Introducing Volume 15 of The Collected Papers of Albert Einstein

From fraudulent science to hope for European reunification, the newest volume of The Collected Papers of Albert Einstein conveys the breakneck speed of Einstein’s personal and professional life. Volume 15 will be out on April 6!

Volume 15: The Berlin Years
Writings & Correspondence, June 1925-May 1927, Documentary Edition

Edited by Diana Kormos Buchwald, József Illy, A. J. Kox, Dennis Lehmkuhl, Ze’ev Rosenkranz & Jennifer Nollar James

Princeton University Press, the Einstein Papers Project at the California Institute of Technology, and the Albert Einstein Archives at the Hebrew University of Jerusalem, are pleased to announce the latest volume in the authoritative COLLECTED PAPERS OF ALBERT EINSTEIN. This volume covers one of the most thrilling two-year periods in twentieth-century physics, as matrix mechanics—developed chiefly by W. Heisenberg, M. Born, and P. Jordan—and wave mechanics—developed by E. Schrödinger—supplanted earlier quantum theory. The almost one hundred writings, a third of which have never before been published, and the more than thirteen hundred letters demonstrate Einstein’s immense productivity at a tumultuous time.

Within this volume, Einstein grasps the conceptual peculiarities involved in the new quantum mechanics; falls victim to scientific fraud while in collaboration with E. Rupp; and continues his participation in the League of Nations’ International Committee on Intellectual Cooperation.


Every document in The Collected Papers of Albert Einstein appears in the language in which it was written, and this supplementary paperback volume presents the English translations of select portions of non-English materials in Volume 15. This translation does not include notes or annotation of the documentary volume and is not intended for use without the original language documentary edition which provides the extensive editorial commentary necessary for a full historical and scientific understanding of the documents.

Translated by Jennifer Nollar James, Ann M. Hentschel, and Mary Jane Teague, Andreas Aebi and Klaus Hentschel, consultants


Diana Kormos Buchwald, General Editor

THE COLLECTED PAPERS OF ALBERT EINSTEIN is one of the most ambitious publishing ventures ever undertaken in the documentation of the history of science.  Selected from among more than 40,000 documents contained in the personal collection of Albert Einstein (1879-1955), and 20,000 Einstein and Einstein-related documents discovered by the editors since the beginning of the Einstein Papers Project, The Collected Papers provides the first complete picture of a massive written legacy that ranges from Einstein’s first work on the special and general theories of relativity and the origins of quantum theory, to expressions of his profound concern with international cooperation and reconciliation, civil liberties, education, Zionism, pacifism, and disarmament.  The series will contain over 14,000 documents as full text and will fill close to thirty volumes.  Sponsored by the Hebrew University of Jerusalem and Princeton University Press, the project is located at and supported by the California Institute of Technology and has made available a monumental collection of primary material. It will continue to do so over the life of the project. The Albert Einstein Archives is located at the Hebrew University of Jerusalem. The open access digital edition of the first 15 volumes of the Collected Papers is available online at

ABOUT THE SERIES: Fifteen volumes covering Einstein’s life and work up to his forty-eighth birthday have so far been published. They present more than 500 writings and 7,000 letters written by and to Einstein. Every document in The Collected Papers appears in the language in which it was written, while the introduction, headnotes, footnotes, and other scholarly apparatus are in English.  Upon release of each volume, Princeton University Press also publishes an English translation of previously untranslated non-English documents.

ABOUT THE EDITORS: At the California Institute of Technology, Diana Kormos Buchwald is professor of history; A. J. Kox is senior editor and visiting associate in history; József Illy and Ze’ev Rosenkranz are editors and senior researchers in history; Dennis Lehmkuhl is research assistant professor and scientific editor; and Jennifer Nollar James is assistant editor.

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:


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.

Jeffrey Bub & Tanya Bub: There are recipes for Pi. But quantum mechanics?

There’s a recipe for Pi, in fact quite a few recipes. Here’s one that dates to the fifteenth century, discovered by the Indian mathematician and astronomer Nilakantha:


For the trillions of decimal places to which the digits have been calculated, each digit in the decimal expansion of Pi occurs about one-tenth of the time, each pair of digits about one-hundredth of the time, and so on. Its still a deep unsolved mathematical problem to prove that this is in fact a feature of Pi—that the digits will continue to be uniformly distributed in this sense as more and more digits are calculated—but the digits aren’t totally random, since there’s a recipe for calculating them.

Quantum mechanics supplies a recipe for calculating the probabilities of events, how likely it is for an event to happen, but the theory doesn’t say whether an individual event will definitely happen or not. So is quantum theory complete, as Einstein thought, in which case we should try to complete the theory by refining the recipe, or are the individual events really totally random?

Einstein didn’t like the idea that God plays dice with the universe, as he characterized the orthodox Copenhagen interpretation of quantum mechanics adopted by Niels Bohr, Werner Heisenberg, and colleagues. He wrote to his friend the physicist Max Born:

I find the idea quite intolerable that an electron exposed to radiation should choose of its own free will, not only its moment to jump off, but also its direction. In that case, I would rather be a cobbler, or even an employee in a gaming house, than a physicist.

