During spring, many peak flights occur ahead of a warm front, as birds heading north use southerly prevailing winds. Most fall flights occur after the passage of a cold front, when northerly winds that assist birds heading south are most prevalent. Geography, however, determines which specific wind directions will lead birds to each site. The most favorable winds for ridge sites are those that strike the ridge at an angle that produces optimal lift. At coastal and shoreline sites, optimal winds are those that “push” birds towards the shorelines. Even during snow squalls or light drizzle, optimal wind conditions can produce significant hawk flights.

**
Hawks from Every Angle
**How to Identify Raptors In Flight

Foreword by David A. Sibley

Identifying hawks in flight is a tricky business. Across North America, tens of thousands of people gather every spring and fall at more than one thousand known hawk migration sites—from New Jersey’s Cape May to California’s Golden Gate. Yet, as many discover, a standard field guide, with its emphasis on plumage, is often of little help in identifying those raptors soaring, gliding, or flapping far, far away. *Hawks from Every Angle* takes hawk identification to new heights. It offers a fresh approach that literally looks at the birds from every angle, compares and contrasts deceptively similar species, and provides the pictures (and words) needed for identification in the field. Jerry Liguori pinpoints innovative, field-tested identification traits for each species at the various angles that they are seen.

Featuring 339 striking color photos on 68 color plates and 32 black & white photos, *Hawks from Every Angle* is unique in presenting a host of meticulously crafted pictures for each of the 19 species it covers in detail—the species most common to migration sites throughout the United States and Canada. All aspects of raptor identification, including plumage, shape, and flight style traits, are discussed. For all birders who follow hawk migration and have found themselves wondering if the raptor in the sky does in fact match the one in the guide, *Hawks from Every Angle*—distilling an expert’s years of experience for the first time into a comprehensive array of truly useful photos and other pointers for each species—is quite simply a must.

**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 am 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.

*This post by director of the University Press of Colorado and president of the Association of American University Presses Darrin Pratt appears concurrently on the University Press of Colorado blog.*

In a previous post, I wrote about the minor miracle continually performed by the membership of the Association of American University Presses, a miracle that involves taking a relatively small annual budget and multiplying that budget until it becomes substantially larger. University presses, I observed, collectively receive an annual budget that would support the publication of roughly 900 scholarly monographs annually, based on an Andrew W. Mellon Foundation–funded ITHAKA S+R study of the average publication cost of a monograph. In reality, university presses create enough additional revenue from the starting budget they are given to produce over 6,000 books annually,^{1} or roughly seven times the number of books supported directly by their institutional budgets.

In this previous post, I acknowledged the fact that not all of those 6,000 or more titles were scholarly monographs in the narrowest sense of the term. There remained some question regarding the proportion of that combined output that comprised the specialized studies that have been at the core of university press programs from the beginning. Fortunately, thanks again to the Andrew W. Mellon Foundation, we can now address that question through the data contained in their recently released study, *Monograph Output of American University Presses, 2009–2013* by Joseph J. Esposito and Karen Barch.

To wit, over the five-year period covered in the study data collected, university presses published 14,619 scholarly monographs, or an average of 2,924 per year.^{2} In terms of mission, we collectively published over three scholarly monographs annually for every one book that we were actually paid to produce.^{3} And as Esposito and Barch’s data indicate, monographic title output as a percentage of total title output was 49 percent from 2009 to 2013 for the sixty-five presses that reported to the study.^{4 }

Our value proposition, however, gets even better if you consider expanding beyond the definition of monograph employed by the study authors. The study defined monographs as “books which are written by scholars and researchers and which are intended primarily for other scholars and researchers” (using John Thompson’s definition in *Books in the Digital Age*),^{5} but excluded books that are “collections of essays, even if the essays are all by a single author.” In certain fields, particularly emerging fields and subfields, edited collections of essays that constitute original scholarship are quite common. If we use Thompson’s original definition, without excluding edited collections, my own press’s 2009–2013 output of original scholarly works as a percentage of the total jumps from 42% to 67% (from data returned to University Press of Colorado by Esposito and Barch).

Although I only have my own press data at hand, most press directors estimate a similar proportion of their list is made up of original scholarship, somewhere between 67 and 75 percent, as noted by Esposito and Barch.^{6} If, let’s say, roughly 70 percent of our combined output is, in fact, original works of scholarship more broadly defined, then we collectively publish almost five works of original scholarship for every one work we are given the budget to produce.^{7}

As for the other 30 percent? Although they are not necessarily publications of primary or original scholarship, most—albeit not quite all—are all nonetheless built upon primary or original scholarship and communicate more broadly to students and the public the knowledge being generated every day by researchers at colleges and universities across the country. This other 30 percent includes textbooks, crossover titles that inform public debate on important policy questions, regional history and natural history titles, and important reference works, all of which do more than their fair share to ensure that we can multiply one paid-for work of original scholarship into five.

