Tag Archives: mathematical art

Sphericon, National Curve Bank, and Cardioid String Art

Welcome to this week’s Math Munch!

Behold the Sphericon!

What is that? Well, it rolls like a sphere, but is made of two cones attached with a twist– hence, the spheri-con! The one in the video is made out of pie (not sure why…), but you can make sphericons out of all kinds of materials.

Wooden sphericonIt was developed by a few people at different times– like many brilliant new objects. But it entered the world of math when mathematician Ian Stewart wrote about it in his column in Scientific American. The wooden sphericon was made by Steve Mathias, an engineer from Sacramento, California, who read Ian’s article and thought sphericons would be fun to make. To learn more about how Steve made those beautiful wooden sphericons, check out his site!

Even if you’re not a woodworker, like Steve, you can still make your own sphericon. You can start with two cones and make one this way, by attaching the cones at their bases, slicing the whole thing in half, rotating one of the halves 90 degrees, and attaching again:How to make a sphericon

Or you can print out this image, cut it out, fold it up, and glue (click on the image for a larger printable size):

Sphericon pattern

If you do make your own sphericon (which I recommend, because they’re really cool), watch the path it makes as it rolls. See how it wiggles? What shape do you think the path is?

ncbmastertitleI found out about the sphericon while browsing through an awesome website– the National Curve Bank. It’s just what it sounds like– an online bank full of curves! You can even make a deposit– though, unlike a real bank, you can take out as many curves as you like. The goal of the National Curve Bank is to provide great pictures and animations of curves that you’d never find in a normal math book. Think of how hard it would be to understand how a sphericon works if you couldn’t watch a video of it rolling?

epicycloidaThere are lots of great animations of curves and other shapes in the National Curve Bank– like the sphericon! Another of my favorites is the “cycloid family.” A cycloid is the curve traced by a point on a circle as the circle rolls– like if you attached a pen to the wheel of your bike and rode it next to a wall, so that the pen drew on the wall. It’s a pretty cool curve– but there are lots of other related curves that are even cooler. The epicycloid (image on the right) is the curve made by the pen on your bike wheel if you rode the bike around a circle. Nice!

You should explore the National Curve Bank yourself, and find your own favorite curve! Let us know in the comments if you find one you like.

String cardioid

String art cardioid

Finally, to round out this week’s post on circle-y curves (pun intended), check out another of my favorite curves– the cardioid. A cardioid looks like a heart (hence the name). There are lots of ways to make a cardioid (some of which we posted about for Valentine’s Day a few years ago). But my favorite way is to make it out of string!

String art is really fun. If you’ve never done any string art, check out the images made by Julia Dweck’s class that we posted last year. Or, try making your own string art cardioid! This site shows you how to draw circles, ovals, cardioids, and spirals using just straight lines– you could follow the same instructions, replacing the straight lines you’d draw with pieces of string attached to tacks! If you’re not sure how the string part would work, check out this site for basic string art instructions.

Bon appetit!

Zippergons, High Fashion, and Really Big Numbers

Welcome to this week’s Math Munch!

Bill Thurston

Bill Thurston

Recently I attended a conference in memory of Bill Thurston. Bill was one of the most imaginative and influential mathematicians of the second half of the twentieth century. He worked with many mathematicians on projects and had many students before he passed away in the fall of 2012 at the age of 65. You can read Bill’s obituary in the New York Times here.

Bill worked where geometry and topology meet. In fact, Bill throughout his career showed that there are rich connections between the two fields that no one thought was possible. For instance, it’s an amazing fact that every surface—no matter how bumpy or holey or twisted—can be given a nice, symmetric curvature. A uniform geometry, it’s called. This was proven by Henri Poincaré in 1907. It was thought that 3D spaces would be far too complicated to be behave according to a similar rule. But Bill had a vision and a conjecture—that every 3D space can be divided into parts that can be given uniform geometries. To give you a flavor of these ideas, here’s a video of Bill describing some unusual and fabulous 3D spaces.

Any surface can be given a nice, symmetric geometry.

Any surface can be given a uniform geometry. Even a bunny. Another video.

