Tag Archives: quotes

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.

Icosadodecahedron.

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!

Light Bulbs, Lanterns, and Lights Out

Welcome to this week’s Math Munch!

thomas-edison

Edison with his light bulb.

On this day in 1880, Thomas Edison was given a patent for his most famous bright idea—the light bulb.

Edison once said, “Genius is one per cent inspiration, ninety-nine per cent perspiration”—a good reminder that putting in some work is important both in math and in life. He also said, “We don’t know a millionth of one percent about anything.” A humbling thought. Also, based on that quote, it sounds like Edison might have had a use for permilles or even permyraids in addition to percents!

Mike's octahedron.

Mike’s octahedron-in-a-light-buld.

In celebration of this illustrious anniversary, I’d like to share some light mathematical fare relating to, well, light bulbs. For starters, J. Mike Rollins of North Carolina has created each of the Platonic solids inside of light bulbs, ship-in-a-bottle style. Getting just the cube to work took him the better part of twelve hours! Talk about perspiration. Mike has also made a number of lovely Escher-inspired woodcuts. Check ’em out!

Evelyn's Schwartz lantern.

Evelyn’s Schwartz lantern.

Next up is a far-out example from calculus that’s also a good idea for an art project. It’s called the Schwartz lantern. I found out about this amazing object last fall when Evelyn Lamb tweeted and blogged about it.

The big idea of calculus is that we can find exact answers to tough problems by setting up a pattern of approximations that get better and better and then—zoop! take the process to its logical conclusion at infinity. But there’s a catch: you have to be careful about how you set up your pattern!

A "nicely" triangulated cylinder.

A “nicely” triangulated cylinder.

For example, if you take a cylinder and approximate its surface with a bunch of triangles carefully, you’ll end up with a surface that matches the cylinder in shape and size. But if you go about the process in a different way, you can end up with a surface that stays right near the cylinder but that has infinite area. That’s the Schwartz lantern, first proposed by Karl Hermann Amandus Schwarz of Cauchy-Schwartz fame. The infinite area happens because of all the crinkles that this devilish pattern creates. For some delightful technical details about the lantern’s construction, check out Evelyn’s post and this article by Conan Wu.

Maybe you’ll try folding a Schwartz lantern of your own. There’s a template and instructions on Conan’s blog to get you started. You’ll be glowing when you finish it up—especially if you submit a photo of it to our Readers’ Gallery. Even better, how about a video? You could make the internet’s first Schwartz lantern short film!

Robert Torrence and his Lights Out puzzle.

Robert and his Lights Out puzzle.

At the MOVES Conference last fall, Bruce Torrence of Randolf-Macon College gave a talk about the math of Lights Out. Lights Out is a puzzle—a close relative of Ray Ray—that’s played on a square grid. When you push one of the buttons in the grid it switches on or off, and its neighbors do, too. Bruce and his son Robert created an extension of this puzzle to some non-grid graphs. Here’s an article about their work and here’s an applet on the New York Times website where you can play Lights Out on the Peterson graph, among others. You can even create a Lights Out puzzle of your own! If it’s more your style, you can try a version of the original game called All Out on Miniclip.

The original Lights Out handheld game from 1995.

The original Lights Out handheld game from 1995.

There’s a huge collection of Lights Out resources on Jaap’s Puzzle Page (previously), including solution strategies, variations, and some great counting problems. Lights Out and Ray Ray are both examples of what’s called a “sigma-plus game” in the mathematical literature. Just as a bonus, there’s this totally other game called Light Up. I haven’t solved a single puzzle yet, but my limitations shouldn’t stop you from trying. Perspiration!

All this great math work might make you hungry, so…bon appetit!