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

by Joshua Holden

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

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

What is a symmetry?

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

Figure1

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

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

Figure 2

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

Figure 3

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

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

Groups are sets of symmetries with structure

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

Figure 4

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

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

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

Figure 5

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

Figure 6

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

Categorizing the Pariah groups

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

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

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

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

Michael Ruse on On Purpose

Can we live without the idea of purpose? Should we even try to? Kant thought we were stuck with purpose, and even Darwin’s theory of natural selection, which profoundly shook the idea, was unable to kill it. Indeed, teleological explanation—what Aristotle called understanding in terms of “final causes”—seems to be making a comeback today, as both religious proponents of intelligent design and some prominent secular philosophers argue that any explanation of life without the idea of purpose is missing something essential. In On Purpose, Michael Ruse explores the history of the idea of purpose in philosophical, religious, scientific, and historical thought, from ancient Greece to the present. Read on to learn more about the idea of “purpose,” the long philosophical tradition around it, and how Charles Darwin fits in.

On Purpose?  So what’s with the smart-alecky title?

It was a friend of Dr. Johnson who said that he had tried to be a philosopher, but cheerfulness always kept breaking in.  Actually, that is a little bit unfair to philosophers.  Overall, we are quite a cheerful group, especially when we think that we might have been born sociologists or geographers.  However, our sense of humor is a bit strained, usually—as in this case—involving weak puns and the like.  My book is about a very distinctive form of understanding, when we do things in terms of the future and not the past.

In terms of the future?  Why not call your book On Prediction?

I am not talking about prediction, forecasting what you think will happen, although that is involved.  I am talking about when the future is brought in to explain things that are happening right now.  Purposeful thinking is distinctive and interesting because normally when we try to explain things we do so in terms of the past or present.  Why do you have a bandage on your thumb?  Because I tried to hang the picture myself, instead of getting a grad student to do it.  Purposeful thinking—involving what Aristotle called “final causes” and what since the eighteenth century has often been labeled “teleological” thinking—explains in terms of future events.  Why are you studying rather than going to the ball game?  Because I want to do well on the GRE exam and go to a good grad school.

Why is this interesting?

In the case of the bandaged thumb, you know that the hammer hit you rather than the nail.  In the case of studying, you may decide that five to ten years of poverty and peonage followed by no job is not worth it, and you should decide to do something worthwhile like becoming a stockbroker or university administrator.  We call this “the problem of the missing goal object.”  Going to grad school never occurred, but it still makes sense to say that you are studying now in order to go to grad school.

Is this something that you thought up, or is it something with a history?

Oh my, does it ever have a history.  One of the great things about my book, if I might show my usual level of modesty, is that I show the whole problem of purpose is one with deep roots in the history of philosophy, starting with Plato and Aristotle, and coming right up to the modern era, particularly the thinking of Immanuel Kant.  In fact, I argue that it is these three very great philosophers who set the terms of the discussion—Plato analyses things in terms of consciousness, Aristotle in terms of principles of ordering whatever that might mean, and Kant opts for some kind of heuristic approach.

If these thinkers have done the spadework, what’s left for you?

I argue that the truth about purposeful thinking could not be truly discovered until Charles Darwin in his Origin of Species (1859) had proposed his theory of evolution through natural selection.  With that, we could start to understand forward-looking thinking about humans—why is he studying on such a beautiful day?  He wants to go to grad school.  About plants and animals—why does the stegosaurus have those funny-looking plates down its back?  To control its temperature.  And why we don’t use such thinking about inanimate objects?  Why don’t we worry about the purpose of the moon?  Perhaps we should.  It really does exist in order to light the way home for drunken philosophers.

Why is it such a big deal to bring up Darwin and his theory of evolution?  Surely, the kind of people who will read your book will have accepted the theory long ago?

Interestingly, no!  The main opposition to evolutionary thinking comes from the extreme ends of the spectrum: evangelical Christians known as Creationists—biblical literalists—and from professional philosophers.  There are days when it seems that the higher up the greasy pole you have climbed, the more likely you are to deny Darwinism and be a bit iffy about evolution generally.  This started just about as soon as the Origin appeared, and the sinister anti-evolutionary effect of Bertrand Russell and G. E. Moore and above all Ludwig Wittgenstein is felt to this day.  A major reason for writing my book was to take seriously Thomas Henry Huxley’s quip that we are modified monkeys rather than modified mud, and that matters.

Given that you are a recent recipient of the Bertrand Russell Society’s “Man of the Year” Award, aren’t you being a bit ungracious?

I have huge respect for Russell.  He was a god in my family when, in the 1940s and 50s, I was growing up in England.  One of my greatest thrills was to have been part of the crowd in 1961 in Trafalgar Square listening to him declaim against nuclear weapons.  But I think he was wrong about the significance of Darwin for philosophy and I think I am showing him great respect in arguing against him.  I feel the same way about those who argue against me.  My proudest boast is that I am now being refuted in journals that would never accept anything by me.

One of the big problems normal people today have about philosophy is that it seems so irrelevant. Initiates arguing about angels on the heads of pins?  Why shouldn’t we say the same about your book?

Three reasons.  First, my style and approach.  It is true that most philosophy produced by Anglophone philosophers today is narrow and boring.  Reading analytic philosophy is like watching paint dry and proudly so.  Against this, on the one hand I am more a historian of ideas using the past to illuminate the present.  That is what being an evolutionist is all about.  Spending time with mega-minds like Plato and Aristotle and Kant is in itself tremendously exciting.  On the other hand, I have over fifty years of teaching experience, at the undergraduate level almost always at the first- and second-year level.  I know that if you are not interesting, you are going to lose your audience.  The trick is to be interesting and non-trivial.

