatable settings, on a planet or spacecraft. The real challenge is to tie the story to human emotions, and human sizes and timescales, while still capturing the enormous scales of the Universe itself.

Just how large the Universe actually is never fails to boggle the mind. We say that the observable Universe extends for tens of billions of light years, but the only way to really comprehend this, as humans, is to break matters down into a series of steps, starting with our visceral understanding of the size of the Earth. A non-stop flight from Dubai to San Francisco covers a distance of about 8,000 miles – roughly equal to the diameter of the Earth. The Sun is much bigger; its diameter is just over 100 times Earth’s. And the distance between the Earth and the Sun is about 100 times larger than that, close to 100 million miles. This distance, the radius of the Earth’s orbit around the Sun, is a fundamental measure in astronomy; the Astronomical Unit, or AU. The spacecraft Voyager 1, for example, launched in 1977 and, travelling at 11 miles per second, is now 137 AU from the Sun.

But the stars are far more distant than this. The nearest, Proxima Centauri, is about 270,000 AU, or 4.25 light years away. You would have to line up 30 million Suns to span the gap between the Sun and Proxima Centauri. The Vogons in Douglas Adams’s The Hitchhiker’s Guide to the Galaxy (1979) are shocked that humans have not travelled to the Proxima Centauri system to see the Earth’s demolition notice; the joke is just how impossibly large the distance is.

Four light years turns out to be about the average distance between stars in the Milky Way Galaxy, of which the Sun is a member. That is a lot of empty space! The Milky Way contains about 300 billion stars, in a vast structure roughly 100,000 light years in diameter. One of the truly exciting discoveries of the past two decades is that our Sun is far from unique in hosting a retinue of planets: evidence shows that the majority of Sun-like stars in the Milky Way have planets orbiting them, many with a size and distance from their parent star allowing them to host life as we know it.

Yet getting to these planets is another matter entirely: Voyager 1 would arrive at Proxima Centauri in 75,000 years if it were travelling in the right direction – which it isn’t. Science-fiction writers use a variety of tricks to span these interstellar distances: putting their passengers into states of suspended animation during the long voyages, or travelling close to the speed of light (to take advantage of the time dilation predicted in Albert Einstein’s theory of special relativity). Or they invoke warp drives, wormholes or other as-yet undiscovered phenomena.

When astronomers made the first definitive measurements of the scale of our Galaxy a century ago, they were overwhelmed by the size of the Universe they had mapped. Initially, there was great skepticism that the so-called ‘spiral nebulae’ seen in deep photographs of the sky were in fact ‘island universes’ – structures as large as the Milky Way, but at much larger distances still. While the vast majority of science-fiction stories stay within our Milky Way, much of the story of the past 100 years of astronomy has been the discovery of just how much larger than that the Universe is. Our nearest galactic neighbour is about 2 million light years away, while the light from the most distant galaxies our telescopes can see has been travelling to us for most of the age of the Universe, about 13 billion years.

We discovered in the 1920s that the Universe has been expanding since the Big Bang. But about 20 years ago, astronomers found that this expansion was speeding up, driven by a force whose physical nature we do not understand, but to which we give the stop-gap name of ‘dark energy’. Dark energy operates on length- and time-scales of the Universe as a whole: how could we capture such a concept in a piece of fiction?

The story doesn’t stop there. We can’t see galaxies from those parts of the Universe for which there hasn’t been enough time since the Big Bang for the light to reach us. What lies beyond the observable bounds of the Universe? Our simplest cosmological models suggest that the Universe is uniform in its properties on the largest scales, and extends forever. A variant idea says that the Big Bang that birthed our Universe is only one of a (possibly infinite) number of such explosions, and that the resulting ‘multiverse’ has an extent utterly beyond our comprehension.

