An Ugly Song That Definitely Won’t Get Stuck In Your Head. I Promise.

If you’ve ever had the hook from Call Me Maybe stuck in your head for three days, you know the power of a well-patterend musical hook. It’s these patterns in music that speak to our human brains and often help us differentiate good music from bad, and beautiful from ugly. Patterns, of course, suggest that math is at play, and this is no surprise. The connection between math and music has been well-studied from the geometric ideas of Pythagoras, the symmetries in music, and even music as topology.

What I want to talk about is how to use math to make the ugliest music possible. If patterns register as “beautiful” in our minds, then a completely pattern free tune would be the least beautiful. But to actually compose a totally pattern free piece of music is not an easy feat. Remember, pattern free is distinctly different from random (the latter being relatively easy). An interesting solution to this problem came to us by way of 1950’s work of engineers John P. Costas and Solomon W. Golomb. Costas and Golomb weren’t trying to write ugly music, rather, they were trying to solve a problem in sonar signaling. Turns out the same principles that make an ugly sound also make a great sonar ping. The solution was a combination of the Golomb Ruler, or its multi-dimensional cousin, the Costas array. These tools give a method for generating points that are not random (obviously), but are completely pattern free.

Using these tools, the mathematician Scott Rickard engineered the ugliest song every written. The basic idea is quite clever. Beginning with an 88 by 88 grid, starting in the far left column, he moves along the columns filling in boxes in rows corresponding to powers of 3. So in each column n he marks the 3n-th block. And when the numbers get too big, he reduces them modulo 88. This gives a totally pattern free sequence of points, and hey, also a handy representation of notes on the 88 key piano! He debuts this hideous (yet mathematically clever) opus in TEDx Talk: The Worlds Ugliest Music. You probably won’t be humming the riffs all day, but it’s cool to hear.

The AMS has a great link-roudup of other blogs and videos about math and music, and since it’s still the month of June, find something beautiful to enjoy and have a happy Fête de la Musique!

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Getting Warmer…

I’m currently teaching a summer school for high school students. Our main focus is number theory and its applications to cryptology, but I like to start each morning with some kind of warm-up math puzzle or game. I know plenty of fun math stuff, but I’ve never worked with high schoolers before, so I took to the blogs to find some good activities for that age group. The Internet has almost as many puzzles and games as it has cat pictures, so the sites I’m highlighting here are just the tip of the iceberg.

An element of {cats}∩{math}. Thanks, Internet! Image: Jimmie, via Flickr.

For the first day, I wanted an activity that would get the kids working together a bit and introducing themselves to each other. A bit of searching, I came upon Dan Meyer’spersonality coordinates” activity. Meyer write the must-read math teacher blog dy/dan (which, by the way, I can’t decide how to pronounce, but I suppose that’s the point). His activity had students in a group label themselves on a coordinate axis by how much of two different traits they had. I didn’t use that activity but one I found in the comments: break people up into groups of size 2n and have them come up with yes-or-no questions so that each person in the group has a different set of answers. I only did it with groups of four students, and I had students mix up a couple of times to meet new people and come up with different traits. The next time I use this activity, I will probably ask them to get into groups of eight after playing once or twice in groups of four.

Sam Shah’s blog Continuous Everywhere but Differentiable Nowhere has some nice problems and puzzles, including the sack problem that nerdsniped me a while ago. Math Munch, “a weekly digest of the mathematical internet,” has also been featured on this blog before. It doesn’t just focus on puzzles and games. There’s a strong art component as well, and the curators usually include some web-based interactive activities. Periodically they run interviews with mathematicians, teachers, and artists. I especially enjoyed the Q&A with Carolyn Yackel, who just sounded so enthusiastic about abstract algebra that I wanted to go find some symmetry groups.

A new-to-me blog that’s been a good puzzle source is Math=Love by high school math teacher Sarah Hagan. Don’t tell my students, but I think I’ll be using the 1-4-5 square puzzle challenge next week, and I might talk about happy numbers at some point. Very helpfully, Hagan often includes logistical information about how she made the puzzles or games work in the classroom and ideas to make them go more smoothly in the future. She also shares links to other sites with math games and puzzles Now that I’ve been reading the blog for a few months, I’m a bit embarrassed that I didn’t start reading it earlier. Hagan is very well known in the math teacher blogging world. Aside from the puzzles and games, she shares a lot of helpful tips about running the classroom and reflections on her teaching practices.

Futility Closet isn’t strictly a math blog, but it has tons of fun puzzles. The jeweler’s observation caught my eye recently. Why must every convex polyhedron have at least two faces with the same number of sides? It’s a simple question with a short, clever answer, but I think students will have fun trying to figure it out.

