It's all about the Washingtons. And that smiley face button. Image: flickr user xJason.Rogersx

When I go to Europe, my pockets rapidly fill up with change. In addition to language barriers that prevent me from quickly understanding how much I owe, I have trouble dealing with the unfamiliar coin denominations. The best way to make 75 cents is to use a fifty cent piece, two twenties, and a five, not three quarters. But I have trouble remembering that on the fly. The one- and two-euro coins further confuse the issue for me, as I reach for a bill rather than coins for a three euro transaction. In the airport on the way home, I generally try to convert as many of my amassed coins to duty-free chocolate as possible, but I often come home with quite a few coins jingling in my pocket anyway.

In the states, I try to avoid using coins in my everyday life. I rarely carry a purse, and my wallet doesn’t have a coin pocket, so they just clang around in my pockets. I let my non-quarters accumulate in a repurposed coffee can, and every few years I count them up and treat myself to a sandwich or something. The quarters are immediately diverted to the laundry fund.

In a paper posted to the preprint site arXiv.org on June 9, mathematicians Lara Pudwell of Valparaiso University and Eric Rowland of the University of Quebec at Montreal tackled the question of how many coins people have sitting around. They used statistical techniques to determine that a typical carrier of US currency is most likely carrying 10 coins at any given time: 1 quarter, 1 dime, 1 nickel, and 7 pennies. If, like me, you save your coins instead of spending them, then it’s likely that about 31.9 percent of your piggybank contents are quarters, 17 percent are dimes, 8.5 percent are nickels, and 42.6 percent are pennies.

But the assumptions and methods that go into the analysis are more interesting than the conclusions by themselves. It’s a pretty fun and readable paper, so I’d encourage you to check it out for yourself. But if you want the Cliffs Notes, read on.

With any mathematical model of real-world behavior, researchers have to decide what assumptions to make. Pudwell and Rowland start the paper with two reasonable assumptions about the everyday use of coins:

“(1) The fractional parts of prices are distributed uniformly between 0 and 99 cents.
(2) Cashiers return change using the fewest possible coins.”

Incidentally, I’m curious about the first assumption. I’d guess that if people always bought items one at a time, prices of their transactions would not be uniformly distributed because a lot of prices end in .99 or .49, and depending on how taxes work in your city, that probably makes certain prices more likely than others. But I’d imagine that for multi-item transactions such as a trip to the grocery store, a uniform distribution of the cents part of prices is pretty likely. It would be interesting to see whether the tax rates and average prices of goods in certain locations bias the prices in favor of certain cents amounts, but this is a matter of going out and collecting a lot of data, not making models and feeding them to computers.

For most of the examples in the paper, the researchers also assume the use of US currency denominations (1-cent penny, 5-cent nickel, 10-cent dime, and 25-cent quarter—half dollars and dollar coins are not included), but their model can handle any set of coin denominations.

Pudwell and Rowland divide the world into two groups depending on coin handling preferences: those who pay for everything with bills and keep coins in a jar at home, and those who do use coins when they pay for things. The situation of the former, the “coin keeper,” is relatively easy to handle. To figure out the proportions of different coins in a coin keeper’s stash, you just have to tally up how many coins you get for every possible price. If the cents amount of prices are truly equally distributed, this will give you the average percentage over a large number of transactions.

The situation for the coin spender is a bit more complicated. With the additional assumption that a coin spender keeps less than $1.00 in coins in her wallet at all times, Pudwell and Rowland note that a spender’s choices can be modeled as a Markov chain, meaning that there are a finite number of possible coin combinations in a wallet (called “wallet states”), and there is an unambiguous process that happens when the spender buys something. For each transaction, the new wallet state only depends on the wallet state directly before it, the amount paid at the store, and an algorithm for how a spender pays for things.

For example, the “big spender” overpays as little as possible and uses as few coins as possible if there are multiple ways to overpay as little as possible. If a big spender has to pay 27 cents, she will use a quarter and a nickel over three dimes if she can’t make exact change. Overpaying as little as possible takes precedence over minimizing the number of coins: if she had only a quarter and three dimes, she would spend the three dimes rather than the quarter and one dime. (In this case, the choice to overpay as little as possible rewards her by decreasing the number of coins she receives as change.)

This is the part of the post where it’s helpful to know a little bit of linear algebra. If you don’t feel like thinking about linear algebra right now, skip the rest of this paragraph. The method that Pudwell and Rowland used to compute the probabilities of different “wallet states” involves making a matrix that keeps track of how likely it is to move from one wallet state to another during one transaction. A particular eigenvector of one of this matrix tells you the likelihood of each wallet state after a long period of time. The only catch is that for real-world currencies and prices, the matrices are ginormous, to use a technical term. There are 6,720 different combinations of coins that add up to under $1.00 in the US currency system, so you end up with a 6720×6720 matrix in the case of the “big spender.” Just running the computations necessary to create the 6720×6720 matrix took 1 day’s worth of CPU time. Pudwell and Rowland use numerical approximation to find the eigenvectors rather than computing them exactly.

My favorite thing about this paper is the thought that went into the different examples Pudwell and Rowland discuss. They analyze several types of coin spending habits based on how people actually spend their cash. In addition to the “big spender” described above, they analyze the “pennyless” (not to be confused with “penniless”) purchaser, who leaves pennies in those “give a penny/take a penny” trays rather than giving or receiving them as change, and the “quarter hoarder.” In their words, “the quarter hoarder is a spending strategy utilized by college students and apartment dwellers who save all their quarters for laundry. All quarters they receive as change are immediately thrown into their laundry fund, so the quarter hoarder’s wallet contains only dimes, nickels, and pennies.” It’s like they’re reading my mind! They also analyze what would happen with a fictional currency that replaces the dime with an 18-cent piece. They chose this particular fictional currency because it is the currency system that minimizes the average number of coins given as change. For each variation, they computed the most likely combinations of coins for people using that system.

