Andrew Granville and Jennifer Granville on Prime Suspects

What inspired you to write this book?

Andrew: I had written quite a few “popular” articles, that had been well received inside the academic community. However I realized, at some point, that the furthest outside this community that read my articles seemed to be very keen high school teachers who organized statewide math competitions. I want to reach a much wider audience.

Jennifer: Andrew’s original idea was to write a screenplay that would be another way of communicating his mathematical ideas.  I brought expertise in screenwriting, Andrew brought the math.  The screenplay was given a rehearsed reading, with some contemporary, illustrative performance elements,at the Wolfson Auditorium at the Institute for Advanced Study in Princeton. In the audience was Vickie Kearn from PUP. Vickie was the one who had the vision to suggest we could turn the work into a graphic novel – so I guess you could say it was Vickie who inspired us to actually ‘write the book’!

Why did you choose to specifically focus on integers and permutations in this graphic novel?

Andrew: I had thought it would make a good subject for a popular article. The extraordinary similarities between their “anatomies” is intriguing and I have been trying to popularize this within the research community. Since we started this project – which was over ten years ago – this area has really taken off.

How did you develop Prime Suspects‘ story, and from where did you draw inspiration?

Jennifer:  Andrew suggested that Integer and Permutation could be personified into murder victims who, forensic mathematical examination would prove via DNA, were twins. This was enough for me to begin to develop a narrative using all the genres I love to read and watch – noir movies, Chandler novels, the TV police procedural.

Prime Suspects is filled with “cameos” from famous mathematicians, as well as pop culture figures like Kevin Smith’s Silent Bob. There are also a lot of interesting ‘props’ and backgrounds peppering the pages. What inspired you to include these fun appearances?

Andrew: A story has more color if there is an interesting background and context. To my mind, a Hitchcock or a Tarantino movie is as intriguing for what, and who, is in the background as for what is in the foreground. I love those extra details. All of the appearances were inspired by the story.

What do you think an average comic book reader will enjoy about Prime Suspects, even if they don’t regularly read trade math books?

Andrew This is an attempt to be very very different. It is the proof of a theorem, developed as a detective story, in which the detective story in prominent, and the mathematics is by metaphor. Any reader can enjoy the art and the story, and try to be comfortable with as much of the mathematics as works for them.

Jennifer:  I am far from being any kind of mathematician, none of the the artists involved – the illustrator, colourist, letterer – are mathematicians, but we are all comic book fans and have all thoroughly enjoyed the process of bringing the story to life without understanding any of the deeper math. One of our characters compares the math to poetry – you don’t have to understand every word, every beat, in order to appreciate the beauty or to feel a concept. As described above, there are masses of cameos and loads of references to movies, books and contemporary culture, so I hope that readers will find plenty to enjoy.

There are so many interactive elements to Prime Suspects, including an original score. What inspired you to include so many creative elements with the text?

Jennifer:  These happened organically. To give one example, very early on in the project’s life, when we were just about to do the reading at Princeton, Andrew was seated at at a conference dinner next to a math hobbyist, Robert Schneider. Andrew told him about the screenplay and Robert was fascinated and excited and explaining that he was indie rock musician, asked if he could compose music to be played live at the reading. As it happened there is a major clue in the story, that involves a piece of music, so Robert ended up composing a real piece of music – that reflects the Sieve of Eratosthenes. There is now a QR code on the relevant page of the graphic novel, that allows the reader to play that piece of music. As a post script to that story, Robert is now a rock musician and a Math professor at UGA.

What did you find most exciting about taking your love for mathematics and putting it into a graphic novel? What did you find most challenging?

Andrew:  I had experience at writing “popular” articles, and my writing in that area has been well received. However, when trying to create a fictional story around my ideas, I found that my writing skills did not translate to this new setting. I had no idea how to develop a story and characters. Although a mathematics article should have a narrative, this is very different from writing dramatic narrative.  Working with Jennifer proved to be exciting, as together we were able to apply dramatic narrative techniques, and I could see ideas I had been thinking about for a long time take shape on the page. A major challenge has been to keep the integrity of the math whilst ensuring the story makes sense.  Where to explain the math and where to allow that the audience will all have different levels of understanding and accept that not everyone will understand everything.  Thus our narrator says, early on, that we are in a world in which you do not need to understand everything to understand something.

Jennifer:  This whole project has been a challenge because I never did have ‘a love for mathematics’ but that is what has made it exciting.  I failed math at school and it has always been a completely mysterious world to me. I could only observe my brother’s world, his passion, from the outside. The opportunity to share that world, to see it from the inside, has been a massive privilege and education. I am a prime example (pun intended) of someone who doesn’t understand everything, but who, now, does understand something.

 

Andrew Granville is the Canada Research Chair in Number Theory at the University of Montreal and professor of mathematics at University College London. Jennifer Granville is an educator, award-winning film and theater producer, writer, and director.

What is Calculus?: The Two Pillars

By Oscar Fernandez

This is the third in a series of short articles exploring calculus. The first article explored the origins of calculus, including the three “big problems” that drove calculus’ development. The second article explored limits, the foundation of calculus. This article discusses how limits help us solve the three “big problems” and introduces two of calculus’ pillars: derivatives and integrals.

In the first article in this series I discussed three Big Problems that drove the development of calculus: the instantaneous speed problem, the tangent line problem, and the area problem. I illustrated these via the figure below.

Reprinted, with permission, from Calculus Simplified (Princeton University Press). Click to expand.

These problems stumped mathematicians for millennia. (We briefly talked about why in the first article.) But their inability to solve these problems—echoing Morpheus in the movie The Matrix—was not due to the techniques they were using; it was due to their mindset.

How a Dynamics Mindset Solves the Three Big Problems

If you’ve read the second article in this series, you’ll remember my first characterization of calculus: calculus is a dynamics mindset. Yet nothing about the figure above says “dynamics.” Every image is a static snapshot of something (e.g., an area). So let’s calculus the figure. (Yep, I’m encouraging you to think of calculus as a verb.)

The figure below takes each Big Problem from the figure above and adds in the dynamics.

Reprinted, with permission, from Calculus Simplified (Princeton University Press). Click to expand.

These images show apples falling, gray lines approaching a blue tangent line, and areas being swept out. Lots of movement (dynamics)! Moreover, notice that as the central change in each row of the figure gets closer to zero —the quantity ∆t in the first row and ∆x in the second and third rows—the resulting diagram approaches the respective diagram in the first figure in this article. We’ve met this “as ∆t  approaches zero” language before—it’s the language of limits we discussed in the second article! Adding this new revelation to the figure above produces…

 

Reprinted, with permission, from Calculus Simplified (Princeton University Press. Click to expand.

Finally, expressing our result in terms of equations involving limits yields the final piece of the puzzle…

Notice how each row employs a dynamics mindset to recast the Big Problem (contained in the “limiting picture” column) as the limit of a sequence of similar quantities (e.g., speeds) involving finite changes, changes which pre-calculus mathematics can handle. Specifically:

  • Row #1: The instantaneous speed of the falling apple is realized as the limit of its average speeds  ∆d / ∆t (ratios of changes in distance to changes in time) as ∆t —> 0.
  • Row #2: The slope of the tangent line is realized as the limit of the secant line slopes ∆y / ∆x (the gray lines in the figure) as ∆x —> 0.
  • Row #3: The area under the curve is realized as the limit as ∆x —> 0 of the area swept out from x = a up to ∆x  past b.

Introducing…Derivatives and Integrals

The limit obtained in the second row of the last figure is called the derivative of f(x) at x = a, the x-value of point P. The limit obtained in the third row of the Figure is called the definite integral of f(x) between x = a and x = b. Derivatives and integrals round out the three most important concepts in calculus (limits are the third).

