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.

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.

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.

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.

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.

Oscar E. Fernandez on The Calculus of Happiness

FernandezIf you think math has little to do with finding a soulmate or any other “real world” preoccupations, Oscar Fernandez says guess again. According to his new book, The Calculus of Happiness, math offers powerful insights into health, wealth, and love, from choosing the best diet, to finding simple “all weather” investment portfolios with great returns. Using only high-school-level math (precalculus with a dash of calculus), Fernandez guides readers through the surprising results. He recently took the time to answer a few questions about the book and how empowering mathematics can be.

The title is intriguing. Can you tell us what calculus has to do with happiness?

Sure. The title is actually a play on words. While there is a sprinkling of calculus in the book (the vast majority of the math is precalculus-level), the title was more meant to convey the main idea of the book: happiness can be calculated, and therefore optimized.

How do you optimize happiness?

Good question. First you have to quantify happiness. We know from a variety of research that good health, healthy finances, and meaningful social relationships are the top contributors to happiness. So, a simplistic “happiness equation” is: health + wealth + love = happiness. This book then does what any good applied mathematician would do (I’m an applied mathematician): quantify each of the “happiness components” on the left-hand side of the equation (health, wealth, and love), and then use math to extract valuable insights and results, like how to optimize each component.

This process sounds very much like the subtitle, how a mathematical approach to life adds up to health, wealth, and love. But just to be sure, can you elaborate on the subtitle?

That’s exactly right. Often we feel like various aspects of our lives are beyond our control. But in fact, many aspects of our lives, including some of the most important ones (like health, wealth, and love), follow mathematical rules. And by studying the equations that emerge from these rules you can quickly learn how to manipulate those equations in your favor. That’s what I do in the book for health, wealth, and love.

Can you give us some examples/applications?

I can actually give you about 30 of them, roughly the number discussed in the book. But let me focus on my three favorite ones. The first is what I called the “rational food choice” function (Chapter 2). It’s a simple formula: divide 100 calories by the weight (say, in grams) of a particular food. This yields a number whose units are calories per gram, the units of “energy density.” Something remarkable then happens when you plot the energy densities of various foods on a graph: the energy densities of nearly all the healthy foods (like fruits and vegetables) are at most about 2 calories per gram. This simple mathematical insight, therefore, helps you instantly make healthier food choices. And following its advice, as I discuss at length in the book, eventually translates to lower risk for developing heart disease and diabetes, weight loss, and even an increase in your life span! The second example comes from Chapter 3; it’s a formula for calculating how many more years you have to work for before you can retire. Among the formula’s many insights is that, in the simplest case, this magic number depends entirely on the ratio of how much you save each year to how much you spend. And the formula, being a formula, tells you exactly how changing that ratio affects your time until retirement. The last example is based on astronomer Frank Drake’s equation for estimating the number of intelligent civilizations in our galaxy (Chapter 5). It turns out that this alien-searching equation can also be used to estimate the number of possible compatible partners that live near you! That sort of equates a good date with an intelligent alien, and I suppose I can see some similarities (like how rare they are to find).

The examples you’ve mentioned have direct relevance to our lives. Is that a feature of the other examples too?

Absolutely. And it’s more than just relevance—the examples and applications I chose are all meant to highlight how empowering mathematics can be. Indeed, the entire book is designed to empower the reader—via math—with concrete, math-backed and science-backed strategies for improving their health, wealth, and love life. This is a sampling of the broader principle embodied in the subtitle: taking a mathematical approach to life can help you optimize nearly every aspect of your life.

Will I need to know calculus to enjoy the book?

Not at all. Most of the math discussed is precalculus-level. Therefore, I expect that nearly every reader will have studied the math used in the book at some point in their K-12 education. Nonetheless, I guide the reader through the math as each chapter progresses. And once we get to an important equation, you’ll see a little computer icon next to it in the margin. These indicate that there are online interactive demonstrations and calculators I created that go along with the formula. The online calculators make it possible to customize the most important formulas in the book, so even if the math leading up to them gets tough, you can still use the online resources to help you optimize various aspects of health, wealth, and love.