I’ve found some activities in other places as well. Last Thursday, we made a level one Menger sponge using leftover supplies from MegaMenger in October. The students had heard about fractals from a guest speaker earlier in the week, so we talked a little more about how something could have a non-integer dimension and figured out the fractal dimensions of the Cantor set and the Menger sponge. The seven penny game from The Proof and the Pudding by Jim Henle was fun (my review of the book is here), and I’ll be using Matt Parker’s Things to Make and Do in the Fourth Dimension later in the program (see my review here). I might even try to make a domino circuit (pdf), but I’m not sure if I have enough patience or dominoes.

Do you have a favorite source for math puzzles, games, or activities?

Posted in K-12 Mathematics, Math Education, Recreational Mathematics | Tagged , , , , , , | 1 Comment

Happy Deep Fried Torus Day

Last Friday was National Doughnut Day and I hope everyone celebrated accordingly. Aside from being delicious with a cup of coffee, these sweet treats are a great way to think about math. From simple brain teasers to advances in quantum teleportation, this shapely pastry has a lot to offer.

How many doughnuts are in this picture?  What sort of things would you need to know to answer that question?

How many doughnuts are in this picture? What sort of things would you need to know to answer that question?

Let’s start out with something fun. The photo above was popping all over the internet this weekend, and it’s a great way for students to work on their understanding of spatial reasoning and geometry. The math teacher focused site YummyMath guides students through a doughnut inspired estimation exercise, and BedtimeMath gives a series of doughnut questions for all ages. Between doughnut flavors, different toppings, and doughnut holes, there are plenty of questions to be asked!

Of course doughnuts as math objects reached their cultural apex around 2006 with the solution to Poincare’s Conjecture, inspiring the popular joke,

A topologist is a mathematician who can’t tell a coffee cup from a doughnut.

Or its lesser known modification

How many topologists does it take to change a light bulb?
Just one. But what’ll you do with the doughnut?

courtesy of the blog MathJokes4MathyFolks. Even Stephen Colbert got in on the action, smooshing a doughnut on the Colbert Report.

But beyond the fun questions and smooshing of baked goods, the doughnut — or torus as we call it in the biz — plays a very important role in advanced mathematics and physics. The special shape of the torus means that it has a relatively large amount of surface area relative to its size. Compare the torus, for example, to a sphere or a cube. Tori are also special because when you rotate them around a central axis, every point moves. But wait, doesn’t that work on any 3-dimensional shape? Not so fast, think about the sphere-like planet earth, it rotates around the axis through the north and south pole, but this means that points exactly at the north and south pole don’t actually move. This torus is special in this way. If you rotate it around an axis through the “doughnut hole,” then every point moves.

This observation was critical in a recent development of new techniques in quantum teleportation. If you imagine sending encoded pieces of data from one torus to another, the large surface area means there are many more points on which to place the data bits, and the nice axis of rotation makes these pieces easier to encode. The math blog and community forum Mathesia gives a nice down-to-earth explanation in donuts, math, and superdense teleportation of quantum information.

Ok, now I want a doughnut.

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Botanical Mathematicians

When I clicked on a blog post called “Bamboo Mathematicians,” I assumed it would be about the bamboo multiplication table recently cleaned up and analyzed by researchers at Tsinghua University in Beijing. Those bamboo strips, dating from approximately 305 BCE, contain the oldest known base 10 multiplication table.

A forest of mathematicians. Image: JFXie, via Flickr.

A forest of mathematicians. Image: JFXie, via Flickr.

But the “Bamboo Mathematicians” I clicked on was a post by Carl Zimmer, a science writer who specializes in biology and evolution, about bamboo plants with decades-long flowering cycles. He reports that researchers have developed mathematical models that explain how a bamboo forest ends up synchronizing to these long cycles. The main idea is that if some plants mutate to have a flowering cycle that is an integral multiple of the dominant flowering cycle, they will tend to outcompete the shorter-cycled plants. Over time, this has led to plants with 32-, 60-, and 120-year cycles, all products of small primes.

On the other hand, periodical cicadas favor larger primes: 13 and 17.This year, broods of both 13- and 17-year cicadas are scheduled to appear in the midwest and southeast US. Cicada Mania reports that they have started emerging in Illinois and should be around for about a month. The cicadas and the bamboo have long life cycles for similar reasons: by appearing at once, they flood the market, so to speak—their predators can’t eat all of them, so the species has a better chance of survival. Steve Mould has a nice Numberphile video about this predator satiation strategy. It’s interesting that the cicadas’ survival strategy led to (relatively) large prime numbers while the bamboo ended up with composite numbers with small prime factors. It’s interesting to think about the evolutionary factors that may have contributed to that difference.