But Einstein was wrong. Consider this puzzle. Could you rig pairs of coins according to some recipe so that if Alice and Bob, separated by any distance, each toss a coin from a rigged pair heads up, one coin lands heads and the other tails, but if they toss the coins any other way (both tails up, or one tails up and the other heads up), they land the same? It turns out that if each coin is designed to land in any way at all that does not depend on the paired coin or how the paired coin is tossed—if each coin has its own “being-thus,” as Einstein put it—you couldn’t get the correlation right for more than 75% of the tosses. This is a version of Bell’s theorem, proved by John Bell in 1964.


What has this got to do with quantum randomness? The coin correlation is actually a “superquantum” correlation called a PR-correlation, after Sandu Popescu and Daniel Rohrlich who came up with the idea. Quantum particles aren’t correlated in quite this way, but measurements on pairs of photons in an “entangled” quantum state can produce a correlation that is close to the coin correlation. If Alice and Bob use entangled photons rather than coins, they could simulate the coin correlation with a success rate of about 85% by measuring the polarizations of the photons in certain directions.

Suppose Alice measures the polarizations of her photons in direction A = 0 or A′ = π/4 instead of tossing her coin tails up or heads up, and Bob measures in the direction B = π/8 or B′ = −π/8 instead of tossing his coin tails up or heads up. Then the angle between Alice’s measurement direction and Bob’s measurement direction is π/8, except when Alice measures in the direction A′ and Bob measures in the direction B′, in which case the angle is 3π/8. According to the quantum recipe for probabilities, the probability that the photon polarizations are the same when they are measured in directions π/8 apart is cos2(π/8), and the probability that the photon polarizations are different when they are measured in directions 3π/8 apart is sin2(3π/8) = cos2(π/8). So the probability that Alice and Bob get outcomes + or − corresponding to heads or tails that mimic the coin correlation is cos2(π/8), which is approximately .85.

Bell’s theorem tells us that this pattern of measurement outcomes is closer to the coin correlation pattern than any possible recipe could produce. So God does play dice, and events involving entangled quantum particles are indeed totally random!

BubTanya Bub is founder of 48th Ave Productions, a web development company. She lives in Victoria, British Columbia. Jeffrey Bub is Distinguished University Professor in the Department of Philosophy and the Institute for Physical Science and Technology at the University of Maryland, where he is also a fellow of the Joint Center for Quantum Information and Computer Science. His books include Bananaworld: Quantum Mechanics for Primates. He lives in Washington, DC. They are the authors of Totally Random: Why Nobody Understands Quantum Mechanics (A Serious Comic on Entanglement).

Celebrate Pi Day with Books about Einstein

Pi Day is coming up! Mathematicians around the world celebrate on March 14th because the date represents the first three digits of π: 3.14.

In Princeton, Pi Day is a huge event even for the non-mathematicians among us, given that March 14 is also Albert Einstein’s birthday. Einstein was born on March 14, 1879, in Ulm, in the German Empire. He turns 139 this year! If you’re in the Princeton area and want to celebrate, check out some of the festivities happening around town:

Saturday, 3/10/18

  • Apple Pie Eating Contest, 9:00 a.m., McCaffrey’s (301 North Harrison Street). Arrive by 8:45 a.m. to participate.
  • Einstein in Princeton Guided Walking Tour, 10:00 a.m. Call Princeton Tour Company at (855) 743-1415 for details.
  • Einstein Look-A-Like Contest, 12:00 p.m., Nassau Inn. Arrive early to get a spot to watch this standing-room-only event!
  • Pi Recitation Contest, 1:30 p.m., Prince William Ballroom, Nassau Inn. Children ages 12 and younger may compete. Register by 1:15 p.m.
  • Pie Throwing Event, 3:14 p.m., Palmer Square. Proceeds to benefit the Princeton Educational Fund Teacher Mini-Grant Program.
  • Cupcake Decorating Competition, 4:00 p.m., House of Cupcakes (34 Witherspoon Street). The winner receives one free cupcake each month for the rest of the year.

Wednesday, 3/14/18

  • Princeton School Gardens Cooperative Fundraiser, 12:00 p.m. to 6:00 p.m., The Bent Spoon (35 Palmer Square West) and Lillipies (301 North Harrison Street). All proceeds from your afternoon treat will be donated to the Princeton School Gardens Cooperative.
  • Pi Day Pop Up Wedding/Vow Renewal Ceremonies, 3:14 p.m. to 6:00 p.m., Princeton Pi (84 Nassau Street). You must pre-register by contacting the Princeton Tour Company.

Not into crowds, or pie? You can also celebrate this multifaceted holiday by picking up one of PUP’s many books about Albert Einstein! In 1922, Princeton University Press published Einstein’s The Meaning of Relativity, his first book produced by an American publisher. Since then, we’ve published numerous works by and about Einstein.

The books and collections highlighted here celebrate not only his scientific accomplishments but also his personal reflections and his impact on present-day scholarship and technology. Check them out and learn about Einstein’s interpersonal relationships, his musings on travel, his theories of time, and his legacy for the 21st century.