You now have a working understanding of what derivatives and definite integrals are, what they measure, and how they arise from the application of a dynamics mindset to pre-calculus mathematics. The next post in this series will explore the derivative in greater details. We’ll discover that it has a nice geometric interpretation and a powerful real-world interpretation. (The last figure above hints to what these are.) Near the end of this series we will return to these interpretations to illustrate the power of derivatives, using them to help us understand phenomena as diverse as the fate of the Universe and, more pragmatically, how to find the best seat in a movie theater. Stay tuned!

 

Calculus Simplified
By Oscar E. Fernandez

Calculus is a beautiful subject that most of us learn from professors, textbooks, or supplementary texts. Each of these resources has strengths but also weaknesses. In Calculus Simplified, Oscar Fernandez combines the strengths and omits the weaknesses, resulting in a “Goldilocks approach” to learning calculus: just the right level of detail, the right depth of insights, and the flexibility to customize your calculus adventure.

Fernandez begins by offering an intuitive introduction to the three key ideas in calculus—limits, derivatives, and integrals. The mathematical details of each of these pillars of calculus are then covered in subsequent chapters, which are organized into mini-lessons on topics found in a college-level calculus course. Each mini-lesson focuses first on developing the intuition behind calculus and then on conceptual and computational mastery. Nearly 200 solved examples and more than 300 exercises allow for ample opportunities to practice calculus. And additional resources—including video tutorials and interactive graphs—are available on the book’s website.

Calculus Simplified also gives you the option of personalizing your calculus journey. For example, you can learn all of calculus with zero knowledge of exponential, logarithmic, and trigonometric functions—these are discussed at the end of each mini-lesson. You can also opt for a more in-depth understanding of topics—chapter appendices provide additional insights and detail. Finally, an additional appendix explores more in-depth real-world applications of calculus.

Learning calculus should be an exciting voyage, not a daunting task. Calculus Simplified gives you the freedom to choose your calculus experience, and the right support to help you conquer the subject with confidence.

  • An accessible, intuitive introduction to first-semester calculus
  • Nearly 200 solved problems and more than 300 exercises (all with answers)
  • No prior knowledge of exponential, logarithmic, or trigonometric functions required
  • Additional online resources—video tutorials and supplementary exercises—provided

What is Calculus?: Limits

This is the second in a series of short articles exploring calculus. The first article explored the origins of calculus. The next few articles explore the mathematics of calculus. This article focuses on the foundation of calculus: limits.

The first article in this series asked the question: What is calculus? I promised then that the second article in the series would explore the substance of calculus, the mathematics of calculus. So let’s dive right in.

Here’s my two-part answer to the “what is calculus?” question:

Calculus is a mindset—a dynamics mindset.
Contentwise, calculus is the
mathematics of infinitesimal change.

The second sentence describes the mathematics of calculus. But I don’t expect you to understand that sentence just yet. That’s where the first sentence comes in. If you ask me, all of calculus flows from a more fundamental—and intuitive—principle, articulated in the first sentence: the notion that calculus is a dynamics mindset. Let me explain.

Calculus: A New Way of Thinking

The mathematics that precedes calculus—often called “pre-calculus,” which includes algebra and geometry—largely focuses on static problems: problems devoid of change. By contrast, change is central to calculus. Calculus is all about dynamics. Example:

  • What’s the perimeter of a square of side length 2 feet? ← Pre-calculus problem.
  • How fast is the square’s perimeter changing if its side length is increasing at the constant rate of 2 feet per second? ← Calculus problem.

Now that I’ve sensitized you to thinking “calculus!” whenever you read about or infer the presence of change, take a quick look at the second sentence in my two-part answer above. What’s the last word? Change. But it’s a new type of change—infinitesimal change—and this requires some explaining. That’s our next stop.

A Philosopher Walks Into a Starbucks

First, a rough definition of “infinitesimal change”:

“Infinitesimal change” means: as close to zero change as you can imagine, but not zero change.

“What?!” I hear you saying. So let me illustrate this definition via my friend, Zeno of Elea (c. 490-430 BC). This ancient Greek philosopher thought up a set of paradoxes arguing that motion is not possible. One such paradox—the Dichotomy Paradox—can be stated as follows:

To travel a certain distance you must first traverse half of it.

Makes perfect sense. Two is one plus one. And one is one-half plus one-half. But don’t be fooled by this seemingly innocent reasoning; it’s a trap! (Admiral Ackbar!) 

To appreciate what’s going on—and connect Zeno’s paradox back to calculus—let’s pretend Zeno is in line at Starbucks, two feet away from the cash register. He’s almost done scanning the menu when the barista yells out “next!” And that’s when poor Zeno panics. He must now walk two feet, but because of his mindset, he walks only half that distance with his first step. He then walks half of the remaining distance with his second step. (Can you imagine how annoying those in line behind him are getting?) The figure below keeps track of the total distance d Zeno has walked and the change in distance Δd after each of his steps.

Fig. 1.2: Zeno trying to walk a distance of 2 feet by traversing half the remaining distance with each step. (Reprinted, with permission, from Calculus Simplified.)

Here’s a tabular representation of the action:

Table 1.1: The distance d and change in distance Δd after each of Zeno’s steps. (Reprinted, with permission, from Calculus Simplified.)

Each change d in Zeno’s distance is half the previous one. So as Zeno continues his walk, d gets closer to zero but never becomes zero.[1] If we checked back in with Zeno after he’s taken an infinite

amount of steps—what a patient barista!—the change d resulting from his next step would be . . . drum roll please . . . an infinitesimal change—as close to zero as you can imagine but not equal to zero.

This example, in addition to illustrating what an infinitesimal change is, also does two more things. First, it illustrates the dynamics mindset of calculus. We discussed Zeno walking; we thought about the change in the distance he traveled; we visualized the situation with a figure and a table that each conveyed movement. (Calculus is full of action verbs!) Second, the example challenges us. Clearly, one can walk 2 feet. But as Table 1.1 suggests, that doesn’t happen during Zeno’s walk—he approaches the 2-foot mark with each step yet never arrives. How do we describe this fact with an equation? (That’s the challenge.) No pre-calculus equation will do. We need a new concept that quantifies our very dynamic conclusion. That new concept is the mathematical foundation of calculus: limits.

Limits: The Foundation of Calculus

In modern calculus speak we paraphrase the main takeaway of Table 1.1 this way: the distance d traveled by Zeno approaches 2 as Δd approaches zero. It’s important to note that d never equals 2 and Δd never equals 0. Today we express these conclusions more compactly by writing

read “the limit of d as Δd approaches zero (but is never equal to zero) is 2.” This new equation—and what we take it to mean—remind us that d is always approaching 2 yet never arrives at 2. (Oh, the dynamics!) The same idea holds for Δd: it is always approaching 0 yet never arrives at 0. Said more succinctly:

Limits approach indefinitely (and thus never arrive).

You’ve now met the foundational concept of calculus—limit. You’ve also gotten a glimpse of what infinitesimal change means and how a limit encodes that notion. Finally, you’ve seen many times how a dynamics mindset is at the core of calculus’ new way of thinking about mathematics. In the next article in this series we’ll employ a dynamics mindset and limits to solve the three Big Problems that drove the development of calculus—instantaneous speed, the tangent line problem, and the area problem (discussed in the first post in this series). See you then!

Footnote: [1] Because each d is always half of a positive number.

 

Calculus Simplified
By Oscar E. Fernandez

Calculus is a beautiful subject that most of us learn from professors, textbooks, or supplementary texts. Each of these resources has strengths but also weaknesses. In Calculus Simplified, Oscar Fernandez combines the strengths and omits the weaknesses, resulting in a “Goldilocks approach” to learning calculus: just the right level of detail, the right depth of insights, and the flexibility to customize your calculus adventure.

