Ken Steiglitz: When Caruso’s Voice Became Immortal

We’re excited to introduce a new series from Ken Steiglitz, computer science professor at Princeton University and author of The Discrete Charm of the Machine, out now. 

The first record to sell a million copies was Enrico Caruso’s 1904 recording of “Vesti la giubba.” There was nothing digital, or even electrical about it; it was a strictly mechanical affair. In those days musicians would huddle around a horn which collected their sound waves, and that energy was coupled mechanically to a diaphragm and then to a needle that traced the waveforms on a wax or metal-foil cylinder or disc. For many years even the playback was completely mechanical, with a spring-wound motor and a reverse acoustical system that sent the waveform from what was often a 78 rpm shellac disc to a needle, diaphragm, and horn. Caruso almost single-handedly started a cultural revolution as the first recording star and became a household name—and millionaire (in 1904 dollars)—in the process. All without the benefit of electricity, and certainly purely analog from start to finish. Digital sound recording for the masses was 80 years in the future.

Enrico Caruso drew this self portrait on April 11, 1902 to commemorate his first recordings for RCA Victor. The process was completely analog and mechanical. As you can see, Caruso sang into a horn; there were no microphones. [Public domain, from Wikimedia Commons]

The 1904 Caruso recording I mentioned is perhaps the most famous single side ever made and is readily available online. It was a sensation and music lovers who could afford it were happy to invest in the 78 rpm (or simply “78”) disc, not to mention the elaborate contraption that played it. In the early twentieth century a 78 cost about a dollar or so, but 1904 dollars were worth about 30 of today’s dollars, a steep price for 2 minutes and 28 seconds of sound full of hisses, pops, and crackles, and practically no bass or treble. In fact the disc surface noise in the versions you’re likely to hear today has been cleaned up and the sound quality greatly improved—by digital processing of course. But being able to hear Caruso in your living room was the sensation of the new century.The poor sound quality of early recordings was not the worst of it. That could be fixed, and eventually it was. The long-playing stereo record (now usually called just “vinyl”) made the 1960s and 70s the golden age of high fidelity, and the audiophile was born. I especially remember, for example, the remarkable sound of the Mercury Living Presence and Deutsche Grammophon labels. The market for high-quality home equipment boomed, and it was easy to spend thousands of dollars on the latest high-tech gear. But all was not well. The pressure of the stylus, usually diamond, on the vinyl disc wore both. There is about a half mile of groove on an LP record, and the stylus that tracks it has a very sharp, very hard tip; records wear out. Not as quickly as the shellac discs of the 20s and 30s, but they wear out.

The noise problem for analog recordings is exacerbated when many tracks are combined, a standard practice in studio work in the recording industry. Sound in analog form is just inherently fragile; its quality deteriorates every time it is copied or played back on a turntable or past a tape head.

Everything changed in 1982 with the introduction of the compact disc (CD), which was digital. Each CD holds about 400 million samples of a 74-minute stereo sound waveform, each sample represented by a 2-byte number (a byte is 8 bits). In this world those 800 million bytes, or 6.4 billion bits (zeros or ones) can be stored and copied forever, absolutely perfectly. Those 6.4 billion bits are quite safe for as long as our civilization endures.

There are 19th century tenors whose voices we will never hear. But Caruso, Corelli, Domingo, Pavarotti… their digital voices are truly immortal.

SteiglitzKen 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. He lives in Princeton, New Jersey.

Browse our 2019 Mathematics Catalog

Our new Mathematics catalog includes an exploration of mathematical style through 99 different proofs of the same theorem; an outrageous graphic novel that investigates key concepts in mathematics; and a remarkable journey through hundreds of years to tell the story of how our understanding of calculus has evolved, how this has shaped the way it is taught in the classroom, and why calculus pedagogy needs to change.

If you’re attending the Joint Mathematics Meetings in Baltimore this week, you can stop by Booth 500 to check out our mathematics titles!


Integers and permutations—two of the most basic mathematical objects—are born of different fields and analyzed with different techniques. Yet when the Mathematical Sciences Investigation team of crack forensic mathematicians, led by Professor Gauss, begins its autopsies of the victims of two seemingly unrelated homicides, Arnie Integer and Daisy Permutation, they discover the most extraordinary similarities between the structures of each body. Prime Suspects is a graphic novel that takes you on a voyage of forensic discovery, exploring some of the most fundamental ideas in mathematics. Beautifully drawn and wittily and exquisitely detailed, it is a once-in-a-lifetime opportunity to experience mathematics like never before.

Ording 99 Variations on a Proof book cover

99 Variations on a Proof offers a multifaceted perspective on mathematics by demonstrating 99 different proofs of the same theorem. Each chapter solves an otherwise unremarkable equation in distinct historical, formal, and imaginative styles that range from Medieval, Topological, and Doggerel to Chromatic, Electrostatic, and Psychedelic. With a rare blend of humor and scholarly aplomb, Philip Ording weaves these variations into an accessible and wide-ranging narrative on the nature and practice of mathematics. Readers, no matter their level of expertise, will discover in these proofs and accompanying commentary surprising new aspects of the mathematical landscape.


Bressoud Calculus Reordered book cover

Exploring the motivations behind calculus’s discovery, Calculus Reordered highlights how this essential tool of mathematics came to be. David Bressoud explains why calculus is credited to Isaac Newton and Gottfried Leibniz in the seventeenth century, and how its current structure is based on developments that arose in the nineteenth century. Bressoud argues that a pedagogy informed by the historical development of calculus presents a sounder way for students to learn this fascinating area of mathematics.

