Tag Archives: music

Lucea, Fiber Bundles, and Hamilton

Welcome to this week’s Math Munch!

The Summer Olympics are underway in Brazil. I have loved the Olympics since I was a kid. The opening ceremony is one of my favorite parts—the celebration of the host country’s history and culture, the athletes proudly marching in and representing their homeland. And the big moment when the Olympic cauldron is lit! This year I was just so delighted by the sculpture that acted as the cauldron’s backdrop.

Isn’t that amazing! The title of this enormous metal sculpture is Lucea, and it was created by American sculptor Anthony Howe. You can read about Anthony and how he came to make Lucea for the Olympics in this article. Here’s one quote from Anthony:

“I hope what people take away from the cauldron, the Opening Ceremonies, and the Rio Games themselves is that there are no limits to what a human being can accomplish.”

Here’s another view of Lucea from Anthony’s website:

Lucea is certainly hypnotizing in its own right, but I think it jumped out at me in part because I’ve been thinking a lot about fiber bundles recently. A fiber bundle is a “twist” on a simpler kind of object called a product space. You are familiar with some examples of products spaces. A square is a line “times” a line. A cylinder is a line “times” a circle. And a torus is a circle “times” a circle.


Square, cylinder, and torus.

So, what does it mean to introduce a “twist” to a product space? Well, it means that while every little patch of your object will look like a product, the whole thing gets glued up in some fancy way. So, instead of a cylinder that goes around all normal, we can let the line factor do a flip as it goes around the circle and voila—a Mobius strip!


Now, check out this image:


It’s two Mobius strips stuck together! Does this remind you of Lucea?! Instead of a line “times” a circle that’s been twisted, we have an X shape “times” a circle.

Do you think you could fill up all of space with an infinity of circles? You might try your hand at it. One answer to this puzzle is a wonderful example of a fiber bundle called the Hopf fibration. Just as you can think about a circle as a line plus one extra point to close it up, and a sphere as a plane with one extra point to close it up, the three-sphere is usual three-dimenional space plus one extra point. The Hopf fibration shows that the three-sphere is a twisted product of a sphere “times” a circle. For a really lovely visualization of this fact, check out this video:

That is some tough but also gorgeous mathematics. Since you’ve made it this far in the post, I definitely think you deserve to indulge and maybe rock out a little. And what’s the hottest ticket on Broadway this summer? I hope you’ll enjoy this superb music video about Hamilton!

William Rowan Hamilton, that is. The inventor of quaternions, explorer of Hamiltonian circuits, and reformulator of physics. Brilliant.

citymapHere are a couple of pages of Hamiltonian circuit puzzles. The goal is to visit every dot exactly once as you draw one continuous path. Try them out! Rio, where the Olympics is happening, pops up as a dot in the first one. You might even try your hand at making some Hamiltonian puzzles of your own.

Happy puzzling, and bon appetit!

Pi Digit, Pi Patterns, and Pi Day Anthem


Painting by Renée Othot for Simon Plouffe’s birthday.

Welcome to this week’s Math Munch!

It’s here—the Pi Day of the Century happens on Saturday: 3-14-15!

How will you celebrate? You might check to see if there are any festivities happening in your area. There might be an event at a library, museum, school, or university near you.

(Here are some pi day events in NYC, Baltimore, San Francisco, Philadelphia, Houston, and Charlotte.)


John Conway at the pi recitation contest in Princeton.

John Conway at the pi recitation contest in Princeton.

There’s a huge celebration here in Princeton—in part because Pi Day is also Albert Einstein’s birthday, and Albert lived in Princeton for the last 22 years of his life. One event involves kids reciting digits of pi and and is hosted by John Conway and his son, a two-time winner of the contest. I’m looking forward to attending! But as has been noted, memorizing digits of pi isn’t the most mathematical of activities. As Evelyn Lamb relays,

I do feel compelled to point out that besides base 10 being an arbitrary way of representing pi, one of the reasons I’m not fond of digit reciting contests is that, to steal an analogy I read somewhere, memorizing digits of pi is to math as memorizing the order of letters in Robert Frost’s poems is to literature. It’s not an intellectually meaningful activity.

I haven’t memorized very many digits of pi, but I have memorized a digit of pi that no one else has. Ever. In the history of the world. Probably no one has ever even thought about this digit of pi.

And you can have your own secret digit, too—all thanks to Simon Plouffe‘s amazing formula.


Simon’s formula shows that pi can be calculated chunk by chunk in base 16 (or hexadecimal). A single digit of pi can be plucked out of the number without calculating the ones that come before it.

Wikipedia observes:

The discovery of this formula came as a surprise. For centuries it had been assumed that there was no way to compute the nth digit of π without calculating all of the preceding n − 1 digits.

Check out some of Simon's math art!

Check out some of Simon’s math art!

Simon is a mathematician who was born in Quebec. In addition to his work on the digits of irrational numbers, he also helped Neil Sloane with his Encyclopedia of Integer Sequences, which soon online and became the OEIS (previously). Simon is currently a Trustee of the OEIS Foundation.

There is a wonderful article by Simon and his colleagues David Bailey, Jonathan Borwein, and Peter Borwein called The Quest for Pi. They describe the history of the computation of digits of pi, as well as a description of the discovery of their digit-plucking formula.

According to the Guinness Book of World Records, the most digits that someone has memorized and recited is 67,890. Unofficial records go up to 100,000 digit. So just to be safe, I’ve used an algorithm by Fabrice Bellard based on Simon’s formula to calculate the 314159th digit of pi. (Details here and here.) No one in the world has this digit of pi memorized except for me.

Ready to hear my secret digit of pi? Lean in and I’ll whisper it to you.

The 314159th digit of pi is…7. But let’s keep that just between you and me!

And just to be sure, I used this website to verify the 314159th digit. You can use the site to try to find any digit sequence in the first 200 million digits of pi.

Aziz and Peter's patterns.

Aziz & Peter’s patterns.

Next up: we met Aziz Inan in last week’s post. This week, in honor of Pi Day, check out some of the numerical coincidences Aziz has discovered in the early digits in pi. Aziz and his colleague Peter Osterberg wrote an article about their findings. By themselves, these observations are nifty little patterns. Maybe you’ll find some more of your own. (This kind of thing reminds me of the Strong Law of Small Numbers.) As Aziz and Peter note at the end of the article, perhaps the study of such little patterns will one day help to show that pi is a normal number.

And last up this week, to get your jam on as Saturday approaches, here’s the brand new Pi Day Anthem by the recently featured John Sims and the inimitable Vi Hart.

Bon appetit!

SquareRoots, Concave States, and Sea Ice

Welcome to this week’s Math Munch!

The most epic Pi Day of the century will happen in just a few weeks: 3/14/15! I hope you’re getting ready. To help you get into the spirit, check out these quilts.

American Pi.

American Pi.

African American Pi.

African American Pi.

There’s an old joke that “pi is round, not square”—a punchline to the formula for the area of a circle. But in these quilts, we can see that pi really can be square! Each quilt shows the digits of pi in base 3. The quilts are a part of a project called SquareRoots by artist and mathematician John Sims.

John Sims.

John Sims.

There’s lots more to explore and enjoy on John’s website, including a musical interpretation of pi and some fractal trees that he has designed. John studied mathematics as an undergrad at Antioch College and has pursued graduate work at Wesleyan University. He even created a visual math course for artists when he taught at the Ringling College of Art and Design in Florida.

I enjoyed reading several articles (1, 2, 3, 4) about John and his quilts, as well as this interview with John. Here’s one of my favorite quotes from it, in response to “How do you begin a project?”

It can happen in two ways. I usually start with an object, which motivates an idea. That idea connects to other objects and so on, and, at some point, there is a convergence where idea meets form. Or sometimes I am fascinated by an object. Then I will seek to abstract the object into different spatial dimensions.


Cellular Forest and Square Root of a Tree, by John Sims.

You can find more of John’s work on his YouTube channel. Check out this video, which features some of John’s music and an art exhibit he curated called Rhythm of Structure.

Next up: Some of our US states are nice and boxy—like Colorado. (Or is it?) Other states have very complicated, very dent-y shapes—way more complicated than the shapes we’re used to seeing in math class.

Which state is the most dent-y? How would you decide?


West Virginia is pretty dent-y. By driving “across” it, you can pass through many other states along the way.

The mathematical term for dent-y is “concave”. One way you might try to measure the concavity of a state is to see how far outside of the state you can get by moving in a straight line from one point in it to another. For example, you can drive straight from one place in West Virginia to another, and along the way pass through four other states. That’s pretty crazy.

But is it craziest? Is another state even more concave? That’s what this study set out to investigate. Click through to find out their results. And remember that this is just one way to measure how concave a state is. A different way of measuring might give a different answer.

Awesome animal kingdom gerrymandering video!

Awesome animal kingdom gerrymandering video!

This puzzle about the concavity of states is silly and fun, but there’s more here, too. Thinking about the denty-ness of geographic regions is very important to our democracy. After all, someone has to decide where to draw the lines. When regions and districts are carved out in a way that’s unfair to the voters and their interests, that’s called gerrymandering.

Karen Saxe

Karen Saxe.