Second, I don’t say that my book is the most important of the past hundred-plus years, but my topic is the most important.  Evolution matters, folks, it really does.  It is indeed scary to think that we are just the product of a random process of change and not the favored product of a Good God—made in His image.  Even atheists get the collywobbles, or at least they should.  It is true all the same.  Fifty years ago, the geneticist and Nobel laureate Hermann J. Muller said that a hundred years without Darwin is enough.  That is still true.  Amen.

Third, deliberately, I have made this book very personal.  At the end, I talk about purpose in my own life.  Why, even though I am a non-believer, I have been able to find meaning in what I think and do.  This ranges from my love of my wife Lizzie and how with dedication and humor we share the challenges of having children—not to mention our love of dogs, most recent addition to the family, Nutmeg a whippet—through cooking on Saturday afternoons while listening to radio broadcasts of Metropolitan Opera matinees, to reading Pickwick Papers yet one more time.  I suspect that many of my fellow philosophers will find this all rather embarrassing.  I mean it to be.  Philosophy matters.  My first-ever class on the subject started with Descartes’ Meditations.  Fifteen minutes into the class, I knew that this was what I was going to do for the rest of my life.  Nearly sixty years later I am still at it and surely this interview tells you that I love it, every moment.

So, why should we read your book?

Because it really does square the circle.  It is cheerful and philosophical.  It is on a hugely important topic and there are some good jokes.  I am particularly proud of one I make about Darwin Day, the celebration by New Atheists, and their groupies of the birthday of Charles Darwin.

Which is?

Oh, hell no.  I am not going to tell you.  Go out and buy the book.  And while you are at it, buy one for your mum and dad and one each for your siblings and multi-copies for your students and….  I am seventy-seven years old.  I need a bestseller so I can retire.  You need a bestseller so I can retire.

RuseMichael Ruse is the Lucyle T. Werkmeister Professor of Philosophy and Director of the Program in the History and Philosophy of Science at Florida State University. He has written or edited more than fifty books, including Darwinism as Religion, The Philosophy of Human Evolution, and The Darwinian Revolution.

Craig Bauer: Attacking the Zodiac Killer

While writing Unsolved! The History and Mystery of the World’s Greatest Ciphers from Ancient Egypt to Online Secret Societies, it soon became clear to me that I’d never finish if I kept stopping to try to solve the ciphers I was covering. It was hard to resist, but I simply couldn’t afford to spend months hammering away at each of the ciphers. There were simply too many of them. If I was to have any chance of meeting my deadline, I had to content myself with merely making suggestions as to how attacks could be carried out. My hope was that the book’s readers would be inspired to actually make the attacks. However, the situation changed dramatically when the book was done.

I was approached by the production company Karga Seven Pictures to join a team tasked with hunting the still unidentified serial killer who called himself the Zodiac. In the late 1960s and early 70s, the Zodiac killed at least five people and terrorized entire cities in southern California with threatening letters mailed to area newspapers. Some of these letters included unsolved ciphers. I made speculations about these ciphers in my book, but made no serious attempt at cracking them. With the book behind me, and its deadline no longer a problem, would I like to join a code team to see if we could find solutions where all others had failed? The team would be working closely with a pair of crack detectives, Sal LaBarbera and Ken Mains, so that any leads that developed could be investigated immediately. Was I willing to take on the challenge of a very cold case? Whatever the result was, it would be no secret, for our efforts would be aired as a History channel mini-series. Was I up for it? Short answer: Hell yeah!

The final code team included two researchers I had corresponded with when working on my book, Kevin Knight (University of Southern California, Information Sciences Institute) and David Oranchak (software developer and creator of Zodiac Killer Ciphers. The other members were Ryan Garlick (University of North Texas, Computer Science) and Sujith Ravi (Google software engineer).

My lips are sealed as to what happened (why ruin the suspense?), but the show premieres Tuesday November 14, 2017 at 10pm EST. It’s titled “The Hunt for the Zodiac Killer.” All I’ll say for now is that it was a rollercoaster ride. For those of you who would like to see how the story began for me, Princeton University Press is making the chapter of my book on the Zodiac killer freely available for the duration of the mini-series. It provides an excellent background for those who wish to follow the TV team’s progress.

If you find yourself inspired by the show, you can turn to other chapters of the book for more unsolved “killer ciphers,” as well challenges arising from nonviolent contexts. It was always my hope that readers would resolve some of these mysteries and I’m more confident than ever that it can be done!

BauerCraig P. Bauer is professor of mathematics at York College of Pennsylvania. He is editor in chief of the journal Cryptologia, has served as a scholar in residence at the NSA’s Center for Cryptologic History, and is the author of Secret History: The Story of Cryptology. He lives in York, Pennsylvania.

Joel Brockner: The Passion Plea

This post originally appears on the blog of Psychology Today

BrocknerIt’s tough to argue with the idea that passion is an admirable aspect of the human condition. Passionate people are engaged in life; they really care about their values and causes and being true to them. However, a big minefield of passion is when people use it to excuse or explain away unseemly behavior. We saw this during the summer of 2017 in how the White House press secretary, Sarah Huckabee Sanders, responded to the infamous expletive-laced attack of Anthony Scaramucci on his then fellow members of the Trump team, Steve Bannon and Reince Priebus. According to The New York Times, (July 27, 2017),  “Ms. Sanders said mildly that Mr. Scaramucci was simply expressing strong feelings, and that his statement made clear that ‘he’s a passionate guy and sometimes he lets that passion get the better of him.’ ” Whereas Ms. Sanders acknowledged that Mr. Scaramucci behaved badly (his passion got the better of him), her meta-message is that it was no big deal, as implied by the words “mildly” and “simply” in the quote above.

