Pariah Moonshine Part II: For Whom the Moon Shines

by Joshua Holden

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

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

Figure 1

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

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

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

Figure 2

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

y2 = x3 – 4 x2 + 16

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

y2 = x3 + ax2 + bx + c

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

Figure 1

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

Figure 2

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

 

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

Modular forms, the Monster, and Moonshine

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

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

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

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

More Moonshine?

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

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

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

by Joshua Holden

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

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

What is a symmetry?

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

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.

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.

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.

Vickie Kearn kicks off Global Math Week

October 10 – 17 marks the first ever Global Math Week. This is exciting for many reasons and if you go to the official website, you’ll find that there are already 736,546—and counting— reasons there. One more: PUP will be celebrating with a series of posts from some of our most fascinating math authors, so check this space tomorrow for the first, on ciphers, by Craig Bauer. Global Math Week provides a purposeful opportunity to have a global math conversation with your friends, colleagues, students, and family.

Mathematics is for everyone, as evidenced in the launch of Exploding Dots, which James Tanton brilliantly demonstrates at the link above. It is a mathematical story that looks at math in a new way, from grade school arithmetic, all the way to infinite sums and on to unsolved problems that are still stumping our brightest mathematicians. Best of all, you can ace this and no longer say “math is hard”, “math is boring”, or “I hate math”.

Vickie Kearn visits the Great Wall during her trip to our new office in Beijing

I personally started celebrating early as I traveled to Beijing in August to attend the Beijing International Book Fair. I met with the mathematics editors at a dozen different publishers to discuss Chinese editions of our math books. Although we did not speak the same language, we had no trouble communicating. We all knew what a differential equation is and a picture in a book of a driverless car caused lots of hand clapping. I was thrilled to be presented with the first Chinese editions of two books written by Elias Stein (Real Analysis and Complex Analysis) from the editor at China Machine Press. Although I love getting announcements from our rights department that one of our math books is being translated into Chinese, Japanese, German, French, etc., there is nothing like the thrill I had of meeting the people who love math as much as I do and who actually make our books come to life for people all over the world.

Because Princeton University Press now has offices in Oxford and Beijing, in addition to Princeton, and because I go to many conferences each year, I am fortunate to travel internationally and experience global math firsthand. No matter where you live, it is possible to share experiences through doing math. I urge you to visit the Global Math Project website and learn how to do math(s) in a global way.

Check back tomorrow for the start of our PUP blog series on what doing math globally means to our authors. Find someone who says they don’t like math and tell them your global math story.

Craig Bauer on unsolved ciphers

In 1953, a man was found dead from cyanide poisoning near the Philadelphia airport with a picture of a Nazi aircraft in his wallet. Taped to his abdomen was an enciphered message. In 1912, a book dealer named Wilfrid Voynich came into possession of an illuminated cipher manuscript once belonging to Emperor Rudolf II, who was obsessed with alchemy and the occult. Wartime codebreakers tried—and failed—to unlock the book’s secrets, and it remains an enigma to this day. In Unsolved, Craig Bauer examines these and other vexing ciphers yet to be cracked. Recently he took the time to answer some questions about his new book.

Why focus on unsolved ciphers?

They’re much more intriguing because they could be concealing anything. Some might reveal the identities of serial killers. Others could unmask spies, rewrite history, expose secret societies, or even give the location of buried treasure worth millions. This sense of mystery is very appealing to me.

Did you try to solve the ciphers yourself first?

There are so many unsolved ciphers that I realized I would never finish writing about them if I kept stopping to try to solve them. There’s one that I’m confident I could solve, but instead of doing so, I simply presented the approach I think will work and am leaving it for a reader to pursue. I expect that several of them will be solved by readers and I look forward to seeing their results!

Does someone who wants to attack these mysteries need to know a lot of mathematics or have computer programming skills?

No. Many of the ciphers were created by people with very little knowledge in either area. Also, past solvers of important ciphers have included amateurs. One of the Zodiac killer’s ciphers was solved by a high school history teacher. Some of the ciphers might be solved in a manner that completely bypasses mathematics. A reader may find a solution through papers the cipher’s creator left behind, perhaps in some library’s archives, in government storage, or in a relative’s possession. I think some may be solved by pursuing a paper trail or some other non-mathematical avenue. Of course, there are mathematical challenges as well, for those who have the skills to take them on. The puzzles span thousands of years, from ancient Egypt to today’s online community. Twentieth century challenges come from people as diverse as Richard Feynman (a world-class physicist) and Ricky McCormick (thought to have been illiterate).