Browse our 2019 Computer Science Catalog

Our new Computer Science catalog includes an introduction to computational complexity theory and its connections and interactions with mathematics; a book about the genesis of the digital idea and why it transformed civilization; and an intuitive approach to the mathematical foundation of computer science.

If you’re attending the Information Theory and Applications workshop in San Diego this week, you can stop by the PUP table to check out our computer science titles!


Mathematics and Computation provides a broad, conceptual overview of computational complexity theory—the mathematical study of efficient computation. Avi Wigderson illustrates the immense breadth of the field, its beauty and richness, and its diverse and growing interactions with other areas of mathematics. With important practical applications to computer science and industry, computational complexity theory has evolved into a highly interdisciplinary field that has shaped and will further shape science, technology, and society. 


Steiglitz Discrete Charm of the Machine book cover

A few short decades ago, we were informed by the smooth signals of analog television and radio; we communicated using our analog telephones; and we even computed with analog computers. Today our world is digital, built with zeros and ones. Why did this revolution occur? The Discrete Charm of the Machine explains, in an engaging and accessible manner, the varied physical and logical reasons behind this radical transformation, and challenges us to think about where its future trajectory may lead.

Lewis Zax Essential Discrete Mathematics for Computer Science

Discrete mathematics is the basis of much of computer science, from algorithms and automata theory to combinatorics and graph theory. This textbook covers the discrete mathematics that every computer science student needs to learn. Guiding students quickly through thirty-one short chapters that discuss one major topic each, Essential Discrete Mathematics for Computer Science can be tailored to fit the syllabi for a variety of courses. Fully illustrated in color, it aims to teach mathematical reasoning as well as concepts and skills by stressing the art of proof.

Calling Girls Who Love Math: Register for Girls’ Angle’s SUMIT 2019!

Get ready for a new mathematical adventure! SUMIT 2019 is coming April 6 and 7 with an all-new plot and math problems galore.

If you’re a 6th-11th grade girl who loves math, you’ll love SUMIT! There will be challenges for all levels and key leadership roles to fulfill. You’ll emerge with an even greater love of math, new friends, and lasting memories.

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.

Registration opens at 2 pm ET on Sunday, February 10 on a first-come-first-served basis and there are limited slots, so register quickly!

For more information, please visit

Edward Burger on Making Up Your Own Mind

BurgerWe solve countless problems—big and small—every day. With so much practice, why do we often have trouble making simple decisions—much less arriving at optimal solutions to important questions? Are we doomed to this muddle—or is there a practical way to learn to think more effectively and creatively? In this enlightening, entertaining, and inspiring book, Edward Burger shows how we can become far better at solving real-world problems by learning creative puzzle-solving skills using simple, effective thinking techniques. Making Up Your Own Mind teaches these techniques—including how to ask good questions, fail and try again, and change your mind—and then helps you practice them with fun verbal and visual puzzles. A book about changing your mind and creating an even better version of yourself through mental play, Making Up Your Own Mind will delight and reward anyone who wants to learn how to find better solutions to life’s innumerable puzzles. 

What are the practical applications of this book for someone who wants to improve their problem-solving skills?

The practicality goes back to the practical elements of one’s own education. Unfortunately, many today view “formal education” as the process of learning, but what they really mean is “knowing”—knowing the facts, dates, methodologies, templates, algorithms, and the like. Once the students demonstrate that newly-found knowledge by reproducing it back to the instructor on a paper or test they quickly let it all go from their short-term memories and move on. Today this kind of “knowledge” can be largely found via any search engine on any smart device. So in our technological information age, what should “formal education” mean?  Instead of focusing solely on “knowing,” it intentionally must also teach “growing”—growing the life of the mind. The practices offered in this volume attempt to do just that: offer readers a way to hone and grow their own thinking while sharpening their own minds. Those practices can then be directly applied to their everyday lives as they try to see the issues around them with greater clarity and creativity to make better decisions. The practical applications certainly will include their enhanced abilities to create better solutions to all the problems they encounter. But from my vantage point as an educator, the ultimate practical application is to help readers flourish and continue along a life-long journey in which they become better versions of themselves tomorrow than they are today. 

How has applying the problem-solving skills described in your book helped you in your everyday life?

In my leadership role as president of Southwestern University, I am constantly facing serious and complex challenges that need to be solved or opportunities to be seized. Those decisions require wisdom, creativity, focus on the macro issues while being mindful of the micro implications. Then action is required along with careful follow-up on the consequences of those decisions moving forward. I use the practices of effective thinking outlined in this book—including my personally favorite: effective failure—in every aspect of my work as president and I believe they have served me well. Effective failure, by the way, is the practice of intentionally not leaving a mistake or misstep until a new insight or deeper understanding is realized.  It is not enough to say, “Oh, that didn’t work, I’ll try something else.” That’s tenacity, which is wonderful, but alone is also ineffective failure.  Before trying that something else, this book offers practical but mindful ways of using one’s own errors to be wise guides to deeper understanding that natural lead to what to consider next. I also believe that through these varied practices of thinking I continue to grow as an educator, as a leader, as a mathematician, and as an individual who has committed his professional life to try to make the world better by inspiring others to be better. 

Can we really train our brains to be better problem solvers?


Would you care to elaborate on that last, one-word response?