As you can probably tell, visualizing and experiencing math was very important to Bill. He even taught a course with John Conway called Geometry and the Imagination. Bill often used computers to help himself see the math he was thinking about, and he enjoyed making hands-on models as well. Beginning in spring of 2010, Bill and Kelly Delp of Ithaca College worked out an idea. Usually all of the curving or turning of a polyhedron is concentrated at the vertices. Most of a cube is flat, but there’s a whole lot of pinch at the corners. What if you could spread that pinching out along the edges? And if you could, wouldn’t longer and perhaps wiggly edges help spread it even better? Yes and yes! You can see some examples of these “zippergons” that Bill and Kelly imagined and made in this gallery and read about them in their Bridges article.

A zippergon based on an octahedron.

A paper octahedron zippergon.


A foam icosadodecahedron zippergon.

One of Bill’s last collaborations happened not with a mathematician but with a fashion designer. Dai Fujiwara, a noted creator of high fashion in Tokyo, got inspired by some of Bill’s illustrations. In collaboration with Bill, Dai created eight outfits. Each one was based on one of the eight Thurston geometries. You can see the result of their work together in this video and read more about it in this article.

Isn’t it amazing how creative minds in very different fields can learn from each other and create something together?

Richard Evan Schwartz (self-portrait)

Richard Evan Schwartz (self-portrait)

Richard Evan Schwartz was one of the speakers at the conference honoring Bill. Rich studied with Bill at Princeton and now is a math professor at Brown University.

Like Bill, Rich’s work can be highly visual and playful, and he often taps the power of computers to visualize and analyze mathematical structures. There’s lots to explore on Rich’s website. Check out these applets he has made, including ones on Poncelet’s Porism, the Euclidean algorithm (previously), and a game called Lucy & Lily (JAVA required). I love how Rich shares some of his earliest applet-making efforts, like Click On A Triangle To Change Its Color. It’s motivating to see that even an accomplished mathematician like Rich began with the basics of programming—a place where any of us can start!

Screen Shot 2014-07-23 at 2.54.37 AMOn Rich’s site you’ll also find information about his project “Counting on Monsters“. And you should definitely make time to read some of the conversations that Rich has had with his five-year-old daughter Lucy.

Recently Rich published a wonderful new book for kids called “Really Big Numbers“. It is a colorful romp through larger and larger numbers and layers of abstraction, with evocative images to light the way. Check out the trailer for “Really Big Numbers” below!

Do you have a question for Rich—about his book, or about the math that he does, or about his life, or about Bill? Then send it to us in the form below and we’ll try to include it in our interview with him!

EDIT: Thanks for all your questions! Our Q&A with Rich will be posted soon.

Diana and Rich

Diana and Rich

Diana and Bill

Diana and Bill

Bill taught Rich, and Rich in turn taught Diana Davis, whose Dance Your PhD video we featured a while back. In fact, Bill’s influence on mathematics can be seen throughout many of our posts on Math Munch. Bill collaborated with Daina Taimina on hyperbolic crochet projects. He taught Jeff Weeks and helped inspire the games and software Jeff created. Bill oversaw the production of the film Outside In about the eversion of a sphere. He even coined the mathematical term “pair of pants.”

Bill’s vision of mathematics will live on in many people. That could include you, if you’d like. It’s just as Bill wrote:

In short, mathematics only exists in a living community of mathematicians that spreads understanding and breaths life into ideas both old and new.

Bon appetit!

The World Cup Group Stage, Math at First Sight, and Geokone

Welcome to this week’s Math Munch! We’ve got some World Cup math from a tremendous recreational mathematics blog and a new mathematical art tool. Get ready to dig in!

Brazuca: The 2014 World Cup Ball

Brazuca: The 2014 World Cup Ball

I’ve been meaning to share the really fantastic Puzzle Zapper Blog, because it’s so full of cool ideas, but the timing is perfect, because IT’S WORLD CUP TIME!!! and the most recent post is about the math of the world cup group stage! It’s called “World Cup Group Scores, and “Birthday Paradox” Paradoxes,” and I hope you’ll give it a read. (For some background on the Birthday Paradox, watch this Numberphile video called 23 and Football Birthdays.)

The thing that got me interested in the article was actually just this chart. I think it’s really cool, probably because I always find myself two games through the group stage, thinking of all the possible outcomes. If you do nothing else with this article, come to understand this chart. I was kind of surprised how many possible outcomes there are.

All Possible World Cup Group Stage Results

All Possible World Cup Group Stage Results

Long story short (though you should read the long story), there’s about a 40% chance that all 8 world cup groups will finish with different scores.