Fernandez begins by offering an intuitive introduction to the three key ideas in calculus—limits, derivatives, and integrals. The mathematical details of each of these pillars of calculus are then covered in subsequent chapters, which are organized into mini-lessons on topics found in a college-level calculus course. Each mini-lesson focuses first on developing the intuition behind calculus and then on conceptual and computational mastery. Nearly 200 solved examples and more than 300 exercises allow for ample opportunities to practice calculus. And additional resources—including video tutorials and interactive graphs—are available on the book’s website.

Calculus Simplified also gives you the option of personalizing your calculus journey. For example, you can learn all of calculus with zero knowledge of exponential, logarithmic, and trigonometric functions—these are discussed at the end of each mini-lesson. You can also opt for a more in-depth understanding of topics—chapter appendices provide additional insights and detail. Finally, an additional appendix explores more in-depth real-world applications of calculus.

Learning calculus should be an exciting voyage, not a daunting task. Calculus Simplified gives you the freedom to choose your calculus experience, and the right support to help you conquer the subject with confidence.

  • An accessible, intuitive introduction to first-semester calculus
  • Nearly 200 solved problems and more than 300 exercises (all with answers)
  • No prior knowledge of exponential, logarithmic, or trigonometric functions required
  • Additional online resources—video tutorials and supplementary exercises—provided

What is Calculus?

By Oscar Fernandez

This is the first of three short articles exploring calculus. This article briefly explores its origins. The second and third articles explore its substance and impact, respectively. They will be published in the coming weeks.

What is calculus? If you were watching Jeopardy on May 31, 2019 you were treated to one whimsical answer: “developed by 2 17th century thinkers & rivals, it’s used to calculate rates of change & to torment high school students.” Funny, Jeopardy. While that answer isn’t totally accurate, what I do like about it is its structure—history, substance, and impact. This is a tried-and-true powerful framework for understanding new concepts that marries context with content. In this three-part series on calculus I’ll give you a short introduction to calculus’ history, substance, and impact to provide you with a more fulfilling answer to the question “what is calculus?” First up: a short tour of the origins of calculus.

Three Big Problems That Drove the Development of Calculus

By the mid-1600s, scientists and mathematicians had spent millennia trying to solve what I’ll call the three Big Problems in mathematics: the instantaneous speed problem, the tangent line problem, and the area problem. The figure below illustrates these.

(Reprinted, with permission, from Calculus Simplified (Princeton University Press))

The instantaneous speed problem (a) popped up in many places, most notably in connection with Isaac Newton’s studies of gravity. You see, gravity continuously accelerates a falling object, changing its velocity from instant to instant. To fully understand gravity, then, requires an understanding of instantaneous velocity. This didn’t exist before calculus. The tangent line problem (b) arose mainly as a mathematical curiosity. The ancient Greeks knew how to calculate tangent lines to circles, but until calculus no one knew how to do that for other curves. The area problem (c) popped up in a variety of places. Ancient Egyptian tax collectors, for example, needed to know how to calculate the area of irregular shapes to accurately tax landowners. Many hundreds of years later, the ancient Greeks found formulas for the areas of certain shapes (e.g., circles) but no one knew how to find the area of any shape.

From understanding gravity to calculating taxes to mathematical curiosities, the three Big Problems illustrate the broad origins of calculus. And for millennia they remained unsolved. What made them so hard was that they could not be solved with pre-calculus mathematics. For example, you’ve been taught that you need two points to calculate the slope of a line. But in the tangent line problem you’re only given one point (point P in (b)). How can one possibly calculate the slope of a line with just one point?! Similarly, we think of speed as “change in distance divided by change in time” (as in “the car zoomed by at 80 miles per hour”). That’s a problem for the instantaneous speed problem (a), because there’s zero change in time during an instant, making the denominator of “change in distance divided by change in time” zero. We can’t divide by zero, so again we’re stuck.

The Two Geniuses That Figured Everything Out

It wasn’t until the mid-1600s that real progress on solving the three Big Problems was made. One thing the Jeopardy answer above got right was the allusion to the two 17th century thinkers credited with making the most progress: Isaac Newton and Gottfried Leibniz. You probably know a few things about Newton—you may have heard about Newton’s Three Laws of Motion, which forms the foundation of much of physics—but you’ve likely heard little if at all about Leibniz. That’s because, in short, Newton used the eventual power and influence he gained after making his many discoveries and advances public to discredit Leibniz’s role in the development of calculus. (Read more about the feud here.) Yet each of these great thinkers made important contributions to calculus. Their frameworks and approaches were very different, yet each provides tremendous insight into the mathematical foundations of calculus and how calculus works.

In the next post in this series we’ll dive into those foundations. We will discuss the ultimate foundation of calculus—limits—and the two pillars erected on that foundation—derivatives and integrals—that altogether constitute the mansion of calculus. And we will discover an amazing fact: all three of the Big Problems can be solved using THE SAME approach. As is true with so many thorny problems, we will see that all that was required was a change in perspective.

 

Calculus Simplified
By Oscar E. Fernandez

Calculus is a beautiful subject that most of us learn from professors, textbooks, or supplementary texts. Each of these resources has strengths but also weaknesses. In Calculus Simplified, Oscar Fernandez combines the strengths and omits the weaknesses, resulting in a “Goldilocks approach” to learning calculus: just the right level of detail, the right depth of insights, and the flexibility to customize your calculus adventure.

Fernandez begins by offering an intuitive introduction to the three key ideas in calculus—limits, derivatives, and integrals. The mathematical details of each of these pillars of calculus are then covered in subsequent chapters, which are organized into mini-lessons on topics found in a college-level calculus course. Each mini-lesson focuses first on developing the intuition behind calculus and then on conceptual and computational mastery. Nearly 200 solved examples and more than 300 exercises allow for ample opportunities to practice calculus. And additional resources—including video tutorials and interactive graphs—are available on the book’s website.

Calculus Simplified also gives you the option of personalizing your calculus journey. For example, you can learn all of calculus with zero knowledge of exponential, logarithmic, and trigonometric functions—these are discussed at the end of each mini-lesson. You can also opt for a more in-depth understanding of topics—chapter appendices provide additional insights and detail. Finally, an additional appendix explores more in-depth real-world applications of calculus.

Learning calculus should be an exciting voyage, not a daunting task. Calculus Simplified gives you the freedom to choose your calculus experience, and the right support to help you conquer the subject with confidence.

  • An accessible, intuitive introduction to first-semester calculus
  • Nearly 200 solved problems and more than 300 exercises (all with answers)
  • No prior knowledge of exponential, logarithmic, or trigonometric functions required
  • Additional online resources—video tutorials and supplementary exercises—provided

Math, Games, and Pizza: Responses from SUMIT 2019

On April 6th and 7th, girls between 6th and 11th grade with a love of math took part in SUMIT 2019, which Princeton University Press proudly sponsors. The event offers one of the most memorable opportunities to do math while forming lasting friendships with like-minded peers. Together, girls build mathematical momentum and frequently surprise themselves with what they’re able to solve. All previous SUMITs have garnered overall ratings of 10 out of 10 by participants.

After the event, organizers spoke with some of the participants about what their role models, experiences at SUMIT, and what they love about math. Take a look at what these girls had to say…

Do you believe you made meaningful connections with other girls and/or staff who share your interest in math?

  • “Yes, I was able to meet some really cool people in a great environment.”
  • “Yes, this was a wonderful experience and I loved meeting new people who shared the same interest as me.”
  • “Yes. I didn’t realize how many girls love math. I actually made friends.”
 