Okay, okay—But I hope I earned some partial credit for being direct and to-the-point. Many believe that their minds are the way they are and cannot be changed. In fact, we are all works-in-progress and capable of change—not the disruptive change that makes us into someone we’re not, but rather incremental change that allows us to be better and better versions of ourselves as we grow and evolve. That change in mindset does not require us to “think harder” (as so many people tell us), but rather to “think differently” (which is not hard at all after we embrace different practices of thinking, analysis, and creativity). Just as we can improve our tennis game, our poker skills, and the playing of the violin, we can improve our thinking and our minds. This book offers practical and straight-forward ways to embraces those enhance practices and puzzles to practice that art in an entertaining but thought-provoking way.

Why do you refer to “puzzle-solving” rather than the more typical phrase, “problem-solving?”

Because throughout our lives we all face challenges and conundrums that need to be faced and resolved as well as opportunities and possibilities that need to be either seized or avoided. Those negative challenges and possibilities are the problems in our lives. But everything we face—positive, negative, or otherwise—are the puzzles that life presents to us. Thus, I do not believe we should call mindful practices that empower us to find innovative or smart solutions “problem-solving.” We should call those practices that enhance our thinking about all the varied puzzles in our lives what they truly are: “puzzle-solving.” Finally, I believe we thrive within an optimistic perspective—and no one likes problems—but most do enjoy puzzles.

How did this book come about?

As with most things, this project natural evolved from a confluence of many previous experiences. My close collaborator, Michael Starbird, and I have been thinking about effective thinking collaboratively and individually for dozens of years. That effort resulted in our book, The 5 Elements of Effective Thinking (published by Princeton University Press and referenced in this latest work). Then when I began my work as president of Southwestern University over five years ago, I wanted to offer a class that was not a “typical” mathematics course, but rather a class that would capture the curiosity of all students who wonder how they can amplify their own abilities to grow and think more effectively—originally, wisely, and creatively. So I created a course entitled Effective Thinking through Creative Puzzle-Solving, and I have been teaching it every year at Southwestern since 2016.

How did your students change through their “puzzle-solving” journey?

Of course that question is best answered by my students at Southwestern University, and I invite you to visit our campus and talk with them to learn more. From my perspective, I have enjoyed seeing them become more open-minded, think in more creative and original ways (“thinking outside the box”), practice a more mindful perspective, and make time for themselves to be contemplative and reflective. Also, I have them write a number of essays (which I personally grade), and over the course of our time together, I have seen their writing and overall communication improve. Obviously, I am very proud of my students.

Edward B. Burger is the president of Southwestern University, a mathematics professor, and a leading teacher on thinking, innovation, and creativity. He has written more than seventy research articles, video series, and books, including The 5 Elements of Effective Thinking (with Michael Starbird) (Princeton), and has delivered hundreds of addresses worldwide. He lives in Georgetown, Texas.

Brian Kernighan on Millions, Billions, Zillions

KernighanNumbers are often intimidating, confusing, and even deliberately deceptive—especially when they are really big. The media loves to report on millions, billions, and trillions, but frequently makes basic mistakes or presents such numbers in misleading ways. And misunderstanding numbers can have serious consequences, since they can deceive us in many of our most important decisions, including how to vote, what to buy, and whether to make a financial investment. In this short, accessible, enlightening, and entertaining book, leading computer scientist Brian Kernighan teaches anyone—even diehard math-phobes—how to demystify the numbers that assault us every day. Giving you the simple tools you need to avoid being fooled by dubious numbers, Millions, Billions, Zillions is an essential survival guide for a world drowning in big—and often bad—data.

Why is it so important to be able to spot “bad statistics?”

We use statistical estimates all the time to decide where to invest, or what to buy, or what politicians to believe. Does a college education pay off financially? Is marijuana safer than alcohol? What brands of cars are most reliable? Do guns make society more dangerous? We make major personal and societal decisions about such topics, based on numbers that might be wrong or biased or cherry-picked. The better the statistics, the more accurately we can make good decisions based on them.

Can you give a recent example of numbers being presented in the media in a misleading way?

“No safe level of alcohol, new study concludes.” There were quite a few variants of this headline in late August. There’s no doubt whatsoever that heavy drinking is bad for you, but this study was actually a meta-analysis that combined the results of nearly 700 studies covering millions of people.  By combining results, it concluded that there was a tiny increase in risk in going from zero drinks a day to one drink, and more risk for higher numbers. But the result is based on correlation, not necessarily causation, and ignores potentially related factors like smoking, occupational hazards, and who knows what else. Fortunately, quite a few news stories pointed out flaws in the study’s conclusion.  To quote from an excellent review at the New York Times, “[The study] found that, over all, harms increased with each additional drink per day, and that the overall harms were lowest at zero. That’s how you get the headlines.”

What is an example of how a person could spot potential errors in big numbers?

One of the most effective techniques for dealing with big numbers is to ask, “How would that affect me personally?” For example, a few months ago a news story said that a proposed bill in California would offer free medical care for every resident, at a cost of $330 million per year. The population of California is nearly 40 million, so each person’s share of the cost would be less than $10. Sounds like a real bargain, doesn’t it? Given what we know about the endlessly rising costs of health care, it can’t possibly be right. In fact, the story was subsequently corrected; the cost of the bill would be $330 *billion* dollars, so each person’s share would be more like $10,000. Asking “What’s my share?” is a good way to assess big numbers.

In your book you talk about Little’s Law. Can you please describe it and explain why it’s useful?

