# 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!

# Zentangle, Graph Paper, and Pancake Art

Some recent doodling, by me.

Welcome to this week’s Math Munch!

As you start a new school year, you might be looking for some new mathy doodle games to play in the margins of your notebooks. Doodling helps me to listen sometimes, and I love making neat patterns. I especially like seeing what new shapes I can make.

This summer I was very happy to run across Zentangle®—”an easy-to-learn, relaxing, and fun way to create beautiful images by drawing structured patterns.” I’ve learned a lot about Zentangle from a blog called Tangle Bucket by Sandy Hunter. She shares how to doodle snircles, snafoozles, and oodles. There’s a whole dictionary of zentangle shapes over at tanglepatterns.com.

My favorite idea in Zentangle is trying to combine two kinds of designs. Sometimes this is described as one pattern “versus” another one. For instance, check out these:

Maybe you’ll pick some tangle patterns to combine with each other. If you try some, maybe you’ll share them in our Readers’ Gallery.

Sandy writes:

It’s so true that the more I tangle, the more I see the potential in patterns all around me. I catch myself mentally deconstructing them (whether I want to or not) to figure out if they can be broken down into simple steps without too much effort. That’s the trademark of a good tangle pattern.

Try some of Sandy’s weekly challenges, or check out Tiffany Lovering’s time-lapse videos—here’s one with music and one with an interview. Can you learn the names of any of the shapes she creates? I spy a Rick’s Paradox. There are lots of ways to begin zentangling—I hope you enjoy giving it a try.

Squares & dots & crosses, oh my!

If zentangling is too freeform for your doodling tastes, then let me share with you one of my longtime favorite websites. I’ve used it for years to help me to do math and to teach math, and it’s great for math doodling, too. I might even call it a trusty friend, one that I met one day through the simple online search: “free online graph paper”.

That’s right, it’s Free Online Graph Paper.

Something I love about the site is that it lets you design different aspects of your graph paper. Then you can print it out. First you get to decide what kind of grid you would like: square? triangular? circular? Then you get to tinker with lots of variables, like how big the grid cells are, how dark the lines are, and what color they are. And more!

Free Online Graph Paper was created by Kevin MacLeod, who composes music and shares it for free. That way other people can use it for creative projects. That’s really awesome! I enjoyed listening to Kevin’s “Winner Winner“. It’s always good to be reminded that everything you use or enjoy was almost certainly made by a person—including custom graph paper websites!

A 7/3 star spirocake.

Last up this week is some doodly math that you can really munch on. Everyone knows that breakfast is the most important meal of the day and that the most important food group is roulette curves.

To get your daily recommended allowance of groovy math, look no further than the edible doodles of Nathan Shields and his family over at Saipancakes.

I can wait until the Shields family tackles the cissoid of Diocles.

Bon appetit!

# Noodles, Flowsnake, and Symmetry

Welcome to this week’s Math Munch!

Gemelli, by Sander Huisman

How much do you like pasta?  Well, these mathematicians and scientists from around the world like pasta so much that they’ve been studying its shape mathematically!  Check out this New York Times article about Sander Huisman, a graduate student in physics from the Netherlands, and Marco Guarnieri and George L. Legendre, two architects from London, who have all taken up making graphs of and equations for pasta shapes.  Sander posts his pasta-graphs on his blog.  Legendre wrote this book about math and pasta, called Pasta By Design.  Legendre has even invented a new type of pasta, shaped like a Mobius strip (see last week’s Math Munch for lots of cool things with Mobius strips), which he named after his baby daughter, Ioli!

Some of Legendre’s pasta plots

Next, here comes the flowsnake.  Wait – don’t run away!  The flowsnake is not a terrifying monster, despite it’s ominous name.  It is a space-filing curve, meaning that the complete curve covers every single point in a part of two-dimensional space.  So if you were to try to draw a flowsnake on a piece of paper, you wouldn’t be able to see any white when you were done.  It’s named flowsnake because it resembles a snowflake.

The flowsnake curve

 A single piece of the flowsnake curve. Units of flowsnake fit together like puzzle pieces to fill the plane

Finally, check out this awesome online symmetry-sketcher, called Symmetry Artist!  Here, you can make doodles of all kinds and then choose how you want to reflect and rotate them.  Fun!

Bon appetit!