Volume 15 of the Collected Papers of Albert Einstein, forthcoming in April 2018, covers one of the most thrilling two-year periods in twentieth-century physics, as matrix mechanics—developed chiefly by W. Heisenberg, M. Born, and P. Jordan—and wave mechanics—developed by E. Schrödinger—supplanted the earlier quantum theory. The almost one hundred writings by Einstein, of which a third have never been published, and the more than thirteen hundred letters show Einstein’s immense productivity and hectic pace of life.

Einstein quickly grasps the conceptual peculiarities involved in the new quantum mechanics, such as the difference between Schrödinger’s wave function and a field defined in spacetime, or the emerging statistical interpretation of both matrix and wave mechanics. Inspired by correspondence with G. Y. Rainich, he investigates with Jakob Grommer the problem of motion in general relativity, hoping for a hint at a new avenue to unified field theory.

Readers can access Volumes 1-14 of the Collected Papers of Albert Einstein online at The Digital Einstein Papers, an exciting new free, open-access website that brings the writings of the twentieth century’s most influential scientist to a wider audience than ever before. This unique, authoritative resource provides full public access to the complete transcribed, annotated, and translated contents of each print volume of the Collected Papers. The volumes are published by Princeton University Press, sponsored by the Hebrew University of Jerusalem, and supported by the California Institute of Technology. Volumes 1-14 of The Collected Papers cover the first forty-six years of Einstein’s life, up to and including the years immediately before the final formulation of new quantum mechanics. The contents of each new volume will be added to the website approximately eighteen months after print publication. Eventually, the website will provide access to all of Einstein’s writings and correspondence accompanied by scholarly annotation and apparatus, which are expected to fill thirty volumes.

The Travel Diaries of Albert Einstein is the first publication of Albert Einstein’s 1922 travel diary to the Far East and Middle East, regions that the renowned physicist had never visited before. Einstein’s lengthy itinerary consisted of stops in Hong Kong and Singapore, two brief stays in China, a six-week whirlwind lecture tour of Japan, a twelve-day tour of Palestine, and a three-week visit to Spain. This handsome edition makes available, for the first time, the complete journal that Einstein kept on this momentous journey.

The telegraphic-style diary entries—quirky, succinct, and at times irreverent—record Einstein’s musings on science, philosophy, art, and politics, as well as his immediate impressions and broader thoughts on such events as his inaugural lecture at the future site of the Hebrew University in Jerusalem, a garden party hosted by the Japanese Empress, an audience with the King of Spain, and meetings with other prominent colleagues and statesmen. Entries also contain passages that reveal Einstein’s stereotyping of members of various nations and raise questions about his attitudes on race. This beautiful edition features stunning facsimiles of the diary’s pages, accompanied by an English translation, an extensive historical introduction, numerous illustrations, and annotations. Supplementary materials include letters, postcards, speeches, and articles, a map of the voyage, a chronology, a bibliography, and an index.

Einstein would go on to keep a journal for all succeeding trips abroad, and this first volume of his travel diaries offers an initial, intimate glimpse into a brilliant mind encountering the great, wide world. 

More than fifty years after his death, Albert Einstein’s vital engagement with the world continues to inspire others, spurring conversations, projects, and research, in the sciences as well as the humanities. Einstein for the 21st Century shows us why he remains a figure of fascination.

In this wide-ranging collection, eminent artists, historians, scientists, and social scientists describe Einstein’s influence on their work, and consider his relevance for the future. Scientists discuss how Einstein’s vision continues to motivate them, whether in their quest for a fundamental description of nature or in their investigations in chaos theory; art scholars and artists explore his ties to modern aesthetics; a music historian probes Einstein’s musical tastes and relates them to his outlook in science; historians explore the interconnections between Einstein’s politics, physics, and philosophy; and other contributors examine his impact on the innovations of our time. Uniquely cross-disciplinary, Einstein for the 21st Century serves as a testament to his legacy and speaks to everyone with an interest in his work. 

The contributors are Leon Botstein, Lorraine Daston, E. L. Doctorow, Yehuda Elkana, Yaron Ezrahi, Michael L. Friedman, Jürg Fröhlich, Peter L. Galison, David Gross, Hanoch Gutfreund, Linda D. Henderson, Dudley Herschbach, Gerald Holton, Caroline Jones, Susan Neiman, Lisa Randall, Jürgen Renn, Matthew Ritchie, Silvan S. Schweber, and A. Douglas Stone.

On April 6, 1922, in Paris, Albert Einstein and Henri Bergson publicly debated the nature of time. Einstein considered Bergson’s theory of time to be a soft, psychological notion, irreconcilable with the quantitative realities of physics. Bergson, who gained fame as a philosopher by arguing that time should not be understood exclusively through the lens of science, criticized Einstein’s theory of time for being a metaphysics grafted on to science, one that ignored the intuitive aspects of time. Jimena Canales tells the remarkable story of how this explosive debate transformed our understanding of time and drove a rift between science and the humanities that persists today.

The Physicist and the Philosopher is a magisterial and revealing account that shows how scientific truth was placed on trial in a divided century marked by a new sense of time.