Little’s Law is a kind of conservation law that can help you assess the accuracy of statements like “every week, 10,000 Americans turn 65.” Little’s Law describes the relationship between the time period (every week), the number of things involved (10,000 Americans), and the event (turning 65). Suppose there are 320 million Americans, each of whom is born, lives to age 80, then dies. Then 4 million people are born each year, 4 million die, and in fact there are 4 million at any particular age. Now divide by 365 days in a year, to see that about 11,000 people turn 65 on any particular day. So the original statement can’t be right—it should have said “per day,” not “per week.” Of course this ignores birth rate, life expectancy, and immigration, but Little’s Law is plenty good enough for spotting significant errors, like using weeks instead of days.

Is presenting numbers in ways designed to mislead more prevalent in the era of “alternative facts” than in the past?

I don’t know whether deceptive presentations are more prevalent today than they might have been, say, 20 years ago, but it’s not hard to find presentations that could mislead someone who isn’t paying attention. The technology for producing deceptive graphs and charts is better than it used to be, and social media makes it all too easy to spread them rapidly and widely.

Brian W. Kernighan is professor of computer science at Princeton University. His many books include Understanding the Digital World: What You Need to Know about Computers, the Internet, Privacy, and Security. He lives in Princeton, New Jersey.

Browse Our 2018 Math Catalog

Our new Mathematics catalog includes the story of ten great ideas about chance and the thinkers who developed them, an introduction to the language of beautiful curves, and a look at how empowering mathematics can be.

If you plan on attending the Joint Mathematics Meeting this week, stop by Booths 504-506 to see our full range of Mathematics titles and more.

In the sixteenth and seventeenth centuries, gamblers and mathematicians transformed the idea of chance from a mystery into the discipline of probability, setting the stage for a series of breakthroughs that enabled or transformed innumerable fields, from gambling, mathematics, statistics, economics, and finance to physics and computer science. This book tells the story of ten great ideas about chance and the thinkers who developed them, tracing the philosophical implications of these ideas as well as their mathematical impact.

Complete with a brief probability refresher, Ten Great Ideas about Chance is certain to be a hit with anyone who wants to understand the secrets of probability and how they were discovered.

Curves are seductive. These smooth, organic lines and surfaces—like those of the human body—appeal to us in an instinctive, visceral way that straight lines or the perfect shapes of classical geometry never could. In this large-format book, lavishly illustrated in color throughout, Allan McRobie takes the reader on an alluring exploration of the beautiful curves that shape our world—from our bodies to Salvador Dalí’s paintings and the space-time fabric of the universe itself.

The Seduction of Curves focuses on seven curves and describes the surprising origins of their taxonomy in the catastrophe theory of mathematician René Thom. In an accessible discussion illustrated with many photographs of the human nude, McRobie introduces these curves and then describes their role in nature, science, engineering, architecture, art, and other areas.  The book also discusses the role of these curves in the work of such artists as David Hockney, Henry Moore, and Anish Kapoor, with particular attention given to the delicate sculptures of Naum Gabo and the final paintings of Dalí, who said that Thom’s theory “bewitched all of my atoms.”

In The Calculus of Happiness, Oscar Fernandez shows us that math yields powerful insights into health, wealth, and love. Using only high-school-level math (precalculus with a dash of calculus), Fernandez guides us through several of the surprising results, including an easy rule of thumb for choosing foods that lower our risk for developing diabetes (and that help us lose weight too), simple “all-weather” investment portfolios with great returns, and math-backed strategies for achieving financial independence and searching for our soul mate. Moreover, the important formulas are linked to a dozen free online interactive calculators on the book’s website, allowing one to personalize the equations.

Fernandez uses everyday experiences—such as visiting a coffee shop—to provide context for his mathematical insights, making the math discussed more accessible, real-world, and relevant to our daily lives. A nutrition, personal finance, and relationship how-to guide all in one, this book invites you to discover how empowering mathematics can be.

SUMIT 2018: A math collaboration

by C. Kenneth Fan
President and Founder of Girls’ Angle, an organization that connects mentors with girls who love math

For decades, math extracurricular activity in the United States has been dominated by the math competition. I, myself, participated in and enjoyed math competitions when I was growing up. Many school math clubs are centered on math contest prep. Today, there are dozens upon dozens of math competitions. While many students gain much from math competitions, many others, for a variety of good reasons, do not find inspiration in math competitions to do more math, and the best way to learn math is to do math.

When I founded Girls’ Angle over ten years ago, a main task was to create new, non-competitive, mathematically compelling avenues into math that appeal to those who, for whatever reason, may not be so inspired by math competitions. To celebrate the end of our first year, we baked a brownie for the girls, but it wasn’t a rectangular brownie—it was a trapezoid, and nobody could have any brownie until members figured out how to split the brownie into equal pieces for all. We were counting on them to succeed because we wanted brownie!

It became a Girls’ Angle tradition to celebrate the conclusion of every semester with a collaborative math Single Digitspuzzle, and every semester the puzzle has grown more elaborate. It finally dawned on me that these collaborative end-of-session math puzzles could well serve as robust, mathematically-intense, but fully collaborative alternatives to the math competition. To directly contrast the concept with that of the math competition, we called these events “math collaborations.” On January 21, 2012, after 4 years of in-house development, we took the concept out of Girls’ Angle with SUMIT 2012, which took place at MIT in conjunction with MIT’s Undergraduate Society of Women in Mathematics. Then, on March 7, 2012, the Buckingham, Browne, and Nichols Middle School became the first school to host a math collaboration. The success of these events led to annual math collaborations at Buckingham, Browne, and Nichols, and, to date, over 100 other math collaborations at schools, libraries, and other venues, such as Girl Scout troops.