At the end of the article, Pudwell and Rowland speculate about some of their assumptions and wonder how well they match the way people actually behave. They assume that people want to minimize the number of coins they spend, but perhaps it makes more sense to maximize the number of coins in a transaction, thereby minimizing the weight in your pocket after the transaction. “Of course, the best way to minimize the number of coins in your wallet is to curtly throw all your coins at the cashier and make them give you change. But…we are strictly interested in civil models of spending.” They suggest some interesting ways to modify the algorithm to take different spending patterns into account.

I decided to dump out my coin jar to see how closely my coin hoarding habits align with what the model predicts. Because I am a hybrid of the coin keeper and the quarter hoarder, I only considered non-quarters. I have 80 dimes (20.3 percent of my total change), 27 nickels (6.8 percent), and 288 pennies (72.9 percent). Ignoring the quarters in the “coin keeper” model, Pudwell and Rowland predict 25 percent dimes, 12.5 percent nickels, and 62.5 percent pennies. So my numbers aren’t too far from the predicted amounts. As disappointing as it is to be so typical, it’s kind of cool that a mathematical model that doesn’t even know me did a pretty good job of guessing what’s in my coin stash.

What are your coin carrying habits? Are you a hybrid of some of the coin types discussed above, or do you use a different algorithm for your cash transactions? Please share in the comments. For more details about what Pudwell and Rowland think your wallet holds, check out the paper on the arXiv.

An image created in the course of Laura DeMarco's dynamical systems research. Image: Laura DeMarco.

This year I’ve been co-writing “Mathematics, Live,” an interview series for the Association for Women in Mathematics newsletter. In my interviews I’m “listening in” on conversations between pairs of female mathematicians. The first interview appeared in the May/June issue of the newsletter (password required). In it, I talked with mathematicians Laura DeMarco of the University of Illinois at Chicago and Amie Wilkinson of the University of Chicago.

Both do research in the field of dynamical systems, the study of how abstract mathematical spaces evolve over time. I got the idea to talk with them when I was at DeMarco’s invited address at the Joint Math Meetings in January. (If you’re interested, Jordan Ellenberg wrote nice post about her talk.) Wilkinson asked a question at the end of the talk, and I realized they would make a great pair for this interview series.

I met with DeMarco and Wilkinson in March, and we talked about how they got interested in math, the importance of female role models for young women in math, and their advice for aspiring mathematicians. This is a slightly abridged version of the interview that appeared in the AWM newsletter. Thank you to DeMarco and Wilkinson for their generosity with their time and advice.

Origins

Evelyn Lamb: Would you like to start by talking about how you got into math?

Amie Wilkinson: I got into math in early infancy. I always liked math.

Laura DeMarco: Early infancy?

AW: I’m exaggerating, but I always liked math.

LD: Did you do stuff outside of school, or was it just in class?

AW: I went to a Montessori kindergarten. I think that’s the first time I actually saw math. What was great about Montessori was that everything was free-form, so you could just spend all your time at one station, all day long. I spent all my time at the math stations, basically. I would just do them all day. Counting base 5 and stuff like that. I think that’s when it was clear that I was passionate about math. You were a physicist, right?

LD: Yes, but not “for real.” In my case, I would say that I definitely always liked math. I always liked class, I always liked learning it and doing it. But my brother, who’s older than me, was always better than me at puzzles and things like that. He was the one who would go into the contests. He was doing MathCounts and whatever the other contests were, and he was really into them. I wasn’t interested in doing the competitions. I sort of found my own path and practiced my flute and did my own thing, but I probably came back to it later than you did.

The first time I thought to myself, “I like math enough to want to do it forever,” was some point in high school, when I thought, “I want to be a math teacher.” The funny thing is, I remember very vividly sitting on the school bus to go home from high school that day and thinking, “I could be a math teacher. I could just do math forever,” thinking that that’s what math means, right, to be a math teacher. I had no idea that there was anything beyond being a teacher.

It was in my second year in college when I learned that professors do research. I had no idea what it meant to do research. I was taking a seminar in social sciences. Each week we went through a different kind of theory with various examples. One day, the professor, who was from the law school, said to us, this group of second-year students, “are you aware that all of your professors are doing research?” And I don’t even know what that means. What does it mean for my math professors to be doing research?

The next day, I went and asked all of my math professors, “What do you do?” I was taking probability at the time, and I went to my probability professor. “I heard you do research. What do you do?” Imagine what it’s like when a student comes and asks you this question. I remember that it was this very awkward conversation. And he said something, and of course I don’t remember what he said, and I’m sure I didn’t understand it anyway. But the moment was very memorable.

At the same time, I was a physics major. I had loved physics classes in high school, and I thought, maybe I’ll just do physics. I knew that scientists do research. That’s obvious, somehow. So learning that mathematicians do research too was eye-opening.

AW: That’s a great story. I have this picture of you walking into the first professor’s office, like: “I’ve heard that you guys do this research thing. That’s not for real, is it?”

EL: Were there any pivotal moments where you knew that you wanted to be a mathematician, beyond learning that math research exists?

AW: My pivotal moment was pretty clear. I went to college, and I was feeling very insecure about my abilities in mathematics, and I hadn’t gotten a lot of encouragement, and I wasn’t really sure this was what I wanted to do, so I didn’t apply to grad school. I came back home to Chicago, and I got a job as an actuary. I enjoyed my work, but I started to feel like there was a hole in my existence. There was something missing. I realized that suddenly my universe had become finite. Anything I had to learn for this job, I could learn eventually. I could easily see the limits of this job, and I realized that with math there were so many things I could imagine that I would never know. That’s why I wanted to go back and do math. I love that feeling of this infinite horizon.

To me, that was a pivotal moment, actually just being away from it. In general, being away from math from time to time has definitely been rejuvenating. Like when I had my kids, and just wasn’t able to do math for a while. Then I would miss it. Then I’d understand why I’m doing it.