The US astronomer Neil deGrasse Tyson once said: ‘The Universe is under no obligation to make sense to you.’ Similarly, the wonders of the Universe are under no obligation to make it easy for science-fiction writers to tell stories about them. The Universe is mostly empty space, and the distances between stars in galaxies, and between galaxies in the Universe, are incomprehensibly vast on human scales. Capturing the true scale of the Universe, while somehow tying it to human endeavours and emotions, is a daunting challenge for any science-fiction writer. Olaf Stapledon took up that challenge in his novel Star Maker (1937), in which the stars and nebulae, and cosmos as a whole, are conscious. While we are humbled by our tiny size relative to the cosmos, our brains can none the less comprehend, to some extent, just how large the Universe we inhabit is. This is hopeful, since, as the astrobiologist Caleb Scharf of Columbia University has said: ‘In a finite world, a cosmic perspective isn’t a luxury, it is a necessity.’ Conveying this to the public is the real challenge faced by astronomers and science-fiction writers alike. Aeon counter – do not remove

UniverseMichael A. Strauss is professor of astrophysics at Princeton University and coauthor with Richard Gott and Neil DeGrasse Tyson of Welcome to The Universe: An Astrophysical Tour.

This article was originally published at Aeon and has been republished under Creative Commons.

Robbert Dijkgraaf on The Usefulness of Useless Knowledge

FlexnerA forty-year tightening of funding for scientific research has meant that resources are increasingly directed toward applied or practical outcomes, with the intent of creating products of immediate value. In such a scenario, it makes sense to focus on the most identifiable and urgent problems, right? Actually, it doesn’t. In his classic essay “The Usefulness of Useless Knowledge,” Abraham Flexner, the founding director of the Institute for Advanced Study in Princeton, describes a great paradox of scientific research. The search for answers to deep questions, motivated solely by curiosity and without concern for applications, often leads not only to the greatest scientific discoveries but also to the most revolutionary technological breakthroughs. This brief book includes Flexner’s timeless 1939 essay alongside a new companion essay by Robbert Dijkgraaf, the Institute’s current director. Read on for Dijkgraaf’s take on the importance of curiosity-driven research, how we can cultivate it, and why Flexner’s essay is more relevant than ever.

The title of the book, The Usefulness of Useless Knowledge, is somewhat enigmatic—what does it mean?

RD: Abraham Flexner, an educational reformer and founding director of the Institute for Advanced Study, wrote an essay with this title for Harper’s magazine in 1939. He believed that there was an indispensable connection between intellectual and spiritual life—“useless forms of activity”—and undreamed-of utility.

Cited as a philanthropic hero by Warren Buffett, Flexner was responsible for bringing Albert Einstein to America to join the Institute’s inaugural Faculty, just when Hitler came to power in 1933.

A true visionary, Flexner was acutely aware that our current conception of what is useful might suffice for the short term but would inevitably become too narrow over time. He believed that the best way to advance understanding and knowledge is by enabling leading scientists and scholars to follow their natural curiosity, intuition, and inquiry, without concern for utility but rather with the purpose of discovering answers to the most fascinating questions of their time.

Flexner’s 1939 article is reprinted in the book along with a companion essay that you have written. What did you realize in revisiting Flexner’s ideas?

RD: One large realization is that while the world has changed dramatically in terms of technological progress since Flexner’s time, human beings still wrestle with the benefits and risks of freedom, with power and productivity versus imagination and creativity, and this dichotomy continues to limit our evolution and sometimes leads to abhorrent behavior as we saw during Flexner’s era and which continues to haunt ours today.

A significant difference is that in the twenty-first century, we are increasingly creating a one-dimensional world determined by external metrics. Why? Our world is becoming ever larger and more complex. In order to provide some clarity, we try to quantify that world with share prices and rankings. In the process, we have exiled our intuition and have lost contact with our environment.

We need to return to timeless values like searching for the truth, while being honest about the things we don’t understand. There is also a great need for passion. I wake up every morning with the thought: I want to do something that I feel good about. As a society, we have largely lost that feeling. We need to reconsider: what kind of world do we want exactly? And what new systems do we need to do good things?

