Recently I’ve been running across tons of neat, slick math applets. I feel like they all go together. What do they have in common? Maybe you’ll be able to tell me.

Ready for some more? Because with these sorts of visualizations, Dan Anderson has been on fire lately. Dan is a high school math teacher in New York state. He and his students had fifteen minutes of fame last year when they investigated whether or not Double Stuf Oreos really have double the stuf.

Here is Dan’s page on OpenProcessing. (Processing is the computer language in which Dan programs his applets.) And check out the images and gifs on Dan’s Tumblr. Here’s a sampling!

That’s a chunk of math to chew on already, but we’re just getting started! Next up, check out the space-filling artwork of John Shier.

John’s artwork places onto the canvas shapes of smaller and smaller sizes. Notice that the circles below fill in gaps, but they don’t touch each other, they way circles do in an Apollonian gasket.

You can learn more about John’s space-filling shapes on this page and find further details in this paper.

Thanks for making us this sweet banner, John!

Last up this week, head to this site to watch an awesome trailer of a film about Julia Robinson. The short clip focuses on Julia’s work on Hilbert’s tenth problem. It includes interviews with a number of people who knew Julia, including her sister Constance Reid. Constance wrote extensively about mathematics and mathematicians. I’ve read her biography of Hilbert and can highly recommend it. You can read more about Julia and Constance here and here.

Julia Robinson

Julia’s sister, Constance Reid

Julia and Constance as young girls.

You might enjoy visiting the site of the Julia Robinson Mathematics Festival. Check to see if a festival will be hosted in your area sometime soon, or find out how you can run one yourself!

With May wrapped up and June getting started, I hope you have a lot of math to look forward to this summer. Bon appetit!

All of our munches this week come from the recent tweets of mathematician, author, and friend of the blog Steven Strogatz. Steve works at Cornell University as an applied mathematician, tackling questions like “If people shared taxis with strangers, how much money could be saved?” and “What caused London’s Millennium Bridge to wobble on its opening day?”

On top of his research, Steve is great at sharing math with others. (This week I learned one great piece of math from him, and then another, and suddenly there was a very clear theme to my post!) Steve has written for the New York Times and was recently awarded the Lewis Thomas Prize as someone “whose voice and vision can tell us about science’s aesthetic and philosophical dimensions, providing not merely new information but cause for reflection, even revelation.”

This Saturday, Steve will be presenting at the first-ever National Math Festival. The free and fun main event is at the Smithsonian in Washington, DC, and there are related math events all around the country this weekend. Check and see if there’s one near you!

Here are a few pieces of math that Steve liked recently. I liked them as well, and I hope you will, too.

First up, check out this lovely image:

It appeared on Numberplay and was created by Hamid Naderi Yeganeh, a student at University of Qom in Iran. Look at the way the smaller and smaller tiles fit together to make the design. It’s sort of like a rep-tile, or this scaly spiral. And do those shapes look familiar? Hamid was inspired by the shapes of the continents of Africa and South America (if you catch my continental drift). Maybe you can create your own Pangaea-inspired tiling.

If you think that’s cool, you should definitely check out Numberplay, where there’s a new math puzzle to enjoy each week!

Next, up check out the Math Explorers’ Club, a collection of great math activities for people of all ages. The Club is a project of Cornell University’s math department, where Steve teaches.

The first item every sold on the auction site eBay. Click through for the story!

One of the bits of math that jumped out to me was this page about auctions. There’s so much strategy and scheming that’s involved in auctions! I remember being blown away when I first learned about Vickrey auctions, where the winner pays not what they bid but what the second-highest bidder did!

And finally this week: have you ever wondered “What will I ever use math for?” Well, SIAM—the Society for Industrial and Applied Mathematics—has just the video for you. They asked people attending one of their meetings to finish the sentence, “I use math for…”. Here are 32 of their answers in just 60 seconds.

Thanks for sharing all this great math, Steve! And bon appetit, everyone!

Painting by Renée Othot for Simon Plouffe’s birthday.

Welcome to this week’s Math Munch!

It’s here—the Pi Day of the Century happens on Saturday: 3-14-15!

How will you celebrate? You might check to see if there are any festivities happening in your area. There might be an event at a library, museum, school, or university near you.

John Conway at the pi recitation contest in Princeton.

There’s a huge celebration here in Princeton—in part because Pi Day is also Albert Einstein’s birthday, and Albert lived in Princeton for the last 22 years of his life. One event involves kids reciting digits of pi and and is hosted by John Conway and his son, a two-time winner of the contest. I’m looking forward to attending! But as has been noted, memorizing digits of pi isn’t the most mathematical of activities. As Evelyn Lamb relays,

I do feel compelled to point out that besides base 10 being an arbitrary way of representing pi, one of the reasons I’m not fond of digit reciting contests is that, to steal an analogy I read somewhere, memorizing digits of pi is to math as memorizing the order of letters in Robert Frost’s poems is to literature. It’s not an intellectually meaningful activity.

I haven’t memorized very many digits of pi, but I have memorized a digit of pi that no one else has. Ever. In the history of the world. Probably no one has ever even thought about this digit of pi.