After completing the final version of his general theory of relativity in November 1915, Albert Einstein wrote a book about relativity for a popular audience. His intention was “to give an exact insight into the theory of relativity to those readers who, from a general scientific and philosophical point of view, are interested in the theory, but who are not conversant with the mathematical apparatus of theoretical physics.” The book remains one of the most lucid explanations of the special and general theories ever written.

This new edition features an authoritative English translation of the text along with an introduction and a reading companion by Hanoch Gutfreund and Jürgen Renn that examines the evolution of Einstein’s thinking and casts his ideas in a broader present-day context.

Published on the hundredth anniversary of general relativity, this handsome edition of Einstein’s famous book places the work in historical and intellectual context while providing invaluable insight into one of the greatest scientific minds of all time.


Omnia El Shakry: Genealogies of Female Writing


Throughout Women’s History Month, join Princeton University Press as we celebrate scholarship by and about women.

by Omnia El Shakry

In the wake of the tumultuous year for women that was 2017, many female scholars have been reflecting upon their experiences in the academy, ranging from sexual harassment to the everyday experiences of listening to colleagues mansplain or even intellectually demean women’s work. Indeed, I can vividly recall, as a young assistant professor, hearing a senior male colleague brush off what has now become a canonical text in the field of Middle East studies as “merely” an example of gender history, with no wider relevance to the region. Gender history rolled off his tongue with disdain and there was an assumption that it was distinct from real history.

Few now, however, would deign to publicly discount the role that female authors have played in the vitality of the field of Middle East studies. In recognition of this, the Middle East Studies Association of North America has inaugurated new book awards honoring the pioneering efforts of two women in the field, Nikkie Keddie and Fatima Mernissi. I can still remember the first time I read Mernissi’s work while an undergraduate at the American University in Cairo. Ever since my freshman year, I had enrolled in Cultural Anthropology courses with Soraya Altorki—a pioneering anthropologist who had written about Arab Women in the Field and the challenges of studying one’s own society. In her courses, and elsewhere, I was introduced to Lila Abu-Lughod’s Veiled Sentiments, an ethnography of poetry and everyday discourse in a Bedouin community in Egypt’s Western desert. Abu-Lughod’s narrative was sensitive to questions of positionality, a lesson she both drew from and imbued with feminism. A second piece of writing, this time an article by Stefania Pandolfo on “Detours of Life” that interpreted the internal logic of imagining space and bodies in a Moroccan village gave me a breathtaking view of ethnography, the heterogeneity of lifeworlds, and the work of symbolic interpretation. 

In hindsight I can see that these early undergraduate experiences of reading, and studying with, female anthropologists profoundly impacted my own writing. Although I would eventually become a historian, I remained interested in the ethnographic question of encounters, and specifically of how knowledge is produced through encounters­—whether the encounter between the colonizer and the colonized or between psychoanalysis and Islam. In my most recent book, The Arabic Freud: Psychoanalysis and Islam in Modern Egypt, I ask what it might mean to think of psychoanalysis and Islam together, not as a “problem” but as a creative encounter of ethical engagement. Rather than conceptualizing modern intellectual thought as something developed in Europe, merely to be diffused at its point of application elsewhere, I imagine psychoanalytic knowledge as something elaborated across the space of human difference.

There is yet another female figure who stands at the door of my entry into writing about the Middle East. My grandmother was a strong presence in my early college years. Every Friday afternoon I would head over to her apartment, just a quick walk away from my dorm in downtown Cairo. We would eat lunch, laugh and talk, and watch the subtitled American soap operas that were so popular back then. Since she could not read or write, we would engage in a collective work of translation while watching and I often found her retelling of the series to be far more imaginative than anything network television writers could ever have produced.

Writing for me is about the creative worlds of possibility and of human difference that exist both within, but also outside, of the written word. As historians when we write we are translating between the living and the dead, as much as between different life worlds, and we are often propelled by intergenerational and transgenerational bonds that include the written word, but also exceed it.

Omnia El Shakry is professor of history at the University of California, Davis. She is the author of The Arabic Freud: Psychoanalysis and Islam in Modern Egypt.

Andrew Scull: On the response to mass shootings

ScullAmerica’s right-wing politicians have developed a choreographed response to the horrors of mass shootings. In the aftermath of Wednesday’s massacre of the innocents, President Trump stuck resolutely to the script. Incredibly, he managed to avoid even mentioning the taboo word “guns.” In his official statement on this week’s awfulness, he offers prayers for the families of the victims—as though prayers will salve their wounds, or prevent the next outrage of this sort; they now fall thick and fast upon us. And he spouted banalities: “No child, no teacher, should ever be in danger in an American school.” That, of course, was teleprompter Trump. The real Trump, as always, had surfaced hours earlier on Twitter. How had such a tragedy come to pass?  On cue, we get the canned answer: the issue was mental health: “So many signs that the Florida shooter was mentally disturbed.”  Ladies and gentlemen, we have a mental health problem don’t you see, not a gun problem.

Let us set aside the crass hypocrisy of those who have spent so much time attempting to destroy access to health care (including mental health care) for tens of millions of people bleating about the need to provide treatment for mental illness. Let us ignore the fact that President Trump, with a stroke of a pen, set aside regulations that made it a little more difficult for “deranged” people to obtain firearms. They have Second Amendment rights too, or so it would seem. Let us overlook the fact that in at least two of the recent mass shootings, the now-dead were worshipping the very deity their survivors and the rest of us are invited to pray to when they were massacred. Let us leave all of that out of account. Do we really just have a mental health problem here, and would addressing that problem make a dent in the rash of mass killings?

