Tag Archives: music

Polyominoes, Clock Calculator, and Nine Bells

Welcome to this week’s Math Munch!

pentominoes!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?

pentahexesWe 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…

whoa square

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.

clock calculatorWhat 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!

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!

N_JoshiTo 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.

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:

CRI_244599 CRI_244605

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.

Taura Auerbach.

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.

0108 MM MM-Tauba-Auerbach-large

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.

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

Temari, Function Families, and Clapping Music

Welcome to this week’s Math Munch!

Carolyn Yackel

Carolyn Yackel

As Justin mentioned last week, the Math Munch team had a blast at the MOVES conference last week.  I met so many lovely mathematicians and learned a whole lot of cool math. Let me introduce you to Carolyn Yackel. She’s a math professor at Mercer University in Georgia, and she’s also a mathematical fiber artist who specializes in the beautiful Temari balls you can see below or by clicking the link. Carolyn has exhibited at the Bridges conference, naturally, and her 2012 Bridges page contains an artist statement and some explanation of her art.

temari15 temari3 temari16

Icosidodecahedron

Icosidodecahedron

Truncated Dodecahedron

Truncated Dodecahedron

Cuboctahedron

Cuboctahedron

Temari is an ancient form of japanese folk art. These embroidered balls feature various spherical symmetries, and part of Carolyn’s work has been figure out how to create and exploit these symmetries on the sphere.  I mean how do you actually make it that symmetric? Can you see in the pictures above how the symmetry of the Temari balls mimic the Archimedean solids? Carolyn has even written about using Temari to teach mathematics, some of which you can read here, if you like.

Read Carolyn Yackel’s Q&A with Math Munch.

 

Edmund Harriss

Edmund Harriss

Up next, you may remember Edmund Harriss from this post, and you might recall Desmos from this post. Well the two have come together! On his blog, Maxwell’s Demon, Edmund shared a whole bunch of interactive graphs from Desmos, in a post he called “Form Follows Function.” Click on the link to read the article, and click on the images to get graphs full of sliders you can move to alter the images. In fact, you can even alter the equations that generate them, so dig in, play some, and see what you can figure out.

graph2 graph1 graph3

Finally, I want to share a piece of music I really love. “Clapping Music” was written by Steve Reich in 1972. It is considered minimalist music, perhaps because it features two performers doing nothing but clapping. If you watch this performance of “Clapping Music” first (and I suggest you do) it might just sound like a bunch of jumbled clapping. But the clapping is actually built out of some very simple and lovely mathematical patterns. Watch the video below and you’ll see what I mean.

Did you see the symmetries in the video? I noticed that even though the pattern shifts, it’s always the same backwards as forwards. And I also noticed that the whole piece is kind of the same forwards and backwards, because of the way that the pattern lines back up with itself. Watch again and see if you get what I mean.

Bonus: Math teacher, Greg Hitt tweeted me about “Clapping Music” and shared this amazing performance by six bounce jugglers!!!  It’s cool how you can really see the patterns in the live performance.

I hope you find something you love and dig in. Bon appetit!

Harmonious Sum, Continuous Life, and Pumpkins

Welcome to this week’s Math Munch!

We’ve posted a lot about pi on Math Munch – because it’s such a mathematically fascinating little number.  But here’s something remarkable about pi that we haven’t yet talked about. Did you know that pi is equal to four times this? Yup.  If you were to add and subtract fractions like this, for ever and ever, you’d get pi divided by 4.  This remarkable fact was uncovered by the great mathematician Gottfried Wilhelm Leibniz, who is most famous for developing the calculus.  Check out this interactive demonstration from the Wolfram Demonstrations Project to see how adding more and more terms moves the sum closer to pi divided by four.  (We’ve written about Wolfram before.)

