# SquareRoots, Concave States, and Sea Ice

Welcome to this week’s Math Munch!

The most epic Pi Day of the century will happen in just a few weeks: 3/14/15! I hope you’re getting ready. To help you get into the spirit, check out these quilts.

 American Pi. African American Pi.

There’s an old joke that “pi is round, not square”—a punchline to the formula for the area of a circle. But in these quilts, we can see that pi really can be square! Each quilt shows the digits of pi in base 3. The quilts are a part of a project called SquareRoots by artist and mathematician John Sims.

John Sims.

There’s lots more to explore and enjoy on John’s website, including a musical interpretation of pi and some fractal trees that he has designed. John studied mathematics as an undergrad at Antioch College and has pursued graduate work at Wesleyan University. He even created a visual math course for artists when he taught at the Ringling College of Art and Design in Florida.

I enjoyed reading several articles (1, 2, 3, 4) about John and his quilts, as well as this interview with John. Here’s one of my favorite quotes from it, in response to “How do you begin a project?”

It can happen in two ways. I usually start with an object, which motivates an idea. That idea connects to other objects and so on, and, at some point, there is a convergence where idea meets form. Or sometimes I am fascinated by an object. Then I will seek to abstract the object into different spatial dimensions.

Cellular Forest and Square Root of a Tree, by John Sims.

You can find more of John’s work on his YouTube channel. Check out this video, which features some of John’s music and an art exhibit he curated called Rhythm of Structure.

Next up: Some of our US states are nice and boxy—like Colorado. (Or is it?) Other states have very complicated, very dent-y shapes—way more complicated than the shapes we’re used to seeing in math class.

Which state is the most dent-y? How would you decide?

West Virginia is pretty dent-y. By driving “across” it, you can pass through many other states along the way.

The mathematical term for dent-y is “concave”. One way you might try to measure the concavity of a state is to see how far outside of the state you can get by moving in a straight line from one point in it to another. For example, you can drive straight from one place in West Virginia to another, and along the way pass through four other states. That’s pretty crazy.

But is it craziest? Is another state even more concave? That’s what this study set out to investigate. Click through to find out their results. And remember that this is just one way to measure how concave a state is. A different way of measuring might give a different answer.

Awesome animal kingdom gerrymandering video!

This puzzle about the concavity of states is silly and fun, but there’s more here, too. Thinking about the denty-ness of geographic regions is very important to our democracy. After all, someone has to decide where to draw the lines. When regions and districts are carved out in a way that’s unfair to the voters and their interests, that’s called gerrymandering.

Karen Saxe.

To find out more about the process of creating congressional districts, you can listen to a talk by Karen Saxe, a math professor at Macalester College. Karen was a part of a committee that worked to draw new congressional districts in Minnesota after the 2010 US Census. (Karen speaks about compactness measures starting here.)

Recently I ran across an announcement for a conference—a conference that was all about the math of sea ice! I never grow tired of learning new and exciting ways that math connects with the world. Check out this video featuring Kenneth Golden, a leading mathematician in the study of sea ice who works at the University of Utah. I love the line from the video: “People don’t usually think about mathematics as a daring occupation.” Ken and his team show that math can take you anywhere that you can imagine.

Bon appetit!

Reflection sheet – SquareRoots, Concave States, and Sea Ice

# Nice Neighbors, Spinning GIFs, and Breakfast

A minimenger.

Welcome to this week’s Math Munch!

Math projects are exciting—especially when a whole bunch of people work together. One example of big-time collaboration is the GIMPS project, where anyone can use their computer to help find the next large prime number. Another is the recent MegaMenger project, where people from all over the world helped to build a giant 3D fractal.

But what if I told you that you can join up with others on the internet to discover some brand-new math by playing a webgame?

Chris Staecker is a math professor at Fairfield University. This past summer he led a small group of students in a research project. Research Experiences for Undergraduates—or REUs, as they’re called—are summer opportunities for college students to be mentored by professors. Together they work to figure out some brand-new math.

The crew from last summer’s REU at Fairfield. Chris is furthest in the back.

The irreducible digital images containing 1, 5, 6, and 7 “chunks”.

Chris and his students Jason Haarmann, Meg Murphy, and Casey Peters worked on a topic in graph theory called “digital images”. Computer images are made of discrete chunks, but we often want to make them smaller—like with pixel art. So how can we make sure that we can make them smaller without losing too much information? That’s an important problem.