LD: You’d get extra excited about it, and really passionate about it.

AW: Yes. And grateful.

LD: I have these moments where I’m kind of overwhelmed by, “Wow, I really like what I’m doing, and isn’t it amazing that I have this job and can live like this!” Of course, I have teaching and other duties, but just the idea that we can be supported, that there is an environment for this. I think that way when it’s going well. When it’s not going well, I think, “What have I gotten myself into?!”

I didn’t know your story, that you had a job the first year after college. I did have some sort of moment that convinced me to go to graduate school. In my last year of undergraduate, my physics professors were very encouraging. There was something about the culture in the physics department that was simply encouraging. Any of their undergraduate students who were doing well were automatically involved in research projects. So I knew most of the faculty members, and it was somehow a natural thing to apply to graduate schools.

The math department didn’t feel like that.  But finally in my very last year, we got our first woman professor in the department. She arrived in my very last year, and that semester I had decided to ask her to be my advisor for my undergraduate thesis project. Just having her around made a big difference to me.

Then it was that fall semester of my last year of undergraduate that the TA of one of my classes said, “Oh, where are you applying for graduate school, Laura?” I said, “I’m not applying to graduate school. I actually have an interview tomorrow for a job.” He said, “What? You’re not applying to graduate school?” He was super encouraging. All of a sudden there was this one graduate student who seemed to care and said, “This is crazy! Why aren’t you applying to graduate school?”

AW: It was serendipity.

LD: It was sort of just by chance that one person had thought through the idea of actually asking me.

AW: Or not thought through it.

LD: That’s right, who had simply asked! My physics advisor had certainly talked about this idea. But I just wasn’t passionate about physics by the end.

Calculus

EL: Are there any math topics that are particularly appealing or beautiful for you?

AW: I like calculus a lot, probably because I learned it when I was young, and I learned it well. To me, it’s always comforting to use calculus to do something. The invention of calculus was certainly revolutionary.

LD: A conceptual breakthrough.

AW: It’s funny, because it’s like we just toss it out there to high school students, and I think a lot of them have no idea of the beauty.

LD: What the ideas really were.

AW: Certainly some of the most beautiful mathematics I’ve learned is just calculus.

LD: It’s funny you mention calculus. I don’t think I really appreciated it until I taught it as a graduate student. I was lecturing to these first-year students. I was just wowed by this subject. I had this moment of, holy cow, this is really beautiful! I remember my grandmother asking me what I was thinking about these days. I said, “Well, I’m teaching calculus right now, and you know what, calculus is really beautiful.” She said, “OK, Laura, what is calculus? Can you just tell me in 20 minutes, what is calculus?” And it was just the greatest thing to have this opportunity to just sit down with my grandmother, of all people, and tell her.

AW: The proverbial grandmother.

LD: That’s right. It’s funny because she says that she liked math when she was young, but it wasn’t something in that era that she could have pursued. She certainly never pursued anything beyond some basic courses. But she sat through and listened to my explanation.

AW: Do you think she got it?

LD: I don’t know. I was speaking more about the philosophy. I wasn’t doing any computations. But the idea of differentiation and then integration, and the fundamental theorem of calculus, how it’s connected. I don’t know if she got it or not. But it was a good conversation.

Women in mathematics

EL: Have you faced any challenges as women in math?

LD: Now I would say it’s an advantage. Once we’re at the stage that we’re at, it’s probably more of an advantage than a disadvantage. People want women speakers and women getting involved at different levels, and a certain amount of women at the top levels. Earlier on, it’s a different story.

AW: I would agree. As long as you’re able to say no, it’s an advantage. I think you’re asked to do more. It’s hard to say no to things that involve young people or women. As you get older, you feel a real responsibility to help the younger people. That’s the only disadvantage. I feel like I get asked to do a lot more.

LW: Yes, definitely.

AW: It’s hard for me to say no to a lot of it because it’s worthwhile. But when I was younger, Laura’s story about having the woman math professor really resonates with me because when I was in college there were no women at all at Harvard. No research faculty, zero.

LD: Not even postdocs?

AW: Not even postdocs. And so I think I craved a role model at that point. I think that if one had shown up it would have made a huge difference.

And having kids for me was difficult. It was scary. Partially because I didn’t really have that many people to look up to, to say it’s doable. Even when Beatrice was born, which was only 13 years ago, it wasn’t quite the norm, it wasn’t quite supported. That is another thing that I think is much harder for women. Hugely harder for some women. I was just lucky, for a lot of reasons, that it worked out OK.

In general, the stereotype threat business kind of held true for me. I think there was a little nagging voice that said, well, do you really think you belong here, when I was younger. When your confidence level is low.

LD: Yes, when you’re not so confident. I wasn’t so confident.

AW: No one’s really confident at that point.

LD: That’s right, nobody is. And people react very differently. But I wasn’t the kind of person to react by speaking louder, or by making myself seen. In fact, what I tended to do was try to play down my femininity in many different ways. I dressed like the boys, and I really went out of my way to be less feminine. Now I feel totally comfortable just being who I am. Certainly then I would make an effort not to stand out in some way. I wasn’t so confident, and being the only girl in my classes didn’t help.

AW: I got certainly some inappropriate off-color comments from people. Those kinds of comments, they were a bit alienating, but I don’t think I found any of those kinds of comments particularly discouraging. It was more the general level of apathy that was hard.

LD: Yes, that is something that probably played a role. Personality-wise, I probably needed encouragement. I would have liked to have gotten some explicit encouragement. If I’m doing well, I want to know!