Merely to pose the question is to suggest how fatuous this whole approach is. Pretend for a moment that all violence of this sort is the product of mental illness, not, as is often the case, the actions of evil, angry, or viciously prejudiced souls. Is there the least prospect that any conceivable investment in mental health care could anticipate and forestall gun massacres? Of course not. Nowhere in recorded history, on no continent, in no country, in no century, has any society succeeded in eliminating or even effectively addressing serious forms of mental illness. Improving the lot of those with serious mental illness is a highly desirable goal. Leaving the mentally disturbed to roam or rot on our sidewalks and in our “welfare” hotels, or using a revolving door to move them in and out of jail—the central elements of current mental health “policy”—constitutes a national disgrace. But alleviating that set of problems (as unlikely as that seems in the contemporary political climate) will have zero effect on gun violence and mass shootings.

Mental illness is a scourge that afflicts all civilized societies. The Bible tells us, “The poor ye shall always have with you.”  The same, sadly, is true of mental illness. Mental distress and disturbance constitute one of the most profound sources of human suffering, and simultaneously constitute one of the most serious challenges of both a symbolic and practical sort to the integrity of the social fabric. Whether one looks to classical Greece and Rome, to ancient Palestine or the Islamic civilization that ruled much of the Mediterranean for centuries, to the successive Chinese empires or to feudal and early modern Europe, everywhere people have wrestled with the problem of insanity, and with the need to take steps to protect themselves against the depredations of the minority of the seriously mentally ill people who pose serious threats of violence. None of these societies, or many more I could mention, ever saw the levels of carnage we Americans now accept as routine and inevitable.

Mental illness is an immutable feature of human existence. Its association with mass slaughter most assuredly has not been. Our ancestors were not so naïve as to deny that madness was associated with violence. The mentally ill, in the midst of their delusions, hallucinations, and fury were sometimes capable of horrific acts: consider the portrait in Greek myth of Heracles dashing out the brains of his children, in his madness thinking them the offspring of his mortal enemy Euryththeus; Lucia di Lammermoor stabbing her husband on their wedding night; or Zola’s anti-hero of La Bete humaine, Jacques Lantier, driven by passions that escape the control of his reason, raping and killing the object of his desire: these and other fictional representations linking mental illness to animality and violence are plausible to those encountering them precisely because they match the assumptions and experience of the audiences toward whom they are directed. And real-life maddened murderers were to be found in all cultures across historical time. Such murders were one of the known possible consequences of a descent into insanity. But repeated episodes of mass killing by deranged individuals, occurring as a matter of routine?  Nowhere in the historical record can precursors of the contemporary American experience be found. It is long past time to stop blaming an immutable feature of human culture—severe mental illness—for routine acts of deadly violence that are instead the produce of a resolute refusal to face the consequences of unbridled access to a deadly form of modern technology.

Claims that the mowing down of unarmed innocents is a mental health problem cannot explain why, in that event, such massacres are exceedingly rare elsewhere in the contemporary world, while they are now routine in the United States. Mental illness, as I have stressed, is a universal feature of human existence. Mass shootings are not. Australia and Britain (to take but two examples) found themselves in the not-too-distant past having to cope with horrendous mass killings that involved guns. Both responded with sensible gun control policies, and have been largely spared a repetition of the horrors routinely visited upon innocent Americans. Our society’s “rational” response, by contrast, is to rush out and buy more guns, inflating the profits of those who profit from these deaths, and ensuring more episodes of mass murder.

The problem in the United States is not crazy people. It is crazy gun laws.

Andrew Scull is Distinguished Professor of Sociology and Science Studies at the University of California, San Diego. He is the author of Masters of Bedlam: The Transformation of the Mad-Doctoring Trade and Madness in Civilization: A Cultural History of Insanity, from the Bible to Freud, from the Madhouse to Modern Medicine.

Michael J. Ryan: A Taste for the Beautiful

Darwin developed the theory of sexual selection to explain why the animal world abounds in stunning beauty, from the brilliant colors of butterflies and fishes to the songs of birds and frogs. He argued that animals have “a taste for the beautiful” that drives their potential mates to evolve features that make them more sexually attractive and reproductively successful. But if Darwin explained why sexual beauty evolved in animals, he struggled to understand how. In A Taste for the Beautiful, Michael Ryan, one of the world’s leading authorities on animal behavior, tells the remarkable story of how he and other scientists have taken up where Darwin left off and transformed our understanding of sexual selection, shedding new light on human behavior in the process. Vividly written and filled with fascinating stories, A Taste for the Beautiful will change how you think about beauty and attraction. Read on to learn more about the evolution of beauty, why the sight of a peacock’s tail made Darwin sick, and why males tend to be the more “beautiful” in the animal kingdom.

What made you interested in the evolution of beauty?

