# Virtual Hyenas, Markov Chains, and Random Knights

Welcome to this week’s Math Munch!

It’s amazing how a small step can lead to a chain reaction of adventure.

Arend Hintze

Recently a reader named Nico left a comment on the Math Munch post where I shared the game Loops of Zen. He asked why the game has that name. Curious, I looked up Dr. Arend Hintze, whose name appears on the game’s title page. This led me to Arend’s page at the Adami Lab at Michigan State University. Arend studies how complex systems—especially biological systems—evolve over time.

Here is a video of one of Arend’s simulations. The black and white square is a zebra. The yellow ones are lions, the red ones are hyenas, and guess who’s hungry?

Arend’s description of the simulation is here. The cooperative behavior in the video—two hyenas working together to scare away a lion—wasn’t programmed into the simulation. It emerged out of many iterations of systems called Markov Brains—developed by Arend—that are based upon mathematical structures called Markov chains. More on those in a bit.

You can read more about how Arend thinks about his multidisciplinary work on biological systems here. Also, it turns out that Arend has made many more games besides Loops of Zen. Here’s Blobs of Zen, and Ink of Zen is coming out this month! Another that caught my eye is Curve, which reminds me of some of my favorite puzzle games. Curve is still in development; here’s hoping we’ll be able to play it soon.

Arend has agreed to do an interview with Math Munch, so share your questions about his work, his games, and his life below!

Eric Czekner

Arend’s simulations rely on Markov chains to model animal behavior. So what’s a Markov chain? It’s closely related to the idea of a random walk. Check out this video by digital artist, musician, and Pure Data enthusiast Eric Czekner. In the video, Eric gives an overview of what Markov chains are all about and shows how he uses them to create pieces of music.

On this page, Eric describes how he got started using Markov chains to make music, along with several of his compositions. It’s fascinating how he captures the feel of a song by creating a mathematical system that “generates new patterns based on existing probabilities.”

Now there’s a big idea: exploring something randomly can capture structures that might be hard to perceive otherwise. Here’s one last variation on the Markov chain theme that involves a pure math question. This blog post ponders the question: what happens when a knight takes a random walk—or random trot?—on a chessboard? It includes some colorful images of chessboards along the way.

 How likely it is that a knight lands on each square after five moves, starting from b1. How likely it is that a knight lands on each square after 200 moves, starting from b1.

The blogger—Leonid Kovalev—shows in his analysis what happens in the long run: the number of times a knight will visit a square will be proportional to the number of moves that lead to that square. For instance, since only two knight moves can reach a corner square while eight knight moves can reach a central square, it’s four times as likely that a knight will finish on a central square after a long, long journey than on a corner square. This idea works because moving a knight around a chessboard is a “reversible Markov chain”—any path that a knight can trace can also be untraced. The author also wrote a follow-up post about random queens.

It’s amazing the things you can find by chaining together ideas or by taking a random walk. Thanks for the inspiration for this post, Nico. Keep those comments and questions coming, everyone—we love hearing from you.

Bon appetit!

# The Numbers Project, Epidemics, and Cut ‘n Slide

Welcome to this week’s Math Munch!

It’s an end-of-the-year group post!

Paul: This week I found Brandon Todd Wilson, a graphic artist who lives in Kansas City. He started a new and ambitious project. He wants to make a design for each of the numbers 0 through 365, making a new one each day of the year. That’s tough, but he’s done some amazing things so far. Check them out over at the numbers project. I’m amazed by the sneaky, clever ways he comes up with to showcase the numbers. Can you tell what numbers these three are below? Click to find out.

Maybe you could try a numeric design of your own. Perhaps for your favorite number or your birthday. If you make something your proud of, email us at mathmunchteam@gmail.com, and we could feature your work on Math Munch!

[Here are some numeric creations inspired by Brandon’s!]

Anna: Next up, it’s probably the end of the school year for most of you readers out there. Our school year is wrapping up, too. It’s sad, but also exciting, because we’re looking forward to what comes in the future. Recently, some of my students, looking to their futures, have been wondering what many students wonder: If I like math, what are some things I can do with it after I leave school? (We’ve posted about this question before – check out this post on the site We Use Math and any of the interviews on our Q&A page.) We here at Math Munch had the honor last week to meet an awesome woman who uses math all the time in her work as a scientist – Nina Fefferman!

Nina works mainly as a biologist at Rutgers University in New Jersey researching all kinds of cool and interesting things relating to epidemiology, or the study of infectious diseases and how they spread into epidemics in groups of people. How does she use math? In everything! Since dealing with infectious diseases is best done before they become epidemics, scientists like Nina make mathematical models to predict how a disease will spread before it hits. These models are really important for governments and hospitals, who use them to figure out how they can prepare for possible epidemics.

Nina loves math and her work – and you can hear all about it in this TEDx talk she did in 2010.

Justin: Finally, check out this short video by Sander Huisman, of mathematical pasta fame:

Sander has some more great videos, too. The shape that Sander’s cut and slide pattern gets closer and closer to is called the twindragon. It’s related to the more famous dragon fractal. Notice how the area of the shape stays the same throughout the video. Thanks to the kind folks at math.stackexchange for helping me to identify this fractal so quickly!

An earlier stage and a later stage of my cut & slide exploration.

In searching about this geometry idea of “cut and slide”, I ran across some great stuff. One thing I found was this neat applet by Frederik Vanhoutte. (Warning: JAVA required.) Frederik is a med­ical radi­a­tion physi­cist who lives in Belgium and who likes to make wonderful graphics in his spare time. Frederik has shared many of these on his site—check out his portfolio.

On his About page, Frederik says this about why he makes his generative graphics:

“When rain hits the wind­screen, I see tracks alpha par­ti­cles trace in cells. When I pull the plug in the bath tub, I stay to watch the lit­tle whirlpool. When I sit at the kitchen table, I play with the glasses to see the caus­tics. At a can­dle light din­ner, I stare into the flame. Sometimes at night, I find myself behind the com­puter. When I finally blink, a mess of code is draw­ing ran­dom struc­tures on the screen. I spend the rest of the night staring.”

Bon appetit!