I think this is amazing for a couple of reasons.  First of all, how can an infinite number of numbers add together to make something that isn’t infinite???  Infinitely long sums, or series, that add to a finite number have a special name in mathematics: convergent series.  Another famous convergent series is this one:

The second reason why I think this sum is amazing is that it adds to pi divided by four.  Pi is an irrational number – meaning it cannot be written as a fraction, with whole numbers in the numerator and denominator.  And yet, it’s the sum of an infinite number of rational numbers.

In this video, mathematician Keith Devlin talks about this amazing series and a group of mathematical musicians (or mathemusicians) puts the mathematics to music.

This video is part of a larger work called Harmonious Equations written by Keith and the vocal group Zambra.  Watch the rest of them, if you have the chance – they’re both interesting and beautiful.

Next up, Conway’s Game of Life is a cellular automaton created by mathematician John Conway.  (It’s pretty fun: check out this to download the game, and this Munch where we introduce it.)  It’s discrete – each little unit of life is represented by a tiny square.  What if the rules that determine whether a new cell is formed or the cell dies were applied to a continuous domain?  Then, it would look like this:

Looks like a bunch of cells under a microscope, doesn’t it?  Well, it’s also a cellular automaton, devised by mathematician Stephan Rafler from Nurnberg, Germany.  In this paper, Stephan describes the mathematics behind the model.  If you’re curious about how it works, check out these slides that compare the new continuous version to Conway’s model.

Finally, I just got a pumpkin.  What should I carve in it?  I spent some time browsing the web for great mathematical pumpkin carvings.  Here’s what I found.

A pumpkin carved with a portion of Escher’s Circle Limit.

A pumpkin tiled with a portion of Penrose tiling.

A dodecapumpkin from Vi Hart.

I’d love to hear any suggestions you have for how I should make my own mathematical pumpkin carving!  And, if you carve a pumpkin in a cool math-y way, send a picture over to MathMunchTeam@gmail.com!

Bon appetit!

Bridges, Meander Patterns, and Water Sports

This past week the Math Munch team got to attend the Bridges 2012. Bridges is a mathematical art conference, the largest one in the world. This year it was held at Towson University outside of Baltimore, Maryland. The idea of the conference is to build bridges between math and the arts.

Participants gave lectures about their artwork and the math that inspired or informed it. There were workshop sessions about mathematical poetry and chances to make baskets and bead bracelets involving intricate patterns. There was even a dance workshop about imagining negative-dimensional space! There were also some performances, including two music nights (which included a piece that explored a Fibonacci-like sequence called Narayana’s Cows) and a short film festival (here are last year’s films). Vi Hart and George Hart talked about the videos they make and world-premiered some new ones. And at the center of it all was an art exhibition with pieces from around the world.

The Zen of the Z-Pentomino by Margaret Kepner

Does this piece by Bernhard Rietzl
remind you of a certain sweater?

5 Rhombic Screens by Alexandru Usineviciu

Pythagorean Proof by Donna Loraine

To see more, you should really just browse the Bridges online gallery.

A shot of the gallery exhibition

I know that Paul, Anna, and I will be sharing things with you that we picked up at Bridges for months to come. It was so much fun!

David Chappell

One person whose work and presentation I loved at Bridges is David Chappell. David is a professor of astronomy at the University of La Verne in California.

David shared some thinking and artwork that involve meander patterns. “Meander” means to wander around and is used to describe how rivers squiggle and flow across a landscape. David uses some simple and elegant math to create curve patterns.

Instead of saying where curves sit in the plane using x and y coordinates, David describes them using more natural coordinates, where the direction that the curve is headed in depends on how far along the curve you’ve gone. This relationship is encoded in what’s called a Whewell equation. For example, as you walk along a circle at a steady rate, the direction that you face changes at a contant rate, too. That means the Whewell equation of a circle might look like angle=distance. A smaller circle, where the turning happens faster, could be written down as angle=2(distance).

Look at how the Cauto River “meanders” across the Cuban landscape.