What was your favorite thing about SUMIT?

  • “I really enjoy how everything fits together to solve the bigger picture. It is so cool! I also really enjoy being in an environment in which it is all girls who are enthusiastic about math.”
  • “I liked solving challenging math problems and working together as a team!!”
  • “The challenging problems and being able to work on them with others.”
  • “Meeting other girls who are enthusiastic about math and collaborating with them or problems…also, the pizza was really good”

What do you love about math?

  • “The satisfaction of solving a problem.”
  • “It’s like solving the world’s greatest puzzle”
  • “Everything except proofs.”
  • “I love how there can be challenging and hard problems that make you think and work harder.”
  • “I love that there are usually multiple ways to solve a problem and that different areas of math connect to one another.”  

Who inspires you to be a mathematician?

  • “Ada Lovelace”
  • “My family”
  • “I usually see other people persevering through math problems and that inspires me.”
  • “Math teachers I have had”
  • “I am pretty self-inspired; my interests in math inspire me. But, seeing other mathematicians around me, especially women, inspires me even more.”  

 

Princeton University Press has been a major sponsor of SUMIT since its inception in 2012, and is always proud to promote this magical escape-the-room-esque event where girls join forces to overcome challenges and become the heroines of an elaborate mathematical saga. The event offers one of the most memorable opportunities to do math while forming lasting friendships with like-minded peers. Together, girls build mathematical momentum and frequently surprise themselves with what they’re able to solve. All previous SUMITs have garnered overall ratings of 10 out of 10 by participants.

Created by Girls’ Angle, a nonprofit math club for girls, together with a team of college students, graduate students, and mathematicians, SUMIT 2019 takes place in Cambridge, MA.

InDialogue with Eelco Rohling and Sean Fleming: Earth’s changing bodies of water

Earth’s bodies of water have gone through considerable changes over time—can these changes tell us anything about climate change—and the future?

Earth’s History and the Oceans

Eelco J. Rohling

Earth’s bodies of water have gone through considerable changes over time—over a lot of time. We have clear geological signs that rivers and lakes have been around for at least 4,400 million years. It never ceases to amaze me that, within 140 million years of its red-hot formation, Earth’s surface had cooled down sufficiently for it to hold fluid water. Then, starting from about 4,000 million years ago, oceans of some shape and form have been around.

Within those ancient bodies of water, life evolved. The earliest signs of life date back to 3,700 million years ago. Then followed a long wait until the first complex life-forms appeared, at around 650 to 700 million years ago. Carbonate coral and shell-reefs became important in shallow waters from about 550 million years ago; many reef systems were formed ever since. And then another major transition took place as late as 125 to 150 million years ago, when carbonate-shelled micro-organisms evolved that rapidly occupied open-ocean surface waters across the world. These organisms are responsible for the formation of geological deposits like the striking white (chalk) cliffs of Dover. Their appearance heralded the start of a fully modern style of operation of the carbon cycle, which includes also atmospheric greenhouse gas concentrations.

Throughout the long history of Earth and its oceans, the carbon cycle and climate changes have been intimately linked. Water is a fantastic substance for absorbing vast quantities of carbon dioxide, and the presence of major bodies of water therefore puts a strong check on greenhouse gas (especially carbon dioxide) concentrations in the atmosphere. Life in the oceans in turn affects the carbon cycle because it involves interaction between dissolved carbon dioxide and both organic matter and carbonate skeletal parts that get (partially) buried and preserved as sediments.

Many ocean sediments eventually get transported into Earth’s hot mantle in subduction zones (think of the Pacific ‘ring of fire,’ where oceanic crust is thrust underneath continental crust). Heating and chemical reactions cause vapour and gas releases, which vent out via volcanoes.  This drives up carbon dioxide levels in the atmosphere and in the oceans. The oceanic part goes directly back into the oceanic carbon cycle. The atmospheric part gets involved in rock weathering on land. This consumes carbon dioxide and releases breakdown products (ions) that flow via rivers back to the oceans, where they help new carbonate formation.

It seems a perfect circle, but it isn’t. There are periods of tiny net carbon dioxide losses or gains. You would not be able to measure these from year to year, but over the multi-million-year timescales of Earth history, they add up to large atmospheric carbon dioxide variations. When this goes up, it gets warmer and weathering increases, which then slowly draws down more carbon dioxide (and vice versa). This way, Earth, with the oceans in a central role, regulates the atmospheric carbon dioxide levels under natural circumstances. It still allows for long, warm periods like the time of the dinosaurs, and excessively cold periods like ice ages or—worse—the exceptional Snowball Earth periods of about 700 million years ago. But overall, the intricately inter-linked long-term carbon cycle processes have held Earth within a ‘habitable’ climate range. Everything changed all the time (and sometimes a lot), but the pace of change was always very slow.

Enter humanity, and our fossil-fuel addiction. We have increased atmospheric carbon dioxide levels in an important manner since the start of the industrial revolution, and especially in the last 60 years. We’re not pushing the actual levels beyond the envelope of where they have been in the natural past—not by a long shot, because we’ve gone from 275 to 410 parts per million, while natural variations have been much greater than that. That’s not the worry. The worry is how fast we’re doing it. We’re doing it easily some 30 to 100 times faster than natural processes have ever done it before (even supervolcanoes cannot get close). Even if all natural removal mechanisms were fired up to 100% their known capacity, then they could offset only about one tenth of our annual emissions.

It is clear, then, that we must drastically reduce our emissions. In addition, it is clear that we must rapidly develop and implement major human-assisted processes of carbon drawdown; that is, we must help nature with the clean-up job. This is important first to deal with ongoing residual emissions that are unavoidable (for example, from cement industry or petrochemical manufacturing), and second to draw down a large part of our past emissions. Both new and existing carbon drawdown approaches are desperately needed at large scales to be able to do this. The sheer amount of carbon removal to be done is enormous.

What can we learn from Ocean and Earth history? That we’re ourselves responsible for the current climate change, and that it’s up to us to deal with it. Mother Nature by itself can and will clean up our greenhouse gases, but don’t wait up for it—it will take her several hundred thousands of years even when working flat-out. We urgently need to lend her a helping hand if we want improvement on societally relevant timescales. Doing so will, incidentally, be a major driver for innovation, development, job creation, and growth potential. What an opportunity!

Eelco J. Rohling is author of The Oceans. He is professor of ocean and climate change in the Research School of Earth Sciences at the Australian National University and at the University of Southampton’s National Oceanography Centre Southampton.

 

Global Warming and our Rivers

Sean W. Fleming

A common misconception about climate change and its impacts on environmental and social systems, like our rivers and water resources, is that it’s something that will only happen at some point in the future.  In reality, climate change is happening now.  Not only does climate vary both naturally due to processes like El Niño, and in response to local-scale human activities like urban heat islands, but global anthropogenic greenhouse gas warming has also been going on for generations – and these signals can be directly detected in actual observational datasets! 

FlemingIn fact, doing so is a crucially important part of understanding the reality and broader impacts of climate change.  When people talk about climate change, they often talk about climate models: math and software that simulate the global climate system.  And because climate models are exactly that – detailed representations of climate, but not of everything climate affects – to understand the impacts on water resources, the predictions of those climate models are taken by hydrologists and other scientists and engineers and run through still other models, such as simulations of watershed hydrology, water quality, habitat quantity and quality, or reservoir operations.  All these models are amazing technical feats, and they’re fantastic for isolating the impacts of human-caused global climate change from other sources of environmental variability.  But by necessity, they contain a lot of simplifications.  Rivers are immensely complex systems that integrate the effects of just about everything in their watersheds, from weather and climate, to forests and icefields, to land use changes like forestry and urban sprawl.  It turns out they’re also full of unexpected surprises. 