Are all of the unsolved ciphers covered in the book?

No, far from it. There are enough unsolved ciphers to fill many volumes. I limited myself to only the most interesting examples, and still there were too many! I originally set out to write a book about half the size of what was ultimately published. The problem was that there was so much fascinating material that I had to go to 600 pages or experience the agony of omitting something fabulous. Also, unsolved ciphers from various eras are constantly coming to light, and new ones are created every year. I will likely return to the topic with a sequel covering the best of these.

Which cipher is your favorite?

I’m the most excited about the Paul Rubin case. It involves a cipher found taped to the abdomen of a teenage whiz-kid who was found dead in a ditch by the Philadelphia airport, way back in 1953. While I like well-known unsolved ciphers like the Voynich Manuscript and Kryptos, I have higher hopes for this one being solved because it hasn’t attracted any attention since the 1950s. The codebreakers have made a lot of progress since then, so it’s time to take another look and see what can be learned about this young man’s death. I felt it was very important to include cases that will be new even to those who have read a great deal about cryptology already and this is one such case.

Should the potential reader have some prior knowledge of the subject?

If he or she does, there will still be much that is new, but for those with no previous exposure to cryptology, everything is explained from the ground up. As a teenager I loved books at the popular level on a wide range of topics. In particular, the nonfiction of Isaac Asimov instilled in me a love for many subjects. He always started at the beginning, assuming his readers were smart, but new to the topic he was covering. This is the approach that I have taken. I hope that the book finds a wide readership among the young and inspires them in the same way Asimov inspired me.

Is there anything that especially qualifies you to write on this topic?

Early work on this book was supported by the National Security Agency through their Scholar-in-Residence program at the Center for Cryptologic History. They wanted me in this role because, while I have a PhD in mathematics and have carried out mathematical research in cryptology, I also have a passion for history and other disciplines. In fact, both of my books have the word “history” in their titles. The journal Cryptologia, for which I serve as the editor-in-chief, is devoted to all aspects of cryptology, mathematical, historical, pedagogical, etc. My love of diverse fields allows me to write with enthusiasm about ciphers in music, art, criminal cases, ancient history, and other areas. The broad approach to the subject is more entertaining and ensures that there’s something in the book for nearly every reader.

BauerCraig 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.

Craig Bauer: The Ongoing Mystery of Unsolved Ciphers (and new hope)

When a civilization first develops writing and few people are literate, simply putting a message down on paper can be all that is required to keep an enemy from understanding it. As literacy spreads, a more sophisticated method is needed, which is why codes and ciphers, a.k.a. “secret writing,” always follow closely on the heels of the discovery of writing. Over the millennia, ciphers have become extremely sophisticated, but so too have the techniques used by those attempting to break them.

In recent decades, everyone from mathematicians and computer scientists to artists and authors have created ciphers as challenges to specialists or the general public, to see if anyone is clever enough to unravel the secrets. Some, like the first three parts of James Sanborn’s sculpture Kryptos and the ciphers appearing in the television show Gravity Falls, have been solved, while others remain mysteries. The highly secretive online society known as Cicada 3301 has repeatedly issued such challenges as a means of talent scouting, though for what purpose such talented individuals are sought remains unknown. One unsolved cipher was laid down as a challenge by former British army intelligence officer Alexander d’Agapeyeff in his book Codes & Ciphers (1939). Sadly, when frustrated letters of enquiry reached the author, he admitted that he had forgotten how to solve it! Another was made by the famous composer Edward Elgar in 1897 as a riddle for a young lady friend of his. She, along with various experts, all failed to ferret out the meaning and Elgar himself refused to reveal it.