Finally, you mention a few other features of the book in the preface. Can you tell us about some of those?

Sure, I’ll mention two particular important ones. Firstly, at least 1/3 of the book is dedicated to personal finance. I wrote that part of the book to explicitly address the low financial literacy in this country. You’ll find understandable discussions of everything from taxes to investing to retirement (in addition to the various formulas derived that will help you optimize those aspects of your financial life). Finally, I organized the book to follow the sequence of math topics covered in a typical precalculus textbook. So if you’re a precalculus student, or giving this book to someone who is, this book will complement their course well. (I also included the mathematical derivations of the equations presented in the chapter appendixes.) This way the youngest readers among us can read about how empowering and applicable mathematics can be. It’s my hope that this will encourage them to continue studying math beyond high school.

Oscar E. Fernandez is assistant professor of mathematics at Wellesley College and 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.

J. Richard Gott: What’s the Value of Pi in Your Universe?

Carl Sagan’s sci-fi novel Contact famously introduced wormholes for rapid transit between the stars. Carl had asked his friend Kip Thorne to tell him if the physics of wormholes was tenable and this led Thorne and his colleagues to investigate their properties. They found that traversable wormholes required exotic matter to prop them open and that, by moving the wormhole mouths one could find general relativity solutions allowing time travel to the past. A quantum state called the Casimir vacuum whose effects have been observed experimentally, could provide the exotic matter. To learn whether such time machines could be constructible in principle, we may have to master the laws of quantum gravity, which govern how gravity behaves on microscopic scales. It’s one of the reasons physicists find these solutions so interesting.

But in Contact there is lurking yet another fantastic sci-fi idea, which gets less publicity because it was not included in the movie version. In the book, the protagonist finds out from the extraterrestrials that the system of wormholes throughout the galaxy was not built by them, but by the long gone “old ones” who could manipulate not only the laws of physics but also the laws of mathematics! And they left a secret message in the digits of pi. In his movie Pi, Darren Aronofsky showed a man driven crazy by his search for hidden meanings in the digits of pi.

This opens the question: could pi have been something else? And if so, does pi depend on the laws of physics? Galileo said: “Philosophy is written in this grand book…. I mean the universe … which stands continually open to our gaze…. It is written in the language of mathematics.” The universe is written in the language of mathematics. Nobel laureate Eugene Wigner famously spoke of the “unreasonable effectiveness of mathematics” in explaining physics. Many philosophers take the Platonic view that mathematics would exist even the universe did not. And cosmologist Max Tegmark goes so far as to say that the universe actually is mathematics.

Yet maybe it is the other way around. The laws of physics are just the laws by which matter behaves. They determine the nature of our universe. Maybe humans have simply developed the mathematics appropriate for describing our universe, and so of course it fits with what we see. The mathematician Leopold Kronecker said, “God created the integers, all the rest is the work of man.” Are the laws of mathematics discovered by us in the same way as we discover the laws of physics? And are the laws of mathematics we discover just those which would have occurred to creatures living in a universe with physics like ours? In our universe, physics produces individual identical particles: all electrons are the same for example. We know about integers because there are things that look the same (like apples) for us to count. If you were some strange creature in a fractal universe containing only one object—yourself—and you thought only recursively, you might not ever think of counting anything and would never discover integers.

What about π = 3.14159265.…? Might it have a different value in a different universe? In our universe we have a fundamental physical dimensionless constant, the fine structure constant α which is related to the square of the value of the electric charge of the proton in natural geometrical Planck units (where the speed of light is 1 and the reduced Planck constant is 1 and Newton’s gravitational constant is 1). Now 1/α = 137.035999… Some physicists hope that one day we may have a mathematical formula for 1/α using mathematical constants such as π and e. If a theory for the fine structure constant could be developed giving a value in agreement with observations but allowing it to be calculated uniquely from pure mathematics, and if more and more digits of the constant were discovered experimentally fulfilling its prediction, it would certainly merit a Nobel Prize. But many physicists feel that no such magic formula will ever be discovered. Inflation may produce an infinite number of bubble universes, each with different laws of physics. Different universes bubbling out of an original inflating sea could have different values of 1/α. As Martin Rees has said, the laws of physics we know may be just local bylaws in an infinite multiverse of universes. String theory, if correct, may eventually give us a probability distribution for 1/α and we may find that our universe is just somewhere in the predicted middle 95% of the distribution, for example. Maybe there could be different universes with different values of π.