For my Masters degree I was studying how male bullfrog set up and defend territories. They have a pretty imposing call that has been described as ‘jug-a-rum;’ it is used to repel neighboring males and to attract females. In those days it was thought that animal sexual displays functioned only to identify the species of the signaler. For example, in the pond where I worked you could easily tell the difference between bullfrogs, leopard frogs, green frogs, and spring peepers by listening to their calls. Females do the same so they can end up mating with the correct species. Variation among the calls within a species was thought of to be just noise, random variation that had little meaning to the females.

But sitting in this swamp night after night I was able to tell individual bullfrogs apart from one another and got used to seeing the same males with the loudest deepest calls in the same parts of the pond. I began to wonder that if I could hear these differences could the female bullfrogs, and could females decide who to mate with based on the male’s call? And also, if some calls sounded more beautiful to me, did female frogs share with me the same aesthetic?

I never got to answer these questions with the bullfrogs but I decided to pursue this general question when I started at Cornell University to work on my PhD degree.

Why did Darwin say that the sight of the peacock’s tail, an iconic example of sexual beauty, made him sick?

Darwin suffered all kinds of physical maladies, some probably brought on by his contraction of Chagas disease during his voyage on the Beagle. But this malady induced by the peacock’s tail probably resulted from cognitive dissonance. He had formulated a theory, natural selection, in which he was able to explain how animals evolve adaptations for survival. Alfred Russel Wallace developed a very similar theory. All seemed right with the world, at least for a while.

But then Darwin pointed out that many animal traits seem to hinder rather than promote survival. These included bright plumage and complex song in birds, flashing of fireflies, male fishes with swords, and of course the peacock with its magnificently long tail. All of these traits presented challenges to his theory of natural selection and the general idea of survival of the fittest. These sexy traits are ubiquitous throughout the animal kingdom but seem to harm rather than promote survival.

Sexual selection is Darwin’s theory that predicts the evolution of sexual beauty. How is this different from Darwin’s theory of natural selection?

The big difference between these two theories is that one focuses on survivorship while the other focuses on mating success. Both are important for promoting evolution, the disproportionate passage of genes from one generation to the next. An animal that survives for a long period of time but never reproduces is in a sense genetically dead. Animals that are extremely attractive but do not live long enough to reproduce also are at a genetic dead-end. It is the proper mix of survivorship and attractiveness that is most favored by selection. But the important point to realize is that natural selection and sexual selection are often opposed to one another; natural selection for example, favoring shorter tails in peacocks and sexual selection favoring longer tails. What the bird ends up with is a compromise between these two opposing selection forces.

In most of evolutionary biology the emphasis is still more on survival than mating success. But sometimes I think that surviving is just nature’s clever trick to keep individuals around long enough so that they can reproduce.

Why is it that in many animals the males are the more beautiful sex?

In most animals there are many differences between males and females. But what is the defining character? What makes a male a male and a female a female? It is not the way they act, the way they look, the way they behave. It is not even defined by the individual’s sex organs, penis versus vagina in many types of animals.

The defining characteristic of the sexes is gamete size. Males have many small gametes and females have fewer large gametes.  The maximum number of offspring that an animal can sire will be limited by the number of gametes. Therefore, males could potentially father many more offspring than a female could mother.

But of course males need females to reproduce. So this sets up competition where the many gametes of the males are competing to hook up with the fewer gametes of the females. Thus in many species males are under selection to mate often, they will never run out of gametes, while females are under selection to mate carefully and make good use of the fewer gametes they have. Thus males are competing for females, either through direct combat or by making themselves attractive to females, and females decide which males get to mate. The latter is the topic of this book.

Why is sexual beauty so dangerous?

The first step in communication is being noticed, standing out against the background. This is true whether animals communicate with sound, vision, or smells. It is especially true for sexual communication. The bind that males face is they need to make themselves conspicuous to females but their communication channel is not private, it is open to exploitation by eavesdroppers. These eavesdroppers can make a quick meal out of a sexually advertising male. One famous example, described in this book, involves the túngara frog and its nemesis, the frog-eating bat. Male túngara frogs add syllables, chucks, to their calls to increase their attractiveness to females. But it also makes them more attractive to bats, so when these males become more attractive they also become more likely to become a meal rather than a mate.

The túngara frog is only one example of the survival cost of attractive traits. When crickets call, for example, they can attract a parasitic fly. The fly lands on the calling male and her larvae crawl off of her onto the calling cricket. The larvae then burrow deep inside the cricket where they will develop. As they develop they eat the male from the inside out, and their first meal is the male’s singing muscle. This mutes the male so he will not attract other flies who would deposit their larvae on the male who would then become competitors.

Another cost of being attractive is tied up with the immune system. Many of the elaborate sexual traits of males develop in response to high levels of testosterone. Testosterone can have a negative effect on the immune system. So as males experience higher testosterone levels that might produce more attractive ornaments, but these males are paying the cost with their ability to resist disease.

How did you come to discover that frog eating bats are attracted to the calls of túngara frogs?

The credit for this initial discovery goes to Merlin Tuttle. Merlin is a well-known bat biologist and he was on BCI the year before I was. He captured a bat with a frog in its mouth. Merlin wondered how common this behavior was and whether the bats could hear the calls of the frogs and use those calls to find the frogs.