To find out more about the process of creating congressional districts, you can listen to a talk by Karen Saxe, a math professor at Macalester College. Karen was a part of a committee that worked to draw new congressional districts in Minnesota after the 2010 US Census. (Karen speaks about compactness measures starting here.)

Recently I ran across an announcement for a conference—a conference that was all about the math of sea ice! I never grow tired of learning new and exciting ways that math connects with the world. Check out this video featuring Kenneth Golden, a leading mathematician in the study of sea ice who works at the University of Utah. I love the line from the video: “People don’t usually think about mathematics as a daring occupation.” Ken and his team show that math can take you anywhere that you can imagine.

Bon appetit!

Reflection sheet – SquareRoots, Concave States, and Sea Ice

Zentangle, Graph Paper, and Pancake Art

My recent doodling.

Some recent doodling, by me.

Welcome to this week’s Math Munch!

As you start a new school year, you might be looking for some new mathy doodle games to play in the margins of your notebooks. Doodling helps me to listen sometimes, and I love making neat patterns. I especially like seeing what new shapes I can make.

This summer I was very happy to run across Zentangle®—”an easy-to-learn, relaxing, and fun way to create beautiful images by drawing structured patterns.” I’ve learned a lot about Zentangle from a blog called Tangle Bucket by Sandy Hunter. She shares how to doodle snircles, snafoozles, and oodles. There’s a whole dictionary of zentangle shapes over at tanglepatterns.com.

My favorite idea in Zentangle is trying to combine two kinds of designs. Sometimes this is described as one pattern “versus” another one. For instance, check out these:


Maybe you’ll pick some tangle patterns to combine with each other. If you try some, maybe you’ll share them in our Readers’ Gallery.

Sandy writes:

It’s so true that the more I tangle, the more I see the potential in patterns all around me. I catch myself mentally deconstructing them (whether I want to or not) to figure out if they can be broken down into simple steps without too much effort. That’s the trademark of a good tangle pattern.

Try some of Sandy’s weekly challenges, or check out Tiffany Lovering’s time-lapse videos—here’s one with music and one with an interview. Can you learn the names of any of the shapes she creates? I spy a Rick’s Paradox. There are lots of ways to begin zentangling—I hope you enjoy giving it a try.

Squares and dots and crosses, oh my!

Squares & dots & crosses, oh my!

If zentangling is too freeform for your doodling tastes, then let me share with you one of my longtime favorite websites. I’ve used it for years to help me to do math and to teach math, and it’s great for math doodling, too. I might even call it a trusty friend, one that I met one day through the simple online search: “free online graph paper”.

That’s right, it’s Free Online Graph Paper.

Something I love about the site is that it lets you design different aspects of your graph paper. Then you can print it out. First you get to decide what kind of grid you would like: square? triangular? circular? Then you get to tinker with lots of variables, like how big the grid cells are, how dark the lines are, and what color they are. And more!

Free Online Graph Paper was created by Kevin MacLeod, who composes music and shares it for free. That way other people can use it for creative projects. That’s really awesome! I enjoyed listening to Kevin’s “Winner Winner“. It’s always good to be reminded that everything you use or enjoy was almost certainly made by a person—including custom graph paper websites!

A 7/3 star spirocake.

A 7/3 star spirocake.

Last up this week is some doodly math that you can really munch on. Everyone knows that breakfast is the most important meal of the day and that the most important food group is roulette curves.

To get your daily recommended allowance of groovy math, look no further than the edible doodles of Nathan Shields and his family over at Saipancakes.

I can wait until the Shields family tackles the cissoid of Diocles.

Bon appetit!

Virtual Hyenas, Markov Chains, and Random Knights

Welcome to this week’s Math Munch!

It’s amazing how a small step can lead to a chain reaction of adventure.

Arend Hintze

Arend Hintze

Recently a reader named Nico left a comment on the Math Munch post where I shared the game Loops of Zen. He asked why the game has that name. Curious, I looked up Dr. Arend Hintze, whose name appears on the game’s title page. This led me to Arend’s page at the Adami Lab at Michigan State University. Arend studies how complex systems—especially biological systems—evolve over time.

Here is a video of one of Arend’s simulations. The black and white square is a zebra. The yellow ones are lions, the red ones are hyenas, and guess who’s hungry?

Arend’s description of the simulation is here. The cooperative behavior in the video—two hyenas working together to scare away a lion—wasn’t programmed into the simulation. It emerged out of many iterations of systems called Markov Brains—developed by Arend—that are based upon mathematical structures called Markov chains. More on those in a bit.

You can read more about how Arend thinks about his multidisciplinary work on biological systems here. Also, it turns out that Arend has made many more games besides Loops of Zen. Here’s Blobs of Zen, and Ink of Zen is coming out this month! Another that caught my eye is Curve, which reminds me of some of my favorite puzzle games. Curve is still in development; here’s hoping we’ll be able to play it soon.