Let’s consider one possible example: taxicab geometry. This was invented by Hermann Minkowski. Now this brilliant mathematician also invented the geometrical interpretation of time as a fourth dimension based on Einstein’s theory of special relativity, so his taxicab geometry merits a serious look. Imagine a city with a checkerboard pattern of equal-sized square blocks. Suppose you wanted to take a taxicab to a location 3 blocks east, and 1 block north of your location, the shortest total distance you would have to travel to get there is 4 blocks. Your taxi has to travel along the streets, it does not get to travel as the crow flies. You could go 1 block east, then 1 block north then 2 blocks east, and still get to your destination, but the total distance you traveled would also be 4 blocks. The distance to your destination would be ds = |dx| + |dy|, where |dx| is the absolute value of the difference in x coordinates and |dy| is the absolute value of the difference in y coordinates. This is not the Euclidean formula. We are not in Kansas anymore! The set of points equidistant from the origin is a set of dots in a diamond shape. See diagram.

Gott

Image showing an intuitive explanation of why circles in taxicab geometry look like diamonds. Wikipedia.

Now if the blocks were smaller, there would be more dots, still in a diamond shape. In the limit where the size of the blocks had shrunk to zero, one would have a smooth diamond shape as shown in the bottom section of the diagram. The set of points equidistant from the origin has a name—a “circle!” If the circle has a radius of 1 unit, the distance along one side of its diamond shape is 2 units: going from the East vertex of the diamond to the North vertex of the diamond along the diagonal requires you to change the x coordinate by 1 unit and the y coordinate by 1 unit, making the distance along one side of the diagonal equal to 2 units (ds = |dx| + |dy| = 1 + 1 units = 2 units). The diamond shape has 4 sides so the circumference of the diamond is 8 units. The diameter of the circle is twice the radius, and therefore 2 units. In the taxicab universe π = C/d = C/2r = 8/2 = 4. If different laws of physics dictate different laws of geometry, you can change the value of π.

This taxicab geometry applies in the classic etch-a-sketch toy (Look it up on google, if you have never seen one). It has a white screen, and an internal stylus that draws a black line, directed by horizontal and vertical control knobs. If you want to draw a vertical line, you turn the vertical knob. If you want to draw a horizontal line you turn the horizontal knob. If you want to draw a diagonal line, you must simultaneously turn both knobs smoothly. If the distance between two points is defined by the minimal amount of total turning of the two knobs required to get from one point to the other, then that is the “taxicab” distance between the two points. In Euclidean geometry there is one shortest line between two points: a straight line between them. In taxicab geometry there can be many different, equally short, broken lines (taxicab routes) connecting two points. Taxicab geometry does not obey the axioms of Euclidean geometry and therefore does not have the same theorems as Euclidean geometry. And π is 4.

Mathematician and computer scientist John von Neumann invented a cellular automaton universe that obeys taxicab geometry. It starts with an infinite checkerboard of pixels. Pixels can be either black or white. The state of a pixel at time step t = n + 1 depends only on the state of its 4 neighbors (with which it shares a side: north, south, east, west of it) on the previous time step t = n. Causal, physical effects move like a taxicab. If the pixels are microscopic, we get a taxicab geometry. Here is a simple law of physics for this universe: a pixel stays in the same state, unless it is surrounded by an odd number of black pixels, in which case it switches to the opposite state on the next time step. Start with a white universe with only 1 black pixel at the origin. In the next time step it remains black while its 4 neighbors also become black. There is now a black cross of 5 pixels at the center. It has given birth to 4 black pixels like itself. Come back later and there will be 25 black pixels in a cross-shaped pattern of 5 cross-shaped patterns.

