Zippergons, High Fashion, and Really Big Numbers

Welcome to this week’s Math Munch!

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 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 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 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!

On 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 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:

Bon appetit!

Stomachion, Toilet Math, and Domino Computer Returns!

Welcome to this week’s Math Munch!

I recently ran across a very ancient puzzle with a very modern solution– and a very funny name. It’s called the Stomachion, and it looks like this:

So, what do you do? The puzzle is made up of these fourteen pieces carved out of a 12 by 12 square– and the challenge is to make as many different squares as possible using all of the pieces. No one is totally sure who invented the Stomachion puzzle, but it’s definite that Archimedes, one of the most famous Ancient Greek mathematicians, had a lot of fun working on it.

Sometimes Archimedes used the Stomachion pieces to make fun shapes, like elephants and flying birds. (If you think that sounds like fun, check out this page of Stomachion critters to try making and this lesson about the Stomachion puzzle from NCTM.) But his favorite thing to do with the Stomachion pieces was to arrange them into squares!

It’s clear that you can arrange the Stomachion pieces into a square in at least one way– because that’s how they start before you cut them out. But is there another way to do it? And, if there’s a second way, is there a third? How about a fourth? Because Archimedes was wondering about how many ways there are to make a square with Stomachion pieces, some mathematicians give him credit for being an inventor of combinatorics, the branch of math that studies counting things.

It turns out that there are many, many ways to make squares (the picture above shows all of them– click on it for greater detail)– and Archimedes didn’t find them all. But someone else did, over 2,000 years later! He used a computer to solve the problem– something Archimedes could never have done– but mathematician Bill Cutler found that there are 536 ways to make a square with Stomachion pieces! That’s a lot! If you’ve tried to make squares with the pieces, you might be particularly surprised– it’s pretty tricky to arrange them into one unique square, let alone 536. This finding was such a big deal that it made it into the New York Times. (Though you may notice that the number reported in the article is different– that’s how many ways there are to make a square if you include all of the solutions that are symmetrically the same.)

Other mathematicians have worked on finding the number of ways to arrange the Stomachion pieces into other shapes– such as triangles and diamonds. Given that it took until 2003 for someone to find the solution for squares, there are many, many open questions about the Stomachion puzzle just waiting to be solved! Who knows– if you play with the Stomachion long enough, maybe you’ll discover something new!

Next up, the mathematicians over at Numberphile have worked out a solution to a problem that plagued me a few weeks ago while I was camping– choosing the best outdoor toilet to use without checking all of them for grossness first. Is there a way to ensure that you won’t end up using the most disgusting toilet without having to look in every single one of them? Turns out there is! Watch this video to learn how:

Finally, a little blast from the past. Almost two years ago I share with you a video of something really awesome– a computer made entirely out of dominoes! Well, this year, some students and I finally got the chance to make one of our own! It very challenging and completely exhausting, but well worth the effort. Our domino computer recently made its debut on the mathematical internet, so I thought I’d share it with all of you! Enjoy!

Bon appetit!

8-bit, Pixel Art, and Aliens

Hello and welcome to this week’s Math Munch! I’m Mai Li, a current college student, and a former student of your regular team. I’m really happy to be making this week’s guest post! Today we’re going to be talking about one of my favourite topics, pixel art!

When I was a kid, the Game Boy Color came out.

15-bit graphics! No idea why 15.

It had impressive 15-bit color graphics, a huge step up from the 2-bit graphics of the original Game Boy. Just looking at the graphical difference between the original MegaMan on the Game Boy, and MegaMan Xtreme on the Game Boy Color, you can tell that 15-bit offers a much larger color variety than the mere four colors available with 2-bit graphics. But what exactly does it mean to be 15-bit, as opposed to 2-bit?

2-bit graphics, in this “pea soup” color scheme.