The passion plea is by no means limited to the world of politics. Executives who are seen as emotionally rough around the edges by their co-workers often defend their behavior with statements like, “I’m just being passionate,” or “I am not afraid to tell it like it is,” or, “My problem is that I care too much.”

The passion plea distorts reality by glossing over the distinction between what is said and how it is said. Executives who deliver negative feedback in a harsh tone are not just being passionate. Even when the content of the negative feedback is factual, harsh tones convey additional messages – notably a lack of dignity and respect. Almost always, there are ways to send the same strong messages or deliver the same powerful feedback in ways that do not convey a lack of dignity and respect. For instance, Mr. Scaramucci could have said something like, “Let me be as clear as possible: I have strong disagreements with Steve Bannon and Reince Priebus.” It may have been less newsworthy, but it could have gotten the same message across. Arguably, Mr. Scaramucci’s 11-day tenure as White House director of communications would have been longer had he not been so “passionate” and instead used more diplomatic language.

Similarly, executives that I coach rarely disagree when it is made evident that they could have sent the same strong negative feedback in ways that would have been easier for their co-workers to digest. Indeed, this is the essence of constructive criticism, which typically seeks to change the behavior of the person on the receiving end. Rarely are managers accused of coming on “too strong” if they deliver negative feedback in the right ways. For example, instead of saying something about people’s traits or characters (e.g., “You aren’t reliable”) it would be far better to provide feedback with reference to specific behavior (e.g., “You do not turn in your work on time”). People usually are more willing and able to respond to negative feedback about what they do rather than who they are. Adding a problem-solving approach is helpful as well, such as, “Some weeks you can be counted on to do a good job whereas other weeks not nearly as much. Why do you think that is happening, and what can we do together to ensure greater consistency in your performance?” Moreover, the feedback has to be imparted in a reasonable tone of voice, and in a context in which people on the receiving end are willing and able to take it in. For instance, one of my rules in discussing with students why they didn’t do well on an assignment is that we not talk immediately after they received the unwanted news. It is far better to have a cooling-off period in which defensiveness goes down and open-mindedness goes up.

If our goal is to alienate people or draw negative attention to ourselves then we should be strong and hard-driving, even passionate, in what we say as well as crude and inappropriate in how we say it. However, if we want to be a force for meaningful change or a positive role model, it is well within our grasp to be just as strong and hard-driving in what we say while being respectful and dignified in how we say it.

Joel Brockner is the Phillip Hettleman Professor of Business at Columbia Business School.

Two PUP Books Longlisted for the 2018 AAAS/Subaru SB&F Prizes

We are delighted that Monarchs and Milkweed by Anurag Agrawal and Welcome to the Universe by Neil DeGrasse Tyson, Michael Strauss, and J. Richard Gott have been longlisted for the AAAS/Subaru SB&F Prizes for Excellence in Science Books!

The Prizes celebrate outstanding science writing and illustration for children and young adults and are meant to encourage the writing and publishing of high-quality science books for all ages. AAAS believes that, through good science books, this generation, and the next, will have a better understanding and appreciation of science.

Agrawal

Welcome to the Universe

Steven S. Gubser: Thunder and Lightning from Neutron Star mergers

As of late 2015, we have a new way of probing the cosmos: gravitational radiation. Thanks to LIGO (the Laser Interferometer Gravitational-wave Observatory) and its new sibling Virgo (a similar interferometer in Italy), we can now “hear” the thumps and chirps of colliding massive objects in the universe. Not for nothing has this soundtrack been described by LIGO scientists as “the music of the cosmos.” This music is at a frequency easily discerned by human hearing, from somewhat under a hundred hertz to several hundred hertz. Moreover, gravitational radiation, like sound, is wholly different from light. It is possible for heavy dark objects like black holes to produce mighty gravitational thumps without at the same time emitting any significant amount of light. Indeed, the first observations of gravitational waves came from black hole merger events whose total power briefly exceeded the light from all stars in the known universe. But we didn’t observe any light from these events at all, because almost all their power went into gravitational radiation.

In August 2017, LIGO and Virgo observed a collision of neutron stars which did produce observable light, notably in the form of gamma rays. Think of it as cosmic thunder and lightning, where the thunder is the gravitational waves and the lightning is the gamma rays. When we see a flash of ordinary lightning, we can count a few seconds until we hear the thunder. Knowing that sound travels one mile in about five seconds, we can reckon how distant the event is. The reason this method works is that light travels much faster than sound, so we can think of the transmission of light as instantaneous for purposes of our estimate.

Things are very different for the neutron star collision, in that the event took place about 130 million light years away, but the thunder and lightning arrived on earth pretty much simultaneously. To be precise, the thunder was first: LIGO and Virgo heard a basso rumble rising to a characteristic “whoop,” and just 1.7 seconds later, the Fermi and INTEGRAL experiments observed gamma ray bursts from a source whose location was consistent with the LIGO and Virgo observations. The production of gamma rays from merging neutron stars is not a simple process, so it’s not clear to me whether we can pin that 1.7 seconds down as a delay precisely due to the astrophysical production mechanisms; but at least we can say with some confidence that the propagation time of light and gravity waves are the same to within a few seconds over 130 million light years. From a certain point of view, that amounts to one of the most precise measurements in physics: the ratio of the speed of light to the speed of gravity equals 1, correct to about 14 decimal places or better.