The upcoming SUMIT 2018 is going to be our biggest and best math collaboration ever. For girls in grades 6-10, participants will be put in a predicament from which they must extricate themselves using the currency of the world they’ll find themselves immersed in: mathematics! They must self-organize and communicate well as there will be no one to help them but themselves. It’ll be an epic journey where participants must become the heroines of their own saga.

Should they succeed, they’ll be rewarded with the knowledge of genuine accomplishment—and gifts, such as Marc Chamberland’s captivating book, Single Digits: In Praise of Small Numbers courtesy of long-time SUMIT sponsor Princeton University Press.

The best way to learn math is to do math, and what better way to do math than to do it while laughing out loud and making new friends?

There are a limited number of spots still available for 9th and 10th graders. Register today!

Pariah Moonshine Part III: Pariah Groups, Prime Factorizations, and Points on Elliptic Curves

by Joshua Holden

This post originally appeared on The Aperiodical. We republish it here with permission. 

In Part I of this series of posts, I introduced the sporadic groups, finite groups of symmetries which aren’t the symmetries of any obvious categories of shapes. The sporadic groups in turn are classified into the Happy Family, headed by the Monster group, and the Pariahs. In Part II, I discussed Monstrous Moonshine, the connection between the Monster group and a type of function called a modular form. This in turn ties the Monster group, and with it the Happy Family, to elliptic curves, Fermat’s Last Theorem, and string theory, among other things. But until 2017, the Pariah groups remained stubbornly outside these connections.

In September 2017, John Duncan, Michael Mertens, and Ken Ono published a paper announcing a connection between the Pariah group known as the O’Nan group (after Michael O’Nan, who discovered it in 1976) and another modular form. Like Monstrous Moonshine, the new connection is through an infinite-dimensional shape which breaks up into finite-dimensional pieces. Also like Monstrous Moonshine, the modular form in question has a deep connection with elliptic curves. In this case, however, the connection is more subtle and leads through yet another set of important mathematical objects: the quadratic fields.

At play in the fields quadratic

What mathematicians call a field is a set of objects which are closed under addition, subtraction, multiplication, and division (except division by zero). The rational numbers form a field, and so do the real numbers and the complex numbers. The integers don’t form a field because they aren’t closed under division, and the positive real numbers don’t form a field because they aren’t closed under subtraction.  (It’s also possible to have fields of things that aren’t numbers, which are useful in lots of other situations; see Section 4.5 of The Mathematics of Secrets for a cryptographic example.)

A common way to make a new field is to take a known field and enlarge it a bit. For example, if you start with the real numbers and enlarge them by including the number i (the square root of -1), then you also have to include all of the imaginary numbers, which are multiples of i, and then all of the numbers which are real numbers plus imaginary numbers, which gets you the complex numbers. Or you could start with the rational numbers, include the square root of 2, and then you have to include the numbers that are rational multiples of the square root of 2, and then the numbers which are rational numbers plus the multiples of the square root of 2. Then you get to stop, because if you multiply two of those numbers you get


which is another number of the same form. Likewise, if you divide two numbers of this form, you can rationalize the denominator and get another number of the same form. We call the resulting field the rational numbers “adjoined with” the square root of 2. Fields which are obtained by starting with the rational numbers and adjoining the square root of a rational number (positive or negative) are called quadratic fields.

Identifying a quadratic field is almost, but not quite, as easy as identifying the square root you are adjoining. For instance, consider adjoining the square root of 8. The square root of 8 is twice the square root of 2, so if you adjoin the square root of 2 you get the square root of 8 for free. And since you can also divide by 2, if you adjoin the square root of 8 you get the square root of 2 for free. So these two square roots give you the same field.  For technical reasons, a quadratic field is identified by taking all of the integers whose square roots would give you that field, and picking out the integer D with the smallest absolute value that can be written in the form b2 – 4ac for integers a, b, and c.  (This is the same b2 – 4ac as in the quadratic formula.)  This number D is called the fundamental discriminant of the field. So, for example, 8 is the fundamental discriminant of the quadratic field we’ve been talking about, not 2, because 8 = 42 – 4 × 2 × 1, but 2 can’t be written in that form.

Prime suspects

After addition, subtraction, multiplication, and division, one of the really important things you can do with rational numbers is factor their numerators and denominators into primes. In fact, you can do it uniquely, aside from the order of the factors. If you have number in a quadratic field, you can still factor it into primes, but the primes might not be unique. For example, in the rational numbers adjoined with the square root of negative 5 we have


where 2, 5, 1 + √–5, and 1 – √–5 are all primes. You’ll have to trust me on that last part, since it’s not always obvious which numbers in a quadratic field are prime. Figures 1 and 2 show some small primes in the rational numbers adjoined with the square roots of negative 1 and negative 3, respectively, plotted as points in the complex plane.

Holden 3

Figure 1. Some small primes in the rational numbers adjoined with the square root of -1 (D = -4), plotted as points in the complex plane. By Wikimedia Commons User Georg-Johann.)



Figure 2. Some small primes in the rational numbers adjoined with the square root of -3 (D = -3), plotted as points in the complex plane. By Wikimedia Commons User Fropuff.)

