Tag Archives: music

Digital Art, Mastermind, and Pythagoras

Welcome to this week’s Math Munch… on (approximately) Math Munch’s second birthday! Hooray!

Check out this video of mathematical art made by artist Nathan Selikoff:

Cool, right? This piece is called “Beautiful Chaos.” The curves on the screen are made from equations (if you’ve ever graphed a line or a parabola you’ll know what I mean). As the viewer waves her hands around, the equations change– and as the equations change, so do the curves! The result is something that might remind you of the images your computer makes when you play music on it or maybe of something you’d make using a spirograph. All in all, a beautiful and interactive piece of mathematical art.

Nathan lives and works as a mathematical artist in Orlando, Florida. As he writes on his website, Nathan uses computer code along with other materials to make art that plays with the mathematical ideas of space, motion, and interaction between objects. To see more of how Nathan does this, check out his giant, interactive marionette or this song that explores the first, second, third, and fourth dimensions:

My school is really lucky to be hosting Nathan this week! We didn’t want any of you, dear readers, to miss out on the excitement, though– so Nathan has kindly agreed to answer your interview questions! Got a question for Nathan? Write it in the box below. He’ll answer seven of your best questions in two weeks!

Next up, who doesn’t love to play Mastermind? It’s a great combination of logic, patterns, and trickery… but I just hate having to use all those tiny pegs. Well, guess what? You can play it online— no pegs (or opponent) necessary!

As I was playing Mastermind, I started wondering about strategy. What’s the best first guess to make? If I were as smart as a computer, is there a number of guesses in which I could guess any Mastermind code? (This kind of question reminds me of God’s Number and the Rubik’s cube…)

Well, it turns out there is a God’s Number for Mastermind – and that number is five. Just five. If you played perfectly and followed the strategy demonstrated by recreational mathematician Toby Nelson on his website, you could guess ANY Mastermind code in five guesses or less. Toby shares many more interesting questions about Mastermind on his website— I suggest you check it out.

What ARE those irrational numbers, so weird that they get their own bubble??

Finally, sometime in your mathematical past you may have heard of irrational numbers. These are numbers like the square-root of 2 or pi or e that can’t be written as a fraction– or so people claim. When you start thinking about this claim, however, it may seem strange. There are A LOT of fractions– and none of them equal the square-root of 2? Really? What kind of number is that? It seems like only an irrational person would believe that, at least without proof.

Vi Hart to the rescue! Irrational numbers were encountered long, long ago by the ancient Greek mathematician (and cult leader) Pythagoras– and he didn’t like them much. In this great video, Vi tells all about Pythagoras and the controversial discovery of numbers that aren’t fractions.

If you didn’t follow her explanation of why the square-root of 2 is irrational on your first watch, don’t worry– it’s a complicated idea that’s worth a second (or third or fourth) run-through.

Thanks for a great two years of Math Munch! Bon appetit!

Posted by Anna Weltman. Categories: Math Munch. Tags: art, computers, dimensions, games, history, interview, music, numbers, Pythagoras, Vi Hart, video. 44 Comments

Sep. ’13

Polyominoes, Clock Calculator, and Nine Bells

Welcome to this week’s Math Munch!

The first thing I have to share with you comes with a story. One day several years ago, I discovered these cool little shapes made of five squares. Maybe you’ve seen these guys before, but I’d never thought about how many different shapes I could make out of five squares. I was trying to decide if I had all the possible shapes made with five squares and what to call them, when along came Justin. He said, “Oh yeah, pentominoes. There’s so much stuff about those.”

Justin proceeded to show me that I wasn’t alone in discovering pentominoes – or any of their cousins, the polyominoes, made of any number of squares. I spent four happy years learning lots of things about polyominoes. Until one day… one of my students asked an unexpected question. Why squares? What if we used triangles? Or hexagons?

We drew what we called polyhexes (using hexagons) and polygles (using triangles). We were so excited about our discoveries! But were we alone in discovering them? I thought so, until…

A square made with all polyominoes up to heptominoes (seven), involving as many internal squares as possible.

… I found the Poly Pages. This is the polyform site to end all polyform sites. You’ll find information about all kinds of polyforms — whether it be a run-of-the-mill polyomino or an exotic polybolo — on this site. Want to know how many polyominoes have a perimeter of 14? You can find the answer here. Were you wondering if polyominoes made from half-squares are interesting? Read all about polyares.

I’m so excited to have found this site. Even though I have to share credit for my discovery with other people, now I can use my new knowledge to ask even more interesting questions.

Next up, check out this clock arithmetic calculator. This calculator does addition, subtraction, multiplication, and division, and even more exotic things like square roots, on a clock.