Arend has agreed to do an interview with Math Munch, so share your questions about his work, his games, and his life below!

Eric Czekner

Eric Czekner

Arend’s simulations rely on Markov chains to model animal behavior. So what’s a Markov chain? It’s closely related to the idea of a random walk. Check out this video by digital artist, musician, and Pure Data enthusiast Eric Czekner. In the video, Eric gives an overview of what Markov chains are all about and shows how he uses them to create pieces of music.

On this page, Eric describes how he got started using Markov chains to make music, along with several of his compositions. It’s fascinating how he captures the feel of a song by creating a mathematical system that “generates new patterns based on existing probabilities.”

Now there’s a big idea: exploring something randomly can capture structures that might be hard to perceive otherwise. Here’s one last variation on the Markov chain theme that involves a pure math question. This blog post ponders the question: what happens when a knight takes a random walk—or random trot?—on a chessboard? It includes some colorful images of chessboards along the way.

How likely it is that a knight lands on each square after five moves, starting from b1.

How likely it is that a knight lands on each square after five moves, starting from b1.

The probabili

How likely it is that a knight lands on each square after 200 moves, starting from b1.

The blogger—Leonid Kovalev—shows in his analysis what happens in the long run: the number of times a knight will visit a square will be proportional to the number of moves that lead to that square. For instance, since only two knight moves can reach a corner square while eight knight moves can reach a central square, it’s four times as likely that a knight will finish on a central square after a long, long journey than on a corner square. This idea works because moving a knight around a chessboard is a “reversible Markov chain”—any path that a knight can trace can also be untraced. The author also wrote a follow-up post about random queens.

It’s amazing the things you can find by chaining together ideas or by taking a random walk. Thanks for the inspiration for this post, Nico. Keep those comments and questions coming, everyone—we love hearing from you.

Bon appetit!

Isomorphisms in Five, Parquet Deformations, and POW!

Welcome to this week’s Math Munch!

Here’s a catchy little video. It’s called “Isomorphisms in Five.” Can you figure out why? The note posted below the video says:

An isomorphism is an underlying structure that unites outwardly different mathematical expressions. What underlying structure do these figures share? What other isomorphisms of this structure will you discover?

One of the reasons I LOVE this video is because I really like how the shapes change with the music– which is played in a very interesting time signature. I also love how you can learn a lot about the different growing shape patterns by comparing them. Watch how they grow as the video flips from pattern to pattern. What do you notice? What does the music tell you about their growth?

This video is by a math educator from North Carolina named Stuart Jeckel. The only thing written about him on his “About” page is, “The Art of Math”– so he’s a bit of a mystery! He has three more beautiful videos, all of which present little puzzles for you to solve. Check them out!

(Five-four isn’t a common time-signature for music, but it makes some great pieces. Check out this particularly awesome one. Anyone want to try making a growing shape pattern video to this tune?)


Here is an example of one of my favorite types of geometric patterns– the parquet deformation. To make one, you start with a tessellation. Then you change it- very gradually- until you’ve made a completely different tessellation that’s connected by many tiny steps to the original one.

I love to draw them. It’s challenging, but full of surprises. I never know what it’s going to look like in the end.

2012_10_31-par5Want to try making your own? Check out this site by the professors/architects Tuğrul Yazar and Serkan Uysal. They had one of their classes map out how some different parquet deformations are made. They mostly used computers, but you could follow their instructions by hand, if you like. The image above is a map for the first deformation I showed.

Click on this link to see some awesome deformations made out of tiles. Aren’t they beautiful? And here’s one made by mathematical artist Craig Kaplan. It has a great fractal quality to it:


Finally, here’s something I’ve been meaning to share with you for ages! Do you ever crave a good puzzle and aren’t sure where to find one? Look no farther than the Saint Ann’s School Problem of the Week! Each week, math teacher Richard Mann writes a new awesome problem and posts it on this website. Here’s this week’s problem:

For November 26, 2013– In the picture below, find the shaded right triangle marked A, the equilateral triangle marked B and the striped regular hexagon marked C. Six students make the following statements about the picture below: Anne says “I can find an equilateral triangle three times the area of B.”  Ben says I can find an equilateral triangle four times the area of B.” Carol says, “I can find a find a right triangle triple the area of A.” Doug says, “I can find a right triangle five times the area of A.” Eloise says, “I can find a regular hexagon double the area of C.” Frank says, “I can find a regular hexagon three times the area of C.” Which students are undoubtedly mistaken?

30- 60-90

If you solve this week’s problem, send us a solution!

Bon appetit!

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.

nathanNathan 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!

565px-MastermindNext 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??

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!