Now, the pixels on a computer screen are in a nice grid, but we could also wonder about the same question on an arbitrary connected network—and that’s what Chris, Jason, Meg, and Casey did. Some networks can be made smaller through one-step “neighbor” moves while still preserving the correct connection properties. Others can’t. By the end of the summer, the team had come up with enough results about digital images with up to eight chunks to write about them in a paper.

To help push their research further, Chris has made a webgame that takes larger networks and offers them as puzzles to solve. Here’s how I solved one of them:

See how the graph “retracts” onto itself, just by moving some of the nodes on top of their neighbors? That’s the goal. And there are lots of puzzles to work on. For many of them, if you solve them, you’ll be the first person ever to do so! Mathematical breakthrough! Your result will be saved, the number at the bottom of the screen will go up by one, and Chris and his students will be one step closer to classifying unshrinkable digital images.

Starting with the tutorial for Nice Neighbors is a good idea. Then you can try out the unsolved experimental puzzles. If you find success, please let us know about in the comments!

Do you have a question for Chris and his students? Then send it to us and we’ll try to include it in our upcoming Q&A with them.

Next up: you probably know by now that at Math Munch, we just can’t get enough of great mathy gifs. Well, Sumit Sijher has us covered this week, with his Tumblr called archery.

Here are four of Sumit’s gifs. There are plenty more where these came from. This is a nice foursome, though, because they all spin. Click to see the images full-sized!

 How many different kinds of cubes can you spot? This one reminds me of the Whitney Music Box. Whoa. Clockwise or counterclockwise?

I really appreciate how Sumit also shares the computer code that he uses to make each image. It gives a whole new meaning to “show your work”!

Through Sumit’s work I discovered that WolframAlpha—an online calculator that is way more than a calculator—has a Tumblr, too. By browsing it you can find some groovy curves and crazy estimations. Sumit won an honorable mention in Wolfram’s One-Liner Competition back in 2012. You can see his entry in this video.

And now for the most important meal of the day: breakfast. Mathematicians eat breakfast, just like everyone else. What do mathematicians eat for breakfast? Just about any kind of breakfast you might name. For some audio-visual evidence, here’s a collection of sound checks by Numberphile.

Sconic sections. Yum!

If that has you hungry for a mathematical breakfast, you might enjoy munching on some sconic sectionsa linked-to-itself bagel, or some spirograph pancakes.

Bon appetit!

# Grothendieck, Circle Packing, and String Art

Welcome to this week’s Math Munch!

This week brought some sad news to the mathematical world. Alexander Grothendieck, known by many as the greatest mathematician of the past century, passed away on November 13th. You may not have heard of him, but many mathematicians say that the work he did in math was as influential as the work Albert Einstein did in physics.

One of the things that make Grothendieck so interesting is, of course, the math he did. Grothendieck was always very creative. When he was in high school, he preferred to do math problems he made up on his own over the problems assigned by his teacher. “These were the book’s problems, and not my problems,” he said.

When he was young, inspired by some gaps he found in definitions in his geometry book about measuring lengths and areas, Grothendieck re-created some of the most important mathematical ideas of the beginning of the twentieth century. Maybe this sounds silly to you– why re-invent something that’s already been done? But, to Grothendieck, the most important part was that he’d done the whole thing by himself. He’d figured out something in his own way. He later wrote that this experience showed him what being a mathematician was like:

Without having been told, I nevertheless knew ‘in
my gut’ that I was a mathematician: someone who
‘does’ math, in the fullest sense of the word…

During his years as a mathematician, Grothendieck worked on connecting different parts of math (a project requiring a lot of creativity)– algebra, geometry, topology, and calculus, among others.

Alexander Grothendieck as a kid

The other thing that makes Grothendieck so interesting is his life story. As a kid, Grothendieck and his parents fled from Germany to France to escape the Nazis. As an adult, Grothendieck spoke out strongly for peace. He used his fame to take a stand against the wars of the second half of the twentieth century. This eventually led him to step away from the world of mathematicians– which many regretted. But he left behind work that changed all of mathematics for the better.

If you’d like to learn more about Grothendieck’s fascinating life and work, check out this great (but long) article from the American Mathematical Society. This article provides a shorter history, including a great statement Grothendieck made about his feelings on creativity in mathematics. Grothendieck was a very private person, so many of his mathematical writings aren’t available online– but the Grothendieck Circle has done their best to collect everything written about him.