What does that mean? Well, a clock only uses the whole numbers 1 through 12. Saying “15 o’clock” doesn’t make a lot of sense (unless you use military time) – but you can figure out what time “15 o’clock” is by determining how much more 15 is than 12. 15 o’clock is 3 hours after 12 – so 15 o’clock is actually 3 o’clock. You can use a similar process to figure out the value of any positive or negative counting number on a 12 clock, or on a clock of any size. This process (called modular arithmetic) can get a bit time consuming (pun time!) – so, give this clock calculator a try!

Finally, here is some wonderful mathemusic by composer Tom Johnson. Tom writes music with underlying mathematics. In this piece (which is almost a dance as well as a piece of music), Tom explores the possible paths between nine bells, hung in a three-by-three square. I think this is an example of mathematical art at its best – it’s interesting both mathematically and artistically. Observe him traveling all of the different paths while listening to the way he uses rhythm and pauses between the phrases to shape the music. Enjoy!

Bon appetit!

Posted by Anna Weltman. Categories: Math Munch. Tags: arithmetic, calculator, combinatorics, music, numbers, polyominoes, puzzles, tessellation, video. 38 Comments

Aug. ’13

MoMA, Pop-Up Books, and A Game of Numbers

Welcome to this week’s Math Munch!

Thank you so much to everyone who participated in our Math Munch “share campaign” over the past two weeks. Over 200 shares were reported and we know that even more sharing happened “under the radar”. Thanks for being our partners in sharing great math experiences and curating the mathematical internet.

Of course, we know that the sharing will continue, even without a “campaign”. Thanks for that, too.

All right, time to share some math. On to the post!

To kick things off, you might like to check out our brand-new Q&A with Nalini Joshi. A choice quote from Nalini:

In contrast, doing math was entirely different. After trying it for a while, I realized that I could take my time, try alternative beginnings, do one step after another, and get to glimpse all kinds of possibilities along the way.

By Philippe Decrauzat.

I hope the math munches I share with you this week will help you to “glimpse all kinds of possibilities,” too!

Recently I went to the Museum of Modern Art (MoMA) in New York City. (Warning: don’t confuse MoMA with MoMath!) On display was an exhibit called Abstract Generation. You can view the pieces of art in the exhibit online.

As I browsed the galley, the sculptures by Tauba Auerbach particularly caught my eye. Here are two of the sculptures she had on display at MoMA:

Just looking at them, these sculptures are definitely cool. However, they become even cooler when you realize that they are pop-up sculptures! Can you see how the platforms that the sculptures sit on are actually the covers of a book? Neat!

Here’s a video that showcases all of Tauba’s pop-ups in their unfolding glory. Why do you think this series of sculptures is called [2,3]?

This idea of pop-up book math intrigued me, so I started searching around for some more examples. Below you’ll find a video that shows off some incredible geometric pop-ups in action. To see how you can make a pop-up sculpture of your own, check out this how-to video. Both of these videos were created by paper engineer Peter Dahmen.

Tauba Auerbach.

Tauba got me thinking about math and pop-up books, but there’s even more to see and enjoy on her website! Tauba’s art gives me new ways to connect with and reimagine familiar structures. Remember our post about the six dimensions of color? Tauba created a book that’s a color space atlas! The way that Tauba plays with words in these pieces reminds me both of the word art of Scott Kim and the word puzzles of Douglas Hofstadter. Some of Tauba’s ink-on-paper designs remind me of the work of Chloé Worthington. And Tauba’s piece Componants, Numbers gives me some new insight into Brandon Todd Wilson’s numbers project.

This piece by Tauba is a Math Munch fave!

For me, both math and art are all about playing with patterns, images, structures, and ideas. Maybe that’s why math and art make such a great combo—because they “play” well together!

Speaking of playing, I’d like to wrap up this week’s post by sharing a game about numbers I ran across recently. It’s called . . . A Game of Numbers! I really like how it combines the structure of arithmetic operations with the strategy of an escape game. A Game of Numbers was designed by a software developer named Joseph Michels for a “rapid” game competition called Ludum Dare. Here’s a Q&A Joseph did about the game.

A Game of Numbers.

If you enjoy A Game of Numbers, maybe you’ll leave Joseph a comment on his post about the game’s release or drop him an email. And if you enjoy A Game of Numbers, then you’d probably enjoy checking out some of the other games on our games page.

Bon appetit!

PS Tauba also created a musical instrument called an auerglass that requires two people to play. Whooooooa!

Reflection Sheet – MoMA, Pop-Up Books, and A Game of Numbers

Posted by Justin Lanier. Categories: Math Munch. Tags: applet, arithmetic, art, building, dimensions, games, geometry, interview, music, numbers, origami, pop-up, sculpture, spirals, video. 63 Comments