 

Elgar's cipher

Elgar’s cipher

 

Many unsolved ciphers appear in much more serious contexts. The serial killer who referred to himself as “The Zodiac” was responsible for at least five murders, as well as the creation of several ciphers sent to San Francisco newspapers. While the first of these ciphers was solved, others remain unbroken. Could a solution to one of these lead to an identification of the killer? Although many have speculated on his identity, it has never been firmly established. The Zodiac is not the only murderer to have left us such mysterious communiques, he is just the best known. Other killers’ secrets have persisted through relative obscurity. How many readers have heard of Henry Debsonys? In 1883, a jury sentenced him to death for the murder of his wife, after deliberating for only nine minutes. But this unfortunate woman was Henry’s third wife and the first two died under strange circumstances. Had Henry killed all of them? Will the ciphers he left behind confirm this? I think his ciphers will be among the first to fall this year, thanks to a major clue I provide in my book, Unsolved: The History and Mystery of the World’s Greatest Ciphers from Ancient Egypt to Online Secret Societies. There are many more such criminal ciphers. One deranged individual even sent threatening letters containing ciphers to John Walsh of America’s Most Wanted fame! The FBI’s codebreakers maintain a list of their top unsolved ciphers. At present, only two of these are known to the public, but many others that didn’t make the top 10 are available for anyone to try to crack.

How do codebreakers, whether amateur or professional, meet the challenges they face? Statistics and other areas of mathematics often help, as do computers, but two of the codebreakers’ most powerful tools are context and intuition. This is why ciphers have often been broken by amateurs with no programming skills and little knowledge of mathematics. Enter Donald Harden, a high school history teacher, who with assistance from his wife Bettye, broke one of the Zodiac killer’s ciphers by guessing that the egotistical killer’s message would begin with “I” and contain the word “KILL.” Context allows the attacker to guess words, sometimes entire phrases, that might appear in the message. These are known as cribs. During World War II, the German word eins (meaning one) appeared in so many Nazi messages that a process known as “einsing” was developed, searching the cipher for the appearance of this word in every possible position. In today’s ciphers, the word President appears frequently.

Of course, time and again cribs and intuition can lead in the wrong direction. Indeed, the single most important attribute for a codebreaker is patience. A good codebreaker will have the ability to work on a cipher for months, for that is sometimes what it takes to reach a solution, ignoring the body’s normal demands for food and sleep; during World War I, the French codebreaker Georges Painvin lost 33 pounds over three months while sitting at a desk breaking the German ADFGX and ADFGVX ciphers.

Fig 2

Fig 3Is it possible that some of the earliest known ciphers, dating from the ancient world, have survived unread by anyone other than those they were created for? I believe this is the case and that they’ve been hiding in plain sight, like the purloined letter in Poe’s classic tale. Those studying ancient cultures have long been aware of so-called “nonsense inscriptions.” These appear on Egyptian sarcophagi, Greek vases, runestones, and elsewhere. They are typically dismissed as the work of illiterates imitating writing, merely because the experts cannot read them. But all of these cultures are known to have made use of ciphers and some of the contexts of the inscriptions are so solemn (e.g. sarcophagi) that it’s hard to believe they could be meaningless. I’d like to see a closer examination of these important objects. I expect some of the messages will be read in the near future, if cryptologists can form collaborations with linguists. These two groups have worked together successfully in military contexts for many decades. It is time that they also join forces for historical studies.

With a very large number of unsolved ciphers, spanning millennia, having been composed by a diverse group of individuals, it seems likely that it will take a diverse group of attackers, with skills ranging over many disciplines, to solve them. Some mysterious texts may reveal themselves to clever computer programmers or linguists, others to those taking the psychological approach, getting into the creator’s head and guessing phrases he or she used in the cipher, and some may be broken by readers who manage to discover related material in government archives or private hands that provides just enough extra information to make the break. I look forward to seeing the results!

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 Unsolved!: The History and Mystery of the World’s Greatest Ciphers from Ancient Egypt to Online Secret Societies. He lives in York, Pennsylvania.

Oscar Fernandez: A Healthier You is Just a Few Equations Away

This post appears concurrently on the Wellesley College Summer blog.

How many calories should you eat each day? What proportion should come from carbohydrates, or protein? How can we improve our health through diets based on research findings?

You might be surprised to find that we can answer all of these questions using math.  Indeed, mathematics is at the heart of nutrition and health research. Scientists in these fields often use math to analyze the results from their experiments and clinical trials.  Based on decades of research (and yes, math), scientists have developed a handful of formulas that have been proven to improve your health (and even help you lose weight!).