We express this by saying the rational numbers have unique factorization, but not all quadratic fields do. The question of which quadratic fields have unique factorization is an important open problem in general. For negative fundamental discriminants, we know that D = ‑3, ‑4, ‑7, ‑8, ‑11, ‑19, ‑43, ‑67, ‑163 are the only such quadratic fields; an equivalent form of this was conjectured by Gauss but fully acceptable proofs were not given until 1966 by Alan Baker and 1967 by Harold Stark. For positive fundamental discriminants, Gauss conjectured that there were infinitely many quadratic fields with unique factorization but this is still unproved.

Furthermore, Gauss identified a number, called the class number, which in some sense measures how far from unique factorization a field is. If the class number is 1, the field has unique factorization, otherwise not. The rational numbers adjoined with the square root of negative 5 (D = -20) have class number 2, and therefore do not have unique factorization. Gauss also conjectured that the class number of a quadratic field went to infinity as its discriminant went to negative infinity; this was proved by Hans Heilbronn in 1934.

Moonshine with class (numbers)

What about Moonshine? Duncan, Mertens, and Ono proved that the O’Nan group was associated with the modular form

F(z) = e -8 π i z + 2 + 26752 e 6 π i z + 143376 e 8 π i z  + 8288256 e 14 π i z  + …

which has the property that the coefficient of e 2 |D| π I z  is related to the class number of the field with fundamental discriminant < 0.  Furthermore, looking at elements of the O’Nan group sometimes gives us very specific relationships between the coefficients and the class number.  For example, the O’Nan group includes a symmetry which is like a 180 degree rotation, in that if you do it twice you get back to where you started.  Using that symmetry, Duncan, Mertens, and Ono showed that for even D < -8, 16 always divides a(D)+24h(D), where a(D) is the coefficient of  e 2 |D| π i z  and h(D) is the class number of the field with fundamental discriminant D.  For the example D = -20 from above, a(D) = 798588584512 and h(D) = 2, and 16 does in fact divide 798588584512 + 48.  Similarly, other elements of the O’Nan group show that 9 always divides a(D)+24h(D) if D = 3k+2 for some integer k and that 5 and 7 always divide a(D)+24h(D) under other similar conditions on And 11 and 19 divide a(D)+24h(D) under (much) more complicated conditions related to points on an elliptic curve associated with each D, which brings us back nicely to the connection between Moonshine and elliptic curves.

How much Moonshine is out there?

Monstrous Moonshine showed that the Monster, and therefore the Happy Family, was related to modular forms and elliptic curves, as well as string theory. O’Nan Moonshine brings in two more sporadic groups, the O’Nan group and its subgroup the “first Janko group”. (Figure 3 shows the connections between the sporadic groups. “M” is the Monster group, “O’N” is the O’Nan group, and “J1” is the first Janko group.) It also connects the sporadic groups not just to modular forms and elliptic curves, but also to quadratic fields, primes, and class numbers. Furthermore, the modular form used in Monstrous Moonshine is “weight 0”, meaning that k = 0 in the definition of a modular form given in Part II. That ties this modular form very closely to elliptic curves.


Figure 3. Connections between the sporadic groups. Lines indicate that the lower group is a subgroup or a quotient of a subgroup of the upper group. “M” is the Monster group and “O’N” is the O’Nan group; the groups connected below the Monster group are the rest of the Happy Family. (By Wikimedia Commons User Drschawrz.)

The modular form in O’Nan Moonshine is “weight 3/2”. Weight 3/2 modular forms are less closely tied to elliptic curves, but are tied to yet more ideas in mathematical physics, like higher-dimensional generalizations of strings called “branes” and functions that might count the number of states that a black hole can be in. That still leaves four more pariah groups, and the smart money predicts that Moonshine connections will be found for them, too. But will they come from weight 0 modular forms, weight 3/2 modular forms, or yet another type of modular form with yet more connections? Stay tuned! Maybe someday soon there will be a Part IV.

Joshua Holden is professor of mathematics at the Rose-Hulman Institute of Technology. He is the author of The Mathematics of Secrets: Cryptography from Caesar Ciphers to Digital Encryption.

Pariah Moonshine Part II: For Whom the Moon Shines

by Joshua Holden

This post originally appeared on The Aperiodical. We republish it here with permission. 

HoldenI ended Part I with the observation that the Monster group was connected with the symmetries of a group sitting in 196883-dimensional space, whereas the number 196884 appeared as part of a function used in number theory, the study of the properties of whole numbers.  In particular, a mathematician named John McKay noticed the number as one of the coefficients of a modular form.  Modular forms also exhibit a type of symmetry, namely if F is a modular form then there is some number k for which

Figure 1

for every set of whole numbers a, b, c, and d such that adbc=1.  (There are also some conditions as the real part of z goes to infinity.)

Modular forms, elliptic curves, and Fermat’s Last Theorem

In 1954, Martin Eichler was studying modular forms and observing patterns in their coefficients.  For example, take the modular form

Figure 2

(I don’t know whether Eichler actually looked at this particular form, but he definitely looked at similar ones.)  The coefficients of this modular form seem to be related to the number of whole number solutions of the equation

y2 = x3 – 4 x2 + 16

This equation is an example of what is known as an elliptic curve, which is a curve given by an equation of the form

y2 = x3 + ax2 + bx + c

Note that elliptic curves are not ellipses!  Elliptic curves have one line of symmetry, two open ends, and either one or two pieces, as shown in Figures 1 and 2. They are called elliptic curves because the equations came up in the seventeenth century when mathematicians started studying the arc length of an ellipse.  These curves are considered the next most complicated type of curve after lines and conic sections, both of which have been understood pretty well since at least the ancient Greeks.   They are useful for a lot of things, including cryptography, as I describe in Section 8.3 of The Mathematics of Secrets.