A pretty circle packing

Next up, a little something for you to play with! We were studying circle packing problems in one of my classes this week. Did you know that you can fit exactly six circles snugly around another circle of the same size? But, if you try to fit circles snugly around a circle twice as large, it doesn’t work? I wonder why that is…

I did it!

Anyway, my class inspired me to look for a circle packing game– and I found one! In this game, simply called Circle Packing, you have to fit all of the smaller circles into the larger circle– without any of them touching! It’s pretty tricky, and really fun.

Finally, the Math Munch team got something wonderful in the mail (email, I guess) this week! Math art made by Julia Dweck’s 5th grade math class! Julia’s class has been working hard to make parabolic curve string art– curves made by drawing (or stringing, in this case) many, many straight lines. They plotted each curve precisely before stringing it, to make sure it was both mathematically and artistically perfect. The pieces they made are so creative and beautiful. We’re proud to feature them on our site!

You can see the whole collection of string art pieces made by Julia’s class on our Readers’ Gallery String Art page. And, want to know more about how the 5th graders made their String Art? Have any questions for Julia and her students about their love of math and the connections they see between math and art? Write your questions here and we’ll send them to Julia’s students!

Have any math art of your own? Send it to mathmunchteam@gmail.com, and we’ll post it in the Readers’ Gallery!

Bon appetit!

# George Washington, Tessellation Kit, and Langton’s Ant

Welcome to this week’s Math Munch!

What will you do with your math notebook at the end of the school year? Keep it as a reference for the future? Save it as a keepsake? Toss it out? Turn it into confetti? Find your favorite math bits and doodles and make a collage?

Lucky for us, our first president kept his math notebooks from when he was a young teenager. And though it’s passed through many hands over the years—including those of Chief Justice John Marshall and the State Department—it has survived to this day. That’s right. You can check out math problems and definitions copied out by George Washington over 250 years ago. They’re all available online at the Library of Congress website.

Or at least most of them. They seem to be out of order, with a few pages missing!

That’s what mathematician and math history detective Fred Rickey has figured out. Fred has long been a fan of math history. Since he retired from the US Military Academy in 2011, Fred has been able to pursue his historical interests more actively. Fred is currently studying the Washington cypher books to help prepare a biography about Washington’s boyhood years. You can see two papers that Fred has co-authored about Washington’s mathematics here.

Fred writes:

Washington valued his cyphering books and kept them as a ready source of reference for the rest of his life. This would seem to be particularly true of his surveying studies.

Surveying played a big role in Washington’s career, and math is important for today’s surveyors, too.

Do you have a question for Fred about the math that George Washington learned? Send it to us and we’ll try to include it in our upcoming Q&A with Fred!

A tessellation, by me!

Next up, check out this Tessellation Kit. It was made by Nico Disseldorp, who also made the geometry construction game we featured recently. The kit is a lot of fun to play with!

One thing I like about this Tessellation Kit is how it’s discrete—it deals with large chunks of the screen at a time. This restriction make me want to explore, because it give me the feeling that there are only so many possible combinations.

I’m also curious about the URL for this applet—the web address for it. Notice how it changes whenever you make a change in your tessellation? What happens when you change some of those letters and numbers—like bababaaaa to bababcccc? Interesting…

For another fun applet, check out this doodling ant:

Langton’s Ant.

Langton’s Ant is following a simple set of rules. In a white square? Turn right. In a black square? Turn left. And switch the color of the square that you leave. This ant is an example of a cellular automaton, and we’ve seen several of these here on Math Munch before. This one is different from others because it changes just one square at a time, and not the whole screen at once.

Breaking out of chaos.

There’s a lot that is unknown about Langton’s ant, and it has some mysterious behavior. For example, after thousands of steps of seeming randomness, the ant goes into a steady pattern, paving a highway out to infinity. What gives? Well, you can try out some patterns of your own in the applets on the Serendip website. (previously). And you can read some amusing tales—ant-ecdotes?—about Langton’s ant in this lovely article.

I learned about Langton’s Ant from Richard Evan Schwartz in our new Q&A. In the interview, Rich shares his thoughts about computers, art, what to pursue in life, and of course: Really Big Numbers.

Check it out, and bon appetit!

# Fields Medal, Favorite Numbers, and The Grapes of Math

Welcome to this week’s Math Munch! And, if you’re a student or teacher, welcome to a new school year!