I remember when I finally decided to apply to graduate schools, I had a very close girlfriend who said, “Well, you should certainly apply to Harvard and Princeton, and all the top schools.” I said, “Oh no, I’ll never get in.” So I didn’t even bother applying. But I did apply to Berkeley, and I got into Berkeley, and I went to Berkeley. And of course in the end I ended up transferring to Harvard, and I ended up with a degree from Harvard, so somehow it ended up happening anyways. And this friend, she’s not a mathematician, so I thought she had no idea what she was talking about, but in the end she was right.

AW: It’s so funny that it seemed obvious to her. An ordinary person would think, “Well, of course. You’re a top student. You should apply to the top schools.” In the math world there’s this huge mystique around these top places, and someone who lacks even just a little bit of confidence, it’s like “no, of course I’m not going to apply to a place like that.” I wonder how many women are kept out of the top places by that kind of attitude.

LD: And not realizing that you should actually go for something.

So you want to be a mathematician…

EL: Do you have advice for young people who might be thinking about doing math?

LD: If you love it, go for it. It is helpful to have some people to talk to. It helps to have an advisor of some sort or a research project to connect you, to learn how to communicate with people.

AW: I’m glad I did math team in high school.

LD: So you did math team?

AW: I did do math team. In junior high school, I was really good at math. I was clearly kind of a math kid, but a bunch of other kids were doing all these gifted programs and taking all these tests, and I was too scared to do that kind of thing, and I probably wouldn’t have done very well. When I got to high school, I don’t know what pushed me to go check it out, but I did. Doing the math team at my high school was really formative. It gave me a community of other math geeks.

LD: Kids that really enjoyed it. At least now these math circles are starting to pop up around various places.

AW: Yes, the math circles are even cooler because it’s not competition. Although, these math competitions get a bad rap. It wasn’t just sitting in a room and filling out these tests. There were oral contests. I remember I presented something on curves of constant width. You’d be given a topic, and you’d read up ahead of time. They’d ask you questions, and you could prepare an answer. Then you’d stand up at the board and present the answer. Girls, even then, happened to do very well. There was this girl named Nadia in our school, this extremely tall Russian volleyball player. She didn’t do anything else in math team, but she was tops at the oral part of this contest.

Then there was this two-person event, and my high school rival and I were the two-person team. There were all sorts of different things. Different talents could take part, and I’m sure there are things like this now.

That’s a piece of advice, to explore. You don’t have to be the very best to get something out of it. And another piece of advice, for young people, is that there really are second chances, and things can change. As an undergraduate, I kind of had a very mixed academic record. I did lots of things I loved that weren’t necessarily math. I did well in a few math classes, and I did badly in a few math classes. So I was lucky to get into Berkeley. But I found graduate school to be an utterly different experience from college. Suddenly there were no distractions, it was all I was doing.

LD: And you were enjoying it.

AW: I was enjoying it, and I felt at the top of my game. It’s worth a shot. That’s not the time to be scared to give it a try. If it doesn’t work out, it’s a year of your life. Big deal, whatever. I just think more people should try.

You made it to the end! You deserve another cool picture from Laura DeMarco's research. Image: Laura DeMarco.

My new 'do, including one of my cowlicks. Image: me.

At this length, I’m much more aware of my hairs as vectors than I was when they were longer. A vector is basically a straight line that points in some direction and has some length, like a strand of hair sticking out from someone’s head. And the hairy ball theorem is all about vectors.

The hairy ball theorem says that you can’t comb a hairy ball. In technical terms, if you have a tangent vector at every point on the surface of a sphere, you can’t make them all line up continuously with their neighbors without having some point where the tangent vector is zero. Or, if all your vectors are nonzero, you end up with a point where the vectors change direction abruptly or stick straight up. In other words, a cowlick. If you were trying to comb a rambutan, there would be one spot where you couldn’t get the hair to lie flat. Your rambutan would have a cowlick.

I would like to say that I have cowlicks because of math, but in truth, my head doesn’t satisfy the hypotheses of the hairy ball theorem. For one, the hair on the top of my head sticks straight up, rather than lying tangent to my scalp. But even when it grows in more and lies flat, the theorem still won’t apply.

The problem isn’t that my head has lumps and bumps like ears and a nose. The hairy ball theorem comes from the field of topology, the branch of mathematics that pretends all objects are made of infinitely stretchy rubber or putty. The theorem applies to any object that can be squished around into the shape of a sphere without making any tears or pinching a hole closed, no matter how much squishing that requires.

The reason my head doesn’t satisfy the hypotheses of the hairy ball theorem is that I don’t have enough hair. The hairy ball theorem requires the hair to cover the entire ball. As soon as you remove just one point from the ball, the hairy ball theorem breaks. An infinitely stretchy sphere with a pinprick in it can be stretched out to look like a circle in the normal 2-dimensional plane, where it’s pretty easy to comb hair. (Think shag carpet.) My head hair only covers part of my head. Even if I count the rest of the hair on my body, I still have eyes and palms and other hairless parts.

According to my 9th grade biology teacher, the lining of my digestive tract, from my mouth all the way to the other end, is an extension of my skin. If you look at it like that, I violate the assumptions of the hairy ball theorem in another way: I’m not topologically equivalent to a sphere at all, but to a torus, the surface of a donut. (I’m not really sure where my ears and sinuses fit into this. I’m not that kind of doctor.) You can comb a hairy donut. The way to see this is to imagine a rectangle covered in hair, maybe a carpet square or some sod. You can definitely comb that, and you can attach opposite sides to make a torus with a well-behaved hairstyle. Although if I had a hairy digestive tract, I don’t think I’d be worried about cowlicks.

I don’t know why I have cowlicks, but sadly, it’s not because of topology. For more on cowlicks, fingerprint whorls, and other natural singularities, check out Steven Strogatz’s New York Times article from last September about Singular Sensations. And for a nice video introduction to the hairy ball theorem, watch this Minute Physics video.

The Lute of Pythagoras is just one of the geometric constructions Hanson uses in her course. Two of Hanson’s students generously shared art they created for her class using the Lute of Pythagoras.

