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.

Ken Steiglitz on The Discrete Charm of the Machine

SteiglitzA 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. Ken Steiglitz examines why our information technology, the lifeblood of our civilization, became digital, and challenges us to think about where its future trajectory may lead.

What is the aim of the book?

The subtitle: To explain why the world became digital. Barely two generations ago our information machines—radio, TV, computers, telephones, phonographs, cameras—were analog. Information was represented by smoothly varying waves. Today all these devices are digital. Information is represented by bits, zeros and ones. We trace the reasons for this radical change, some based on fundamental physical principles, others on ideas from communication theory and computer science. At the end we arrive at the present age of the internet, dominated by digital communication, and finally greet the arrival of androids—the logical end of our current pursuit of artificial intelligence. 

What role did war play in this transformation?

Sadly, World War II was a major impetus to many of the developments leading to the digital world, mainly because of the need for better methods for decrypting intercepted secret messages and more powerful computation for building the atomic bomb. The following Cold War just increased the pressure. Business applications of computers and then, of course, the personal computer opened the floodgates for the machines that are today never far from our fingertips.

How did you come to study this subject?

I lived it. As an electrical engineering undergraduate I used both analog and digital computers. My first summer job was programming one of the few digital computers in Manhattan at the time, the IBM 704. In graduate school I wrote my dissertation on the relationship between analog and digital signal processing and my research for the next twenty years or so concentrated on digital signal processing: using computers to process sound and images in digital form.

What physical theory played—and continues to play—a key role in the revolution?

Quantum mechanics, without a doubt. The theory explains the essential nature of noise, which is the natural enemy of analog information; it makes possible the shrinkage and speedup of our electronics (Moore’s law); and it introduces the possibility of an entirely new kind of computer, the quantum computer, which can transcend the power of today’s conventional machines. Quantum mechanics shows that many aspects of the world are essentially discrete in nature, and the change from the classical physics of the nineteenth century to the quantum mechanics of the twentieth is mirrored in the development of our digital information machines.

What mathematical theory plays a key role in understanding the limitations of computers?

Complexity theory and the idea of an intractable problem, as developed by computer scientists. This theme is explored in Part III, first in terms of analog computers, then using Alan Turing’s abstraction of digital computation, which we now call the Turing machine. This leads to the formulation of the most important open question of computer science, does P equal NP? If P equals NP it would mean that any problem where solutions can just be checked fast can be solved fast. This seems like asking a lot and, in fact, most computer scientists believe that P does not equal NP. Problems as hard as any in NP are called NP-complete. The point is that NP-complete problems, like the famous traveling problem, seem to be intrinsically difficult, and cracking any one of them cracks them all.  Their essential difficulty manifests itself, mysteriously, in many different ways in the analog and digital worlds, suggesting, perhaps, that there is an underlying physical law at work. 

What important open question about physics (not mathematics) speaks to the relative power of digital and analog computers?

The extended Church-Turing thesis states that any reasonable computer can be simulated efficiently by a Turing machine. Informally, it means that no computer, even if analog, is more powerful (in an appropriately defined way) than the bare-boned, step-by-step, one-tape Turing machine. The question is open, but many computer scientists believe it to be true. This line of reasoning leads to an important conclusion: if the extended Church-Turing thesis is true, and if P is not equal to NP (which is widely believed), then the digital computer is all we need—Nature is not hiding any computational magic in the analog world.

What does all this have to do with artificial intelligence (AI)?

The brain uses information in both analog and digital form, and some have even suggested that it uses quantum computing. So, the argument goes, perhaps the brain has some special powers that cannot be captured by ordinary computers.

What does philosopher David Chalmers call the hard problem?

We finally reach—in the last chapter—the question of whether the androids we are building will ultimately be conscious. Chalmers calls this the hard problem, and some, including myself, think it unanswerable. An affirmative answer would have real and important consequences, despite the seemingly esoteric nature of the question. If machines can be conscious, and presumably also capable of suffering, then we have a moral responsibility to protect them, and—to put it in human terms—bring them up right. I propose that we must give the coming androids the benefit of the doubt; we owe them the same loving care that we as parents bestow on our biological offspring.

Where do we go from here?

