Tag Archives: graphs

Tsoro Yematatu, Fano’s Plane, and GIFs

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 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.

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?

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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.

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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

MOVES, the Tower of Hanoi, and Mathigon

Welcome to this week’s Math Munch!

MOVESThe Math Munch team just wrapped up attending the first MOVES Conference, which was put on by the Museum of Math in NYC. MOVES is a recreational math conference and stands for Mathematics of Various Entertaining Subjects. Anna coordinated the Family Activities track at the conference and Paul gave a talk about his imbalance problems. I was just there as an attendee and had a blast soaking up wonderful math from some amazing people!

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Who all was there? Some of our math heroes—and familiar faces on Math Munch—like Erik Demaine, Tanya Khovanova, Tim and Tanya Chartier, and Henry Segerman, just to name just a few. I got to meet and learn from many new people, too! Even though I know it’s true, it still surprises me how big and varied the world of math and mathematicians is.

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Suzanne Dorée at MOVES.

One of my favorite talks at MOVES was given by Suzanne Dorée of Augsburg College. Su spoke about research she did with a former student—Danielle Arett—about the puzzle known as the Tower of Hanoi. You can try out this puzzle yourself with this online applet. The applet also includes some of the puzzle’s history and even some information about how the computer code for the applet was written.

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Danielle Arett

A pice of a Tower of Hanoi graph with three pegs and four disks.

A piece of a Tower of Hanoi graph with three pegs and four disks.

But back to Su and Danielle. If you think of the different Tower of Hanoi puzzle states as dots, and moving a disk as a line connecting two of these dots, then you can make a picture (or graph) of the whole “puzzle space”. Here are some photos of the puzzle space for playing the Tower of Hanoi with four disks. Of course, how big your puzzle space graph is depends on how many disks you use for your puzzle, and you can imagine changing the number of pegs as well. All of these different pictures are given the technical name of Tower of Hanoi graphs. Su and Danielle investigated these graphs and especially ways to color them: how many different colors are needed so that all neighboring dots are different colors?

Images from Su and Danielle's paper. Towers of Hanoi graphs with four pegs.

Images from Su and Danielle’s paper. Tower of Hanoi graphs with four pegs.

 

Su and Danielle showed that even as the number of disks and pegs grows—and the puzzle graphs get very large and complicated—the number of colors required does not increase quickly. In fact, you only ever need as many colors as you have pegs! Su and Danielle wrote up their results and published them as an article in Mathematics Magazine in 2010.

Today Danielle lives in North Dakota and is an analyst at Hartford Funds. She uses math every day to help people to grow and manage their money. Su teaches at Ausburg College in Minnesota where she carries out her belief “that everyone can learn mathematics.”

Do you have a question for Su or Danielle—about their Tower of Hanoi research, about math more generally, or about their careers? If you do, send them to us in the form below for an upcoming Q&A!

UPDATE: We’re no longer accepting questions for Su and Danielle. Their interview will be posted soon! Ask questions of other math people here.

MathigonLast up, here’s a gorgeous website called Mathigon, which someone shared with me recently. It shares a colorful and sweeping view of different fields of mathematics, and there are some interactive parts of the site as well. There are features about graph theory—the field that Su and Danielle worked in—as well as combinatorics and polyhedra. There’s lots to explore!

Bon appetit!

World’s Oldest Person, Graphing Challenge, and Escher Sketch

265282-jiroemon-kimura-the-world-s-oldest-living-man-celebrated-his-115th-birOn April 19th, Jiroeman Kimura celebrated his 116th birthday. He was – and still is – the world’s oldest person, and the world’s longest living man – ever. (As far as researchers know, that is. There could be a man who has lived longer that the public doesn’t know about.) The world’s longest living woman was Jeanne Calment, who lived to be 122 and a half!

Most people don’t live that long, and, obviously, only one person can hold the title of “Oldest Person in the World” at any given time. So, you may  be wondering… how often is there a new oldest person in the world? (Take a few guesses, if you like. I’ll give you the answer soon!)

stackSome mathematicians were wondering this, too, and they went about answering their question in the way they know best: by sharing their question with other mathematicians around the world! In April, a mathematician who calls himself Gugg, asked this question on the website Mathematics Stack Exchange, a free question-and-answer site that people studying math can use to share their ideas with each other. Math Stack Exchange says that it’s for “people studying math at any level.” If you browse around, you’ll see mathematicians asking for help on all kinds of questions, such as this tricky algebra problem and this problem about finding all the ways to combine coins to get a certain amount of money.  Here’s an entry from a student asking for help on trigonometry homework. You might need some specialized math knowledge to understand some of the questions, but there’s often one that’s both interesting and understandable on the list.

Anyway, Gugg asked on Math Stack Exchange, “How often does the oldest person in the world die?” and the community of mathematicians around the world got to work! Several mathematicians gave ways to calculate how often a new person becomes the oldest person in the world. You can read about how they worked it out on Math Stack Exchange, if you like, or on the Smithsonian blog – it’s a good example of how people use math to model things that happen in the world. Oh, and, in case you were wondering, a new person becomes the world’s oldest about every 0.65 years. (Is that around what you expected? It was definitely more often than I expected!)

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Next, check out this graph! Yes, that’s a graph – there is a single function that you can make so that when you graph it, you get that.  Crazy – and beautiful! This was posted by a New York City math teacher named Michael Pershan to a site called Daily Desmos, and he challenges you to figure out how to make it!  (He challenged me, too. I worked on this for days.)

qod0nxgctfMichael made this graph using an awesome free, online graphing program called Desmos. Michael and many other people regularly post graphing challenges on Daily Desmos. Some of them are very difficult (like the one shown above), but some are definitely solvable without causing significant amounts of pain. They’re marked with levels “Basic” and “Advanced.” (See if you can spot contributions from a familiar Math Munch face…)

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Here are more that I think are particularly beautiful. If you’re feeling more creative than puzzle-solvey, try making a cool graph of your own! You can submit a graphing challenge of your own to Daily Desmos.

escher 3If you’ve got the creative bug, you could also check out a new MArTH tool that we just found called Escher Web Sketch. This tool was designed by three Swiss mathematicians, and it helps you to make intricate tessellations with interesting symmetries – like the ones made by the mathematical artist M. C. Escher. If you like Symmetry Artist and Kali, you’ll love this applet.

Be healthy and happy! Enjoy graphing and sketching! And, bon appetit!