A drawing by Joseph Koch incorporates the Lute of Pythagoras into a portrait of Pythagoras himself. Image copyright Joseph Koch. Used with permission.

Both superimpose the Lute on another picture, highlighting the proportions of the underlying images.

The Lute of Pythagoras superimposed on a photograph of a forest. Image copyright Tanya Nikolic. Used with permission.

The Lute of Pythagoras is based on the “golden” isosceles triangle, a triangle with two equal sides and an apex angle of 36 degrees.

A golden triangle. The ratio a:b is the Golden ratio. The angle θ is 36 degrees, or π/5 radians. Image: Krishnavedala, via Wikimedia commons.

Each of the bottom angles is twice the size of the top angle, and with liberal use of sine and cosine addition formulas, you can check for yourself that the ratio a:b is indeed (1+√5)/2, the famous Golden ratio. Some compass and straightedge steps give you a cool pentagon-y, triangle-y, starry figure, the Lute of Pythagoras.

The Lute of Pythagoras, built from the golden triangle with vertices ABC. Image: Ann Hanson.

The ancient mathematical/musical Pythagorean cult is a bit mysterious, and apparently one of those mysteries is why this construction is called a lute! I don’t even know whether the figure was known to the Pythagoreans. I’d be thrilled if a math history buff educated me about the origin of the name.

In my correspondence with Hanson, I focused on the Lute of Pythagoras, but her students have also created quilts, origami, and tessellations for the class, in addition to learning to recognize mathematical inspiration when it appears in other people’s artwork. “One of their assignments is to go to the Art Institute [of Chicago],” Hanson says. “They say, ‘I never thought about all these painting and artwork having a relationship to mathematics.’”

Hanson, who is herself an artist as well as a math instructor, says that her math-art class is useful for students who are anxious about their math skills. “I’m not saying it’s a cure-all. This is just one approach that seems to help.” she says. “They come away with a different appreciation of math.” If you’d like to shake out the geometry cobwebs and get creative with the Lute of Pythagoras, full instructions for making the figure are here (pdf, kindly provided by Ann Hanson).

Harald Helfgott, who earlier this week posted a proof of the ternary Goldbach conjecture. Image: Harald Helfgott.

On Monday, Harald Helfgott of the École Normale Supériure in Paris posted a proof of one of the oldest open problems in number theory to the preprint repository arxiv. The ternary Goldbach conjecture, like so many questions in number theory, is easy to state but hard to prove. Every odd number greater than 5 can be written as the sum of three prime numbers. (Prime numbers have no factors other than themselves and the number 1.) For example, 7=2+2+3 and 91=7+41+43.

The ternary Goldbach conjecture is sometimes called the weak Goldbach conjecture. The strong Goldbach conjecture states that every even number greater than 2 can be written as the sum of two primes. Both conjectures were formulated in correspondence between Christian Goldbach and Leonhard Euler in 1742, hence the name. Logically enough, if you prove the strong Goldbach conjecture, you get the weak one for free: if you have an odd number greater than 5, subtract 3 from it. Now you have an even number greater than 2. So if you know that every even number greater than 2 is the sum of two primes, you can add 3 (a prime) to it to get your odd number, decomposed into the sum of three primes.

Sadly, it doesn’t work the other way. If you have an odd number written as the sum of 3 primes and subtract one of the odd primes, you’re left with an even number written as the sum of two primes, but there’s no guarantee that all even numbers will show up this way. But the ternary Goldbach conjecture does establish that every even number can be written as the sum of at most 4 primes: just subtract any odd prime number (for example, 3, or 2^57885161-1 ) from the even number you want to split up, and you’re left with another odd number, which we now know can be written as the sum of three primes. This improves Olivier Ramaré’s 1995 theorem that every even number is the sum of at most 6 primes.

Helfgott’s result is a big deal, but it didn’t come as a lightning bolt from the heavens. His work is part of a long line of papers using a technique called the Hardy-Littlewood-Vinogradov circle method. (Catchy, huh?) The very general idea of the circle method is that we turn a question about a set of numbers, in this case the primes, into a question about integrals over circles using techniques originally coming from analysis in the complex plane. It seems kind of miraculous that it’s even possible to convert questions about integers, which are spaced out discretely on the number line, into questions about continuous functions. “Questions of distribution of primes, or integers, can be expressed naturally in terms of the properties of continuous functions defined in terms of them,” Helfgott wrote in an email. A more concrete explanation of the circle method is beyond me, but if you want to dig into it and its limitations a bit more, you can check out this post by Terence Tao. It’s not for the equation-averse.

In the 1930s, Soviet mathematician Ivan Vinogradov established that the ternary Goldbach conjecture holds for all but finitely many odd numbers, so if someone could “just” check the odd numbers below a certain huge number C, everything would be good. There was just the pesky problem that Vinogradov’s bound was on the order of 10^6846168, an impossibly huge number for today’s computational resources, much less those available to Vinogradov. Over the next 70 or so years, the upper bound was whittled down to around 10¹³⁴⁶ as of 2002, but it was still far too large to handle.

Helfgott started working on Goldbach’s conjecture in 2006 as a postdoc in Montréal. “I had been trying to see what some different ways of proving Vinogradov’s theorem would be,” he wrote in an email. “I realized that one could prove it without the circle method,” he wrote, but, “it didn’t seem possible to give reasonable bounds C by the alternative proofs.” But papers and conversations with other researchers gave him hints of how to improve the bounds coming from the circle method.

Helfgott eventually managed to wrestle the upper bound down to 10³⁰, a much more manageable size, and with David Platt of the University of Bristol, he verified the conjecture for all numbers below that bound by computer. But the heavy computational resources were dedicated to proving the Generalized Riemann Hypothesis (GRH) for a large but finite number of cases. The GRH is one of the most important unsolved problems in mathematics. If solved, it would help us understand the distribution of prime numbers much better than we do. In fact, if the GRH were proved, the ternary Goldbach conjecture would be a corollary. But for the time being, computer-assisted checks of GRH for certain numbers are the best we can do.