So, while the physics-based virtual realities of process simulation models are great tests of what we know about the world and are our best bet for making predictions based on that knowledge, they can only contain what we know, not what we don’t know.  In contrast, drawing sophisticated data analytics algorithms from statistics, digital signal processing, information theory, and artificial intelligence, and applying them to actual measurements of climate and the things it affects, provides a valuable “ground truth” – giving direct empirical evidence for the impacts of climate change on rivers, and often revealing previously unknown patterns that the next generation of models must then seek to explain and, ultimately, predict.

My favorite example is how mountain glaciers control water resource responses to climate change.  My doctoral studies began in 2001 with a glacier science expedition to the high peaks of the Yukon-Alaska-British Columbia border region.  Hearing summer melt water run deep in the crevasses of Trapridge Glacier and watching white-water streams gushing from its terminus, I decided to focus my research on statistical and machine learning studies of decades-long historical streamflow data in glacial watersheds.  The goal was to understand how these gigantic ice cubes modify the downstream expressions of climate change – specifically, by comparing climate variability and change responses in several glacier-fed rivers to a control population of nearby rivers that didn’t have glaciers in their headwaters. 

We made a few discoveries.  One revelation was that recent global warming affected the net downstream flow of glacial rivers in a completely different way from non-glacial watersheds: glacial rivers grew larger while non-glacial rivers shrank.   It was solid evidence of the present reality of climate change, but at the same time, specific patterns like this were poorly represented, if at all, in environmental models.  With further refinement by many scientists worldwide, this knowledge has since become part of a standard model of how water resources downstream of mountain glaciers – which lie at the heart of the continental “water towers” of the Rockies, Andes, Alps, and Himalayas, in turn feeding the headwaters of the Columbia, Amazon, Danube, Brahmaputra, and Yangtze rivers, among others – are affected by climate change.

Sean W. Fleming is author of Where the River Flows. He 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.

A Celebration of Mathematics Editor Vickie Kearn

This month, across the world, we have celebrated the enduring contributions of all women. For those of us at PUP, it is a chance as well to focus on a particularly generous, intelligent, and dynamic publisher, Vickie Kearn. In April, Vickie will retire from the Press after 18 years of synergistic and inspiring collaborations in math and computer science publishing, leaving us with a library of books that have educated and entertained millions, billions, and zillions of readers (borrowing from the title of one of her recent acquisitions).

Vickie has also been a powerful role model for women in STEM publishing, and one who empowered a population of publishers, myself included, and our new math editor Susannah Shoemaker as another. Vickie’s strength as a competitive publisher set the bar dauntingly high, but in that competition was also always an admirable collaboration, knowing that a cohort of us were changing the face of scholarly STEM publishing. It has been such a great privilege to be a colleague of Vickie’s since 2017, to travel to a math meeting with her, to meet incredibly creative authors with whom she has worked, and to learn from her at weekly project meetings. The PUP math list, particularly the popular math list, has grown exponentially and in multiple dimensions under Vickie’s leadership. If there are theorems or rules in math publishing, I would attribute these to Vickie’s rule: be smart, be curious, be generous, and be strong.

–Christie Henry

CH: Some say math is its own language. How did you learn to speak it?

I grew up in Venezuela and the English school only went through the 9th grade, so when I was 15, I went away to boarding school in North Carolina. There were only 125 girls in the whole school and there were two math teachers. One taught the girls who liked math and another taught those who did not like math. My class was very small since fewer of us liked math. Elsie Nunn was my teacher for three years and she made me fall in love with math. Before she taught anything new, she taught us about the person responsible for what we were about to learn. There was always a face behind the numbers, a person who had a family and hobbies. I found I could connect with these people. We had math club every day after school and she always had wonderful stories to tell. When I went to the University of Richmond, I knew I was going to major in math. This led to an unexpected benefit and a bit of a surprise. In the late 1960’s, University of Richmond was a Baptist school, and the classes for the men and women were held on separate sides of a lake. The one exception was that the upper level math classes were on the men’s side. Men and women were only allowed to talk with one another on Wednesday, Saturday, and Sunday, but I was able to talk with them every day because we had math class together. The surprise for me was that I was the only female math major. This felt strange at times, but Ms. Nunn had prepared me well and I got along fine with my classmates. The classes were small and we stuck together because unlike many people at UR we were more interested in math and less interested in parties.

CH: How can we continue to empower girls and women in STEM- as authors and publishers?

Based on conversations I have had with other women my age, I have had a very easy time in my career. This could be because I only have an undergraduate degree and did not experience the problems that arise in graduate school and a career as a mathematician. However, I would advise young women to join an organization that focuses on confidence building, like the Girl Scouts. I would also recommend finding a mentor—someone to look up to who can advise about a field that has long been male dominated. After I got my undergraduate degree, I taught school for 8 years, five of them in elementary school and 3 teaching math in junior high school. Most of the elementary teachers were female and the math teachers were both men and women. Although all of my college classmates in my math courses were male, it wasn’t until I went into publishing and attended my first mathematics meeting that I realized how gender specific math was.  I believe that as more women with math PhD degrees publish books and give plenary talks at conferences, the more visible they will be, and in turn, young women majoring in math will feel more a part of the mathematics community. It is critical for publishers to encourage female mathematicians to write scholarly books and ask them to review books under consideration for publication. We need more women who are advising publishers on the decisions we are making about the books we are publishing and not rely only on male scholars to help us make these decisions. Publishers need to ask female scholars to blurb books and endorse scholarly publications. There are many terrific female mathematicians and we need to increase their visibility in the book publishing community.

CH: You have published textbooks, popular math books, graphic works, works of magic, and monographs, all successful. What are the 5 essentials of a great math book?

A great book is not always measured by the number of copies it sells. It is sometimes measured by the impact it makes on a small community of scholars. Did it provide that one missing piece of information that led to the solution of an unsolved problem? Did it inspire a high school student to major in math? Did it turn a “math hater” into someone curious about math? Nevertheless, they all can benefit from some essential advice.

First, I feel that the most essential thing is that the author writes on something that she or he is passionate about. If this is the case, the reader will be engaged and love reading the book. Second, the author needs to clearly define the audience. No book can be for everyone. If the author defines the audience that way, then the book will be for no one. Third, the author needs to write for the audience and keep the mathematical level consistent throughout the book. One problem I have had with authors writing for audiences without an advanced math degree is over and under explaining math concepts. Fourth on my list is authors often introduce terms without defining them or define them by introducing other terms that need elaboration but instead lead to further confusion. Always provide examples that clarify definitions. Finally, if you have included any jokes or explanation marks in your manuscript, please delete them before sending the manuscript to your editor.

CH: What are the 5 math books you would gift to every aspiring female mathematician to learn about the art and science of math? 

Before I reveal my suggestions, I would like to say that I think that the books I have suggested would make anyone want to learn about the art and science of math. They are particularly important to me because they point out the personal relationships that can develop out of the love of a subject. It is so hard for me to select only five because each book one selects to publish is special. Each one has a backstory. Most of my choices are, OF COURSE, Princeton University Press books because they are the ones I know the best and ones I have the time to read. 

My first suggestion is not a book but a wonderful website, MacTutor History of Mathematics. I have spent many hours there and there is a link to Female Mathematicians, which is updated regularly.