Well, what’s a bit? A bit is a single piece of information that can be stored by a computer, either a 1 or a 0. A 1-bit system can have up to two whole colors! Either color 1, or color 0. Take Pong, for instance. Let’s say the color scheme is black and white. Now, white can be color 0, and black can be color 1. The pixels making up the paddles, the ball, the board, and the score are color 0, and the background is color 1. Pretty simple! But what if we want more than that?

Pong, the oldest game many people are familiar with. 1-bit colors!

In comes 2-bit, to the rescue! The Game Boy had a 2-bit color system, usually four shades of green. As you might have guessed, this mean that each color had two pieces of information, two “bits,” for a total of four possible combinations- 00, 01, 10, 11. And there you go! Four combination, four colors, just like that. For each bit, there are two possibilities, so the number of total colors available is 2^2. That means that for a 15-bit system like the Game Boy Color, there are 2^15 available colors! That comes out to a palette of 32,768 colors! Although the Game Boy Color was only physically capable of displaying 56 different colors simultaneously, you can understand now why 15-bit looks so much nicer than it’s earlier 2-bit counterpart. Now that you know what 8-bit means, you probably want to make your own pixel art, so here are some programs to help you do just that: Piq is a simple program that is available online, without downloading. GraphicsGale is favorite of mine, however it is only for windows. If all else fails. GIMP is a free Photoshop alternative.

One of my all time favourite artists, Fool.

8-bit is a popular art style these days, and one I often work in myself. 8-bit is 2^8, or 256 different colors. Now days, this rule of 256 colors or less is entirely a stylistic choice, as computers and consoles can work with a much higher color resolution. Many artists, however, will limit themselves to even less than 256 colors, for aesthetic and color theory reasons. In addition, artists might also use a space constraint, like using a canvas that is only 256 pixels high and 256 pixels wide. Besides the limited number of colors, many people consider works to be pixel art only if each of the pixels was hand placed by the artist, read: no Photoshop filters. Because of this, pixel art is often limited in size, simply due to the amount of time it takes to hand place each pixel. Two of my favourite pixel artists are Fool, and Pixelatedcrown. My own artwork can be found here.

 By Pixelatedcrown, who’s work I adore. She also does 3D modeling and game dev stuff. My own work.

Retro graphics are making a comeback, and I have to admit I love it. I’m going to shine a spotlight on one of the new games that I think has some of the best retro graphics I’ve seen in a while – Shovel Knight.

Shovel Knight must rescue his friend Shield Knight in a timeless tale of shovelry.

Looking and playing like something akin to an SNES platformer, Shovel Knight explores an attractive 8-bit world to find his partner, Shield Knight. Although the game itself is ten dollars (and totally worth it), a first impressions video by one of my favorite Youtubers, Rockleesmile, is completely free. The video is part of his Indie Impressions series, which covers a new indie game daily, and which I adore. If that’s not enough, he has a playthrough of the entire game available, if you just need to see all the graphics right now. And if you do, who could blame you?

Finally, an unexpected use of pixel graphics: to contact aliens. No, I’m not kidding.

We sent this into space. No kidding.

The Arecibo Message was a radio message sent into space in 1974 after the remodeling of the Arecibo Radio Telescope in Puerto Rico. The message, aimed at the globular star cluster M13, which is approximately 25,000 light years away, was mostly sent to prove that we could. Although scientists don’t really expect to hear anything back (and even it we did, it would spend 50,000 years in transit alone!), the message contained information that we thought would be important for aliens to know about us. The entire message is about 210 bits, and is in 1-bit resolution! Try and see if you can figure out what it’s trying to explain. If you can’t, the wiki page explains it all. And check out this Mental Floss article all about it! Speaking of which, if you want to de-code more alien messages, check out the Cosmic Call, a longer message sent by telescopes in Texas in 1999. I was given the message as a sixth grader, and with some friends, was able to decode the first few pages so I highly suggest giving it a try!

The Cosmic Call! Are you smarter than an alien?

It’s generally the same idea as the Arecibo message, but it’s easier to de-code, and they sent it to many more star systems, in hopes of a response. If you were an alien and you received this message, could you understand it? I think so, but you should really see for yourself.