Of course, making substantial progress on a problem that some of the most brilliant mathematicians of the past century have worked on was not an easy task. “There were several blind alleys-at one point I had to throw away a 50-page manuscript,” Helfgott wrote. “It was difficult to tell down the middle whether the plan would truly succeed. After all, if I had brought C down to 10¹⁰⁰, that would still have been larger than the number of subatomic particles in the universe multiplied by the number of seconds since the Big Bang-there would have been no hope of checking things that far!” Helfgott wrote that keeping track of the bounds explicitly was one of the most difficult parts of the work. “One annoying thing about the problem was that it turned out not to be the kind of thing I could work on in my head while at the movies or at a concert (not that I should),” he wrote. “I did get some good ideas in the shower, though.”

Helfgott’s paper has not been peer reviewed yet, but number theorists seem to be optimistic that the theorem will hold up to scrutiny. Unfortunately, it doesn’t provide much illumination on the strong Goldbach conjecture. Terence Tao, who proved last year that every odd number can be written as the sum of at most five primes, wrote on Google Plus that “the circle method is very unlikely to be able to settle the even Goldbach conjecture by itself.” Helfgott wrote that the problem is essentially that the strong Goldbach conjecture would require asymptotic estimates—more refined information about the values of certain quantities—at key points, rather than the coarse upper bounds available through current methods.

I asked Helfgott how he had celebrated his accomplishment. “Well, I gave a talk on this yesterday, and then I was taken out for dinner by the locals, as usually happens when one visits a place to give a talk. My parents are coming to visit now, so it will be a very good time to take a short break.” Helfgott is understandably relieved to have this major project finished and get back to his normal routine, which includes non-mathematical study as well. “I sometimes faced the difficult dilemma of whether to work into the night or prepare for a Russian test instead,” he wrote. “Hopefully I will be catching up on languages now that this is done.” Helfgott is fluent in English, French, Spanish, German and Esperanto, and according to his blog “badly needs to practice” Polish, Quechua (an indigenous language in his home country Peru), and Russian.*

The title of this post is an allusion to Bach’s Goldberg Variations. I can only hope that Vi Hart or another talented person is writing a song about the Goldbach conjecture to the tune of the theme from the Goldberg Variations. In the meantime, here’s a Wired article from 2012 about a cool visualization of the notes.

*This sentence was edited after publishing. Helfgott emailed me to correct the list of languages he speaks. He also sent me a more recent photo of himself, which I have added to the top of the post.

Patrick Honner (runner-up), Saul Rosenthal (Trustee and Sponsor), and Scott Goldthorp (winner) pose for a photo at the announcement of the winner of the first annual Rosenthal Prize for Innovation in Mathematics Teaching. Image: Museum of Mathematics

Many math teachers have a hands-on approach to their subject, but those hands aren’t usually covered in finger paint. Scott Goldthorp, however, sometimes teaches messy math classes. Goldthorp, a teacher at Rosa International Middle School in Cherry Hill, New Jersey, was the grand prize winner of the inaugural Rosenthal Prize for innovation in math teaching, sponsored by the recently opened Museum of Mathematics and awarded during MoMath’s opening week last December.

The Rosenthal Prize aims both to provide incentives to outstanding math teachers with new ideas and to disseminate those ideas as widely as possible. In addition to cash prizes, Goldthorp and runner-up Patrick Honner will have their winning lessons distributed to schools around the country later this year.

Goldthorp’s lesson plan covers some basic concepts in statistics. To make the lesson exciting, Goldthorp has the students create their own data set. He hangs butcher paper on the wall and has students roll up their sleeves and paint their palms. Each student places two handprints on the butcher paper: one from a standing position, and one at the top of a jump. Students measure the heights of each handprint and the distances between the standing and jumping prints. “The beginning part is a little messy,” Goldthorp says. “I think that’s necessary to get the students excited about it. Once they get excited about it, they can do anything they want.”

Goldthorp, who teaches science in addition to math, welcomes the non-mathematical inquiry that the lesson fosters. “It was surprising for the students that the taller people didn’t necessarily jump higher than the shorter students in the class.” They talk about why this might be and have some interesting discussions on anatomy and physiology in addition to statistics. “It’s fun having those conversations with the students,” Goldthorp says.

To me, part of the beauty of this lesson is how simple and easy it is to implement. Pretty much every middle school in the country can get some paint and butcher paper, and the math topics it teaches are part of the standard curriculum, so teachers won’t have to add a new topic to their schedules.

Goldthorp says that students respond very positively to hands-on lessons like this. “When I first started doing it, the students seemed a little apprehensive: This is math class, we should just be calculating numbers!” says Goldthorp. “At the beginning of the year, they might be out of their comfort zone. But after they get used to it, they love it.”

Runner-up Patrick Honner, a teacher at Brooklyn Technical High School in New York, submitted a very different lesson. He has students work in groups make hats for spheres. As a sometimes wearer of mathematically inspired hats, I approve. Honner’s lesson emphasizes concepts in solid geometry while allowing kids to let their creative, artistic sides show too. “Students reflected that it was nice to have not just an image of certain solids, but this direct experience with how you put them together.”

A sphere wears a hexagonal hat. Image: Patrick Honner

In Honner’s classroom, the hat lesson was very open-ended. But for the competition he created a more structured set of activities so that teachers will be able to modify the projects for students of different ages and levels of mathematical sophistication. Honner’s classes are interactive and flexible whenever possible. “I try to create opportunities to explore ideas with students,” he says. “I want the classroom to be a place where we explore ideas together, where students can play around, experiment, collaborate, argue, create, and reflect on everything.” Honner is active in Math for America and writes for the New York Times Learning Network in addition to his personal blog.