In his artwork, David explores curves whose equations are more complicated—ones that involve multiple sine functions. The interactions of the components of his equations allow for complex but rhythmic behavior. You can create meander patterns of your own by tinkering with an applet that David designed. You can find both the applet and more information about the math of meander patterns on David’s website.

David Chappell’s Meander #6
Make your own here!

When I asked David about how being a scientist affects his approach to making art, and vice versa, he said:

My research focuses on nonlinear dynamics and pattern formation in fluid systems. That is, I study the spatial patterns that arise when fluids are agitated (i.e. shaken or stirred). I think I was attracted to this area because of my interest in the visual arts. I’ve always been interested in patterns. The science allows me to study the underlying physical systems that generate the patterns, and the art allows me to think about how and why we respond to different patterns the way we do.  Is there a connection between how we respond to a visual image and the underlying “rules” that produced the image?  Why to some patterns look interesting, but others not so much?

For more of my Q&A with David, click here. In addition, David will be answering questions in the comments below, so ask away!

Since bridges and meandering rivers are both water-related, I thought I’d round out this post with a couple of interesting links about water sports and the Olympics. My springboard was a site called Maths and Sport: Countdown to the Games.

No wiggle rigs

Arrangements of rowers that are “wiggle-less”

Here’s an article that explores different arrangements of rowers in a boat, focusing on finding ones where the boat doesn’t “wiggle” as the rowers row. It’s called Rowing has its Moments.

Next, here’s an article about the swimming arena at the 2008 Beijing games, titled Swimming in Mathematics.


Paul used to be a competitive diver, and he says there’s an interesting code for the way dives are numbered.  For example, the “Forward 1 ½ Somersaults in Tuck Position” is dive number 103C.  How does that work?  You can read all about it here.  (Degree of difficulty is explained as well.)

Finally, enjoy these geometric patterns inspired by synchronized swimming!

Stay cool, and bon appetit!

Math Cats, Frieze Music, and Numbers

Welcome to this week’s Math Munch!

I just ran across a website that’s chock full of cool math applets, links, and craft ideas – and perfect for fulfilling those summer math cravings!  Math Cats was created by teacher and parent Wendy Petti to, as she says on her site, “promote open-ended and playful explorations of important math concepts.”

Math Cats has a number of pages of interesting mathematical things to do, but my favorite is the Math Cats Explore the World page.  Here you’ll find links to cool math games and explorations made by Wendy, such as…

… the Crossing the River puzzle!  In this puzzle, you have to get a goat, a cabbage, and a wolf across a river without any of your passengers eating each other!  And…

… the Encyclogram!  Make beautiful images called harmonograms, spirographs, and lissajous figures with this cool applet.  Wendy explains some of the mathematics behind these images, too. And, one of my favorites…

Scaredy Cats!  If you’ve ever played the game NIM, this game will be very familiar.  Here you play against the computer to chase cats away – but don’t be left with the last cat, or you’ll lose!

These are only a few of the fun activities to try on Math Cats.  If you happen to be a teacher or parent, I recommend that you look at Wendy’s Idea Bank.  Here Wendy has put together a very comprehensive and impressive list of mathematics lessons, activities, and links contributed by many teachers.

Next, Vi Hart has a new video that showcases one of my favorite things in mathematics – the frieze.  A frieze is a pattern that repeats infinitely in one direction, like the footsteps of the person walking in a straight line above.  All frieze patterns have translation symmetry – or symmetry that slides or hops – but some friezes have additional symmetries.  The footsteps above also have glide reflection symmetry – a symmetry that flips along a horizontal line and then slides.  Frieze patterns frequently appear in architecture.  You can read more about frieze patterns here.

Reading about frieze patterns is all well and good – but what if you could listen to them?  What would a translation sound like?  A glide reflection?  The inverse of a frieze pattern?  Vi sings the sounds of frieze patterns in this video.