A funny thing happens on the way from chapter 1 to 12. I begin with the modest plan of describing, in the simplest way I can, the ideas behind the analog-to-digital revolution.  We visit along the way some surprising tourist spots: the Antikythera mechanism, a 2000-year old analog computer built by the ancient Greeks; Jacquard’s embroidery machine with its breakthrough stored program; Ada Lovelace’s program for Babbage’s hypothetical computer, predating Alan Turing by a century; and B. F. Skinner’s pigeons trained in the manner of AI to be living smart bombs. We arrive at a collection of deep conjectures about the way the universe works and some challenging moral questions.

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

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.

Gift Guide: Biographies and Memoirs!

Not sure what to give the reader who’s read it all? Biographies, with their fascinating protagonists, historical analyses, and stranger-than-fiction narratives, make great gifts for lovers of nonfiction and fiction alike! These biographies and memoirs provide glimpses into the lives of people both famous and forgotten:

Galawdewos Life of Walatta-Petros book coverThe radical saint: Walatta-Petros

Walatta-Petros was an Ethiopian saint who lived from 1592 to 1642 and led a successful nonviolent movement to preserve African Christian beliefs in the face of European protocolonialism. Written by her disciple Galawdewos in 1672, after Walatta-Petros’s death, and translated and edited by Wendy Laura Belcher and Michael Kleiner, The Life of Walatta-Petros praises her as a friend of women, a devoted reader, a skilled preacher, and a radical leader, providing a rare picture of the experiences and thoughts of Africans—especially women—before the modern era.

This is the oldest-known book-length biography of an African woman written by Africans before the nineteenth century, and one of the earliest stories of African resistance to European influence. This concise edition, which omits the notes and scholarly apparatus of the hardcover, features a new introduction aimed at students and general readers.

 

Devlin_Finding Fibonacci book coverThe forgotten mathematician: Fibonacci

The medieval mathematician Leonardo of Pisa, popularly known as Fibonacci, is most famous for the Fibonacci numbers—which, it so happens, he didn’t invent. But Fibonacci’s greatest contribution was as an expositor of mathematical ideas at a level ordinary people could understand. In 1202, his book Liber abbaci—the “Book of Calculation”—introduced modern arithmetic to the Western world. Yet Fibonacci was long forgotten after his death.

Finding Fibonacci is Keith Devlin’s compelling firsthand account of his ten-year quest to tell Fibonacci’s story. Devlin, a math expositor himself, kept a diary of the undertaking, which he draws on here to describe the project’s highs and lows, its false starts and disappointments, the tragedies and unexpected turns, some hilarious episodes, and the occasional lucky breaks.

 

The college president: Hanna Gray Gray_Academic Life book cover

Hanna Holborn Gray has lived her entire life in the world of higher education. The daughter of academics, she fled Hitler’s Germany with her parents in the 1930s, emigrating to New Haven, where her father was a professor at Yale University. She has studied and taught at some of the world’s most prestigious universities. She was the first woman to serve as provost of Yale. In 1978, she became the first woman president of a major research university when she was appointed to lead the University of Chicago, a position she held for fifteen years. In 1991, Gray was awarded the Presidential Medal of Freedom, the nation’s highest civilian honor, in recognition of her extraordinary contributions to education.

Gray’s memoir An Academic Life is a candid self-portrait by one of academia’s most respected trailblazers.

 

The medieval historian: Ibn Khaldun Irwin_Ibn Khaldun book cover

Ibn Khaldun (1332–1406) is generally regarded as the greatest intellectual ever to have appeared in the Arab world—a genius who ranks as one of the world’s great minds. Yet the author of the Muqaddima, the most important study of history ever produced in the Islamic world, is not as well known as he should be, and his ideas are widely misunderstood. In this groundbreaking intellectual biography, Robert Irwin presents an Ibn Khaldun who was a creature of his time—a devout Sufi mystic who was obsessed with the occult and futurology and who lived in a world decimated by the Black Death.

Ibn Khaldun was a major political player in the tumultuous Islamic courts of North Africa and Muslim Spain, as well as a teacher and writer. Irwin shows how Ibn Khaldun’s life and thought fit into historical and intellectual context, including medieval Islamic theology, philosophy, politics, literature, economics, law, and tribal life.

 

The novelist and philosopher: Iris Murdoch Murdoch_Living on Paper book cover

Iris Murdoch was an acclaimed novelist and groundbreaking philosopher whose life reflected her unconventional beliefs and values. Living on Paper—the first major collection of Murdoch’s most compelling and interesting personal letters—gives, for the first time, a rounded self-portrait of one of the twentieth century’s greatest writers and thinkers. With more than 760 letters, fewer than forty of which have been published before, the book provides a unique chronicle of Murdoch’s life from her days as a schoolgirl to her last years.

The letters show a great mind at work—struggling with philosophical problems, trying to bring a difficult novel together, exploring spirituality, and responding pointedly to world events. We witness Murdoch’s emotional hunger, her tendency to live on the edge of what was socially acceptable, and her irreverence and sharp sense of humor. Direct and intimate, these letters bring us closer than ever before to Iris Murdoch as a person.

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?

Yes!

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.

Eli Maor on Music by the Numbers

MaorThat music and mathematics are somehow related has been known for centuries. Pythagoras, around the 5th century BCE, may have been the first to discover a quantitative relation between the two: experimenting with taut strings, he found out that shortening the effective length of a string to one half its original length raises the pitch of its sound by an agreeable interval—an octave. Other ratios of string lengths produced smaller intervals: 2:3 corresponds to a fifth (so called because it is the fifth note up the scale from the base note), 3:4 corresponded to a fourth, and so on. Moreover, Pythagoras found out that multiplying two ratios corresponds to adding their intervals: (2:3) x (3:4) = 1:2, so a fifth plus a fourth equals an octave. In doing so, Pythagoras discovered the first logarithmic law in history.

The relations between musical intervals and numerical ratios have fascinated scientists ever since. Johannes Kepler, considered the father of modern astronomy, spent half his lifetime trying to explain the motion of the known planets by relating them to musical intervals. Half a century later, Isaac Newton formulated his universal law of gravitation, thereby providing a rational, mathematical explanation for the planetary orbits. But he too was obsessed with musical ratios: he devised a “palindromic” musical scale and compared its intervals to the rainbow colors of the spectrum. Still later, four of Europe’s top mathematicians would argue passionately over the exact shape of a vibrating string. In doing so, they contributed significantly to the development of post-calculus mathematics, while at the same time giving us a fascinating glimpse into their personal relations and fierce rivalries. As Eli Maor points out in Music by the Numbers, the “Great String Debate” of the eighteenth century has some striking similarities to the equally fierce debate over the nature of quantum mechanics in the 1920s.

What brought you to write a book on such an unusual subject? 

The ties between music and mathematics have fascinated me from a young age. My grandfather played his violin for me when I was five years old, and I still remember it quite clearly. He also spent many hours explaining to me various topics from his physics book, from which he himself had studied many years earlier. In the chapter on sound there was a musical staff showing the note A with a number under it: 440, the frequency of that note. It may have been this image that first triggered my fascination with the subject. I still have that physics book and I treasure it immensely. My grandfather must have studied it thoroughly, as his penciled annotations appear on almost every page.

Did you study the subject formally?

Yes. I did my master’s and later my doctoral thesis in acoustics at the Technion – Israel Institute of Technology. There was just one professor who was sufficiently knowledgeable in the subject, and he agreed to be my advisor. But first we had to find a department willing to take me under its wing, and that turned out to be tricky. To me acoustics was a branch of physics, but the physics department saw it as just an engineering subject. So I applied to the newly-founded Department of Mechanics, and they accepted me. The coursework included a heavy load of technical subjects—strength of materials, elasticity, rheology, and the theory of vibrations—all of which I did as independent studies. In the process I learned a lot of advanced mathematics, especially Fourier series and integrals. It served me well in my later work.

What about your music education?

I started my musical education playing Baroque music on the recorder, and later I took up the clarinet. This instrument has the unusual feature that when you open the thumb hole on the back side of the bore, the pitch goes up not by an octave, as with most woodwind instruments, but by a twelfth—an octave and a fifth. This led me to dwell into the acoustics of wind instruments. I was—and still am—intrigued by the fact that a column of air can vibrate and produce an agreeable sound just like a violin string. But you have to rely entirely on your ear to feel those vibrations; they are totally invisible to the eye.

When I was a physics undergraduate at the Hebrew University of Jerusalem, a group of students and professors decided to start an amateur orchestra, and I joined. At one of our performances we played Mozart’s overture to The Magic Flute. There is one bar in that overture where the clarinet plays solo, and it befell upon me to play it. I practiced for that single bar again and again, playing it perhaps a hundred times simultaneously with a vinyl record playing on a gramophone. Finally the evening arrived and I played my piece—all three seconds of it. At intermission I asked a friend of mine in the audience, a concert pianist, how did it go. “Well,” she said, “you played it too fast.”  Oh Lord!  I was only glad that Mozart wasn’t present!

