# 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!

# 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!

# Zippergons, High Fashion, and Really Big Numbers

Welcome to this week’s Math Munch!

Bill Thurston

Recently I attended a conference in memory of Bill Thurston. Bill was one of the most imaginative and influential mathematicians of the second half of the twentieth century. He worked with many mathematicians on projects and had many students before he passed away in the fall of 2012 at the age of 65. You can read Bill’s obituary in the New York Times here.

Bill worked where geometry and topology meet. In fact, Bill throughout his career showed that there are rich connections between the two fields that no one thought was possible. For instance, it’s an amazing fact that every surface—no matter how bumpy or holey or twisted—can be given a nice, symmetric curvature. A uniform geometry, it’s called. This was proven by Henri Poincaré in 1907. It was thought that 3D spaces would be far too complicated to be behave according to a similar rule. But Bill had a vision and a conjecture—that every 3D space can be divided into parts that can be given uniform geometries. To give you a flavor of these ideas, here’s a video of Bill describing some unusual and fabulous 3D spaces.

Any surface can be given a uniform geometry. Even a bunny. Another video.

As you can probably tell, visualizing and experiencing math was very important to Bill. He even taught a course with John Conway called Geometry and the Imagination. Bill often used computers to help himself see the math he was thinking about, and he enjoyed making hands-on models as well. Beginning in spring of 2010, Bill and Kelly Delp of Ithaca College worked out an idea. Usually all of the curving or turning of a polyhedron is concentrated at the vertices. Most of a cube is flat, but there’s a whole lot of pinch at the corners. What if you could spread that pinching out along the edges? And if you could, wouldn’t longer and perhaps wiggly edges help spread it even better? Yes and yes! You can see some examples of these “zippergons” that Bill and Kelly imagined and made in this gallery and read about them in their Bridges article.

 A paper octahedron zippergon. A foam icosadodecahedron zippergon.

One of Bill’s last collaborations happened not with a mathematician but with a fashion designer. Dai Fujiwara, a noted creator of high fashion in Tokyo, got inspired by some of Bill’s illustrations. In collaboration with Bill, Dai created eight outfits. Each one was based on one of the eight Thurston geometries. You can see the result of their work together in this video and read more about it in this article.

Isn’t it amazing how creative minds in very different fields can learn from each other and create something together?

Richard Evan Schwartz (self-portrait)

Richard Evan Schwartz was one of the speakers at the conference honoring Bill. Rich studied with Bill at Princeton and now is a math professor at Brown University.

Like Bill, Rich’s work can be highly visual and playful, and he often taps the power of computers to visualize and analyze mathematical structures. There’s lots to explore on Rich’s website. Check out these applets he has made, including ones on Poncelet’s Porism, the Euclidean algorithm (previously), and a game called Lucy & Lily (JAVA required). I love how Rich shares some of his earliest applet-making efforts, like Click On A Triangle To Change Its Color. It’s motivating to see that even an accomplished mathematician like Rich began with the basics of programming—a place where any of us can start!

On Rich’s site you’ll also find information about his project “Counting on Monsters“. And you should definitely make time to read some of the conversations that Rich has had with his five-year-old daughter Lucy.

Recently Rich published a wonderful new book for kids called “Really Big Numbers“. It is a colorful romp through larger and larger numbers and layers of abstraction, with evocative images to light the way. Check out the trailer for “Really Big Numbers” below!

Do you have a question for Rich—about his book, or about the math that he does, or about his life, or about Bill? Then send it to us in the form below and we’ll try to include it in our interview with him!

EDIT: Thanks for all your questions! Our Q&A with Rich will be posted soon.

 Diana and Rich Diana and Bill

Bill taught Rich, and Rich in turn taught Diana Davis, whose Dance Your PhD video we featured a while back. In fact, Bill’s influence on mathematics can be seen throughout many of our posts on Math Munch. Bill collaborated with Daina Taimina on hyperbolic crochet projects. He taught Jeff Weeks and helped inspire the games and software Jeff created. Bill oversaw the production of the film Outside In about the eversion of a sphere. He even coined the mathematical term “pair of pants.”

Bill’s vision of mathematics will live on in many people. That could include you, if you’d like. It’s just as Bill wrote:

Bon appetit!