[youtube http://www.youtube.com/watch?v=Av_Us6xHkUc&feature=BFa&list=UUOGeU-1Fig3rrDjhm9Zs_wg]

Do you have your own take on frieze music?  Send us your musical compositions at MathMunchTeam@gmail.com .

Finally, if I were to ask you to name the number directly in the middle of 1 and 9, I bet you’d say 5.  But not everyone would.  What would these strange people say – and why would they also be correct?  Learn about this and some of the history, philosophy, and psychology of numbers – and hear some great stories – in this podcast from Radiolab.  It’s called Numbers.

Bon appetit!

P.S. – Paul made a new Yoshimoto video!  The Mega-Monster Mesh comes alive!  Ack!

[youtube https://www.youtube.com/watch?v=PMpr8pA5lJw&feature=player_embedded]

P.P.S. – Last week – June 28th, to be exact – was Tau Day.  For more information about Tau Day and tau, check out the last bit of this old Math Munch post.  In honor of the occasion, Vi Hart made this new tau video.  And there’s a remix.

Music Box, FatFonts, and the Yoshimoto Cube

Welcome to this week’s Math Munch!

The Whitney Music Box

Jim Bumgardner

Solar Beat

With the transit of Venus just behind us and the summer solstice just ahead, I’ve got the planets and orbits on my mind. I can’t believe I haven’t yet shared with you all the Whitney Music Box. It’s the brainchild of Jim Bumgardner, a man of many talents and a “senior nerd” at Disney Interactive Labs. His music box is one of my favorite things ever–so simple, yet so mesmerizing.

It’s actually a bunch of different music boxes–variations on a theme. Colored dots orbit in circles, each with a different frequency, and play a tone when they come back to their starting points. In Variation 0, for instance, within the time it takes for the largest dot to orbit the center once, the smallest dot orbits 48 times. There are so many patterns to see–and hear! There are 21 variations in all. Go nuts! In this one, only prime dots are shown. What do you notice?

You can find a more astronomical version of this idea at SolarBeat.

Above you’ll find a list of the numerals from 1 to 9. Or is it 0 to 9?

Where’s the 0 you ask? Well, the idea behind FatFonts is that the visual weight of a number is proportional to its numerical size. That would mean that 0 should be completely white!

FatFonts can also be nested. The first number below is 64. Can you figure out the second?

This is 64 in FatFonts.

What number is this?
Click to zoom!

FatFonts was developed by the team of Miguel NacentaUta Hinrichs, and Sheelagh Carpendale. You can see some uses that FatFonts has been put to on their Gallery page, and even download FatFonts to use in your word processor. Move over, Times New Roman!

This past week, Paul pointed me to this cool video by George Hart about interlocking complementary polyhedra that together form a cube. It reminded me of something I saw for the first time a few years ago that just blew me away. You have to see the Yoshimoto Cube to believe it:

In addition to its more obvious charms, something that delights me about the Yoshimoto Cube is how it was found so recently–only in 1971, by Naoki Yoshimoto.  (That other famous cube was invented in 1974 by Ernő Rubik.) How can it be that simple shapes can be so inexhaustible? If you’re feeling inspired, Make Magazine did a short post on the Yoshimoto Cube a couple of years that includes a template for making a Yoshimoto Cube out of paper. Edit: These template and instructions aren’t great. See below for better ones!

Since it’s always helpful to share your goals to help you stick to them, I’ll say that this week I’m going to make a Yoshimoto Cube of my own. Begone, back burner! Later in the week I’ll post some pictures below. If you decide to make one, share it in the comments or email us at

MathMunchTeam@gmail.com

We’d love to hear from you.

Bon appetit!

Update:

Here are the two stellated rhombic dodecahedra that make the Yoshimoto Cube that Paul and I made! Templates, instructions, and video to follow!

Here are two different templates for the Yoshimoto cubelet. You’ll need eight cubelets to make one star.

And here’s how you tape them together: