Tag Archives: topology

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!

“Happy Birthday, Euler!”, Project Euler, and Pants

Welcome to this week’s Math Munch!

Did you see the Google doodle on Monday?

Leonhard Euler Google doodleThis medley of Platonic solids, graphs, and imaginary numbers honors the birthday of mathematician and physicist Leonhard Euler. (His last name is pronounced “Oiler.” Confusing because the mathematician Euclid‘s name is not pronounced “Oiclid.”) Many mathematicians would say that Euler was the greatest mathematician of all time – if you look at almost any branch of mathematics, you’ll find a significant contribution made by Euler.

480px-Leonhard_Euler_2Euler was born on April 15, 1707, and he spent much of his life working as a mathematician for one of the most powerful monarchs ever, Frederick the Great of Prussia. In Euler’s time, the kings and queens of Europe had resident mathematicians, philosophers, and scientists to make their countries more prestigious.  The monarchs could be moody, so mathematicians like Euler had to be careful to keep their benefactors happy. (Which, sadly, Euler did not. After almost 20 years, Frederick the Great’s interests changed and he sent Euler away.) But, the academies helped mathematicians to work together and make wonderful discoveries.

Want to read some of Euler’s original papers? Check out the Euler Archive. Here’s a little bit of an essay called, “Discovery of a Most Extraordinary Law of Numbers, Relating to the Sum of Their Divisors,” which you can find under the subject “Number Theory”:

Mathematicians have searched so far in vain to discover some order in the progression of prime numbers, and we have reason to believe that it is a mystery which the human mind will never be able to penetrate… This situation is all the more surprising since arithmetic gives us unfailing rules, by means of which we can continue the progression of these numbers as far as we wish, without however leaving us the slightest trace of any order.

Mathematicians still find this baffling today! If you’re interested in dipping your toes into Euler’s writings, I’d suggest checking out other articles in “Number Theory,” such as “On Amicable Numbers,” or some articles in “Combinatorics and Probability,” like “Investigations on a New Type of Magic Square.”

pe_banner_lightWant to work, like Euler did, on important math problems that will stretch you to make connections and discoveries? Check out Project Euler, an online set of math and computer programming problems. You can join the site and, as you work on the problems, talk to other problem-solvers, contribute your solutions, and track your progress. The problems aren’t easy – the first one on the list is, “Find the sum of all the multiples of 3 and 5 below 1000” – but they build on one another (and are pretty fun).


Pants made from a crocheted model of the hyperbolic plane, by Daina Taimina.

Finally, if someone asked you what a pair of pants is, you probably wouldn’t say, “a sphere with three open disks removed.” But maybe you also didn’t know that pants are important mathematical objects!

I ran into a math problem involving pants on Math Overflow (previously). Math Overflow is a site on which mathematicians can ask and answer each other’s questions. The question I’m talking about was asked by Tony Huynh. He knew it was possible to turn pants inside-out if your feet are tied together. (Check out the video below to see it done!) Tony was wondering if it’s possible to turn your pants around, so that you’re wearing them backwards, if your feet are tied together.

Is this possible? Another mathematician answered Tony’s question – but maybe you want to try it yourself before reading about the solution. Answering questions like this about transformations of surfaces with holes in them is part of a branch of mathematics called topology – which Euler is partly credited with starting. A more mathematical way of stating this problem is: is it possible to turn a torus (or donut) with a single hole in it inside-out? Here’s another video, by James Tanton, about turning things inside-out mathematically.

Bon appetit!

MMteam-240x240P.S. – The Math Munch team will be speaking next weekend, on April 27th, at TEDxNYED! We’re really excited to get to tell the story of Math Munch on the big stage. Thank you for being such enthusiastic and curious readers and allowing us to share our love of math with you. Maybe we’ll see some of you there!

The Museum of Math, Shapes That Roll, and Mime-matics

Welcome to this week’s Math Munch!  We have so many exciting things to share with you this week – so let’s get started!

Something very exciting to math lovers all over the world happened on Saturday.  The Museum of Mathematics opened its doors to the public!

MoMath entranceThe Museum of Mathematics (affectionately called MoMath – and that’s certainly what you’ll get if you go there) is in the Math Munch team’s hometown, New York City.

human treeThere are so many awesome exhibits that I hardly know where to start.  But if you go, be sure to check out one of my favorite exhibits, Twist ‘n Roll.  In this exhibit, you roll some very interestingly shaped objects along a slanted table – and investigate the twisty paths that they take.  And you can’t leave without seeing the Human Tree, where you turn yourself into a fractal tree.

coaster rollersOr going for a ride on Coaster Rollers, one of the most surprising exhibits of all.  In this exhibit, you ride in a cart over a track covered with shapes that MoMath calls “acorns.”  The “acorns” aren’t spheres – and yet your ride over them is completely smooth!  That’s because these acorns, like spheres, are surfaces of constant width.  That means that if you pick two points on opposite ends of the acorn – with “opposite” meaning points that you could hold between your hands while your hands are parallel to each other – the distance between those points is the same regardless of the points you choose.  See some surfaces of constant width in action in this video:

Rouleaux_triangle_AnimationOne such surface of constant width is the shape swept out by rotating a shape called a Reuleaux triangle about one of its axes of symmetry.  Much as an acorn is similar to a sphere, a Reuleaux triangle is similar to a circle.  It has constant diameter, and therefore rolls nicely inside of a square.  The cart that you ride in on Coaster Rollers has the shape of a Reuleaux triangle – so you can spin around as you coast over the rollers!

Maybe you don’t live in New York, so you won’t be able to visit the museum anytime soon.  Or maybe you want a little sneak-peek of what you’ll see when you get there.  In any case, watch this video made by mathematician, artist, and video-maker George Hart on his first visit to the museum.  George also worked on planning and designing the exhibits in the museum.

We got the chance to interview Emily Vanderpol, the Outreach Exhibits coordinator for MoMath, and Melissa Budinic, the Assistant Exhibit Designer for MoMath.  As Cindy Lawrence, the Associate Director for MoMath says, “MoMath would not be open today if it were not for the efforts” of Emily and Melissa.  Check out Melissa and Emily‘s interviews to read about their favorite exhibits, how they use math in their jobs for MoMath, and what they’re most excited about now that the museum is open!

mimematicsLogo (1)Finally, meet Tim and Tanya Chartier.  Tim is a math professor at Davidson College in North Carolina, and Tanya is a language and literacy educator.  Even better, Tim and Tanya have combined their passion for math and teaching with their love of mime to create the art of Mime-matics!  Tim and Tanya have developed a mime show in which they mime about some important concepts in mathematics.  Tim says about their mime-matics, “Mime and math are a natural combination.  Many mathematical ideas fold into the arts like shape and space.  Further, other ideas in math are abstract themselves.  Mime visualizes the invisible world of math which is why I think math professor can sit next to a child and both get excited!”

One of my favorite skits, in which the mime really does help you to visualize the invisible world of math, is the Infinite Rope.  Check it out:

slinkyIn another of my favorite skits, Tanya interacts with a giant tube that twists itself in interesting topological ways.  Watch these videos and maybe you’ll see, as Tanya says, how a short time “of positive experiences with math, playing with abstract concepts, or seeing real live application of math in our world (like Google, soccer, music, NASCAR, or the movies)  can change the attitude of an audience member who previously identified him/herself as a “math-hater.””  You can also check out Tim’s blog, Math Movement.

Tim and Tanya kindly answered some questions we asked them about their mime-matics.  Check out their interview by following this link, or visit the Q&A page.

Bon appetit!