Come back still later and you can find 125 black pixels in 5 cross-shaped patterns (of 5 cross-shaped patterns). All these new black pixels lie inside a diamond-shaped region whose radius grows larger by one pixel per time step. In our universe, drop a rock in a pond, and a circular ripple spreads out. In the von Neumann universe, causal effects spread out in a diamond-shaped pattern.

If by “life” you mean a pattern able to reproduce itself, then this universe is luxuriant with life. Draw any pattern (say a drawing of a bicycle) in black pixels and at a later time you will find 5 bicycles, and then 25 bicycles, and 125 bicycles, etc. The laws of physics in this universe cause any object to copy itself. If you object that this is just a video game, I must tell you that some physicists seriously entertain the idea that we are living in an elaborate video game right now with quantum fuzziness at small scales providing the proof of microscopic “pixelization” at small scales.

Mathematicians in the von Neumann universe would know π = 4 (Or, if we had a taxicab universe with triangular pixels filling the plane, causal effects could spread out along three axes instead of two and a circle would look like a hexagon, giving π = 3.). In 1932, Stanislaw Golab showed that if we were clever enough in the way distances were measured in different directions, we could design laws of physics so that π might be anything we wanted from a low of 3 to a high of 4.

Back to the inhabitants of the von Neumann universe who think π = 4. Might they be familiar with number we know and love, 3.14159265…? They might:

3.14159265… = 4 {(1/1) – (1/3) + (1/5) – (1/7) + (1/9) + …} (Leibnitz)

If they were familiar with integers, they might be able to discover 3.14159265… But maybe the only integers they know are 1, 5, 25, 125, … and 4 of course. They would know that 5 = SQRT(25), so they would know what a square root was. In this case they could still find a formula for

3.14159265. . . =
SQRT(4) {SQRT(4)/SQRT(SQRT(4))}{SQRT(4)/SQRT(SQRT(4) + SQRT(SQRT(4)))}{SQRT(4)/ SQRT(SQRT(4) + SQRT(SQRT(4) + SQRT(SQRT(4))))} …

This infinite product involving only the integer 4 derives from one found by Vieta in 1594.

There are indeed many formulas equal to our old friend 3.14159265… including a spectacular one found by the renowned mathematician Ramanujan. Though every real number can be represented by such infinite series, products and continued fractions, these are particularly simple. So 3.14159265… does seem to have a special intimate relationship with integers, independent of geometry. If physics creates individual objects that can be counted, it seems difficult to avoid learning about 3.14159265… eventually—“If God made the integers,” as Kronecker suggested. So 3.14159265… appears not to be a random real number and we are still left with the mystery of the unreasonable effectiveness of mathematics in explaining the physics we see in our universe. We are also left with the mystery of why the universe is as comprehensible as it is. Why should we lowly carbon life forms be capable of finding out as much about how the universe works as we have done? Having the ability as intelligent observers to ask questions about the universe seems to come with the ability to actually answer some of them. That’s remarkable.

UniverseGottJ. Richard Gott is professor of astrophysics at Princeton University. His books include The Cosmic Web: Mysterious Architecture of the Universe. He is the coauthor of Welcome to the Universe: An Astrophysical Tour with Neil DeGrasse Tyson and Michael A. Strauss.