The whole story adds up much more easily when we remember that gravitational waves are not sound at all. In fact, they’re nothing like ordinary sound, which is a longitudinal wave in air, where individual air molecules are swept forward and backward just a little as the sound waves pass them by. Gravitational waves instead involve transverse disturbances of spacetime, where space is stretched in one direction and squeezed in another—but both of those stretch-squeeze directions are at right angles to the direction of the wave. Light has a similar transverse quality: It is made up of electric and magnetic fields, again in directions that are at right angles to the direction in which the light travels. It turns out that a deep principle underlying both Maxwell’s electromagnetism and Einstein’s general relativity forces light and gravitational waves to be transverse. This principle is called gauge symmetry, and it also guarantees that photons and gravitons are massless, which implies in turn that they travel at the same speed regardless of wavelength.

It’s possible to have transverse sound waves: For instance, shearing waves in crystals are a form of sound. They typically travel at a different speed from longitudinal sound waves. No principle of gauge symmetry forbids longitudinal sound waves, and indeed they can be directly observed, along with their transverse cousins, in ordinary materials like metals. The gauge symmetries that forbid longitudinal light waves and longitudinal gravity waves are abstract, but a useful first cut at the idea is that there is extra information in electromagnetism and in gravity, kind of like an error-correcting code. A much more modest form of symmetry is enough to characterize the behavior of ordinary sound waves: It suffices to note that air (at macroscopic scales) is a uniform medium, so that nothing changes in a volume of air if we displace all of it by a constant distance.

In short, Maxwell’s and Einstein’s theories have a feeling of being overbuilt to guarantee a constant speed of propagation. And they cannot coexist peacefully as theories unless these speeds are identical. As we continue Einstein’s hunt for a unified theory combining electromagnetism and gravity, this highly symmetrical, overbuilt quality is one of our biggest clues.

The transverse nature of gravitational waves is immediately relevant to the latest LIGO / Virgo detection. It is responsible for the existence of blind spots in each of the three detectors (LIGO Hanford, LIGO Livingston, and Virgo). It seems like blind spots would be bad, but they actually turned out to be pretty convenient: The signal at Virgo was relatively weak, indicating that the direction of the source was close to one of its blind spots. This helped localize the event, and localizing the event helped astronomers home in on it with telescopes. Gamma rays were just the first non-gravitational signal observed: the subsequent light-show from the death throes of the merging neutron stars promises to challenge and improve our understanding of the complex astrophysical processes involved. And the combination of gravitational and electromagnetic observations will surely be a driver of new discoveries in years and decades to come.

 

BlackSteven S. Gubser is professor of physics at Princeton University and the author of The Little Book of String TheoryFrans Pretorius is professor of physics at Princeton. They both live in Princeton, New Jersey. They are the authors of The Little Book of Black Holes.

Announcing the trailer for The Seduction of Curves by Allan McRobie

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

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

Steven S. Gubser & Frans Pretorius: The Little Book of Black Holes

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 by Steven S. Gubser and Frans Pretorius 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. Read on to learn a bit more about black holes and what inspired the authors to write this book.

Your book tells the story of black holes from a physics perspective. What are black holes, really? What’s inside?

Black holes are regions of spacetime from which nothing can escape, not even light. In our book, we try to live up to our title by getting quickly to the heart of the subject, explaining in non-technical terms what black holes are and how we use Einstein’s theory of relativity to understand them. What’s inside black holes is a great mystery. Taken at face value, general relativity says spacetime inside a black hole collapses in on itself, so violently that singularities form. We need something more than Einstein’s theory of relativity to understand what these singularities mean. Hawking showed that quantum effects cause black holes to radiate very faintly. That radiation is linked with quantum fluctuations inside the black hole. But it’s a matter of ongoing debate whether these fluctuations are a key to resolving the puzzle of the singularity, or whether some more drastic theory is needed.

How sure are we that black holes exist?

A lot more certain than we were a few years ago. In September 2015, the LIGO experiment detected gravitational waves from the collision of two black holes, each one about thirty times the mass of the sun. Everything about that detection fit our expectations based on Einstein’s theories, so it’s hard to escape the conclusion that there really are black holes out there. In fact, before the LIGO detection we were already pretty sure that black holes exist. Matter swirling around gigantic black holes at the core of distant galaxies form the brightest objects in the Universe. They’re called quasars, and the only reason they’re dim in our sight is that they’re so far away, literally across the Universe. Similar effects around smaller black holes generate X-rays that we can detect relatively nearby, mere thousands of light years away from us. And we have good evidence that there is a large black hole at the center of the Milky Way.

Can you talk a bit about the formation of black holes?

Black holes with mass comparable to the sun can form when big stars run out of fuel and collapse in on themselves. Ordinarily, gravity is the weakest force, but when too much matter comes together, no force conceivable can hold it up against the pull of gravity. In a sense, even spacetime collapses when a black hole forms, and the result is a black hole geometry: an endless inward cascade of nothing into nothing. All the pyrotechnics that we see in distant quasars and some nearby X-ray sources comes from matter rubbing against itself as it follows this inward cascade.

How have black holes become so interesting to non-specialists? How have they been glorified in popular culture?