When Stan Rand and I discovered that túngara frogs become more attractive when they add chucks to their calls, we wondered why they didn’t produce chucks all the time. We were both convinced that there were some cost of producing chucks and we both thought it was likely the ultimate cost imposed by a predator.

Merlin contacted Stan about collaborating on research with the frog-eating bat and frog calls, and Stan then introduced Merlin to me. The rest is history as this research has blossomed into a major research program for a number of people.

Is beauty really in the eye of the beholder?

Yes, but it is also in the ears, the noses, the toes and any other sense organ recruited to check out potential mates. All of these sense organs forward information to the brain where judgements about beauty are made. So it is more accurate to say beauty is in the brain of the beholder. It might be true that the brain is our most important sex organ, but the brain has other things on its mind besides sex. It evolves under selection to perform a number of functions, and adaptations in one function can lead to unintended consequences for another function. For example, studies of some fish show that the color sensitivity of the eyes evolves to facilitate the fish’s ability to find its prey. Once this happens though, males evolve courtship colors to which their females’ eyes are particularly sensitive. This is called sensory exploitation.

A corollary of ‘beauty is in the brain of the beholder’ is that choosers, usually females, define what is beautiful. Females are not under selection to find out which males are attractive, by determining which males are attractive. They are in the driver’s seat when it comes to the evolution of beauty.

What is sensory exploitation?

We have probably all envisioned the perfect sexual partner. And in many cases those visions do not exist in reality. In a sense, the same might be true in animals. Females can have preferences for traits that do not exist. Or at least do not yet exist. When males evolve traits that elicit these otherwise hidden preferences this is called sensory exploitation. We can think of the evolution of sexual beauty as evolutionary attempts to probe the ‘preference landscape’ of the female. When a trait matches one of these previously unexpressed preferences, the male trait is immediately favored by sexual selection because it increases his mating success.

A good example of this occurs in a fish called the swordtail. In these fishes males have sword-like appendages protruding from their tails. Female swordtails prefer males with swords to those without swords, and males with longer swords to males with shorter swords. Swordtails are related to platyfish, the sword of swordtails evolved after the platyfish and swordtails split off from one another thousands of years ago. But when researchers attach a plastic sword to a male platyfish he becomes more attractive to female platyfish. These females have never seen a sworded male but they have a preference for that trait nonetheless. Thus it appears that when the first male swordtail evolved a sword the females already had a preference for this trait.

Do the girls really get prettier at closing time, as Mickey Gilly once sang?

They sure do, and so do the boys. A study showed that both men and women in a bar perceive members of the opposite sex as more attractive as closing time approaches. This classic study was repeated in Australia where they measured blood alcohol levels and showed that the ‘closing time’ effect was not only due to drinking but to the closing time of the bar.

The interpretation is of these results is that if an individual wants to go home with a member of the opposite sex but none of the individuals meet her or his expectations of beauty, the individual has two choices. They can lower their standards of beauty or they can deceive themselves and perceive the same individuals as more attractive. They seem to do the latter.

Although we do not know what goes on in an animal’s head, they show a similar pattern of behavior. Guppies and roaches are much more permissive in accepting otherwise unattractive mates as they get closer to the ultimate closing time, the end of their lives. In a similar example, early in the night female túngara frogs will reject certain calls that are usually unattractive, but later in the night when females become desperate to find a mate they become more than willing to be attracted to these same calls.  It is also noteworthy that middle-aged women think about sex more and have sex more often than do younger women.

Deception seems to be widespread in human courtship. What about animals?

Males have a number of tricks to deceive females for the purpose of mating. One example involves moths in which males make clicking sounds to court females. When males string together these clicks in rapid succession it sounds like the ‘feeding buzz’ of a bat, the sound a bat makes as it zeros in on its prey. At least this is what the female moths think. When they hear these clicks they freeze and the male moth is then able to mount the female and mate with her with little resistance as she appears to be scared to death, not of the male moth but of what she thinks is a bat homing in on her for the kill.

Other animals imitate food to drive female’s attention. Male mites beat their legs on the water surface imitating vibrations caused by copepods, the main source of food for the water mites. When females approach the source of these were vibrations they find a potential mate rather than a potential meal.

What about peer pressure? We know this plays a role in human in interactions, and the influence our perceptions of beauty? What about animals, can they be subjected to peer pressure?

Suppose a woman looks at my picture and is told rate my attractiveness on the scale of 1 to 10. Another woman is asked to do the same but in this picture I am standing next to an attractive woman. Almost certainly I will get a higher score the second time; my attractiveness increases although nothing about my looks have changed, only that I was consorting with a good-looking person.  This is referred to as mate choice and it is widespread in the animal kingdom.

Mate choice copying was first experimentally demonstrated in guppies. Female guppies prefer males who have more orange over those who have less orange. In a classic experiment, females were given a choice between a more than a less colorful male. They preferred the more colorful male and then were returned to the center of the tank for another experiment. In this instance they saw the less colorful and less preferred male courting a female. That female was removed and the test female was tested for her preference for the same two males once again. Now the female changes her preference and prefers what previously had been the less preferred male. She too seems to be employing mate choice copying.