Bamboo isn’t the only mathematical plant. Two years ago, there was a flurry of articles claiming that plants do math when they change their starch consumption at night. The Aperiodical mentioned it, and Christina Agapakis had a nice post about it at her blog, Oscillator

The plants in question aren’t spitting out numerical answers to word problems on their leaves, but doing normal plant stuff: using energy stored as starch at different rates depending on environmental conditions. Plants get their energy from sunlight, so at night the rate of starch consumption has to be smooth in order to maintain energy until dawn and prevent a “sugar crash.” The researchers found in a previous study that that plants will consume their starch almost completely every night and that the rate of consumption will stay mostly constant after “sunset,” regardless of whether the lights go out earlier or later than the plant “expects” based on their circadian rhythm. Based on these results, the researchers proposed a mathematical model whereby the plants are “dividing” the level of starch stores by the number of hours until dawn in order to determine the proper rate of consumption.

So plants can multiply, at least by small numbers, and divide! I wonder what other mathematical tasks they’ve been doing in secret.

Posted in Applied Math, Biomath | Tagged , , , , , , | Leave a comment

How To Optimize Summer Travel And Not Get Blown Up

The only thing more fun than traveling to a math conference is optimizing the route you'll take to get to it. Photo Jo@net via Flickr

The only thing more fun than traveling to a math conference is optimizing the route you’ll take to get there. Photo: Jo@net via Flickr

Every year I promise myself that I’ll just stay in one place for the summer, and every year that simply doesn’t happen. Today I’m posting from CIRM in Marseille, France. Next week I’m headed to Hong Kong to visit with a collaborator, and eventually will make my way back to the US for an IBL workshop in San Luis Obispo, CA, a conference to work on the LMFDB in Portland, OR, and MAA MathFest in Washington, D.C.. (I know, so many acronyms, life is tough.) All of this has gotten me thinking about the fun mathematical questions that come up in transportation and travel.

For travel by car, Laura McLay has some great posts on her blog Punk Rock Operations Research. She talks about the some statistics behind traffic jams, and why women are more likely to cause congestion (it’s not because we’re worse drivers, so wipe that smirk off your face). How to use Operations Research to optimize your search for a parking space. And in one post she answers that question we’ve all had at some point: how likely are you to actually blow yourself up pumping gas? (Not very.)

In OR By the Beach, Tallys Yunes blogs about traveling by air, discussing the apparent strategy behind the unrelenting and seemingly arbitrary gate changes at airports. Is there a better way to do this? In a similar turn, Michael Trick’s Operations Research Blog laments the annoying practice of overbooking hotels and discusses a more quantified approach to accommodating guests.

One undeniable downside of so much travel is the resulting carbon footprint. Particia Randall, who blogs for Reflections on Operations Research, writes about optimizing carbon emissions for her corporate clients. And while you are likely not bringing any sort of payload with you to your summer conferences, it is a good way to think about your own carbon footprint.

Do you have some favorite OR or transportation math blogs? Tweet them at me @extremefriday, I’d love to hear what you’ve got.

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Money, Money, Money

Ben Bernanke is blogging. (For some reason I find that funny: former Federal Reserve chairmen— they’re just like us!) In March, he started a blog at the Brookings Institution website. Several of the early posts are about explaining (and defending) the Fed’s low interest rates. He includes a response by Larry Summers to a post about “secular stagnation.” I think it would be interesting to see more different viewpoints on the blog.

Money. Image: Nick Eres, via flickr.

Money. Image: Nick Eres, via flickr.

Like many mathematicians, I have some of the tools to understand finance and economics, but I’m naive about both subjects. After reading some of Bernanke’s blog, I wanted to look at some math blogs that focus on finance. I’ve had the Mathematical Investor in my feed for a while. It’s written by David H. Bailey, Jonathan M. Borwein, Marcos Lopez de Prado, and Qiji Jim Zhu and is a result of their “growing concern with the usage of less-than-fully rigorous mathematical and statistical methodologies in the financial/investment world.”

Recent posts have been about the 2010 “flash crash,” conflicted financial advice, and an assessment of 2014 market predictions. The authors are especially concerned about “backtest overfitting,” which is basically the error of making investment decisions that are based too heavily on historical data; one of their posts announces an online tool that demonstrates the problem.