And you can have your own secret digit, too—all thanks to Simon Plouffe‘s amazing formula.

Simon’s formula shows that pi can be calculated chunk by chunk in base 16 (or hexadecimal). A single digit of pi can be plucked out of the number without calculating the ones that come before it.

The discovery of this formula came as a surprise. For centuries it had been assumed that there was no way to compute the nth digit of π without calculating all of the preceding n − 1 digits.

Check out some of Simon’s math art!

Simon is a mathematician who was born in Quebec. In addition to his work on the digits of irrational numbers, he also helped Neil Sloane with his Encyclopedia of Integer Sequences, which soon online and became the OEIS (previously). Simon is currently a Trustee of the OEIS Foundation.

There is a wonderful article by Simon and his colleagues David Bailey, Jonathan Borwein, and Peter Borwein called The Quest for Pi. They describe the history of the computation of digits of pi, as well as a description of the discovery of their digit-plucking formula.

According to the Guinness Book of World Records, the most digits that someone has memorized and recited is 67,890. Unofficial records go up to 100,000 digit. So just to be safe, I’ve used an algorithm by Fabrice Bellard based on Simon’s formula to calculate the 314159th digit of pi. (Details here and here.) No one in the world has this digit of pi memorized except for me.

Ready to hear my secret digit of pi? Lean in and I’ll whisper it to you.

The 314159th digit of pi is…7. But let’s keep that just between you and me!

And just to be sure, I used this website to verify the 314159th digit. You can use the site to try to find any digit sequence in the first 200 million digits of pi.

Aziz & Peter’s patterns.

Next up: we met Aziz Inan in last week’s post. This week, in honor of Pi Day, check out some of the numerical coincidences Aziz has discovered in the early digits in pi. Aziz and his colleague Peter Osterberg wrote an article about their findings. By themselves, these observations are nifty little patterns. Maybe you’ll find some more of your own. (This kind of thing reminds me of the Strong Law of Small Numbers.) As Aziz and Peter note at the end of the article, perhaps the study of such little patterns will one day help to show that pi is a normal number.

And last up this week, to get your jam on as Saturday approaches, here’s the brand new Pi Day Anthem by the recently featured John Sims and the inimitable Vi Hart.

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.

Squares and circles are pretty different. Squares are boxy and have their feet firmly on the ground. Circles are round and like to roll all over the place.

Superellipses.

Since they’re so different, people have long tried to bridge the gap between squares and circles. There’s an ancient problem called “squaring the circle” that went unsolved for thousands of years. In the 1800s, the gap between squares and circles was explored by Gabriel Lamé. Gabriel invented a family of curves that both squares and circles belong to. In the 20th century, Danish designer Piet Hein gave Lamé’s family of curves the name superellipses and used them to lay out parts of cities. One particular superellipse that’s right in the middle is called a squircle. Squircles have been used to design everything from dinner plates to touchpad buttons.

The space of superellipsoids.

Piet had the following to say about the gap between squares and circles:

Things made with straight lines fit well together and save space. And we can move easily — physically or mentally — around things made with round lines. But we are in a straitjacket, having to accept one or the other, when often some intermediate form would be better. … The super-ellipse solved the problem. It is neither round nor rectangular, but in between. Yet it is fixed, it is definite — it has a unity.

“Squaring the Circle” by Troika.

These circles aren’t what they seem to be.

There’s another kind of squircular object that I ran across recently. It’s a sculpture called “Squaring the Circle”, and it was created by a trio of artists known as Troika. Check out the images on this page, and then watch a video of the incredible transformation. You can find more examples of room-sized perspective-changing objects in this article.

Next up: it’s been a snowy week here on the east coast, so I thought I’d share some ideas for a great indoor activity—coloring!

Marshall and Violet.

Marshall Hampton is a math professor at University of Minnesota, Duluth. Marshall studies n-body problems—a kind of physics problem that goes all the way back to Isaac Newton and that led to the discovery of chaos. He also uses math to study the genes that cause mammals to hibernate. Marshall made a coloring book full of all kinds of lovely mathematical images for his daughter Violet. He’s also shared it with the world, in both pdf and book form. Check it out!

Inspired by Mashrall’s coloring book, Alex Raichev made one of his own, called Contours. It features contour plots that you can color. Contour plots are what you get when you make outlines of areas that share the same value for a given function. Versions of contour plots often appear on weather maps, where the functions are temperature, atmospheric pressure, or precipitation levels.

Contour plots are useful. Alex shows that they can be beautiful, too!

And there are even more mathematical patterns to explore in the coloring sheets at Patterns for Colouring.

Last up, that’s not a typo in this week’s post title. I really do want to share some snowfakes with you—some artificial snowflake models created with math by Janko Gravner and David Griffeath. You can find out more by reading this paper they authored, or just skim it for the lovely images, some of which I’ve shared below.

I ran across these snowfakes at the Mathematical Imagery page of the American Mathematical Society. There are lots more great math images to explore there.

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 pixelart. 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 greatmathygifs. 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!

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.

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.

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!

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.