The Calculus of Friendship: What a Teacher and a Student Learned about Life while Corresponding about Math by Steven Strogatz (Princeton University Press) is a book about a teacher and a student and their love of calculus as chronicled over thirty years through their letters. As you know by now, my love of math came from my high school math teacher. This author tried to help me find her. Unfortunately, we were unsuccessful. Later, at my 50th high school reunion I found out that she had passed away but it was the act of trying to find her that is illustrative of how tightly knit and wonderful I find the math community to be.

The Housekeeper and the Professor by Yoko Ogawa (Picador) was translated from Japanese. This is a novel about a math professor whose memory, due to an accident, is reset every 80 minutes, his housekeeper, and her young son. It is a wonderful story about how mathematics can bind three very different people.

Mathematics and Art: A Cultural History by Lynn Gamwell (Princeton University Press) covers the history of mathematics through exquisite works of art from antiquity to the present. I believe that learning about the history of mathematics is as important as the mathematics itself because you understand the time and place in which it is set and the math takes on more meaning.

The Seduction of Curves: The Lines of Beauty that Connect Mathematics, Art and the Nude by Allan McRobie (Princeton University Press) connects mathematics with art and engineering. This book focuses on the seven curves that are the basis of the catastrophe theory of mathematician René Thom. It is an accessible discussion of their role in nature, science, engineering, architecture, art, and other areas. Also included are their use in the work of David Hockney, Henry Moore, Anish Kapoor, and the delicate sculptures of Naum Gabo. The final two chapters focus on the collaborative work and friendship of Thom and Salvador Dalí. I searched for a book that could explain the work of René Thom for over twenty years before I found this one so it is pretty special.

CH: If you could invite five historic women mathematicians to join you at a dinner, who would they be, and why?

There are so many wonderful women mathematicians, historical and modern, that it is hard to choose just five. There are also many women who have made terrific contributions to mathematics who do not have advanced math degrees. See the references at the end of this post for additional resources.

At the top of my list would be Olga Taussky-Todd. Early in my career, I had the privilege of working with her on a book and got to know her a bit. I would love to spend more time with her. Not only was she smart, she had a great sense of humor. She made many contributions to the field of linear algebra, as did her husband, John, and we spent many hours talking about results in which, at the time, was one of my favorite topics in math. After Olga died, John gave me the poster from which the photo here was taken.

Emmy Noether is very important to me as I published a biography of her in my first position as an acquiring editor. I learned a lot about her work and would like to know more about her as a person. She has been described by many as the most important woman in the history of math. She developed the theories of rings, fields, and algebras.

Sophie Germain and I share a birthday, so of course I have to have dinner with her. Due to the great opposition against women in mathematics Sophie was not able to have a career in mathematics. Even her parents opposed her. She learned from books in her father’s library, often secretly after everyone was asleep. In spite of this she made many contributions to math such as her work on Fermat’s Last Theorem.

CH: What are five of your favorite mathematical puzzles?

Instead of listing single puzzles, I’ve chosen my favorite puzzles as types or groups. The following are some illustrations.

Word logic puzzles are fascinating and can also drive you crazy. Here is an example from Brain Food:

At a family reunion were the following people: one grandfather, one grandmother, two fathers, two mothers, four children, three grandchildren, one brother, two sisters, two sons, two daughters, one father-in-law, one mother-in-law, and one daughter-in-law. But not as many people attended as it sounds. How many were there, and who were they? Go to Rinkworks.com for more excellent puzzles and the answer to this one. However, you should try to solve it first.

Kakuro is like a crossword puzzle with numbers. Each word” must add up to the number provided in the clue above it or to the left. Words can only use the numbers 1 through 9, and a given number can only be used once in a word. Every kakuro puzzle has one and only one solution and can be solved through logic alone.

Martin Gardner was a master puzzler. If you don’t know who he is, or his puzzles (like cutting the pie, twiddled bolts, and the mutilated chessboard) head over to martin-gardner.org You will be glad you did.

I love playing Yahtzee which is more a game of logic, luck, and chance but always a lot of fun.
Jenga also does not strictly fall into the category of math but a lot of my math friends love playing it and it often appears at math meetings.

CH: how should we best compute the impact of mathematical publishing on the world?

From teaching in rural and inner-city schools for 8 years, I learned that there were so many students and adults who knew nothing about surviving in an increasingly complicated world that depends on a mastery of basic math skills. Over the past 42 years, I have seen the publication of numerous wonderful books for this very audience. These are books coming from university presses, commercial presses and society presses. These are books that have been published for the “math haters” and those who think math is hard. They present math through music and art and in graphic novels, detective stories, and puzzle books. There are ancillary materials posted on websites where readers can manipulate equations and discover new math of their own invention. As the number of books being published continues to increase, more people are clearly reading them. I am finding that there is much more enthusiasm for mathematics than there was four decades ago. There has been an increase in math clubs, math circles are very active, and the Girl Scouts announce many new STEM badges each year. I believe that publishers will continue to produce high quality books from mathematical writers around the world. This includes books that are being translated from one language into another, fostering an understanding of cultural differences through books about mathematics. I take every opportunity I can to tell people about the cool factor of math. If you are reading this post and have not discovered the wonder and empowerment of math, I’d advise you to go find a mathematician or anyone who has and ask them to let you in on the secret.

Additional Resources for inspiring information on women in STEM
MacTutor
Grandma Got STEM
A Mighty Girl

In Dialogue: Christopher Phillips and Tim Chartier on Sports & Statistics

Question: How would you describe the intersection between statistics and sports? How does one inform the other?

Christopher Phillips, author of Scouting and Scoring: Sports have undoubtedly become one of the most visible and important sites for the rise of data analytics and statistics. In some respects, sports seem to be an easy, even inevitable place to apply new statistical tools: most sports produce a lot of data across teams and seasons; games have fixed rules and clear measures of success (e.g., wins or points); players and teams have incentives to adjust in order to gain a competitive edge.

But as I discuss in my new book Scouting and Scoring: How We Know What We Know About Baseball, it is also easy to fall prey to myths about the use of statistics in sports. Though these myths apply across many sports, it is easiest to hone in on baseball, as that has been one of the most consequential areas for statistics.

Perhaps the most persistent and pernicious myth is that data emerge naturally from sporting events. There is no doubt that new video-, Doppler-, and radar-based technologies, especially when combined with increasingly cheap computing power and storage capability, have dramatically expanded the amount of data that can be collected. But it takes a huge about of labor to create, collect, clean, and curate data, even before anyone tries to analyze them. Moreover, some data, like errors in baseball, are inescapably the product of individual judgment which has to be standardized and monitored.

The second myth is that sport statistics emerged only recently, particularly after the rise of the electronic computer. In fact, statistical analysis in sports goes back decades: in baseball, playing statistics were being used to evaluate players for year-end awards and negotiate contracts for as long as professional baseball has existed. (And statistics were collected and published for cricket decades before baseball’s rules were formalized.) As new methods of statistical analysis emerged in the early twentieth century in fields like psychology and physiology, some observers immediately tried to apply them to sports. In the 1910 book Touching Second, the authors promoted the use of data for shifting around fielders and for scouting prospects, two of the most important uses of statistical data in the modern era as well. There’s certainly been a flurry of new statistics over the last twenty years, but the general idea isn’t new—consider that Allen Guttmann’s half-century-old book From Ritual to Record, highlights the “numeration of achievement” and the “quantification of the aesthetic” as defining features of modern sport.

Finally, it’s a myth that there is a fundamental divide between those who look at performance statistics (i.e., scorers) and those who evaluate bodies (i.e., scouts). The usual gloss is that scouts are holistic, subjective judges of quality whereas scorers are precise, objective measurers. In reality, baseball scouts have long used methods of quantification, whether for the pricing of amateur prospects, or for the grading of skills, or the creation of single metrics like the Overall Future Potential that reduce a player to a single number. There’s a fairly good case to be made that scouts and other evaluators of talent are even more audacious quantifiers than scorers in that the latter mainly analyze things that can be easily counted.