Cipher challenge #3 from Joshua Holden: Binary ciphers

The Mathematics of Secrets by Joshua Holden takes readers on a tour of the mathematics behind cryptography. Most books about cryptography are organized historically, or around how codes and ciphers have been used in government and military intelligence or bank transactions. Holden instead focuses on how mathematical principles underpin the ways that different codes and ciphers operate. Discussing the majority of ancient and modern ciphers currently known, The Mathematics of Secrets sheds light on both code making and code breaking. Over the next few weeks, we’ll be running a series of cipher challenges from Joshua Holden. The last post was on subliminal channels. Today’s is on binary ciphers:

Binary numerals, as most people know, represent numbers using only the digits 0 and 1.  They are very common in modern ciphers due to their use in computers, and they frequently represent letters of the alphabet.  A numeral like 10010 could represent the (1 · 24 + 0 · 23 + 0 · 22 + 1 · 2 + 0)th = 18th letter of the alphabet, or r.  So the entire alphabet would be:

 plaintext:   a     b     c     d     e     f     g     h     i     j
ciphertext: 00001 00010 00011 00100 00101 00110 00111 01000 01001 01010

 plaintext:   k     l     m     n     o     p     q     r     s     t
ciphertext: 01011 01100 01101 01110 01111 10000 10001 10010 10011 10100

 plaintext:   u     v     w     x     y     z
ciphertext: 10101 10110 10111 11000 11001 11010

The first use of a binary numeral system in cryptography, however, was well before the advent of digital computers. Sir Francis Bacon alluded to this cipher in 1605 in his work Of the Proficience and Advancement of Learning, Divine and Humane and published it in 1623 in the enlarged Latin version De Augmentis Scientarum. In this system not only the meaning but the very existence of the message is hidden in an innocuous “covertext.” We will give a modern English example.

Suppose we want to encrypt the word “not” into the covertext “I wrote Shakespeare.” First convert the plaintext into binary numerals:

   plaintext:   n      o     t
  ciphertext: 01110  01111 10100

Then stick the digits together into a string:

    011100111110100

Now we need what Bacon called a “biformed alphabet,” that is, one where each letter can have a “0-form” and a “1-form.”We will use roman letters for our 0-form and italic for our 1-form. Then for each letter of the covertext, if the corresponding digit in the ciphertext is 0, use the 0-form, and if the digit is 1 use the 1-form:

    0 11100 111110100xx
    I wrote Shakespeare.

Any leftover letters can be ignored, and we leave in spaces and punctuation to make the covertext look more realistic. Of course, it still looks odd with two different typefaces—Bacon’s examples were more subtle, although it’s a tricky business to get two alphabets that are similar enough to fool the casual observer but distinct enough to allow for accurate decryption.

Ciphers with binary numerals were reinvented many years later for use with the telegraph and then the printing telegraph, or teletypewriter. The first of these were technically not cryptographic since they were intended for convenience rather than secrecy. We could call them nonsecret ciphers, although for historical reasons they are usually called codes or sometimes encodings. The most well-known nonsecret encoding is probably the Morse code used for telegraphs and early radio, although Morse code does not use binary numerals. In 1833, Gauss, whom we met in Chapter 1, and the physicist Wilhelm Weber invented probably the first telegraph code, using essentially the same system of 5 binary digits as Bacon. Jean-Maurice-Émile Baudot used the same idea for his Baudot code when he invented his teletypewriter system in 1874. And the Baudot code is the one that Gilbert S. Vernam had in front of him in 1917 when his team at AT&T was asked to investigate the security of teletypewriter communications.

Vernam realized that he could take the string of binary digits produced by the Baudot code and encrypt it by combining each digit from the plaintext with a corresponding digit from the key according to the rules:

0 ⊕ 0 = 0
0 ⊕ 1 = 1
1 ⊕ 0 = 1
1 ⊕ 1 = 0

For example, the digits 10010, which ordinarily represent 18, and the digits 01110, which ordinarily represent 14, would be combined to get:

1 0 0 1 0
0 1 1 1 0


1 1 1 0 0

This gives 11100, which ordinarily represents 28—not the usual sum of 18 and 14.

Some of the systems that AT&T was using were equipped to automatically send messages using a paper tape, which could be punched with holes in 5 columns—a hole indicated a 1 in the Baudot code and no hole indicated a 0. Vernam configured the teletypewriter to combine each digit represented by the plaintext tape to the corresponding digit from a second tape punched with key characters. The resulting ciphertext is sent over the telegraph lines as usual.

