Tag Archives: games

TesselManiac, Zeno’s Paradox, and Platonic Realms

Welcome to this week’s Math Munch!

Before we begin, we’d like to thank all of you who have checked out the site in the past week. Since we’ve kicked off our share campaign, we’ve had so many new visitors and heard from many of them, too! Reading your feedback – whether a recommendation, some praise, a question, or just a brief, “Hello!” – brings us great joy and helps us to know that you all are out there.

Whether you’re a regular reader or visiting the site for the first time, we’d like to ask you for a little favor. If you see some math you like, share it with someone who you think would like it, too! Do you love the burst of excitement that you get from reading about a new mathematical idea, seeing an original piece of math artwork, or trying out a new game? Do you know someone who would love that, too? Then tell them about Math Munch – we’d love to spread the joy.

If you enjoy Math Munch, join in our “share campaign” this week.

You can read more about the share campaign here. There are lots of ways to participate, and you can let us know about your sharing through this form. We’d love to see the share total rise up to 1000 over the course of the next week.

Now for the post!

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This beautiful tessellated wooden box was made by computer scientist and mathematical artist Kevin Lee. I met Kevin two weeks ago at the MOVES conference (which Justin and Paul have both written about already). Kevin teaches computer science at Normandale Community College in Minnesota. He makes woodcut tessellations (which won an award for the “Best Textile, Sculpture, or Other Medium” at the Joint Mathematics Meetings art exhibition this year!). He’s also used a combination of his knowledge of computer science and his love of Escher-type tessellations to make software that helps you create tessellations. His new software, TesselManiac!, is due out soon (watch this short movie Kevin made about it for the Bridges conference), but you can download an older version of the software here and play a preview version of The Flipping Tile Game.

To play this game, you must fill in an outline of a tessellation with the piece given. You can use any of four symmetry motions – translation (or shift), rotation, reflection, or glide reflection (which reflects the tile and then translates it along a line parallel to the line of reflection). You get points for each correct tile placed (and lose points if you have to delete). Translations are the simplest, and only give you 5 points each. Reflections are the most difficult – you get 20 points for each one used!

While you’re downloading The Flipping Tile Game, try one of Kevin’s Dot-to-Dot puzzles. These are definitely not your typical dot-to-dot. Only the portion of the image corresponding to one tile in the tessellation is numbered. Once you figure out the shape of that single tile, you have to figure out how to number the rest of the puzzle!

Lucky for us, Kevin has agreed to answer some questions about his life and work as a math artist and computer scientist. Leave a question for Kevin here. (We’ll take questions for the next two weeks.)

I’ve recently been thinking about a paradox that has puzzled mathematicians for centuries. Maybe you’ve heard of it. It’s one of the ancient Greek philosopher Zeno‘s paradoxes of motion, and it goes like this: Achilles (a really fast Greek hero) and a tortoise are going to run a race. Achilles agrees to give the tortoise a head-start, because the tortoise is so slow. Achilles then starts to run. But as Achilles catches up with the tortoise, the tortoise moves a little further. So the tortoise is still ahead. And as Achilles moves to catch up again, the tortoise moves even further! Sounds like Achilles will never catch up to the tortoise, let alone pass him… But that doesn’t make sense…

Will Achilles lose the race??? Check out this great video from Numberphile that explains both the paradox and the solution.

While I was looking for information about this paradox, I stumbled across a great math resource site called Platonic Realms. The homepage of this site has a daily historical fact, mathematical quote, and puzzle.

The site hosts a math encyclopedia with explanations of all kinds of math terms and little biographies of famous mathematicians. You can also read “mini-texts” about different mathematical topics, such as this one about M. C. Escher (the inspiration behind the art at the beginning of this post!) or this one about coping with math anxiety.

I hope we here at Math Munch have given you something to tantalize your mathematical taste buds this week! If so, we’d love it if you would pass it along.

Thank you for reading, and bon appetit!

P.S. – We’ve posted a new game, suggested to us by one of our readers! It’s an online version of Rush Hour. Check it out!

Posted by Anna Weltman. Categories: Math Munch. Tags: art, computers, escher, games, history, infinity, interview, math anxiety, math encyclopedia, paradox, puzzles, recreational mathematics, software, symmetry, tessellation, video. 16 Comments

May. ’13

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.

Nalini Joshi lecturing about solitons.

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, kind of looks like a four-body problem. (click for video)

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!

Visual_Topography_of_a_Generation_Gap_Brooklyn_2

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!

Posted by Justin Lanier. Categories: Math Munch. Tags: applet, art, cellular automata, chess, differential equations, diversity, dynamical systems, games, history, interview, kinetic sculpture, video. 6 Comments

Mar. ’13

Sam Loyd, Weight Problems, and Exercises

Welcome to this week’s Math Munch!

Chess master, puzzlist, and recreational mathematician Sam Loyd. GREAT mustache.

Chess composer, puzzlist, and recreational mathematician Sam Loyd. GREAT mustache.

First up, remember Sam Loyd? (We’ve featured him twice before.) He was an american chess player and recreational mathematician who lived from 1841-1911. He was also a chess composer, someone who writes endgame strategies and chess puzzles. In fact, he wrote all sorts of puzzles, which his son published in a book called Sam Loyd’s Cyclopedia of 5000 Puzzles, Tricks, and Conundrums. (That link will take you to a scan of all 385 pages!) By the way, those 5000 puzzles are only about half of the ones he wrote in his lifetime. It’s no wonder Martin Gardner called him “America’s greatest puzzler.” An interesting anecdote: Sam Loyd claimed until his death to have invented the 15 puzzle, but in fact he did not. The actual inventor was Noyes Chapman, the Postmaster of Canastota, NY.

I wanted to show you some of Sam’s “Puzzling Scales” problems. Why don’t you stop reading now and just solve them both?

These different weights balance because of the torque they apply

There are lots of puzzles like this, based on different weights balancing with each other. A friend sent me this page of weight puzzles based on the idea of torque. The farther out an object is placed, the more torque it applies to the balance, so it’s possible for a 1 pound weight to balance a 2 pound weight if you set them at the right distances. The distance and wight multiply to give the torque applied.

These problems come from a massive bank of puzzles over on Erich’s Puzzle Palace. If you like, you can also play this torque game I found.

Place 1 through 5 to balance the weights.

Place 1 through 6 to balance the weights.

I love problems like this, but I started to wonder, “what if the scales don’t balance? Maybe you could make a puzzle out of that.” I did exactly that, creating a series of imbalance puzzles. Your job is to order the shapes by weight. They start out easy, but there are some tricky ones. I especially like #6.

In each case, order the three objects by weight.

I’m also hosting an imbalance puzzle-writing contest. My two favorite puzzlists will win a print of their choosing from my Stars of the Mind’s Sky series of mathematical art. You should try your hand at writing one. Just email it to Lost in Recursion.

Finally – we all love great problems and puzzles, but skill building is an important aspect of mathematics as well, and exercises help us build skill. Exercises are often dull, but I found a website with some exercises I quite like, and I wanted to share them with you. Check out the Coffee Break section over on StudyMaths.co.uk.

Detention Dash

Find the Primes

Odd One Out

Detention Dash, for example, is just a timed multiplication chart, but typing the answers in on my computer really made me feel some of the patterns in the numbers. You should try it. Odd One Out also keeps you on your toes and makes you think about different kinds of numbers. I find them surprisingly fun. I hope you agree.

Bon appetit!

Posted by Paul Salomon. Categories: Math Munch. Tags: balances, exercises, games, puzzles, sam loyd, torque, weights. 8 Comments