Cathy O’Neil is my other main source of blog posts about finance, and I’m eagerly awaiting Weapons of Math Destruction, her forthcoming book about big data and the dangers it poses to democracy.

I poked around for some other financial math blogs and stumbled on Fermat’s Last Spreadsheet, a blog that hasn’t been updated in a while but has posts on coding, normal subgroups, and poker in addition to the main focus on fixed-income trading. Fermat’s Last Spreadsheet also introduced me to Magic, Maths and Money by Timothy Johnson of Heriot-Watt University. He concentrates on moral/ethical aspects of finance and the recent financial crisis.

Do you have any recommendations for financial math blogs?

Posted in Applied Math | Tagged , , | 1 Comment

Turns Out You Can Be Diverse and Segregated At the Same Time

A swirly-whirly picture of diversity and segregation in the northeastern USA.  But does this tell the whole story? Courtesy of The Racial Dot Map

A swirly-whirly picture of diversity and segregation in the northeastern USA. But does this tell the whole story? Courtesy of The Racial Dot Map

Well, it’s official, I’m an unrelenting fangirl for Dustin Cable’s Racial Dot Map and everything it stands for. If you’re not yet familiar, it’s one of the coolest data visualization projects to come out of the census data. The map does the following simple thing: every person in the country is represented by a dot on the map, and every dot has a color based on the person’s race. Black, white, asian, hispanic, and other.

The map, a non-trivial feat in data handling, paints a beautiful picture of the racial breakdown of the USA with little commentary beyond what you see in front of you. But it is a rich source for discussion, especially if you are someone who likes to analyze statistical trends, and let’s be real, who doesn’t love a good regression line?

Most recently, Nate Silver on the data science blog FiveThirtyEight, coined the index of dissimilarity, a measure of diversity vs. segregation of American cities with the racial dot map as inspiration. The basic idea is that a city can be simultaneous diverse and segregated. How does this happen? On a city-wide level there can be many people of all different races living in the city, but on a neighborhood level they night be totally distinct.

The beauty of this measure is that it is totally quantifiable, by counting people in cities/neighborhoods and then counting the percentage of their neighbors that are of a different race. If that number is high on both the city and neighborhood level, that means city is both diverse and non-segregated. I always find it satisfying when you can really apply a metric to these types of questions. You can check out the search function in post to see how your city stacks up. Pittsburgh — the Paris of Appalachia that I call home — has a citywide diversity of 50%, which basically means that half of the people in the city are different from you. But the neighborhood diversity is only 35.5%, which means that on the more local scale, things look pretty segregated.

It’s fun to see how different cities and regions of the country look under this metric. As always, a good handling of data and statistics is a great way to start a conversation about the deeper implications that this has for out communities and lives.

Posted in Applied Math, Data Science, Statistics | Tagged , , , , , | Leave a comment

Prepare to Be Nerdsniped

You have a lot of bags, and you want to store them by stuffing all of them into one of the bags. For n bags, how many ways are there to do this?

Nerdsniping. Image: xkcd by Randall Munroe.

Nerdsniping. Image: xkcd by Randall Munroe.

I’ve spent a good amount of time thinking about that question recently, and I figured I’d inflict it on you as well. It’s not my fault, though. Sam Shah, a high school math teacher, wrote about the question on his blog, Continuous everywhere but differentiable nowhere, after Matt Enlow shared it. I’ve been reading Shah’s blog since 2012, when he posted a really interesting calculus class project investigating the Gini index, a measure of wealth inequality.

The stuffing sacks problem appears in the good math problems tag on his blog, which certainly lives up to its name. Plenty of nerdsniping there. Some of the problems are fun brain teasers like stuffing sacks, but some of them are more geared towards making sure his math class is about thinking instead of formulas. For example, what’s the derivative of log(log(sin(x)))? It’s easy to use the chain rule to come up with an answer without stopping to consider the small detail that log(log(sin(x))) doesn’t make sense for any real numbers. Oops.

In addition to puzzles that will help you pass the time waiting for the bus, Shah often writes about how he cultivates deep understanding in the classroom. He recently wrote about the way he has students make conjectures at the beginning of the semester and then the eventual payoff when they finally have the tools to prove the conjectures. Outside of the classroom, he helped launch a student math and science journal at his school. Even if you don’t teach high school, a trip to his blog will probably give you some ideas for your classroom or just for talking about math with people.

So how many ways are there to stuff sacks? It’s not hard to figure out the values for the first few numbers n. But holy overcounting, Batman, things get complicated as n increases! There’s an OEIS sequence that will get you the answer, but that would take all the fun away.