Throughout your book there runs a common thread—the parallels between musical and mathematical frames of reference. Can you elaborate on this comparison? 

For about 300 years—roughly from 1600 to 1900—classical music was based on the principle of tonality: a composition was always tied to a given home key, and while deviating from it during the course of the work, the music was invariably related to that key. The home key thus served as a musical frame of reference in which the work was set, similar to a universal frame of reference to which the laws of classical physics were supposed to be bound.

But in the early 1900s, Arnold Schoenberg set out to revolutionize music composition by proposing his tone row, or series, consisting of all twelve semitones of the octave, each appearing exactly once before the series is completed. No more was each note defined by its relation to the tonic, or base note; in Schoenberg’s system a complete democracy reigned, each note being related only to the note preceding it in the series. This new system bears a striking resemblance to Albert Einstein’s general theory of relativity, in which no single frame of reference has a preferred status over others. Music by the Numbers expands on this fascinating similarity, as well as on the remarkable parallels between the lives of Schoenberg and Einstein.

You also touch on some controversial subjects. Can you say a few words about them?

It is generally believed that over the ages, mathematics has had a significant influence on music. Attempts to quantify music and subject it to mathematical rules began with Pythagoras himself, who invented a musical scale based entirely on his three “perfect intervals”—the octave, the fifth, and the fourth. From a mathematical standpoint it was a brilliant idea, but it was out of sync with the laws of physics; in particular, it ignored other important intervals such as the major and minor thirds. Closer to our time, Schoenberg’s serial music was another attempt to generate music by the numbers. It aroused much controversy, and after half a century during which his method was the compositional system to follow, enthusiasm for atonal music has waned.

But it is much less known that the attraction between the two disciplines worked both ways. I have already mentioned the Great String Debate of the eighteenth century—a prime example of how a problem originating in music has ended up advancing a new branch of mathematics: post-calculus analysis. It is also interesting to note that quite a few mathematical terms have their origin in music, such as harmonic series, harmonic mean, and harmonic functions, to name but a few.

Perhaps the most successful collaboration between the two disciplines was the invention of the equal-tempered scale—the division of the octave into twelve equally-spaced semitones. Although of ancient origins, this new tuning method has become widely known through Johann Sebastian Bach’s The Well-Tempered Clavier— his two sets of keyboard preludes and fugues covering all 24 major and minor scales. Controversial at the time, it has become the standard tuning system of Western music.

In your book there are five sidebars, one of which with the heading “Music for the Record Books: The Lowest, the Longest, the Oldest, and the Weirdest.”  Can you elaborate on them?

Yes. The longest piece of music ever performed—or more precisely, is still being performed—is a work for the organ at the St. Burkhardt Church in the German town of Halberstadt. The work was begun in 2003 and is an ongoing project, planned to be unfolding for the next 639 years. There are eight movements, each lasting about 71 years. The work is a version of John Cages’ composition As Slow as Possible. As reported by The New York Times, “The organ’s bellows began their whoosh on September 5, 2001, on what would have been Cage’s 89th birthday. But nothing was heard because the score begins with a rest—of 20 months. It was only on February 5, 2003, that the first chord, two G-sharps and a B in between, was struck.” It will be interesting to read the reviews when the work finally comes to an end in the year 2640.

I’ll mention one more piece for the record books: in 2012, astronomers discovered the lowest known musical note in the universe. Why astronomers?  Because the source of this note is the galaxy cluster Abell 426, some 250 million light years away. The cluster is surrounded by hot gas at a temperature of about 25,000,000 degrees Celsius, and it shows concentric ripples spreading outward—acoustic pressure waves. From the speed of sound at that temperature—about 1,155 km/sec—and the observed spacing between the ripples—some 36,000 light years—it is easy to find the frequency of the sound, and thus its pitch: a B-flat nearly 57 octaves below middle C. Says the magazine Sky & Telescope, “You’d need to add 635 keys to the left end of your piano keyboard to produce that note!  Even a contrabassoon won’t go that low.”

Eli Maor has taught the history of mathematics at Loyola University Chicago until his recent retirement. He is the author of six previous books by Princeton University Press: To Infinity and Beyonde: the Story of a NumberTrigonometric DelightsThe Pythagorean TheoremVenus in Transit; and Beautiful Geometry (with Eugen Jost). He is also an active amateur astronomer, has participated in over twenty eclipse and transit expeditions, and is a contributing author to Sky & Telescope.