One of the most exciting events in the world of math happened this August– the awarding of the Fields Medal! This award honors young mathematicians who have already done awesome mathematical work and who show great promise for the future. It also only happens every four years, at the beginning of an important math conference called the International Congress of Mathematicians, so it’s a very special occasion when it does!

Maryam Mirzakhani, first woman ever to win a Fields Medal

This year’s award was even more special than usual, though. Not only were there four winners (more than the usual two or three), but one of the winners was a woman!

Now, if you’re like me, you probably heard about the Fields Medal and thought, “There’s no way I’ll understand the math that these Field Medalists do.” But this couldn’t be more wrong! Thanks to these great articles from Quanta Magazine, you can learn a lot about the super-interesting math that the Fields Medalists study– and why they study it.

Manjul Bhargava

One thing you’ll immediately notice is that each Fields Medalist has non-math interests that inspire their mathematical work. Take Manjul, for instance. When he was a kid, his grandfather introduced him to Sanskrit poetry. He was fascinated by the patterns in the rhythms of the poems, and the number patterns that he found inspired him to study the mathematics of number patterns– number theory!

But, don’t just take my word for it– you can read all about Manjul and the others in these great articles! And did I mention that they come with videos about each mathematician?

… What’s your favorite number? Is it 7? If it is, then you’re in good company! Alex polled more than 30,000 people about their favorite number, and the most popular was 7. But why? What’s so special about 7? Here’s why Alex thinks 7 is such a favorite:

Why do you like your favorite number? People gave Alex all kinds of different reasons. One woman said about 3, her favorite number, “3 wishes. On the count of 3. 3 little pigs… great triumvirates!” Alex made these questions the topic of the first chapter of his new book, The Grapes of Math. (Get the reference?) In this book, Alex shares many curious ways that math appears in our world. Did you know that a weird pattern in numbers can be used to catch criminals? Or that the Game of Life, a simple computer program, shares surprisingly many characteristics with real life? These are only a few of the hundreds of topics Alex covers in his book. Whether you’re a math whiz or a newbie, you’ll learn something new on every page.

Alex currently writes about math for The Guardian in a blog called, “Alex’s Adventures in Numberland”— but he also loves and writes about soccer (or futbol, as it’s called in his native Brazil)! He even wrote a few articles for his blog about math and soccer.

Do you have any questions for Alex? (About math, soccer, or their intersection?) Write them here and you might find them featured in our interview with Alex!

Good writing about math is hard to find. If you’ve ever picked up a standard math textbook, you’ll know what I mean. But reading something fascinating, that grabs your interest from the first page and leads you through the most complex ideas like they’re as natural as anything you’ve observed, is a great way to learn. The Grapes of Math and “Alex’s Adventures in Numberland” do just that. Give them a go!

Bon appetit!

# Girls’ Angle, Spiral Tilings, and Coins

Welcome to this week’s Math Munch!

Girls’ Angle is a math club for girls. Since 2007 it has helped girls to grow their love of math through classes, events, mentorship, and a vibrant mathematical community. Girls’ Angle is based in Cambridge, Massachusetts, but its ideas and resources reach around the world through the amazing power of the internet. (And don’t you worry, gentlemen—there’s plenty for you to enjoy on the site as well.)

Amazingly, the site contains an archive of every issue of Girls’ Angle Bulletin, a wonderful bimonthly journal to “foster and nurture girls’ interest in mathematics.” In their most recent issue, you’ll find an interview with mathematician Karen E. Smith, along with several articles and puzzles about balance points of shapes.

There’s so much to dig into at Girls’ Angle! In addition to the Bulletins, there are two pages of mathematical videos. The first page shares a host of videos of women in mathematics sharing a piece of math that excited them when they were young. The most recent one is by Bridget Tenner, who shares about Pick’s Theorem. The second page includes several videos produced by Girls’ Angle, including this one called “Summer Vacation”.

Girls’ Angle can even help you buy a math book that you’d like, if you can’t afford it. For so many reasons, I hope you’ll find some time to explore the Girls’ Angle site over your summer break. (And while you’ve got your explorer’s hat on, maybe you’ll tour around Math Munch, too!)

I did a Google search recently for “regular tilings.” I needed a few quick pictures of the usual triangle, square, and hexagon tilings for a presentation I was making. As I scrolled along, this image jumped out at me:

What is that?! It certainly is a tiling, and all the tiles are the “same”—even if they are different sizes. Neat!