There’s so much poetry in black hole physics. Black hole horizons are where time stands still—literally! Black holes are the darkest things that exist in Nature, formed from the ultimate ashes of used-up stars. But they create brilliant light in the process of devouring yet more matter. The LIGO detection was based on a black hole collision that shook the Universe, with a peak power greater than all stars combined; yet we wouldn’t even have noticed it here on earth without the most exquisitely sensitive detector of spacetime distortions ever built. Strangest of all, when stripped of surrounding matter, black holes are nothing but empty space. Their emptiness is actually what makes them easy to understand mathematically. Only deep inside the horizon does the emptiness end in a terrible, singular core (we think). Horrendous as this sounds, black holes could also be doorways into wormholes connecting distant parts of the Universe. But before packing our bags for a trip from Deep Space Nine to the Gamma Quadrant, we’ve got to read the fine print: as far as we know, it’s impossible to make a traversable wormhole.

What inspired you to write this book? Was there a point in life where your interest in this topic was piqued?

We both feel extremely fortunate to have had great mentors, including Igor Klebanov, Curt Callan, Werner Israel, Matthew Choptuik, and Kip Thorne who gave us a lot of insight into black holes and general relativity. And we owe a big shout-out to our editor, Ingrid Gnerlich, who suggested that we write this book.

GubserSteven S. Gubser is professor of physics at Princeton University and the author of The Little Book of String Theory. Frans Pretorius is professor of physics at Princeton.

Kip Thorne & Roger Blandford on Modern Classical Physics

PhysicsThis 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 S. Thorne, winner of the 2017 Nobel Prize in Physics, and Roger D. 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.

This book emerged from a course you both began teaching nearly 4 decades ago. What drove you to create the course, and ultimately to write this book?

KST: We were unhappy with the narrowness of physics graduate education in the United States. We believed that every masters-level or PhD physicist should be familiar with the basic concepts of all the major branches of classical physics and should have some experience applying them to real world phenomena. But there was no obvious route to achieve this, so we created our course.

RDB: Of course we had much encouragement from colleagues who helped us teach it and students who gave us invaluable feedback on the content.

The title indicates that the book is a “modern” approach to classical physics (which emphasizes physical phenomena at macroscopic scales). What specifically is “modern” in your book’s approach to this subject?

KST: Classical-physics ideas and tools are used extensively today in research areas as diverse as astrophysics, high-precision experimental physics, optical physics, biophysics, controlled fusion, aerodynamics, computer simulations, etc. Our book draws applications from all these modern topics and many more. Also, these modern applications have led to powerful new viewpoints on the fundamental concepts of classical physics, viewpoints that we elucidate—for example, quantum mechanical viewpoints and language for purely classical mode-mode coupling in nonlinear optics and in nonlinear plasma physics.

Why do you feel that it is so important for readers to become more familiar with classical physics, beyond what they may have been introduced to already?

KST: In their undergraduate and graduate level education, most physicists have been exposed to classical mechanics, electromagnetic theory, elementary thermodynamics, and little classical physics beyond this. But in their subsequent careers, most physicists discover that they need an understanding of other areas of classical physics (and this book is a vehicle for that).

In many cases they may not even be aware of their need. They encounter problems in their research or in R&D where powerful solutions could be imported from other areas of classical physics, if only they were aware of those other areas. An example from my career: in the 1970s, when trying to understand recoil of a binary star as it emits gravitational waves, I, like many relativity physicists before me, got terribly confused. Then my graduate student, Bill Burke—who was more broadly educated than I—said “we can resolve the confusion by adopting techniques that are used to analyze boundary layers in fluid flows around bodies with complicated shapes.” Those techniques (matched asymptotic expansions), indeed, did the job, and through Bill, they were imported from fluid mechanics into relativity.

RDB: Yes. To give a second example, when I was thinking about ways to accelerate cosmic rays, I recalled graduate lectures on stellar dynamics and found just the tools I needed.

You also mention in the book that geometry is a deep theme and important connector of ideas. Could you explain your perspective, and how geometry is used thematically throughout the book?

KST: The essential point is that, although coordinates are a powerful, and sometimes essential, tool in many calculations, the fundamental laws of physics can be expressed without the aid of coordinates; and, indeed, their coordinate-free expressions are generally elegant and exceedingly powerful. By learning to think about the laws in coordinate-free (geometric) language, a physicist acquires great power. For example, when one searches for new physical laws, requiring that they be geometric (coordinate-free) constrains enormously the forms that they may take. And in many practical computations (for example, of the relativistic Doppler shift), a geometric route to the solution can be faster and much more insightful than one that uses coordinates. Our book is infused with this.

RDB: We are especially keen on presenting these fundamental laws in a manner which makes explicit the geometrically formulated conservation laws for mass, momentum, energy, etc. It turns out that this is often a good starting point when one wants to solve these equations numerically. But ultimately, a coordinate system must be introduced to execute the calculations and interpret the output.

One of the areas of application that you cover in the book is cosmology, an area of research that has undergone a revolution over the past few decades. What are some of the most transformative discoveries in the field’s recent history? How does classical physics serve to underpin our modern understanding of how the universe formed and is evolving? What are some of the mysteries that continue to challenge scientists in the field of cosmology?   