Figure 1

Figure 1. The elliptic curve y2= x3 + x has one line of symmetry, two open ends, and one piece.

Figure 2

Figure 2. The elliptic curve y2 = x3 – x has one line of symmetry, two open ends, and two pieces.


In the late 1950’s it was conjectured that every elliptic curve was related to a modular form in the way that the example above is.  Proving this “Modularity Conjecture” took on more urgency in the 1980’s, when it was shown that showing the conjecture was true would also prove Fermat’s famous Last Theorem.  In 1995 Andrew Wiles, with help from Richard Taylor, proved enough of the Modularity Conjecture to show that Fermat’s Last Theorem was true, and the rest of the Modularity Conjecture was filled in over the next six years by Taylor and several of his associates.

Modular forms, the Monster, and Moonshine

Modular forms are also related to other shapes besides elliptic curves, and in the 1970’s John McKay and John Thompson became convinced that the modular form

J(z) = e -2 π i z + 196884 e 2 π i z + 21493760 e 4 π i z  + 864299970 e 6 π i z  + …

was related to the Monster.  Not only was 196884 equal to 196883 + 1, but 21493760 was equal to 21296876 + 196883 + 1, and 21296876 was also a number that came up in the study of the Monster.  Thompson suggested that there should be a natural way of associating the Monster with an infinite-dimensional shape, where the infinite-dimensional shape broke up into finite-dimensional pieces with each piece having a dimension corresponding to one of the coefficients of J(z).   This shape was (later) given the name V♮, using the natural sign from musical notation in a typically mathematical pun.  (Terry Gannon points out that there is also a hint that the conjectures “distill information illegally” from the Monster.) John Conway and Simon Norton formulated some guesses about the exact connection between the Monster and V♮, and gave them the name “Moonshine Conjectures” to reflect their speculative and rather unlikely-seeming nature. A plausible candidate for V♮ was constructed in the 1980’s and Richard Borcherds proved in 1992 that the candidate satisfied the Moonshine Conjectures.  This work was specifically cited when Borcherds was awarded the Fields medal in 1998.

The construction of V♮ turned out also to have a close connection with mathematical physics.  The reconciliation of gravity with quantum mechanics is one of the central problems of modern physics, and most physicists think that string theory is likely to be key to this resolution.  In string theory, the objects we traditionally think of as particles, like electrons and quarks, are really tiny strings curled up in many dimensions, most of which are two small for us to see.  An important question about this theory is exactly what shape these dimensions curl into.  One possibility is a 24-dimensional shape where the possible configurations of strings in the shape are described by V♮.  However, there are many other possible shapes and it is not known how to determine which one really corresponds to our world.

More Moonshine?

Since Borcherds’ proof, many variations of the original “Monstrous Moonshine” have been explored.  The other members of the Happy Family can be shown to have Moonshine relationships similar to those of the Monster.  “Modular Moonshine” says that certain elements of the Monster group should have their own infinite dimensional shapes, related to but not the same as V♮.  (The “modular” in “Modular Moonshine” is related to the one in “modular form” because they are both related to modular arithmetic, although the chain of connections is kind of long. )  “Mathieu Moonshine” shows that one particular group in the Happy Family has its own shape, entirely different from V♮, and “Umbral Moonshine” extends this to 23 other related groups which are not simple groups.  But the Pariah groups remained outsiders, rejected by both the Happy Family and by Moonshine — until September 2017.

Joshua Holden is professor of mathematics at the Rose-Hulman Institute of Technology. He is the author of The Mathematics of Secrets: Cryptography from Caesar Ciphers to Digital Encryption.

Pariah Moonshine Part I: The Happy Family and the Pariah Groups

by Joshua Holden

This post originally appeared on The Aperiodical. We republish it here with permission. 

HoldenBeing a mathematician, I often get asked if I’m good at calculating tips. I’m not. In fact, mathematicians study lots of other things besides numbers. As most people know, if they stop to think about it, one of the other things mathematicians study is shapes. Some of us are especially interested in the symmetries of those shapes, and a few of us are interested in both numbers and symmetries. The recent announcement of “Pariah Moonshine” has been one of the most exciting developments in the relationship between numbers and symmetries in quite some time. In this blog post I hope to explain the “Pariah” part, which deals mostly with symmetries. The “Moonshine”, which connects the symmetries to numbers, will follow in the next post.

What is a symmetry?

First I want to give a little more detail about what I mean by the symmetries of shapes. If you have a square made out of paper, there are basically eight ways you can pick it up, turn it, and put it down in exactly the same place. You can rotate it 90 degrees clockwise or counterclockwise. You can rotate it 180 degrees. You can turn it over, so the front becomes the back and vice versa. You can turn in over and then rotate it 90 degrees either way, or 180 degrees. And you can rotate it 360 degrees, which basically does nothing. We call these the eight symmetries of the square, and they are shown in Figure 1.


Figure 1. The square can be rotated into four different positions, without or without being flipped over, for eight symmetries total.

If you have an equilateral triangle, there are six symmetries. If you have a pentagon, there are ten. If you have a pinwheel with four arms, there are only four symmetries, as shown in Figure 2, because now you can rotate it but if you turn it over it looks different. If you have a pinwheel with six arms, there are six ways. If you have a cube, there are 24 if the cube is solid, as shown in Figure 3. If the cube is just a wire frame and you are allowed to turn it inside out, then you get 24 more, for a total of 48.

Figure 2

Figure 2. The pinwheel can be rotated but not flipped, for four symmetries total.