So, back to our first question: How many calories should we eat each day?  Let’s find out…

Each of us has a “total daily energy expenditure” (TDEE), the total number of calories your body burns each day. Theoretically, if you consume more calories than your TDEE, you will gain weight. If you consume less, you will lose weight. Eat exactly your TDEE in calories and you won’t gain or lose weight.

“Great! So how do I calculate my TDEE?” I hear you saying. Good question. Here’s a preliminary answer:

TDEE = RMR + CBE + DIT         (1)                                                                                                                                                                  

Here’s what the acronyms on the right-hand side of the equation mean.

  • RMR: Your resting metabolic rate, roughly defined as the number of calories your body burns while awake and at rest
  • CBE: The calories you burned during the day exercising (including walking)
  • DIT: Your diet’s diet-induced thermogenesis, which quantifies what percentage of calories from dietary fat, protein, and carbohydrates are left over for your body to use after you ingest those calories

So, in order to calculate TDEE, we need to calculate each of these three components. This requires very precise knowledge of your daily activities, for example: what exercises you did, how many minutes you spent doing them, what foods you ate, and how much protein, carbohydrates, and dietary fat these foods contained. Luckily, nutrition scientists have developed a simpler formula that takes all of these factors into account:

    TDEE = RMR(Activity Factor) + 0.1C.         (2)

Here C is how many calories you eat each day, and the “Activity Factor” (below) estimates the calories you burn through exercise:

 

Level of Activity Activity Factor
Little to no physical activity 1.2
Light-intensity exercise 1-3 days/week 1.4
Moderate-intensity exercise 3-5 days/week 1.5
Moderate- to vigorous-intensity exercise 6-7 days/week 1.7
Vigorous daily training 1.9

 

As an example, picture a tall young man named Alberto. Suppose his RMR is 2,000 calories, that he eats 2,100 calories a day, and that his Activity Factor is 1.2. Alberto’s TDEE estimate from (2) would then be

TDEE = 2,000(1.2) + 0.1(2,100) = 2,610.

Since Alberto’s caloric intake (2,100) is lower than his TDEE, in theory, Alberto would lose weight if he kept eating and exercising as he is currently doing.

Formula (2) is certainly more user-friendly than formula (1). But in either case we still need to know the RMR number. Luckily, RMR is one of the most studied components of TDEE, and there are several fairly accurate equations for it that only require your weight, height, age, and sex as inputs. I’ve created a free online RMR calculator to make the calculation easier: Resting Metabolic Heart Rate. In addition, I’ve also created a TDEE calculator (based on equation (2)) to help you estimate your TDEE: Total Daily Energy Expenditure.

I hope this short tour of nutrition science has helped you see that mathematics can be empowering, life-changing, and personally relevant. I encourage you to continue exploring the subject and discovering the hidden math all around you.

Oscar E. Fernandez is assistant professor of mathematics at Wellesley College. He is the author of Everyday Calculus: Discovering the Hidden Math All around Us and 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.

 

Joshua Holden: Quantum cryptography is unbreakable. So is human ingenuity

Two basic types of encryption schemes are used on the internet today. One, known as symmetric-key cryptography, follows the same pattern that people have been using to send secret messages for thousands of years. If Alice wants to send Bob a secret message, they start by getting together somewhere they can’t be overheard and agree on a secret key; later, when they are separated, they can use this key to send messages that Eve the eavesdropper can’t understand even if she overhears them. This is the sort of encryption used when you set up an online account with your neighbourhood bank; you and your bank already know private information about each other, and use that information to set up a secret password to protect your messages.

The second scheme is called public-key cryptography, and it was invented only in the 1970s. As the name suggests, these are systems where Alice and Bob agree on their key, or part of it, by exchanging only public information. This is incredibly useful in modern electronic commerce: if you want to send your credit card number safely over the internet to Amazon, for instance, you don’t want to have to drive to their headquarters to have a secret meeting first. Public-key systems rely on the fact that some mathematical processes seem to be easy to do, but difficult to undo. For example, for Alice to take two large whole numbers and multiply them is relatively easy; for Eve to take the result and recover the original numbers seems much harder.