Jason Rosenhouse: Yummy, Delicious Pi!

RosenhouseHere is a classic bar bet for you: take a wine glass, the kind with a really long stem. Ask whoever is near you to guess whether the height of the glass or the circumference at the top is greater. Most people will choose the height. In fact, they will regard it as obvious that the height is greater. But they will be wrong! Unless it is a very oddly-shaped glass, the circumference will be significantly greater. (Of course, you will need a piece of string to convince your mark of that.) It is a remarkably effective optical illusion.

As we all learned in grade school, the circumference of a circle is pi times the diameter, and pi is just a little greater than three. So the circumference at the top will be three times longer than the diameter. Any glass taller than that would be unpleasant to drink from.

Apparently knowing something about pi can make you money. Who said math isn’t practical?

I remember being fascinated by pi as a kid. When my father—a chemical engineer—first told me about it, I asked him if there was also a number called cake. The number pi is typically defined as a sort of geometrical object: it is the ratio of the circumference of a circle to its diameter. We could also say that pi is the area of a circle whose diameter is one. Yet somehow it keeps appearing in the most unexpected of places.

For example, suppose you pick two whole numbers at random, by which I mean the usual numbers like 1, 2, 3, 4, and so on. Sometimes the two numbers will share a common factor, like 4 and 6, which share a common factor of 2. Other times the two numbers will share no common factor (other than 1), like 3 and 7. Pairs like the second are said to be relatively prime. It turns out the probability that a pair of randomly chosen numbers is relatively prime is 6 divided by pi squared. Not a circle in sight, yet there is pi!

Or imagine that you have a very large sheet of notebook paper whose lines are one inch apart. Suppose you take a one-inch needle and drop it from a height onto the paper. The probability that the needle hits a line is 2 divided by pi. Only lines this time. Still no circles. This is called the Buffon needle problem, if you were curious.

One of the first things you learn about pi is that it is an irrational number, which means it is an infinite, non-repeating decimal. My sixth grade math teacher told me it was just crazy that a number should behave like that, and that is why it is called irrational. You can imagine my disappointment when I later learned that it is irrational only in the sense that it cannot be expressed as a ratio of whole numbers. I like my teacher’s explanation better. You can find fractions that are good approximations, like 22/7 or 355/113, but approximations are not the real thing.

The fact that pi is an infinite, non-repeating decimal, and that it cannot be written simply in terms of whole numbers, makes it difficult to write down at all. That is why we just give it a name, pi, and call it a day. We could as easily have called it Harry the number if we wanted to, but perhaps that lacks gravitas.

Pi is one of the special numbers of mathematics. Another is e, which is typically defined in ways that require calculus, and which have nothing to do with circles. This is another of those strange, irrational numbers that seems to keep popping up in unexpected places. Still another is i, which is defined to be the square root of minus 1, a number so bizarre it is commonly said to be imaginary. And we certainly should not forget the two most special numbers of them all, by which I mean 1 and 0.

Perhaps having experienced social ostracism at the hands of more normal numbers, the five special numbers have gotten together to create one of the most remarkable equations in mathematics. It is called Euler’s identity, and says:

e+1=0

It is remarkable that these five special numbers, defined in contexts entirely separate from one another, should play together so well. At the risk of seeming melodramatic, religions have started over less.

So take a moment this March 14 to give some thought to the most delicious number we have: pi. We will not have another perfect square day until May 5, 2025 (a date that will be written 5/5/25). And since e is 2.72 when rounded to two decimal places, we will never have an e day until February is granted 72 days. Or perhaps someday we will dramatically increase the size of the calendar, and then we will have e day on the second day of the twenty-seventh month.

But pi day comes every year. Enjoy it!

Jason Rosenhouse is a professor of mathematics at James Madison University in Harrisonburg, Virginia. He is the author or editor of six books, including The Monty Hall Problem: The Remarkable Story of Math’s Most Contentious Brainteaser, and Among the Creationists: Dispatches From the Anti-Evolutionist Frontline. His book Taking Sudoku Seriously, coauthored with Laura Taalman, received the 2012 Prose award, from the American Association of Publishers, for popular science and mathematics. With Jennifer Beineke, he is the editor of the Mathematics of Various Entertaining Subjects series, published by Princeton University Press and the Museum of Mathematics in New York. He is currently working on a book about logic puzzles, to be published by Princeton.

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

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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

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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.

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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.)

 

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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.

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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.