Posted in K-12 Mathematics, Math Education | Tagged , , , , | 2 Comments

Caring For The Community

Photo courtesy of Scott Traczyk, via Flickr CC

Photo courtesy of Scott Traczyk, via Flickr CC

As academic mathematicians, we spend a great deal of our days performing deeds of service to the mathematical community. Editing papers, organizing workshops, contributing to open-source software initiatives. One could even argue that it is out of sheer benevolence to the mathematical community that we even write papers at all. In a blog I just discovered by mathematician and Clay Prize winner, Michael Harris, the dynamics of this community get a very thorough and thoughtful analysis.

Mathematics without Apologies is, in the words of Michael Harris, “an unapologetic guided tour of the mathematical life.” Harris takes us on a tour of some of the very compelling — and highly fraught — issues that arise when you start to think about the mathematical profession, and its community, from a sociological point of view. Michael Harris is a professor of mathematics at Université Paris-Diderot and Columbia University.

In a multi-part post, Harris explains the much debated Elsevier boycott initiated by fellow mathematician and blogger, Tim Gowers. The boycott was proposed in 2012 in opposition to the extremely high price tag the Elsevier puts on its research papers that are essentially donated to them. Harris addresses some of the alternative publishing models that people have proposed. One idea is that the math community should be doing our own reviewing and publishing through Yelp-style real time user reviews. Harris is not so keen on this idea.

I don’t like the fact that unpaid Customer Reviews have undermined professions (as Tom Waits pointed out). I’m wary of replacing a practice that has evolved over several centuries to serve the needs of the profession by a model of sociability that in less than a decade has led to the creation of massive fortunes and an enormous shift of power with practically no democratic oversight.

What I really appreciate about Harris’ blog is that it is very mindful of the community aspect of the mathematical profession. In one post he writes about Mathematics as a gift community, where we are defined by the gifts that we bring to it. He reflects on the amount of time and effort that it takes to be a good community member, and that it should be in all of our best interest to improve our community through deeds of service. I suppose this is true across all disciplines in academia, but it’s nice to think of things that way. It makes me happy to mow my mathematical lawn and take out the trash to help my whole mathematical neighborhood look a bit more sparkly and clean.

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How to Celebrate Math Poetry Month

April is the intersection of Math Awareness Month and National Poetry Month, so obviously we are all celebrating Math Poetry Month. Some of my favorite posts on Roots of Unity have been about poetry. This year, I posted “In Praise of Fractals” by Emily Grosholz. Last year, I shared Sandra Beasley’s poem “Unit of Measure,” in which all is measured by the standard of the capybara. And the year before, I wrote about Sandra DeLozier Coleman’s poem “Group: n. collection, set, assembly.

Christine Rueter, who writes the astropoetry blog Tychogirl (featured here last November), is celebrating National Poetry Month by writing a poem a day. I’m partial to “The LM (after Blake)” and “color leaked in,” which was based on the first color images of Pluto and Ceres from the New Horizons spacecraft.

color leaked in by Christine Rueter.

Image: Christine Rueter.

I’ve written previously about JoAnne Growney’s blog, Intersections–Poetry with Mathematics, but it is still my favorite math poetry source and a great place to visit this April. If you live in the DC area, you can even see Growney at a poetry reading tonight (if it’s still April 20 when you read this)!

Earlier this month, Growney shared a link to an interview with Enriqueta Carrington at the Art Works blog. Carrington is a mathematician at Rutgers and a poet in both Spanish and English. She currently has an NEA translation fellowship to support her work translating the work of Sor Juana Inés de la Cruz, a 17th-century poet from New Spain (now Mexico). In the interview, Carrington talks about how mathematics and poetry are linked for her and about the challenges of translation.

The other math poetry blog I have in my feed, appropriately named Mathematical Poetry, is by Kaz Maslanka. He primarily creates and shares visual poetry that uses mathematical symbols to express relationships between ideas. Two common types of these poems are orthogonal space poems and congruent triangle or proportional poems, where a fairly simple equation expressing proportion can be interpreted in several subtly different ways. (There is a whole blog devoted to proportional poetry here.) I also recommend his post about various types of mathematical poetry. In addition to his own poetry, he shares works by other visual mathematical poets. While I am more comfortable with verbal poetry, it’s interesting to see the way words, pictures, and mathematical symbols flow together in these works. Maslanka’s most recent posts share the sad news of the passing of Bob Grumman, another visual mathematical poet who wrote a series called M@h*(pOet)?ica for Scientific American a few years ago.

Do you have a favorite poem or poet inspired by math?

Posted in Mathematics and the Arts | Tagged , | 2 Comments