At the other end, Bob feeds an identical copy of the tape through the same circuitry. Notice that doing the same operation twice gives you back the original value for each rule:

(0 ⊕ 0) ⊕ 0 = 0 ⊕ 0 = 0
(0 ⊕ 1) ⊕ 1 = 1 ⊕ 1 = 0
(1 ⊕ 0) ⊕ 0 = 1 ⊕ 0 = 1
(1 ⊕ 1) ⊕ 1 = 0 ⊕ 1 = 1

Thus the same operation at Bob’s end cancels out the key, and the teletypewriter can print the plaintext. Vernam’s invention and its further developments became extremely important in modern ciphers such as the ones in Sections 4.3 and 5.2 of The Mathematics of Secrets.

But let’s finish this post by going back to Bacon’s cipher.  I’ve changed it up a little — the covertext below is made up of two different kinds of words, not two different kinds of letters.  Can you figure out the two different kinds and decipher the hidden message?

It’s very important always to understand that students and examiners of cryptography are often confused in considering our Francis Bacon and another Bacon: esteemed Roger. It is easy to address even issues as evidently confusing as one of this nature. It becomes clear when you observe they lived different eras.

Answer to Cipher Challenge #2: Subliminal Channels

Given the hints, a good first assumption is that the ciphertext numbers have to be combined in such a way as to get rid of all of the fractions and give a whole number between 1 and 52.  If you look carefully, you’ll see that 1/5 is always paired with 3/5, 2/5 with 1/5, 3/5 with 4/5, and 4/5 with 2/5.  In each case, twice the first one plus the second one gives you a whole number:

2 × (1/5) + 3/5 = 5/5 = 1
2 × (2/5) + 1/5 = 5/5 = 1
2 × (3/5) + 4/5 = 10/5 = 2
2 × (4/5) + 2/5 = 10/5 = 2

Also, twice the second one minus the first one gives you a whole number:

2 × (3/5) – 1/5 = 5/5 = 1
2 × (1/5) – 2/5 = 0/5 = 0
2 × (4/5) – 3/5 = 5/5 = 1
2 × (2/5) – 4/5 = 0/5 = 0

Applying

to the ciphertext gives the first plaintext:

39 31 45 45 27 33 31 40 47 39 28 31 44 41
 m  e  s  s  a  g  e  n  u  m  b  e  r  o
40 31 35 45 46 34 31 39 31 30 35 47 39
 n  e  i  s  t  h  e  m  e  d  i  u  m

And applying

to the ciphertext gives the second plaintext:

20  8  5 19  5  3 15 14  4 16 12  1  9 14 
 t  h  e  s  e  c  o  n  d  p  l  a  i  n
20  5 24 20  9 19  1 20 12  1 18  7  5
 t  e  x  t  i  s  a  t  l  a  r  g  e

To deduce the encryption process, we have to solve our two equations for C1 and C2.  Subtracting the second equation from twice the first gives:


so

Adding the first equation to twice the second gives:


so

Joshua Holden is professor of mathematics at the Rose-Hulman Institute of Technology.

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Fibonacci helped to revive the West as the cradle of science, technology, and commerce, yet he vanished from the pages of history. Finding Fibonacci is Keith Devlin’s compelling firsthand account of his ten-year quest to tell Fibonacci’s story.

Devlin Fibonacci cover

This annual anthology brings together the year’s finest mathematics writing from around the world. Featuring promising new voices alongside some of the foremost names in the field, The Best Writing on Mathematics 2016 makes available to a wide audience many articles not easily found anywhere else—and you don’t need to be a mathematician to enjoy them.

Pitici Best writing on Maths

In The Calculus of Happiness, Oscar Fernandez shows us that math yields powerful insights into health, wealth, and love. Using only high-school-level math, he guides us through several of the surprising results, including an easy rule of thumb for choosing foods that lower our risk for developing diabetes, simple “all-weather” investment portfolios with great returns, and math-backed strategies for achieving financial independence and searching for our soul mate.

Fernandez Calculus of Happiness

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