Many animals learn by observing others. Mate choice copying seems to be a type of observational learning that is common in many animals in many domains. It might suggest that we be careful with whom we hang out.

What is the link between sexual attraction in animals and pornography in humans?

Animal sexual beauty is often characterized by being extreme: long tails, complex songs, brilliant colors, and outrageous dances. The same is often true of sexual beauty in humans. Female supermodels, for example, tend to be much longer and thinner than most other women in the population, male supermodels are super-buff—hardly normal. Furthermore, in animals we can create sexual traits that are more extreme than what exists in males of the population, and in experiments females often prefer these artificially exaggerated traits, such as: even longer tales, more complex songs, and more brilliant colors than exhibited by their own males. These are called supernormal stimuli. Pornography also creates supernormal stimuli not only in showcasing individuals with extreme traits but also in creating social settings that hardly exist in most societies, this manufactured social setting is sometimes referred to as Pornotopia.

RyanMichael J. Ryan is the Clark Hubbs Regents Professor in Zoology at the University of Texas and a Senior Research Associate at the Smithsonian Tropical Research Institute in Panama. He is a leading researcher in the fields of sexual selection, mate choice, and animal communication. He lives in Austin, Texas.


Browse our 2018 Computer Science & Information Science Catalog

Our new Computer Science & Information Science catalog includes an accessible and rigorous textbook for introducing undergraduates to computer science theory, a fascinating account of the breakthrough ideas that transformed probability and statistics, and an amazing tour of many of history’s greatest unsolved ciphers.

If you’re attending the ITA Workshop-Information Theory and Its Application conference this week, please stop by our table to browse our full range of titles.

What Can Be Computed? is a uniquely accessible yet rigorous introduction to the most profound ideas at the heart of computer science. Crafted specifically for undergraduates who are studying the subject for the first time, and requiring minimal prerequisites, the book focuses on the essential fundamentals of computer science theory and features a practical approach that uses real computer programs (Python and Java) and encourages active experimentation. It is also ideal for self-study and reference.

Throughout, the book recasts traditional computer science concepts by considering how computer programs are used to solve real problems. Standard theorems are stated and proven with full mathematical rigor, but motivation and understanding are enhanced by considering concrete implementations. The book’s examples and other content allow readers to view demonstrations of–and to experiment with—a wide selection of the topics it covers. The result is an ideal text for an introduction to the theory of computation.

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.

Unsolved! begins by explaining the basics of cryptology, and then explores the history behind an array of unsolved ciphers. It looks at ancient ciphers, ciphers created by artists and composers, ciphers left by killers and victims, Cold War ciphers, and many others. Some are infamous, like the ciphers in the Zodiac letters, while others were created purely as intellectual challenges by figures such as Nobel Prize–winning physicist Richard P. Feynman. Bauer lays out the evidence surrounding each cipher, describes the efforts of geniuses and eccentrics—in some cases both—to decipher it, and invites readers to try their hand at puzzles that have stymied so many others.

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!

Browse Our 2018 Physics & Astrophysics Catalog

Our new Physics & Astrophysics catalog includes two new graduate-level textbooks from Kip S. Thorne, Co-Winner of the 2017 Noble Prize in Physics, as well as a look into the physics behind black holes.

If you plan on attending AAS 2018 in National Harbor, MD this weekend, please stop by Booth 1003 to see our full range of Physics and Astrophysics titles and more.

Black holes, predicted by Albert Einstein’s general theory of relativity more than a century ago, have long intrigued scientists and the public with their bizarre and fantastical properties. Although Einstein understood that black holes were mathematical solutions to his equations, he never accepted their physical reality—a viewpoint many shared. This all changed in the 1960s and 1970s, when a deeper conceptual understanding of black holes developed just as new observations revealed the existence of quasars and X-ray binary star systems, whose mysterious properties could be explained by the presence of black holes. Black holes have since been the subject of intense research—and the physics governing how they behave and affect their surroundings is stranger and more mind-bending than any fiction.

The Little Book of Black Holes takes readers deep into the mysterious heart of the subject, offering rare clarity of insight into the physics that makes black holes simple yet destructive manifestations of geometric destiny.

Modern Classical Physics is a long-awaited, first-year, graduate-level text and reference book covers the fundamental concepts and twenty-first-century applications of six major areas of classical physics that every masters- or PhD-level physicist should be exposed to, but often isn’t: statistical physics, optics (waves of all sorts), elastodynamics, fluid mechanics, plasma physics, and special and general relativity and cosmology. Growing out of a full-year course that the eminent researchers Kip Thorne and Roger Blandford taught at Caltech for almost three decades, this book is designed to broaden the training of physicists. Its six main topical sections are also designed so they can be used in separate courses, and the book provides an invaluable reference for researchers.

First published in 1973, Gravitation is a landmark graduate-level textbook that presents Einstein’s general theory of relativity and offers a rigorous, full-year course on the physics of gravitation. Upon publication, Science called it “a pedagogic masterpiece,” and it has since become a classic, considered essential reading for every serious student and researcher in the field of relativity. This authoritative text has shaped the research of generations of physicists and astronomers, and the book continues to influence the way experts think about the subject.

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.