Honner has gotten positive feedback on hands-on lessons like this one. “Students really enjoy opportunities to collaborate in a meaningful way,” he says. “They enjoy seeing how creative their classmates are.”

This sphere is thirsty 4 math. Image: Patrick Honner

It’s easy for me to get discouraged by the seemingly constant barrage of negative stories about mathematics education. But the Rosenthal Prize reminds me that there are a lot of great teachers in our classrooms who are reaching students in fun ways. If you are one of those teachers, you have until May 10 to submit a lesson plan for this year’s Rosenthal Prize.

Image: Design Shack

In the current issue of the Association for Women in Mathematics newsletter (password required), Anne Carlill asks where the female mathematicians are on Twitter:

“I found that the only female mathematicians or math educators I followed were Nalini Joshi in Sydney and Fawn Nguyen in California. In contrast there are about 15 males, including Marcus du Sautoy and Simon Singh….I am sure there are great female mathematicians around who do tweet; I just need help finding them.”

Of course I humbly nominate myself, but here are some other mathy (mathsy?) women I follow on Twitter. I thought about separating this list into mathematicians and math educators, but on Twitter, I don’t think that dichotomy really makes sense. No need to create artificial divisions when we’re all trying to tell people how cool math is!

@AnnaWeltman, who co-runs @MathMunch
@bettynlove, a math professor at the University of Nebraska-Omaha
@DainaTaimina, godmother of hyperbolic crochet
@danicamckellar, actress and author of math books for girls
@DrJStockton, math professor at Sacred Heart University
@faroop, math professor at Lesley University, also runs @liberationmath, which focuses on math shame/frustration and how to move past it
@fawnpnguyen, middle school math teacher in California
@haggismaths, math educator who does some cool mathematical textile stuff
@hildabast, who focuses on statistics in medicine
@ilaba, math professor at the University of British Columbia
@JenLucPiquant, a science writer who focuses on physics but tweets about a lot of fun math stuff
@julierehmeyer, freelance math writer. She hasn’t tweeted in a while, but we can hope!
@katemath, math professor at the College of Charleston
@lauramclay, professor of operations research at Virginia Commonwealth University
@MadeleineS, mathematical textile creator
@MariaDroujkova, math educator
@mathbabedotorg, former math professor, current data scientist with lots to say about the financial system
@mathcirque, math professor at Harvey Mudd College
@mathinyourfeet, who uses dance to teach mathematics
@monsoon0, math professor at the University of Sydney
@Ms_A_Geometry, a math teacher in Hawaii
@nattyover, a writer for Simons Science News who focuses on physics, math, and computer science
@NoelAnn, who co-runs @mathshistory
@quietannie1, retired maths lecturer from Leeds City College and the person who asked about female mathematicians on Twitter
@sc_k, a physicist who sometimes tweets about math
@stecks, who co-runs @aperiodical
@suevanhattum, community college math teacher
@TheEtymologyofZ, maybe a stretch. It’s the Twitter feed for a delightful looking film about a princess who loves math.
@vihartvihart, mathemusician and entertaining YouTuber

I’ve only been on Twitter for about 10 months, so I know this list is incomplete. Please add your suggestions in the comments. Self-nomination is encouraged! I’ve put together a “Mathy Ladies” list on Twitter, and I’ll keep updating it.

Added since original posting:

@ReginaNuzzo, freelance science journalist and statistics professor
@ilovemathsgames, who runs a maths game website and blog
@CathDoesMath, math and statistics teacher
@Supernetworks, a network research center feed, run by Anna Nagurney of UMass Amherst
@lpudwell, math professor at Valparaiso University
@mathemalicious
@tchmathculture, professor of math education at Peabody College
@pegcagle, math teacher and education policy person
@joboaler,  math education professor at Stanford University
@mathhistory, who is studying the history of math education at Berkeley
@popthatmatrix
@DrEugeniaCheng, mathematician at the University of Sheffield
@Mythagon, math educator in Maine
@jreulbach, middle school math teacher
@k8nowak, “mathalician,” who is on the @mathalicious team
@teachteKBeck, middle school math teacher
@crstn85, high school math teacher
@algebrainiac1, middle school math teacher
@wahedahbug, math teacher
@alllison_krasno, math teacher
@pamjwilson, math teacher
@RogoNic, stats app developer and YouTuber
@Pure_Numbers, mathematical observations about each day’s date
@a_mcsquared, math educator in Montréal
@CUMATMathDrB, math education professor at Clemson University
@NicoraPlaca, math education researcher
@KMorrowleong, math educator
@BridgetTenner, math professor at DePaul University
@MuirMath240, math instructor at Colorado University Boulder
@PeggyONeillDr, online math teacher
@CyberAuntie, freelance mathematician
@mathbrief, mathematician and blogger
@hspter, biostatistician

As of May 1, I’ll no longer be updating the list here on the blog, but you can always find the up-to-date list on my Twitter profile.

SOS, 181418, appears starting at the 1,377,767th digit of pi. Image: xkcd.

Today I have a piece in Slate about that pi meme that’s been going around. According to the meme, your life story is encoded in pi somewhere. My life story would probably include the word “Evelyn” at some point. (I’m going out on a limb, but stay with me.) In a code that assigns the string 00 to A and 25 to Z, EVELYN is 042104112413. It does not occur in the first 2 billion digits of pi, according to this pi search page.

At first, I was a bit surprised. 2 billion is a lot of digits, and I was only trying to match 12 of them. But there are 10¹²—1 trillion—12-digit strings, so only about 0.2 percent of them are present in the first 2 billion digits. In other words, at least 99.8 percent of possible 6-letter words won’t occur in the first 2 billion digits of pi. When it comes to encoding entire sentences, the numbers quickly get out of hand.