Why is curiosity-driven basic research important today and how can we cultivate it?

RD: The progress of our modern age, and of the world of tomorrow, depends not only on technical expertise, but also on unobstructed curiosity and the benefits of traveling far upstream, against the current of practical considerations. Much of the knowledge developed by basic research is made publicly accessible and so benefits society as a whole, spreading widely beyond the narrow circle of individuals who, over years and decades, introduce and develop the ideas. Fundamental advances in knowledge cannot be owned or restricted by people, institutions, or nations, certainly not in the current age of the Internet. They are truly public goods.

But driven by an ever-deepening lack of funding, against a background of economic uncertainty, global political turmoil, and ever-shortening time cycles, research criteria are becoming dangerously skewed towards conservative short-term goals that may address more immediate problems, but miss out on the huge advances that human imagination can bring in the long term.

The “metrics” used to assess the quality and impact of research proposals—even in the absence of a broadly accepted framework for such measurements—systematically undercut pathbreaking scholarship in favor of more predictable goal-directed research. It can easily take many years, even decades, or sometimes, a century, as in the case of the gravitational waves predicted by Einstein’s theory of relativity that were only detected last year, for the societal value of an idea to come to light.

In order to enable and encourage the full cycle of scientific innovation, we need to develop a solid portfolio of research in much the same way as we approach well-managed financial resources. Such a balanced portfolio would contain predictable and stable short-term investments, as well as long-term bets that are intrinsically more risky but can potentially earn off-the-scale rewards. The path from exploratory basic research to practical applications is not one-directional and linear, but rather complex and cyclic, with resultant technologies enabling even more fundamental discoveries. Flexner and I give many examples of this in our book, from the development of electromagnetic waves that carry wireless signals to quantum mechanics and computer chips.

How do curiosity and imagination enable progress?

RD: An attitude aimed at learning and investigating, wherein imagination and creativity play an important role, is essential not only in scientific institutions but in every organization. Companies and institutions themselves need to develop the inquisitive and explorative approach they would like to see in their employees. Organizations are often trapped in the framework of their own thinking. Out-of-the-box thinking is very hard, because one doesn’t know where the box is. At the basis of progress lies a feeling of optimism: problems can be solved. Organizations need to cultivate the capacity to visualize the future and define their position in it.

What conditions are necessary for the spark of a new idea or theory?

RD: If we want more imagination, creativity, and curiosity, we need to accept that people occasionally run in the wrong direction. As a business, institution, or society, we need to allow once again for failure. Encourage workers to spend a certain percentage of their time on the process of exploration. A brilliant idea never appears out of the blue, but is generated simply by allowing people to try out things. Nine times out of ten, nothing results, but something may emerge suddenly and unexpectedly. That free space and those margins of error are increasingly under pressure in our head, our role, our organization, and our society. I am worried about the loss of that exploratory force.

What don’t we know, and how does uncertainty drive advancement?

RD: How did the universe begin and how does it end? What is the origin of life on Earth and possibly elsewhere in the cosmos? What in our brain makes us conscious and human? In addition to these fundamental questions and many others, we are struggling with major issues about time and space, about matter and energy. What are our ideas on this and what questions are we trying to answer? In science, a long process precedes any outcome. In general, the media only has time and space to pay attention to outcomes. But for scientists it’s precisely the process that counts, walking together down that path. It’s the questions that engage us, not the answers.

Abraham Flexner (1866–1959) was the founding director of the Institute for Advanced Study, one of the world’s leading institutions for basic research in the sciences and humanities. Robbert Dijkgraaf, a mathematical physicist who specializes in string theory, is director and Leon Levy Professor at the Institute for Advanced Study. A distinguished public policy adviser and passionate advocate for science and the arts, he is also the cochair of the InterAcademy Council, a global alliance of science academies, and former president of the Royal Netherlands Academy of Arts and Sciences. They are the authors of The Usefulness of Useless Knowledge.