Public-key cryptography was invented by researchers at the Government Communications Headquarters (GCHQ) – the British equivalent (more or less) of the US National Security Agency (NSA) – who wanted to protect communications between a large number of people in a security organisation. Their work was classified, and the British government neither used it nor allowed it to be released to the public. The idea of electronic commerce apparently never occurred to them. A few years later, academic researchers at Stanford and MIT rediscovered public-key systems. This time they were thinking about the benefits that widespread cryptography could bring to everyday people, not least the ability to do business over computers.

Now cryptographers think that a new kind of computer based on quantum physics could make public-key cryptography insecure. Bits in a normal computer are either 0 or 1. Quantum physics allows bits to be in a superposition of 0 and 1, in the same way that Schrödinger’s cat can be in a superposition of alive and dead states. This sometimes lets quantum computers explore possibilities more quickly than normal computers. While no one has yet built a quantum computer capable of solving problems of nontrivial size (unless they kept it secret), over the past 20 years, researchers have started figuring out how to write programs for such computers and predict that, once built, quantum computers will quickly solve ‘hidden subgroup problems’. Since all public-key systems currently rely on variations of these problems, they could, in theory, be broken by a quantum computer.

Cryptographers aren’t just giving up, however. They’re exploring replacements for the current systems, in two principal ways. One deploys quantum-resistant ciphers, which are ways to encrypt messages using current computers but without involving hidden subgroup problems. Thus they seem to be safe against code-breakers using quantum computers. The other idea is to make truly quantum ciphers. These would ‘fight quantum with quantum’, using the same quantum physics that could allow us to build quantum computers to protect against quantum-computational attacks. Progress is being made in both areas, but both require more research, which is currently being done at universities and other institutions around the world.

Yet some government agencies still want to restrict or control research into cryptographic security. They argue that if everyone in the world has strong cryptography, then terrorists, kidnappers and child pornographers will be able to make plans that law enforcement and national security personnel can’t penetrate.

But that’s not really true. What is true is that pretty much anyone can get hold of software that, when used properly, is secure against any publicly known attacks. The key here is ‘when used properly’. In reality, hardly any system is always used properly. And when terrorists or criminals use a system incorrectly even once, that can allow an experienced codebreaker working for the government to read all the messages sent with that system. Law enforcement and national security personnel can put those messages together with information gathered in other ways – surveillance, confidential informants, analysis of metadata and transmission characteristics, etc – and still have a potent tool against wrongdoers.

In his essay ‘A Few Words on Secret Writing’ (1841), Edgar Allan Poe wrote: ‘[I]t may be roundly asserted that human ingenuity cannot concoct a cipher which human ingenuity cannot resolve.’ In theory, he has been proven wrong: when executed properly under the proper conditions, techniques such as quantum cryptography are secure against any possible attack by Eve. In real-life situations, however, Poe was undoubtedly right. Every time an ‘unbreakable’ system has been put into actual use, some sort of unexpected mischance eventually has given Eve an opportunity to break it. Conversely, whenever it has seemed that Eve has irretrievably gained the upper hand, Alice and Bob have found a clever way to get back in the game. I am convinced of one thing: if society does not give ‘human ingenuity’ as much room to flourish as we can manage, we will all be poorer for it.Aeon counter – do not remove

Joshua Holden is professor of mathematics at the Rose-Hulman Institute of Technology and the author of The Mathematics of Secrets.

This article was originally published at Aeon and has been republished under Creative Commons.

A peek inside The Calculus of Happiness

What’s the best diet for overall health and weight management? How can we change our finances to retire earlier? How can we maximize our chances of finding our soul mate? In The Calculus of Happiness, Oscar Fernandez shows us that math yields powerful insights into health, wealth, and love. Moreover, the important formulas are linked to a dozen free online interactive calculators on the book’s website, allowing one to personalize the equations. A nutrition, personal finance, and relationship how-to guide all in one, The Calculus of Happiness invites you to discover how empowering mathematics can be. Check out the trailer to learn more:

The Calculus of Happiness: How a Mathematical Approach to Life Adds Up to Health, Wealth, and Love, Oscar E. Fernandez from Princeton University Press on Vimeo.

FernandezOscar E. Fernandez is assistant professor of mathematics at Wellesley College and the author of Everyday Calculus: Discovering the Hidden Math All around Us. He also writes about mathematics for the Huffington Post and on his website, surroundedbymath.com.