Tim Chartier, author of Math Bytes: Data surrounds us. The rate at which data is produced can make us seem like specks in the cavernous expanse of digital information.  Each day 3 billion photos and videos are shared on Snapchat.  In the last minute, 300 hours of video were uploaded to YouTube.  Data is offering new possibilities for insight. Sports is an area where data has a traditional role and newfound possibilities, in part, due to the enlarging datasets. 

For years, there are a number of constants in baseball that include the ball, bat, bases, and statistics like balls, strikes, hits and outs.  Statistics are and have simply been a part of the game.  You can find from the 1920 box score that Babe Ruth got 2 hits in 4 at-bats in his first game as a Yankee. While new metrics have emerged with analytical advances, the game has been well studied for some time. As Ford C. Frick stated in Games, Asterisks and People,

“Baseball is probably the world’s best documented sport.”

While this is true, the prevalence of data does not necessarily result in trusting the recommendations of those who study it.  For example, Manager Bobby Bragen stated, “Say you were standing with one foot in the oven and one foot in an ice bucket. According to the percentage people, you should be perfectly comfortable.”  This underscores an important aspect of data and analytics.  Data, inherently, can lead to insight but it becomes actionable when one trusts in how accurately it reflects our world. 

Other sports, while not as statistically robust as baseball also have an influx of data.  In basketball, cameras positioned in the rafters report the (x,y) position of every player on the court and the (x,y,z) position of the ball throughout the entire game every fraction of a second.  As such, we can replay aspects of games via this data for years to come.  With such information comes new information.  For example, we know that Steph Curry, while averaging just over 34 minutes a game, runs, on average, just over 2.6 miles per game. He also runs almost a quarter of a mile more on offense than defense. 

While such data can be stunning with its size and detail, it also comes with challenges. How do you recognize a pick and roll versus an isolation play simply from essentially dots moving in a plane?  Further, basketball, like football but unlike baseball, generally involves multiple players at a time.  How much credit do players get for a basket on offense?  A player’s position may open up possibilities for scoring, even if that player didn’t touch the ball.  As such, metrics have been and continued to be created in order to better understand the game.

Sports are played with a combination of analytics, gut and experience.  What combination depends on the sport, player, coach and context.  Nonetheless, data is here and will continue to give insight on the game. 

Anna Frebel on women in science who paved the way

As a young girl growing up in Germany, I always felt drawn to the idea of discovery. Noticing my expanding interest in science, my mother cultivated my curiosity about the world and our place in the universe. She repeatedly gifted me biographies of women scientists who defied the odds to pioneer discoveries in their respective fields. Indeed, these stories of accomplishment and determination greatly fueled my desire to become an astronomer.

As I spent countless hours reading and exploring on my own, I would find myself alone but never lonely in my educational pursuits. Little did I know, this form of self-reliance would serve me well as I completed my advanced degrees and research into finding ancient stars to learn about the cosmic origin of the chemical elements — published in my book Searching for the Oldest Stars: Ancient Relics from the Early Universe.

These days, I fly to Chile to use large telescopes once or twice per year. This work means long hours spent in solitude carrying out our observations. It is usually then that I most strongly feel it again: a sense of fulfillment and pride in this discovery work which I was lucky to gain a long time ago by reading the life stories of women in science.

I fondly remember learning about the thrill of traveling across continents with inspiring naturalist and scientific illustrator Maria Sybilla Merian (1647-1717) as she was researching and illustrating caterpillars and insects and their various life stages in the most detailed of ways. I met fierce and gifted mathematician Sofia Kovalevskaya (1850-1891) who was the first woman in math to obtain a PhD (coincidentally from the university in my hometown) and who later became the first woman math professor in Sweden. One of the most profound role models remains two time Nobel prize winner Marie Curie (1867-1934), a remarkably persistent physicist and chemist who discovered radioactivity and new chemical elements. Reading about her years of long work in the lab to eventually isolate 1/10th of a gram of radium, I too could imagine becoming a scientists. Curie’s immense dedication to science and humanity encapsulated everything I wanted to do with my life. Finally, atomic physicist Lise Meitner (1878-1968) showed me how groundbreaking discoveries can be made when daring to invoke unconventional ideas to explain experimental results. She realized that atoms cannot be arbitrarily large. If too heavy, they fission, break apart, and thus produce various heavy elements from the bottom half of the periodic table.

Throughout the years, these stories have stayed with me. Their impact and insight gave me comfort and guidance during the many phases of my academic and professional life. It was more than a question of gender. It was the confidence in knowing the women who came before me had created a path for the next generation to travel, myself included.

Some of these books have traveled with me as I moved from Germany to Australia to the US for my career and my path to professorship. In many ways, I’ve incorporated central aspects from the lives and research of these giants in science into my own work. Hence, these women remain in my heart and soul – and by knowing their stories, I never feel alone. From my perspective, reading biographies thus remains one of the most important forms of personal and professional mentorship and growth.

Recently, through a collaboration with STEM on Stage, I became a science adviser to the living history film “Humanity Needs Dreamers: A Visit With Marie Curie”. I also rekindled my love for these ladies and their stories by crafting a short play in which I portray Lise Meitner as she recalls her discovery of nuclear fission in 1938/39. The play “Pursuit of Discovery” is followed by a slide presentation about my research and how Meitner’s work provided the theoretical framework for my current studies into the formation of the heaviest elements in the periodic table.

I’m often asked about the challenges facing women in science. Although we have made significant progress, one of the main challenges is providing mentorship and role models. In astronomy, the number of senior level women remains small compared to our male counterparts. To help change this ratio, I’ve devoted time to help mentor undergraduate and graduate women in physics and astronomy.

Whether reading biographies of women in science, mentoring, or becoming Meitner on stage, it is important to give credit to those who paved the way for the next generation, and to highlight the amazing and inspiring accomplishments of women in science. As I write in my book, “we stand on the shoulders of giants.” And by knowing their stories, we can better know ourselves.

Anna Frebel is an Associate Professor in the Department of Physics at the Massachusetts Institute of Technology. She has received numerous international honors and awards for her discoveries and analyses of the oldest stars. She lives in Cambridge, Massachusetts.

 

 

90 Years Ago Today: Einstein’s 50th Birthday

This post is made available by the Einstein Papers Project

Einstein’s fiftieth birthday appears to have been more of a cause for celebration by others than for himself. Having lived under intense scrutiny from the (mostly) adoring public and intrusive journalists for 10 years already, Einstein made valiant efforts to avoid attention from the press on this momentous occasion. He was particularly keen to avoid the hullabaloo ratcheting up for his fiftieth in Berlin. The day before his birthday, a New York Times article, Einstein Flees Berlin to Avoid Being Feted reported that: “To evade all ceremonies and celebrations, he suddenly departed from Berlin last night and left no address. Even his most intimate friends will not know his whereabouts.”

Einstein’s decision allowed him and his family relative respite. While Einstein hid in a countryside retreat, “[t]elegraph messengers, postmen and delivery boys had to wait in line hours today in front of the house No. 5 Haberland Strasse, delivering congratulations and gifts to Albert Einstein on the occasion of his fiftieth birthday today,” according to the March 15 issue of the Jewish Daily Bulletin. Above is one card of the many that Einstein received on and around his birthday; it was made by a pupil at the Jüdische Knabenschule, Hermann Küchler.

