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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 points.

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 sections, a linked-to-itself bagel, or some spirograph pancakes.

Bon appetit!

Welcome to this week’s Math Munch!

Board and pieces for tsoro yematatu.

Here’s a little game with a big name: tsoro yematatu. If you enjoyed Paul’s recent post about tic-tac-toe, I think you’ll like tsoro yematatu a lot.

I ran across this game on a website called Behind the Glass. The site is run by the Cincinnati Art Museum. (What is it with me and art museums lately?) The museum uses Behind the Glass to curate many pieces of African art and culture, including four mathematical games that are played in Africa.

The simplest of these is tsoro yematatu. It has its origin in Zimbabwe. Like tic-tac-toe, the goal is to get three of your pieces in a row, but the board is “pinched” and you can move your pieces. Here’s an applet where you can play a modified version of the game against a computer opponent. While the game still feels similar to tic-tac-toe, there are brand-new elements of strategy.

Tsoro yematatu reminds me of one that I played as a kid called Nine Men’s Morris. I learned about it and many other games—including go—from a delightful book called The Book of Classic Board Games. Kat Mangione—a teacher, mom, and game-lover who lives in Tennessee—has compiled a wonderful collection of in-a-row games. And wouldn’t you know, she includes Nine Men’s Morris, tsoro yematatu, tic-tac-toe, and dara—another of the African games from Behind the Glass.

The Fano plane.

The board for tsoro yematatu also reminds me of the Fano plane. This mathematical object is very symmetric—even more than meets the eye. Notice that each point is on three lines and that each line passes through three points. The Fano plane is one of many projective planes—mathematical objects that are “pinched” in the sense that they have vanishing points. They are close cousins of perspective drawings, which you can check out in these videos.

Can you invent a game that can be played on the Fano plane?

Closely related to the Fano plane is an object called the Klein quartic. They have the same symmetries—168 of them. Felix Klein discovered not only the Klein quartic and the famous Klein bottle, but also the gorgeous Kleinian groups and the Beltrami-Klein model. He’s one of my biggest mathematical heroes.

The Klein quartic.

This article about the Klein quartic by mathematician John Baez contains some wonderful images. The math gets plenty tough as the article goes on, but in a thoughtfully-written article there is something for everyone. One good way to learn about new mathematics is to read as far as you can into a piece of writing and then to do a little research on the part where you get stuck.

If you’ve enjoyed the animation of the Klein quartic, then I bet my last find this week will be up your alley, too. It’s a Tumblr by David Whyte and Brian Fitzpatrick called Bees & Bombs. David and Brian create some fantastic GIFs that can expand your mathematical imagination.

This one is called Pass ‘Em On. I find it entrancing—there’s so much to see. You can follow individual dots, or hexagons, or triangles. What do you see?

This one is called Blue Tiles. It makes me wonder what kind of game could be played on a shape-shifting checkerboard. It also reminds me of parquet deformations.

A few of my other favorites are Spacedots and Dancing Squares. Some of David and Brian’s animations are interactive, like Pointers. They have even made some GIFs that are inspired by Tilman Zitzmann’s work over at Geometry Daily (previously).

I hope you enjoy checking out all of these new variations on some familiar mathematical objects. Bon appetit!

Reflection Sheet – Tsoro Yematatu, Fano’s Plane, and GIFs

Math Munch

A Weekly Digest of the Mathematical Internet

Tag Archives: gifs

Nice Neighbors, Spinning GIFs, and Breakfast

Tsoro Yematatu, Fano’s Plane, and GIFs