Alexandre Owen Muñiz, Author of Puzzle Zapper.  (click for an interview video about Alexandre's interactive fiction)

Alexandre Owen Muñiz, Author of Puzzle Zapper.  (click for an interview video about Alexandre’s interactive fiction)

Puzzle Zapper is the recreational mathematics blog of Alexandre Owen Muñiz. You can also find much of his work on his Math at First Sight site. He has a lot of great stuff with polyominoes and other polyforms (see the nifty pics below). Alexandre is also a writer of interactive fiction, which is basically a sort of text-based video game. Click on Alexandre’s picture to learn more.

The Complete Set of "Hinged Tetriamonds"

The complete set of “hinged tetrominoes”

A lovely family portrait of the hinged tetriamonds.

A lovely, symmetric family portrait of the “hinged tetriamonds”

I hope you’ll poke around Alexandre’s site and find something interesting to learn about.

For our last item this week, I’ve decided to share a new mathematical art tool called Geokone. This app is a recursive, parametric drawing tool. It’s recursive, because it is based on a repeating structure, similar to those exhibited by fractals, and it’s parametric, because the tool bar on the right has a number of parameters that you can change to alter the image. The artistic creation is in playing with the parameter values and deciding what is pleasing. Below are some examples I created and exported.

geokone2 geokone1


I have to say, Geokone is not the easiest thing in the world to use, but if you spend some time playing AND thinking, you can almost certainly figure some things out! As always, if you make something cool, please email it to us!

Now go create something!  Click to go to Geokone.net.

I hope you find something tasty this week. Bon appetit!

TED, Bridges, and Silk

Welcome to this week’s Math Munch!


The Math Munch team at TEDxNYED

Marjorie Rice

Marjorie Rice | click to watch her interview video

On Saturday, the Math Munch team gave a 16-minute presentation at TEDxNYED about Math Munch!  (Eventually there will be a video, and we’ll be sure to share it with you right away, but you’ll have to wait a month, maybe.)

We started with the story of Marjorie Rice, and in searching for a good picture of her, we came across this wonderful interview in a documentary about Martin Gardner.  It’s so neat to hear her speak about her discoveries.  You can see how proud she is and how much she truly loves math.  Feel free to watch the whole documentary if you like.  I haven’t gotten a chance yet, but I know it’s full of incredible stuff.

In the spirit of TED, I decided to share a few mathematical TED talks.  This one is absolutely fascinating.  In it, mathematician Ron Eglash describes how fractals underly the african designs.  You know how we love fractals.

If you’re hungry for another TED talk, here’s one about connections between music, mathematics, and sonar.

Up next, remember when we wrote about attending last year’s Bridges conference?  Well it happens every year, of course, and this year’s gallery of mathematical art is available online!  Click on one of those images and you get to more of the artists work.  I could easily spend hours staring at this art, trying to understand them, and reading the descriptions and artist statements.  Seriously, there is just way too much cool stuff there, so I’ve picked out a few of my favorites.  Also, I have great news to announce: Chloé Worthington (previously featured) had some of her art accepted to the exhibition!  Congratulations, Chloe!  If you look closely, you’ll see some of my art in there too.  🙂

Bjarne Jespersen

Bjarne Jespersen

Marc Chamberland

Marc Chamberland

Bob Rollings

Bob Rollings

Chloe Worthington

Chloé Worthington

Mehrdad Garousi

Mehrdad Garousi

By the way, if you ever create any mathematical art of your own, we’d love to see it!  Send us an email at mathmunchteam@gmail.com, and maybe we’ll feature your work in an upcoming Math Munch. (Only if you want us too, of course.)

Yuri Vishnevsky

Silk creator Yuri Vishnevsky

Finally, I know many of you like playing around with Symmetry Artist, which can be found on our page of Math Art Tools.  If you like that, then you’ll love Silk!  It’s much the same, but generates a certain kind of whispiness as you draw that looks really cool.  It also lets you spiral your designs toward the center, a feature which Symmetry Artist lacks.  You can download the Silk app for iPad or iPhone, if you like.  Silk was designed by Yuri Vishnevsky, with sound design by Mat Jarvis.  Yuri has agreed to do a Q&A for us, but we haven’t quite finished it just yet.  I’ll upload it as soon as possible, but for now, you can read an interview Mat and Yuri did with a website called Giant Fire Breathing Dragon.

Bon appetit!

Silk1 Silk4 Silk2