After all, an intrepid reporter did find Einstein – in a leafy neighborhood of Berlin called Gatow, half an hour from the city center. A report for avid fans, Einstein Found Hiding on his Birthday, in the March 15 edition of The New York Times provides a gamut of details from the color of his sweater to the menu for his birthday dinner and the array of gifts found on a side table. Happy reading, on this, the 140th anniversary of Einstein’s birth!

03-07-19

Einstein’s 50th will be covered in Volume 16 of The Collected Papers of Albert Einstein. Of the many and various resources we refer to for historical research, the two used for this web post were: The New York Times archive: Times Machine and the Jewish Telegraphic Agency Archive. Access to the Times Machine requires a subscription to The New York Times. The card, item number 30-349, is held at the Albert Einstein Archives at HUJI.

Pi: A Window into the World of Mathematics

Mathematicians have always been fascinated by Pi, the famous never-ending never-repeating decimal that rounds to 3.14. But why? What makes Pi such an interesting number? Every mathematician has their own answer to that question. For me, Pi’s allure is that it illustrates perfectly the arc of mathematics. Let me explain what I mean by taking you on a short mathematical adventure.

Picture yourself in a kitchen, rummaging the pantry for two cans of food. Let’s say you’ve found two that have circular bases of different diameters d1 and d2. Associated with each circle is a circumference value, the distance you’d measure if you walked all the way around the circle.

Were you to perfectly measure each circle’s circumference and diameter you would discover an intriguing relationship:

In other words, the ratio of each circle’s circumference to its diameter doesn’t change, even though one circle is bigger than the other. (This circumference-to-diameter number is  (“Pi”), the familiar 3.14-ish number.) This is the first stop along the arc of mathematics: the discovery of a relationship between two quantities.

Where this story gets very interesting is when, after grabbing even more cans and measuring the ratio of their circumferences to their diameters—you seem to have lots of free time on your hands—you keep finding the same ratio. Every. Time. This is the second stop along the arc of mathematics: the discovery of a pattern. Shortly after that, you begin to wonder: does every circle, no matter its size, have the same circumference-to-diameter ratio? You have reached the third stop along the arc of mathematics: conjecture. (Let’s call our circumference-to-diameter conjecture The Circle Conjecture.)

At first you consider proving The Circle Conjecture by measuring the ratio C/d for every circle. But you soon realize that this is impossible. And that’s the moment when you start truly thinking like a mathematician and begin to wonder: Can I prove The Circle Conjecture true using mathematics? You have now reached the most important stop along the arc of mathematics: the search for universal truth.

One of the first thinkers to make progress on The Circle Conjecture was the Greek mathematician Euclid of Alexandria. Euclid published a mammoth 13-book treatise text called Elements circa 300 BC in which he, among other accomplishments, derived all the geometry you learned in high school from just five postulates. One of Euclid’s results was that the ratio of a circle’s area A to the square of its diameter d2 is the same for all circles:

This is close to what we are trying to prove in The Circle Conjecture, but not the same. It would take another giant of mathematics—the Greek mathematician Archimedes of Syracuse—to move us onto what is often the last stop on the arc of mathematics: thinking outside the box.

Archimedes went back to Euclid’s five postulates, all but one of which dealt with lines, and extended some of Euclid’s postulates to handle curves. With these new postulates Archimedes was able to prove in his treatise Measurement of a Circle (circa 250 BC) that the area, circumference, and radius r of a circle are related by the equation:

(You may recognize this as the area of a triangle with base C and height r. Indeed, Archimedes’ proof of the formula effectively “unrolls” a circle to produce a triangle and then calculates its area.) Combining Archimedes’ formula with Euclid’s result, and using the fact that r = d/2, yields:

Et Voilà! The Circle Conjecture is proved! (To read more about the mathematical details involved in proving The Circle Conjecture, I recommend this excellent article.)

This little Pi adventure illustrated the core arc of mathematics: discovery of a relationship between to quantities; discovery of a more general pattern; statement of a conjecture; search for a proof of that conjecture; and thinking outside the box to help generate a proof. Let me end our mathematical adventure by encouraging you to embark on your own. Find things you experience in your life that are quantifiable and seem to be related (e.g., how much sleep you get and how awake you feel) and follow the stops along the arc of mathematics. You may soon afterward discover another universal truth: anyone can do mathematics! All it takes is curiosity, persistence, and creative thinking. Happy Pi Day!

 

Oscar E. Fernandez is associate professor of mathematics at Wellesley College. He is the author of Calculus Simplified, Everyday Calculus, and The Calculus of Happiness (all Princeton).

Ken Steiglitz: Happy π Day!

As every grammar school student knows, π is the ratio of the circumference to the diameter of a circle. Its value is approximately 3.14…, and today is March 14th, so Happy π Day! The digits go on forever, and without a pattern. The number has many connections with computers, some obvious, some not so obvious, and I’ll mention a few.

The most obvious connection, I suppose, is that computers have allowed enthusiasts to find the value of π to great accuracy. But how accurately do we really need to know its value? Well, if we knew the diameter of the Earth precisely, knowing π to 14 or 15 decimal places would enable us to compute the length of the equator to within the width of a virus. This accuracy was achieved by the Persian mathematician Jamshīd al-Kāshī in the early 15th century. Of course humans let loose with digital computers can be counted on to go crazy; the current record is more than 22 trillion digits. (For a delightful and off-center account of the history of π, see A History of Pi, third edition, by Petr Beckmann, St. Martin’s Press, New York, 1971. The anti-Roman rant in chapter 5 alone is worth the price of admission.)

A photo of a European wildcat, Felis silvestris silvestris. The original photo is on the left. On the right is a version where the compression ratio gradually increases from right to left, thereby decreasing the image quality. The original photograph is by Michael Ga¨bler; it was modified by AzaToth to illustrate the effects of compression by JPEG. [Public domain, from Wikimedia Commons]

Don’t condemn the apparent absurdity of setting world records like this; the results can be useful. Running the programs on new hardware or software and comparing results is a good test for bugs. But more interesting is the question of just how the digits of π are distributed. Are they essentially random? Do any patterns appear? Is there a message from God hidden in this number that, after all, God created? Alas, so far no pattern has been found, and the digits appear to be “random” as far as statistical tests show. On the other hand, mathematicians have not been able to prove this one way or another.

Putting aside these more or less academic thoughts, the value of π is embedded deep in the code on your smartphone or computer and plays an important part in storing the images that people are constantly (it seems to me) scrolling through. Those images take up lots of space in memory, and they are often compressed by an algorithm like JPEG to economize on that storage. And that algorithm uses what are called “circular functions,” which, being based on the circle, depend for their very life on… π. The figure shows how the quality of an original image (left) degrades as it is compressed more and more, as shown on the right.

I’ll close with an example of an analog computer which we can use to find the value of π. The computer consists of a piece of paper that is ruled with parallel lines 3 inches (say) apart, and a needle 3 inches long. Toss the needle so that it has an equal chance of landing anywhere on the paper, and an equal chance of being at any angle. Then it turns out that the chance of the needle intersecting a line on the piece of paper is 2/π, so that by repeatedly tossing the needle and counting the number of times it does hit a line we can estimate the value of π. Of course to find the value of π to any decent accuracy we need to toss the needle an awfully large number of times. The problem of finding the probability of a needle tossed this way was posed and solved by Georges-Louis Leclerc, Comte de Buffon in 1777, and the setup is now called Buffon’s Needle. This is just one example of an analog computer, in contrast to our beloved digital computers, and you can find much more about them in The Discrete Charm of the Machine.

Ken Steiglitz is professor emeritus of computer science and senior scholar at Princeton University. His books include The Discrete Charm of the MachineCombinatorial OptimizationA Digital Signal Processing Primer, and Snipers, Shills, and Sharks (Princeton). He lives in Princeton, New Jersey.