Tag Archives: algebra

Bridges, Unfolding the Earth, and Juggling

Welcome to this week’s Math Munch – from the Netherlands!

I’m at the Bridges Mathematical Art Conference, which this year is being held in Enschede, a city in the Netherlands. I’ve seen so much beautiful mathematical artwork, met so many wonderful people, and learned so many interesting new things that I can’t wait to start sharing them with you! In the next few weeks, expect many more interviews and links to sites by some of the world’s best mathematical artists.

But first, have a look at some of the artwork from this year’s art gallery at Bridges.

Hyperbolic lampshade

By Gabriele Meyer


By Henry Segerman and Craig Kaplan

Here are three pieces that I really love. The first is a crocheted hyperbolic plane lampshade. I love to crochet hyperbolic planes (and we’ve posted about them before), and I think the stitching and lighting on this one is particularly good. The second is a bunny made out of the word bunny! (Look at it very closely and you’ll see!) It was made by one of my favorite mathematical artists, Henry Segerman. Check back soon for an interview with him!

Hexagonal flower

By Francisco De Comite

This last is a curious sculpture. From afar, it looks like white arcs surrounding a metal ball, but up close you see the reflection of the arcs in the ball – which make a hexagonal flower! I love how this piece took me by surprise and played with the different ways objects look in different dimensions.

Jack van WijkMathematical artists also talk about their work at Bridges, and one of the talks I attended was by Jack van Wijk, a professor from Eindhoven University of Technology in the Netherlands. Jack works with data visualization and often uses a mixture of math and images to solve complicated problems.

One of the problems Jack tackled was the age-old problem of drawing an accurate flat map of the Earth. The Earth, as we all now know, is a sphere – so how do you make a map of it that fits on a rectangular piece of paper that shows accurate sizes and distances and is simple to read?

myriahedronTo do this, Jack makes what he calls a myriahedral projection. First, he draws many, many polygons onto the surface of the Earth – making what he calls a myriahedron, or a polyhedron with a myriad of faces. cylindrical mapThen, he decides how to cut the myriahedron up. This can be done in many different ways depending on how he wants the map to look. If he wants the map to be a nice, normal rectangle, maybe he’ll cut many narrow, pointed slits at the North and South Poles to make a map much like one we’re used to. But, maybe he wants a map that groups all the continents together or does the opposite and emphasizes how the oceans are connected…crazy maps

Jack made a short movie that he submitted to the Bridges gallery. He animates the transformation of the Earth to the map projections beautifully.

Jack’s short movie wasn’t the only great film I saw at Bridges. The usual suspects – Vi Hart and her father, George Hart – also submitted movies. George’s movie is about a math topic that I find particularly fascinating: juggling! The movie stars professional juggler Rod Kimball. Click on the picture below to watch:


This is only the tip of the iceberg that is the gorgeous and interesting artwork I saw at Bridges. Check out the gallery to see more (including artwork by our own Paul and a video by Paul and Justin!), or visit Math Munch again in the coming weeks to learn more about some of the artists.

Bon appetit!

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

265282-jiroemon-kimura-the-world-s-oldest-living-man-celebrated-his-115th-birOn 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!)

stackSome 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!)

advanced 4

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.)

qod0nxgctfMichael 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.

escher 3If 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!

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!