If you’re looking for, say, the works of Shakespeare in pi, you’ll need the 618 characters of Sonnet 18 in there, which will require 1236 digits to encode. There are 10¹²³⁶ possible 618-character strings, and 10¹²³⁶-1 are impostor sonnets. The amount of time it would take to sift out all the garbage and find that comparison to a summer’s day is simply unfathomable. To give you a frame of reference, the universe has been around for about 10¹⁷ seconds and contains approximately 10⁸⁰ atoms.

So if you’re worried that human creativity is worthless because every possible piece of literature, the MP3 of every song anyone will ever sing, the choreography of every ballet, along with the DNA of every person who will ever perform it, is encoded in pi, don’t despair. No one with any sense will be switching to a pi-mining strategy to write the next great American novel or record the next “Gangnam Style.” The messy, complicated, frustrating, exhilarating inner workings of the human brain are still the most efficient tools for creating profound new art, or meaningful new ideas in any field. At least until we become the sniveling slaves of our robot masters.

A fabric design I made using a picture of the Himalayas. (Pictureof the Himalayas by NASA.) Don't you need a sheath dress with this design? Or perhaps a cravat?

I had been aware of Spoonflower for a while, but I really got excited about fabric design at the Joint Mathematics Meetings in January. I attended a talk by Frank Farris, a mathematician at Santa Clara University, in which he described how he used Spoonflower to create mathematically sophisticated textiles for a recent art installation. To read more about how Farris uses mathematics to make beautiful, “impossible” designs, check out this excellent article by Erica Klarreich.

One of my hobbies is sewing (or as I prefer to think of it, applied geometry), and as a result I have been slightly reluctant to succumb to the siren song of Spoonflower. Without careful self-regulation, I’m sure I could spend all my money buying custom fabric and all my time sewing with it! But with slight trepidation I’ve started to dip my toe in the water. I only have two public designs so far, but the geek chic contest has me fired up about making more.

I haven’t quite decided what my contest entry will be yet. The geek landscape is almost unfathomably large, extending from the forests of Middle Earth to the most distant quasars in the universe, but of course I will be looking to mathematics for my inspiration. I’m really looking forward to seeing what all the other designers come up with as well, so if you’re a chic geek, you should enter the contest and decrease my chances of winning!

7 × 13 pieces of beach glass found on the shore of Lake Michigan and arranged on my coffee table.

Rather appropriately, April 1st is the 91st day of the year, at least in non-leap years such as 2013. 91 might look innocent, but it’s a sneaky little number because 91=7×13.

That might not seem sneaky to you, but I’m here to tell you why it is. Every whole number can be broken down into some finite number of prime factors. For instance, 10=2×5, and 54=2×3×3×3. When you learn about factoring, you usually learn a few tricks to figure out a number’s factors. A number is divisible by 2 if it ends in an even digit, and by 5 if it ends in 0 or 5. And of course if a number’s digits add up to a multiple of 3, then the number is divisible by 3.

There is also an easy divisibility test for 11, although it’s a bit more involved than the tests for 2, 3, and 5. To test for divisibility by 11, first you alternate adding and subtracting digits of the number. The original number is divisible by 11 if the alternating sum of digits is divisible by 11. (This includes 0.) For example, 2013 gives us 2-0+1-3=0, so 2013 is divisible by 11.

I think of these tests as filters with different mesh sizes. Numbers are trapped by the 2 filter if they’re even and flow through if they’re odd. Then they go on to the 3 filter, where the multiples of 3 are weeded out, and so on. If a number is not prime, there’s a pretty good chance that the 2, 3, 5, or 11 filter will pick it out. Which means that it’s easy for me to be lazy and assume a number that passes through all four filters is prime, and in fact, that’s exactly what happens to me with the number 91. Except for squares, which many people, including me, have memorized up to 15 or 20 anyway, 91 is the first non-prime that makes it through all four of my filters. I call those numbers “fake primes” because I want to transfer responsibility onto the numbers rather than my imperfect factoring skills. 91 is especially fake because it is so small. As numbers get larger, there are more and more possibilities for factors, so I get less and less surprised when a composite number makes it through my filters.

I came up with the name “fake prime” for smallish numbers that aren’t divisible by 2, 3, 5, or 11, but in fact there is a real mathematical term for numbers that “look” prime. A pseudoprime is a number that satisfies some property that all primes satisfy, but isn’t itself a prime. Pseudoprimes are generally divided into different classes based on what properties they satisfy. I guess I could define a class called  Lamb pseudoprimes as are all non-squares that aren’t divisible by 2, 3, 5, or 11, but most classes of pseudoprimes are quite a bit more sophisticated than that.

91 is a Fermat pseudoprime. This is the same Fermat of Fermat’s last theorem, and the pseudoprimes in question are related to what is called Fermat’s little theorem. That theorem states that if a number p is prime, then for any other number a, the number a^p -a is divisible by p. For example, 5 is prime, and 2⁵-2=30, which is divisible by 5. And for that matter, because 2 is prime as well, we can flip it around. 5²-5=20, which is divisible by 2.

Any non-prime number x that satisfies this property for some base number a, that is, a^x-a is divisible by x, is called a Fermat pseudoprime to base a, and 91 is the smallest Fermat pseudoprime to base 3. You’re not going to be able to verify it on your calculator because 3⁹¹ has 44 digits, but 3⁹¹-3 is divisible by 91. (But you can check it on Wolfram Alpha if you think I’m trying to pull an April Fool’s prank on you.)

You can generate your own fake primes, or Lamb pseudoprimes, by finding any two or more prime numbers other than 2, 3, 5, and 11 and multiplying them together. It’s all fun and games with little numbers such as 91 and 119 (7×17), but they’re actually pretty serious business in the form of RSA encryption. That’s right, information you send over the Internet is often protected by souped-up fake primes! So maybe 91 isn’t trying to pull an April fool’s prank on unsuspecting algebra students—it’s just trying to keep your credit card safe.