Figure 3

Figure 3. The cube can be rotated along three different axes, resulting in 24 different symmetries.

These symmetries don’t just come with a count, they also come with a structure. If you turn a square over and then rotate it 90 degrees, it’s not the same thing as if you rotate it first and then flip it over. (Try it and see.) In this way, symmetries of shapes are like the permutations I discuss in Chapter 3 of my book, The Mathematics of Secrets: you can take products, which obey some of the same rules as products of numbers but not all of them. These sets of symmetries, which their structures, are called groups.

Groups are sets of symmetries with structure

Some sets of symmetries can be placed inside other sets. For example, the symmetries of the four-armed pinwheel are the same as the four rotations in the symmetries of the square. We say the symmetries of the pinwheel are a subgroup of the symmetries of the square. Likewise, the symmetries of the square are a subgroup of the symmetries of the solid cube, if you allow yourself to turn the cube over but not tip it 90 degrees, as shown in Figure 4.

Figure 4

Figure 4. The symmetries of the square are contained inside the symmetries of the cube if you are allowed to rotate and flip the cube but not tip it 90 degrees.

In some cases, ignoring a subgroup of the symmetries of a shape gets us another group, which we call the quotient group. If you ignore the subgroup of how the square is rotated, you get the quotient group where the square is flipped over or not, and that’s it. Those are the same as the symmetries of the capital letter A, so the quotient group is really a group. In other cases, for technical reasons, you can’t get a quotient group. If you ignore the symmetries of a square inside the symmetries of a cube, what’s left turns out not to be the symmetries of any shape.

You can always ignore all the symmetries of a shape and get just the do nothing (or trivial) symmetry, which is the symmetries of the capital letter P, in the quotient group. And you can always ignore none of the nontrivial symmetries, and get all of the original symmetries still in the quotient group. If these are the only two possible quotient groups, we say that the group is simple. The group of symmetries of a pinwheel with a prime number of arms is simple. So is the group of symmetries of a solid icosahedron, like a twenty-sided die in Dungeons and Dragons. The group of symmetries of a square is not simple, because of the subgroup of rotations. The group of symmetries of a solid cube is not simple, not because of the symmetries of the square, but because of the smaller subgroup of symmetries of a square with a line through it, as shown in Figures 5 and 6. The quotient group there is the same as the symmetries of the equilateral triangle created by cutting diagonally through a cube near a corner.

Figure 5

Figure 5. The symmetries of a square with line through it. We can turn the square 180 degrees and/or flip it, but not rotate it 90 degrees, so there are four.

Figure 6

Figure 6. The symmetries of the square with a line through it inside of the symmetries of the cube.

Categorizing the Pariah groups

As early as 1892, Otto Hölder asked if we could categorize all of the finite simple groups. (There are also shapes, like the circle, which have an infinite number of symmetries. We won’t worry about them now.)  It wasn’t until 1972 that Daniel Gorenstein made a concrete proposal for how to make a complete categorization, and the project wasn’t finished until 2002, producing along the way thousands of pages of proofs. The end result was that almost all of the finite simple groups fell into a few infinitely large categories: the cyclic groups, which are the groups of symmetries of pinwheels with a prime number of arms, the alternating groups, which are the groups of symmetries of solid hypertetrahedra in 5 or more dimensions, and the “groups of Lie type”, which are related to matrix multiplication over finite fields and describe certain symmetries of objects known as finite projective planes and finite projective spaces. (Finite fields are used in the AES cipher and I talk about them in Section 4.5 of The Mathematics of Secrets.)

Even before 1892, a few finite simple groups were discovered that didn’t seem to fit into any of these categories. Eventually it was proved that there were 26 “sporadic” groups, which didn’t fit into any of the categories and didn’t describe the symmetries of anything obvious — basically, you had to construct the shape to fit the group of symmetries that you knew existed, instead of starting with the shape and finding the symmetries. The smallest of the sporadic groups has 7920 symmetries in it, and the largest, known as the Monster, has over 800 sexdecillion symmetries. (That’s an 8 with 53 zeros after it!) Nineteen of the other sporadic groups turn out to be subgroups or quotient groups of subgroups of the Monster. These 20 became known as the Happy Family. The other 6 sporadic groups became known as the ‘Pariahs’.

The shape that was constructed to fit the Monster lives in 196883-dimensional space. In the late 1970’s a mathematician named John McKay noticed the number 196884 turning up in a different area of mathematics. It appeared as part of a function used in number theory, the study of the properties of whole numbers. Was there a connection between the Monster and number theory? Or was the idea of a connection just … moonshine?

Joshua Holden is professor of mathematics at the Rose-Hulman Institute of Technology. He is the author of The Mathematics of Secrets: Cryptography from Caesar Ciphers to Digital Encryption.

Announcing the trailer for The Seduction of Curves by Allan McRobie

CurvesCurves are seductive. These smooth, organic lines and surfaces—like those of the human body—appeal to us in an instinctive, visceral way that straight lines or the perfect shapes of classical geometry never could. In this large-format book, lavishly illustrated in color throughout, Allan McRobie takes the reader on an alluring exploration of the beautiful curves that shape our world—from our bodies to Salvador Dalí’s paintings and the space-time fabric of the universe itself. A unique introduction to the language of beautiful curves, this book may change the way you see the world.

Allan McRobie is a Reader in the Engineering Department at the University of Cambridge, where he teaches stability theory and structural engineering. He previously worked as an engineer in Australia, designing bridges and towers.