RDB: There have indeed been great strides in understanding the large scale structure and evolution of the universe, and there is good observational support for a comparatively simple description. Cosmologists have found that 26 percent of the energy density in the contemporary, smoothed-out universe is in the form of “dark matter,” which only seems to interact through its gravity. Meanwhile, 69 percent is associated with a “cosmological constant,” as first introduced by Einstein and which causes the universe to accelerate. The remaining five percent is the normal baryonic matter which we once thought accounted for essentially all of the universe. The actual structure that we observe appears to be derived from almost scale-free statistically simple, random fluctuations just as expected from an early time known as inflation. Fleshing out the details of this description is almost entirely an exercise in classical physics. Even if this description is validated by future observations, much remains to be understood, including the nature of dark matter and the cosmological constant, what fixes the normal matter density, and the great metaphysical question of what lies beyond the spacetime neighborhood that we can observe directly.

KST: Remarkably, in fleshing out the details in the last chapter of our book, we utilize classical-physics concepts and results from every one of the other chapters. ALL of classical physics feeds into cosmology!

The revolution in cosmology that you describe depends upon many very detailed observations using telescopes operating throughout the entire electromagnetic spectrum and beyond. How do you deal with this in the book?

RDB: We make no attempt to describe the rich observational and experimental evidence, referring the reader to many excellent texts on cosmology that describe these in detail. However, we do describe some of the principles that underlie the design and operation of the radio and optical telescopes that bring us cosmological data.

There is has also been a lot of excitement regarding the recent observation by LIGO of gravitational waves caused by merging black holes. How is this subject covered in the book, and how, briefly, are some of the concepts of classical physics elucidated in your description of this cutting-edge research area?   

KST: LIGO’s gravitational wave detectors rely on an amazingly wide range of classical physics concepts and tools, so time and again we draw on LIGO for illustrations. The theory of random processes, spectral densities, the fluctuation-dissipation theorem, the Fokker-Planck equation; shot noise, thermal noise, thermoelastic noise, optimal filters for extracting weak signals from noise; paraxial optics, Gaussian beams, the theory of coherence, squeezed light, interferometry, laser physics; the interaction of gravitational waves with light and with matter; the subtle issue of the conservation or non conservation of energy in general relativity—all these and more are illustrated by LIGO in our book.

What are some of the classical physics phenomena in every day life that you are surprised more people do not fully understand—whether they are lay people, students, or scientists?

KST: Does water going down a drain really have a strong preference for clockwise in the northern hemisphere and counterclockwise in the south? How strong? What happens as you cross the equator? How are ocean waves produced? Why do stars twinkle in the night sky, and why doesn’t Jupiter twinkle? How does a hologram work? How much can solid objects be stretched before they break, and why are there such huge differences from one type of solid (for example thin wire) to another (a rubber band)?

RDB: I agree and have to add that I am regularly humbled by some every day phenomenon that I cannot explain or for which I have carried around for years a fallacious explanation. There is, rightly, a lot of focus right now on climate change, energy, hurricanes, earthquakes, and so on. We hear about them every day. We physicists need to shore up our understanding and do a better job of communicating this.

Do you believe that some of your intended readers might be surprised to discover the deep relevance of classical physics to certain subject areas?

KST: In subjects that physicists think of as purely quantum, classical ideas and classical computational techniques can often be powerful. Condensed matter physics is an excellent example—and accordingly, our book includes a huge number of condensed-matter topics. Examples are Bose-Einstein condensates, the van der Waals gas, and the Ising model for ferromagnetism.

RDB: Conversely, quantum mechanical techniques are often used to simplify purely classical problems, for example in optics.

Writing a book is always an intellectual journey. In the preparation of this tremendously wide-ranging book, what were some of the most interesting things you learned along the way?

KST: How very rich and fascinating is the world of classical physics—far more so than we thought in 1980 when we embarked on this venture. And then there are the new inventions, discoveries, and phenomena that did not exist in 1980 but were so important or mind-boggling that we could not resist including them in our book. For example, optical-frequency combs and the phase-locked lasers that underlie them, Bose-Einstein condensates, the collapse of the World Trade Center buildings on 9/11/01, the discovery of gravitational waves and the techniques that made it possible, laser fusion, and our view of the universe at large.

Kip S. Thorne is the Feynman Professor Emeritus of Theoretical Physics at Caltech. His books include Gravitation and Black Holes and Time Warps. Roger D. Blandford is the Luke Blossom Professor of Physics and the founding director of the Kavli Institute of Particle Astrophysics and Cosmology at Stanford University. Both are members of the National Academy of Sciences.

 

Global Math Week: The Universal Language

by Oscar Fernandez

FernandezFill in the blank: Some people speak English, some speak French, and some speak ____. I doubt you said “math.” Yet, as I will argue, the thought should have crossed your mind. And moreover, the fact that mathematics being a language likely never has, speaks volumes about how we think of math, and why we should start thinking of it—and teaching it—as a language.

To make my point, consider the following fundamental characteristics shared by most languages:

  •  A set of words or symbols (the language’s vocabulary)
  •  A set of rules for how to use these words or symbols (the language’s rules of grammar)
  •  A set of rules for combining these words or symbols to make statements (the language’s syntax)

Now think back to the math classes you have taken. I bet you will soon remember each of these characteristics present throughout your courses. (For instance, when you learned that 𝑎2 means 𝑎 × 𝑎, you were learning how to combine some of the symbols used in mathematics to make a statement—that the square of a number is the number multiplied by itself.) Indeed, viewed this way, every mathematics lesson can be thought of as a language lesson: new vocabulary, rules of grammar, or syntax is introduced; everyone then practices the new content; and the cycle repeats. By extension, every mathematics course can be thought of as a language course.