Clicking on the image, I found myself transported to a page all about spiral tilings at the Geometry Junkyard. The site is a whole heap of geometrical odds and ends—and a place that I’ve stumbled across many times over the years. Here are a few places to get started. I’m sure you’ll enjoy poking around the site to find some favorite “junk” of your own.

 Spirals Circles & spheres Coloring

Last up this week, you may have seen this coin puzzle before. Can you make the triangle point downwards by moving just three pennies?

There are lots of variants of this puzzle. You can find some in an online puzzle game called Coins. In the game you have to make arrangements of coins, but the twist is that you can only move a coin to a spot where would it touch at least two other coins. I’m enjoying playing Coins—give it a try!

I solved this Coins puzzle in four moves. Can you? Can you do better?

That’s it for this week’s Math Munch. Bon appetit!

# Fullerenes, Fibonacci Walks, and a Fourier Toy

Welcome to this week’s Math Munch!

Stan and James

Earlier this month, neuroscientists Stan Schein and James Gayed announced the discovery of a new class of polyhedra. We’ve often posted about Platonic solids here on Math Munch. The shapes that Stan and James found have the same symmetries as the icosahedron and dodecahedron, and they also have all equal edge lengths.

One of Stan and James’s shapes, made of equilateral pentagons and hexagons.

These new shapes are examples of fullerenes, a kind of shape named after the geometer, architect, and thinker Buckminster Fuller. In the 1980s, chemists discovered that molecules made of carbon can occur in polyhedral shapes, both in the lab and in nature. Stan and James’s new fullerenes are modifications of some existing shapes first described in 1937 by Michael Goldberg. The faces of Goldberg’s shapes were warped, not flat, and Stan and James showed that flattening can be achieved—thus turning Goldberg’s shapes into true polyhedra—while also having all equal edge lengths. There’s great coverage of Stan and James’s discovery in this article at Science News and a fascinating survey of the media’s coverage of the discovery by Adam Lore on his blog. Adam’s post includes an interview with Stan!

Next up—how much fun is it to find a fractal that’s new to you? That happened to me recently when I ran across the Fibonacci word fractal.

A portion of a Fibonacci word curve.

Fibonacci “words”—really just strings of 0’s and 1’s—are constructed kind of like the numbers in the Fibonacci sequence. Instead of adding numbers previous numbers to get new ones, we link up—or “concatenate”—previous words. The first few Fibonacci words are 1, 0, 01, 010, 01001, and 01001010. Do you see how new words are made out of the two previous ones?

Here’s a variety of images of Fibonacci word fractals, and you can find more details about the fractal in this article. The infinite Fibonacci word has an entry at the OEIS, and you can find a Fibonacci word necklace on Etsy. Dale Gerdemann, a linguist at the University of Tübingen, has a whole series of videos that show off patterns created out of Fibonacci words. Here is one of my favorites:

Last but not least this week, check out this groovy applet!

Lucas’s applet showing the relationship between epicycles and Fourier series

A basic layout of Ptolemy’s model, including epicycles.

Sometime around the year 200 AD, the astronomer Ptolemy proposed a way to describe the motion of the sun, moon, and planets. Here’s a video about his ideas. Ptolemy relied on many years of observations, a new geometrical tool we call “trigonometry”, and a lot of ingenuity. He said that the sun, moon, and planets move around the earth in circles that moved around on other circles—not just cycles, but epicycles. Ptolemy’s model of the universe was incredibly accurate and was state-of-the-art for centuries.

Joseph Fourier

In 1807, Joseph Fourier turned the mathematical world on its head. He showed that periodic functions—curves with a repeated pattern—can be built by adding together a very simple class of curves. Not only this, but he showed that curves created in this way could have breaks and gaps even though they are built out of continuous curves called “sine” and “cosine”. (Sine and cosine are a part of the same trigonometry that Ptolemy helped to found.) Fourier series soon became a powerful tool in mathematics and physics.

A Fourier series that converges to a discontinuous function.

And then in the early 21st century Lucas Vieira created an applet that combines and sets side-by-side the ideas of Ptolemy and Fourier. And it’s a toy, so you can play with it! What cool designs can you create? We’ve featured some of Lucas’s work in the past. Here is Lucas’s short post about his Fourier toy, including some details about how to use it.

Bon appetit!