# Solitons, Contours, and Thinking Sdrawkcab

Welcome to this week’s Math Munch!

Meet Nalini Joshi, a mathematician at the University of Sydney in Australia. I’ll let her introduce herself to you.

Nalini has an amazing story and amazing passion. What does her video make you think? To hear more from Nalini, you can watch this talk she gave last month at the Women in Mathematics conference at the Isaac Newton Institute in Cambridge, England. Her talk is called “Mathematics and life: a personal journey.” You might also enjoy reading this interview or others on her media page.

I’d like to share three clumps of ideas that might give you a flavor for the math that Nalini enjoys doing. Most of it is way over my head, but I’m reaching for it! You can, too, if you try.

Here’s clump number one. Two of the main objects that Nalini studies are dynamical systems and differential equations. You can think of a dynamical system as some objects that interact with each other and evolve over time. Think of the stars that Nalini described in the video, heading toward each other and tugging on each other. Differential equations are one way of describing these interactions in a mathematically precise way. They capture how tiny changes in one amount affect tiny changes in another amount.

Vlasov billiards.

To play around with some simple dynamical systems that can still produce some complex behaviors, check out dynamical-systems.org. Vlasov billiards was new to me. I think it’s really cool. The three-body problem is one of the oldest and most famous dynamical systems, and you can tinker around with examples of it here and here. There’s even a three-body problem game you can try playing. I’m not too crazy about it, but maybe you’ll enjoy it. It certainly gives you a sense for how chaotic the a three-body system can be!

Nalini doesn’t study just any old dynamical systems. She’s particularly interested in ones where the chaotic parts of the system cancel each other out. Remember in the video how she described the stars that go past each other and don’t destroy each other, that are “transparent to each other”? Places where this happens in dynamical systems are called soliton solutions. They’re like steady waves that can pass through each other. Check out these four videos on solitons, each of which gives a different perspective on them. If you’re feeling adventurous, you could try reading this article called What is a Soliton?

 Making a water wave soliton in the Netherlands. A computer animation of interacting solitons. Japanese artist Takashi Suzuki tests a soliton to be used in a piece of performance art. Students studying and building solitons in South Africa.

Level curves that are generalized Cassini curves.
Also, it kind of looks like a four-body problem.
(click for video)

The second idea that Nalini uses that I’d like to share is level curves, or contours. Instead of studying complicated differential equations directly, it’s possible to get at them geometrically by studying families of curves—contours—that are produced by related algebraic equations. They’re just like the lines on a topographic map that mark off areas of equal elevation.

Here’s a blog post by our friend Tim Chartier about colorful contour lines that arise from the differential equation governing heat flow. The temperature maps by Zachary Forest Johnson from a few weeks ago also used contour lines. And I found some great pieces of art that take contours as their inspiration. Click to check these out!

The last idea clump I’ll share involves integrable systems. In an integrable system, it’s possible to uniquely “undo” what has happened—the rules are such that there’s only one possible past that could lead to the present. Most systems don’t work this way—you can’t tell what was in your refrigerator a week ago by looking at it now! Nalini mentions on her research page that “ideas on integrable differential equations also extend to difference equations, and even to extended versions of cellular automata.” I enjoyed reading this article about reversible cellular automata, especially the section about Critters.

What move did Black just play?
A puzzle by Raymond Smullyan.

And this made me think of a really nifty kind of chess puzzle called retrograde analysis—a fancy way of saying “thinking backwards”. Instead of trying to find the best chess move to play next, you instead have to figure out what move was made to get to the position in the puzzle. Most chess positions could be arrived at through multiple moves, but the positions in these puzzles are specially designed so that only one move will work. There’s a huge index of this kind of problem at The Retrograde Analysis Corner, and there are some great starter problems on this page.

Maurice Ashley

And perhaps you’d like to hear a little bit about thinking backwards from one of the greatest teachers of chess, Grandmaster Maurice Ashley. Check out his TED video here.

I hope you’ve enjoyed finding out about Nalini Joshi and the mathematics that she loves. I asked Nalini if she would do a Q&A with us, and she said yes! Do you have a question you’d like to ask her? Send it to us below and we’ll include it in the interview, which I send to Nalini in about a week.

UPDATE: We’re no longer accepting questions for Nalini, because the interview has happened! Check it out!

Bon appetit!

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

On 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!)

Some 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!)

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.)

Michael 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…)

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.

If 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!

# Circling, Squaring, and Triangulating

Welcome to this week’s Math Munch!

How good are you at drawing circles? To find out, try this circle drawing challenge. There are adorable cat pictures for prizes!

What’s the best score you can get? And hey—what’s the worst score you can get? And how is your score determined? Well, no matter how long the path you draw is, using that length to make a circle would surround the most area. How close your shape gets to that maximum area determines your score.

Do you think this is a good way to measure how circular a shape is? Can you think of a different way?

Dido, Founder and Queen of Carthage.

This idea that a circle is the shape that has the biggest area for a fixed perimeter reminds me of the story of Dido and her famous problem. You can find a retelling of it at Mathematica Ludibunda, a charming website that’s home to all sorts of mathematical stories and puzzles. The whole site is written in the voice of Rapunzel, but there’s a team of authors behind it all. Dido’s story in particular was written by a girl named Christa.

If you have any trouble drawing circles in the applet, you might try using pencil and paper or a chalkboard. I bet if you practice your circling and get good at it, you might even be able to challenge this fellow:

The simple perfect squared square
of smallest order.

Next up is squaring and the incredible Squaring.Net. The site is run by Stuart Anderson, who works at the Reserve Bank of Australia and lives in Sydney.

The site gathers together all of the research that’s been done about breaking up squares and rectangles into squares. It’s both a gallery and an encyclopedia. I love getting to look at the timelines of discovery—to see the progress that’s been made over time and how new things have been discovered even this year! Just within the last month or so, Stuart and Lorenz Milla used computers to show that there are 20566 simple perfect squared squares of order 30. Squaring.Net also has a wonderful links page that can connect you to more information about the history of squaring, as well as some of the delightful mathematical art that the subject has inspired.

Last up this week is triangulating. There are lots of ways to chop up a shape into triangles, and so I’ll focus on one particular way known as a Delaunay triangulation. To make one, scatter some points on the plane. Then connect them up into triangles so that each triangle fits snugly into a circle that contains none of the scattered points.

Fun Fact #1: Delaunay triangulations are named for the Soviet mathematician Boris Delaunay. What else is named for him? A mountain! That’s because Boris was a world-class mountain climber.

Fun Fact #2: The idea of Delaunay triangulations has been rediscovered many times and is useful in fields as diverse as computer animation and engineering.

Here are two uses of Delaunay triangulations I’d like to share with you. The first comes from the work of Zachary Forest Johnson, a cartographer who shares his work at indiemaps.com. You can check out a Delaunay triangulation applet that he made and read some background about this Delaunay idea here. To see how Zach uses these triangulations in his map-making, you’ve gotta check out the sequence of images on this page. It’s incredible how just a scattering of local temperature measurements can be extended to one of those full-color national temperature maps. So cool!

 Zachary Forest Johnson A Delaunay triangulation used to help create a weather map.

Finally, take a look at these images that Jonathan Puckey created. Jonathan is a graphic artist who lives in Amsterdam and shares his work on his website. In 2008 he invented a graphical process that uses Delaunay triangulations and color averaging to create abstractions of images. You can see more of Jonathan’s Delaunay images here.

I hope you find something to enjoy in these circles, squares, and triangles. Bon appetit!