Now that I have you thinking of mathematics as a language, let me point out the many benefits of this new viewpoint. For one, this viewpoint helps dispel many myths about the subject. For instance, travel to any country and you will find a diverse set of people speaking that country’s language. Some are smarter than others; some are men and some women; perhaps some are Latino and some Asian. Group them as you wish, they will all share the capacity to speak the same language. The same is true of mathematics. It is not a subject accessible only to people of certain intelligence, sex, or races; we all have the capacity to speak mathematics. And once we start thinking of the subject as a language, we will recognize that learning mathematics is like learning any other language: all you need are good teachers, and lots of practice. And while mastering a language is often the endpoint of the learning process, mastering the language that is mathematics will yield much larger dividends, including the ability to express yourself precisely, and the capacity to understand the Universe. As Alfred Adler put it: “

Mathematics is pure language – the language of science. It is unique among languages in its ability to provide precise expression for every thought or concept that can be formulated in its terms.” Galileo—widely regarded the father of modern science—once wrote that Nature is a great book “written in the language of mathematics” (The Assayer, 1623). Centuries later, Einstein, after having discovered the equation for gravity using mathematics, echoed Galileo’s sentiment, writing: “pure mathematics is, in its way, the poetry of logical ideas” (Obituary for Emmy Noether, 1935). Most of us today wouldn’t use words like “language” and “poetry” to describe mathematics. Yet, as I will argue, we should. And moreover, we should start thinking of—and teaching—math as a language.

Oscar E. Fernandez is assistant professor of mathematics at Wellesley College and the author of The Calculus of Happiness: How a Mathematical Approach to Life Adds Up to Health, Wealth, and Love. He also writes about mathematics for the Huffington Post and on his website, surroundedbymath.com.

Global Math Week: Counting on Math

by Tim Chartier

The Global Math Project has a goal of sharing the joys of mathematics to 1 million students around the world from October 10th through the 17th. As we watch the ever-increasing number of lives that will share in math’s wonders, let’s talk about counting, which is fundamental to reaching this goal.

Let’s count. Suppose we have five objects, like the plus signs below. We easily enough count five of them.

 

 

 

You could put them in a hat and mix them up.

 

 

 

 

 

 

 

If you take them out, they might be jumbled but you’d still have five.

 

 

 

 

 

 

 

 

 

 

Easy enough! Jumbling can induce subtle complexities, even to something as basic as counting.

Counting to 14 isn’t much more complicated than counting to five. Be careful as it depends what you are counting and how you jumble things! Verify there are 14 of Empire State Buildings in the picture below.

 

 

 

 

 

 

 

If you cut out the image along the straight black lines, you will have three pieces to a puzzle. If you interchange the left and right pieces on the top row, then you get the configuration below. How many buildings do you count now? Look at the puzzle carefully and see if you can determine how your count changed.

 

 

 

 

 

 

 

Can you spot any changes in the buildings in the first versus the second pictures? How we pick up an additional image is more easily seen if we reorder the buildings. So, let’s take the 14 buildings and reorder them as seen below.

 

 

 

 

 

 

 

Swapping the pieces on the top row of the original puzzle has the same effect as shifting the top piece in the picture above. Such a shift creates the picture below. Notice how we pick up that additional building. Further, each image loses 1/14th of its total height.

 

 

 

 

 

 

 

Let’s look at the original puzzle before and after the swap.

 

 

 

 

 

 

 

 

 

 

 

 

 

This type of puzzle is called a Dissection Puzzle. Our eyes can play tricks on us. We know 14 doesn’t equal 15 so something else must be happening when a puzzle indicates that 14 = 15. Mathematics allows us to push through assumptions that can lead to illogical conclusions. Math can also take something that seems quite magical and turn it into something very logical — even something as fundamental as counting to 14.

Want to look at counting through another mathematical lens? A main topic of the Global Math Project will be exploding dots. Use a search engine to find videos of James Tanton introducing exploding dots. James is a main force behind the Global Math Project and quite simply oozes joy of mathematics. You’ll also find resources at the Global Math Project web page. Take the time to look through the Global Math Project resources and watch James explain exploding dots, as the topic can be suitable from elementary to high school levels. You’ll enjoy your time with James. You can count on it!

ChartierTim Chartier is associate professor of mathematics at Davidson College. He is the coauthor of Numerical Methods and the author of Math Bytes: Google Bombs, Chocolate-Covered Pi, and Other Cool Bits in Computing.

Global Math Week: Around the World from Unsolved to Solved

by Craig Bauer

BauerWhat hope do we have of solving ciphers that go back decades, centuries, or even all the way back to the ancient world? Well, we have a lot more hope than we did in the days before the Internet. Today’s mathematicians form a global community that poses a much greater threat to unsolved problems, of every imaginable sort, than they have every faced before.

In my Princeton University Press book, Unsolved! The History and Mystery of the World’s Greatest Ciphers from Ancient Egypt to Online Secret Societies, I collected scores of the most intriguing unsolved ciphers. It’s a big book, in proper proportion to its title, and I believe many of the ciphers in it will fall to the onslaught the book welcomes from the world’s codebreakers, both professionals and amateurs. Why am I making this prediction with such confidence? Well, I gave a few lectures based on material from the book, while I was still writing it, and the results bode well for the ciphers falling.

Here’s what happened.