Of course, the previous paragraph suggests that the titles that are in this “other” 30 percent are published more for money than mission, with the further implication that they have little to contribute to research agendas in their fields. And, truth be told, university presses will occasionally publish coffee table books, cookbooks, or the like with the primary intention of bringing in revenue to support their scholarly publishing programs. But the vast majority of all books published by university presses are mission-driven products that have been rigorously peer-reviewed, including our text, crossover, and regional titles that sometimes make substantial scholarly contributions. As Peter Dougherty and Al Bertrand have written elsewhere, in 1922 Princeton University Press published a book of public lectures delivered the previous year. The work was not a monograph in the strict sense: rather, it was a scholarly work since read by generations of scientists and nonscientists alike. The lecturer was Albert Einstein, and the book was *The Meaning of Relativity*.

In the final analysis, whether we consider monographs as only those works narrowly defined by Esposito and Barch, expand our definition of original scholarship (following Thompson), or include other publications like crossover books, textbooks, or regional titles, the fact is that university presses play a vital role in cultivating and distributing works of serious scholarship. In a world of alternative facts and fake news, we continue to carry the torch for research, for scholarship, for facts, and for truth.

1. *The source of the figure cited here is the 2012–2015 Annual Operating Statistics Survey of the Association of American University Presses and compiles data from sixty-seven reporting presses excluding Cambridge and Oxford. Esposito and Barch’s report also excludes title output data from Cambridge and Oxford. Return to text.*

2. *Joseph J. Esposito and Karen Barch, Monograph Output of American University Presses, 2009–2013: A Report Prepared for the Andrew W. Mellon Foundation (2017), 32 (data table). Return to text.*

3. *2,924 ÷ 900 = 3.25. Return to text.*

4. *Don’t go looking for the 49 percent figure in the Esposito and Barch report, because you will not find it there. The number can be calculated, however, using the data they present. I derived the 49 percent by dividing the total number of monograph editions published (28,625) by the total number of all editions published (58,555). I excluded Esposito and Barch’s extrapolations from the original data in making this calculation. See Esposito and Barch, Monograph Output, 32 (data table). Return to text.*

5. *John Thompson, *Books in the Digital Age: The Transformation of Academic and Higher Education Publishing in Britain and the United States* (Cambridge: Polity Press, 2005), 103. Return to text.*

6. *See the discussion of university press estimates in Esposito and Barch, Monograph Output, 41. Return to text.*

7. *(6,000 × 70%) ÷ 900 = 4.67. Note that the Esposito and Barch do not de-duplicate total editions to drill down to the total books published in the same fashion that they de-duplicate editions to derive unique (“primary”) monographs published. Their report therefore contains no total unique/primary books published figure. That said, their data strongly suggests that number to be an average of roughly 6,000 (5,984) unique books per year. This number is an estimate drawing from the data table on page 32, where the proportion of unique monographs to monograph editions is 14,619/28,625, or 51.1%. Presuming a similar proportion of books/editions in the total figure, 51.1% × 58,555 total editions = 29,922 unique books ÷ 5 years = 5,984 unique books annually. Return to text.*

A snow goose can be very loquacious, even noisy, especially in flight, when taking off or landing. It produces loud, raucous, barking calls *gwok* or *ga-ik*, as well as other sounds more like those of grey geese, lower and hoarser: *gung*, *wa-iir* or *hun-hrr*. Large flocks utter these calls continuously and at different pitches, linked to the birds’ size.

**Waterfowl of North America, Europe, and Asia**

An Identification Guide

**Sébastien Reeber**

This is the ultimate guide for anyone who wants to identify the ducks, geese, and swans of North America, Europe, and Asia. With 72 stunning color plates (that include more than 920 drawings), over 650 superb photos, and in-depth descriptions, this book brings together the most current information on 84 species of Eurasian and North American waterfowl, and on more than 100 hybrids. The guide delves into taxonomy, identification features, determination of age and sex, geographic variations, measurements, voice, molt, and hybridization. In addition, the status of each species is treated with up-to-date details on distribution, population size, habitats, and life cycle. Color plates and photos are accompanied by informative captions and 85 distribution maps are also provided. Taken together, this is an unrivaled, must-have reference for any birder with an interest in the world’s waterfowl.

]]>Let us know which ones you’re reading on Twitter and Instagram!

Ireland’s Immortals: A History of the Gods of Irish Myth

**Mark Williams**

Revolutionary Lives: Constance and Casimir Markievicz

**Lauren Arrington**

The Princeton History of Modern Ireland

Edited by **Richard Bourke** & **Ian McBride**

Empire and Revolution: The Political Life of Edmund Burke

**Richard Bourke**

On Elizabeth Bishop

**Colm Tóibín**

**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.

**Sean 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.

**Craig Clunas** is Professor of the History of Art at the University of Oxford. His books include *Screen of Kings: Royal Art and Power in Ming China*, *Empire of Great Brightness: Visual and Material Culture and Social Status in Early Modern China*, and *Art in China*.

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.

]]>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.

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.

**J. 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.

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

]]>**Reason 1**

**It provides accuracy for scientific measurements**

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**

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**

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**

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**

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.

**Marc 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.