Early in the writing process, I was invited to give a lecture on unsolved ciphers at the United States Naval Academy. I was surprised, when I got there, by the presence of a video camera. I was asked if I was okay with the lecture being filmed and placed on YouTube. I said yes, but inside I was cursing myself for not having gotten a much needed haircut before the talk. Oh well. Despite my rough appearance, the lecture went well.[1] I surveyed some of the unsolved ciphers that I was aware of at the time, including one that had been put forth by a German colleague and friend of mine, Klaus Schmeh. It was a double transposition cipher that he had created himself to show how difficult it is to solve such ciphers. He had placed it in a book he had written on unsolved ciphers, a book which is unfortunately only available in German.[2] But to make the cipher as accessible as possible, he assured everyone that that particular bit of writing was in English.

 

VESINTNVONMWSFEWNOEALWRNRNCFITEEICRHCODEEA

HEACAEOHMYTONTDFIFMDANGTDRVAONRRTORMTDHE

OUALTHNFHHWHLESLIIAOETOUTOSCDNRITYEELSOANGP

VSHLRMUGTNUITASETNENASNNANRTTRHGUODAAARAO

EGHEESAODWIDEHUNNTFMUSISCDLEDTRNARTMOOIREEY

EIMINFELORWETDANEUTHEEEENENTHEOOEAUEAEAHUHI

CNCGDTUROUTNAEYLOEINRDHEENMEIAHREEDOLNNIRAR

PNVEAHEOAATGEFITWMYSOTHTHAANIUPTADLRSRSDNOT

GEOSRLAAAURPEETARMFEHIREAQEEOILSEHERAHAOTNT

RDEDRSDOOEGAEFPUOBENADRNLEIAFRHSASHSNAMRLT

UNNTPHIOERNESRHAMHIGTAETOHSENGFTRUANIPARTAOR

SIHOOAEUTRMERETIDALSDIRUAIEFHRHADRESEDNDOION

ITDRSTIEIRHARARRSETOIHOKETHRSRUAODTSCTTAFSTHCA

HTSYAOLONDNDWORIWHLENTHHMHTLCVROSTXVDRESDR

Figure 1. Klaus Schmeh’s double transposition cipher challenge.

When the YouTube video went online, it was seen by an Israeli computer scientist, George Lasry, who became obsessed with it. He was not employed at the time, so he was able to devote a massive amount of time to seeking the solution to this cipher. As is natural for George, he attacked it with computer programs of his own design. He eventually found himself doing almost nothing other than working on the cipher. His persistence paid off and he found himself reading the solution.

I ended up being among the very first to see George’s solution, not because I’m the one who introduced him to the challenge via the YouTube video, but because I’m the editor-in-chief of the international journal (it’s owned by the British company Taylor and Francis) Cryptologia. This journal covers everything having to do with codes and ciphers, from cutting edge cryptosystems and attacks on them, to history, pedagogy, and more. Most of the papers that appear in it are written by men and women who live somewhere other than America and it was to this journal that George submitted a paper describing how he obtained his solution to Klaus’s challenge.

George’s solution looked great to me, but I sent it to Klaus to review, just to be sure. As expected, he was impressed by the paper and I queued it up to see print. The solution generated some media attention for George, which led to him being noticed by people at Google (an American company, of course). They approached him and, after he cleared the interviewing hurdles, offered him a position, which he accepted. I was very happy that George found the solution, but of course that left me with one less unsolved cipher to write about in my forthcoming book. Not a problem. As it turns out there are far more intriguing unsolved ciphers than can be fit in a single volume. One less won’t make any difference.

Later on, but still before the book saw print, I delivered a similar lecture at the Charlotte International Cryptologic Symposium held in Charlotte, North Carolina. This time, unlike at the Naval Academy, Klaus Schmeh was in the audience.

One of the ciphers that I shared was fairly new to me. I had not spoken about it publicly prior to this event. It appeared on a tombstone in Ohio and seemed to be a Masonic cipher. It didn’t look to be sophisticated, but it was very short and shorter ciphers are harder to break. Brent Morris, a 33rd degree Mason with whom I had discussed it, thought that it might be a listing of initials of offices, such as PM, PHP, PIM (Past Master, Past High Priest, Past Illustrious Master), that the deceased had held. This cipher was new to Klaus and he made note of it and later blogged about it. Some of his followers collaborated in an attempt to solve it and succeeded. Because I hadn’t even devoted a full page to this cipher in my book, I left it in as a challenge for readers, but also added a link to the solution for those who want to see the solution right away.

Bauer

Figure 2. A once mysterious tombstone just south of Metamora, Ohio.

So, what was my role in all of this? Getting the ball rolling, that’s all. The work was done by Germans and an Israeli, but America and England benefited as well, as Google gained yet another highly intelligent and creative employee and a British owned journal received another great paper.

I look forward to hearing from other people from around the globe, as they dive into the challenges I’ve brought forth. The puzzles of the past don’t stand a chance against the globally networked geniuses of today!

Craig P. Bauer is professor of mathematics at York College of Pennsylvania. He is editor in chief of the journal Cryptologia, has served as a scholar in residence at the NSA’s Center for Cryptologic History, and is the author of Unsolved!: The History and Mystery of the World’s Greatest Ciphers from Ancient Egypt to Online Secret Societies. He lives in York, Pennsylvania.

 

[1] It was split into two parts for the YouTube channel. You can see them at https://www.youtube.com/watch?v=qe0JhEajfj8 (Part 1) and https://www.youtube.com/watch?v=5L12gjgMOMw (Part 2). A few years later, I got cleaned up and delivered an updated version of the talk at the International Spy Museum. That talk, aimed at a wider audience, may be seen at https://www.youtube.com/watch?v=rsdUDdkjdQg.

[2] Schmeh, Klaus, Nicht zu Knacken, Carl Hanser Verlag, Munich, 2012.