Monthly Archives: November 2014

Grothendieck, Circle Packing, and String Art

Welcome to this week’s Math Munch!

Grothendieckportra_3107171cThis week brought some sad news to the mathematical world. Alexander Grothendieck, known by many as the greatest mathematician of the past century, passed away on November 13th. You may not have heard of him, but many mathematicians say that the work he did in math was as influential as the work Albert Einstein did in physics.

One of the things that make Grothendieck so interesting is, of course, the math he did. Grothendieck was always very creative. When he was in high school, he preferred to do math problems he made up on his own over the problems assigned by his teacher. “These were the book’s problems, and not my problems,” he said.

When he was young, inspired by some gaps he found in definitions in his geometry book about measuring lengths and areas, Grothendieck re-created some of the most important mathematical ideas of the beginning of the twentieth century. Maybe this sounds silly to you– why re-invent something that’s already been done? But, to Grothendieck, the most important part was that he’d done the whole thing by himself. He’d figured out something in his own way. He later wrote that this experience showed him what being a mathematician was like:

Without having been told, I nevertheless knew ‘in
my gut’ that I was a mathematician: someone who
‘does’ math, in the fullest sense of the word…

During his years as a mathematician, Grothendieck worked on connecting different parts of math (a project requiring a lot of creativity)– algebra, geometry, topology, and calculus, among others.

Grothendieck kid

Alexander Grothendieck as a kid

The other thing that makes Grothendieck so interesting is his life story. As a kid, Grothendieck and his parents fled from Germany to France to escape the Nazis. As an adult, Grothendieck spoke out strongly for peace. He used his fame to take a stand against the wars of the second half of the twentieth century. This eventually led him to step away from the world of mathematicians– which many regretted. But he left behind work that changed all of mathematics for the better.

If you’d like to learn more about Grothendieck’s fascinating life and work, check out this great (but long) article from the American Mathematical Society. This article provides a shorter history, including a great statement Grothendieck made about his feelings on creativity in mathematics. Grothendieck was a very private person, so many of his mathematical writings aren’t available online– but the Grothendieck Circle has done their best to collect everything written about him.

Fractal-Apollonian-Gasket-Variations-02

A pretty circle packing

Next up, a little something for you to play with! We were studying circle packing problems in one of my classes this week. Did you know that you can fit exactly six circles snugly around another circle of the same size? But, if you try to fit circles snugly around a circle twice as large, it doesn’t work? I wonder why that is…

 

I did it!

I did it!

Anyway, my class inspired me to look for a circle packing game– and I found one! In this game, simply called Circle Packing, you have to fit all of the smaller circles into the larger circle– without any of them touching! It’s pretty tricky, and really fun.

String art wall long

String art circleFinally, the Math Munch team got something wonderful in the mail (email, I guess) this week! Math art made by Julia Dweck’s 5th grade math class! Julia’s class has been working hard to make parabolic curve string art– curves made by drawing (or stringing, in this case) many, many straight lines. They plotted each curve precisely before stringing it, to make sure it was both mathematically and artistically perfect. The pieces they made are so creative and beautiful. We’re proud to feature them on our site!

String art parabolaYou can see the whole collection of string art pieces made by Julia’s class on our Readers’ Gallery String Art page. And, want to know more about how the 5th graders made their String Art? Have any questions for Julia and her students about their love of math and the connections they see between math and art? Write your questions here and we’ll send them to Julia’s students!

Have any math art of your own? Send it to mathmunchteam@gmail.com, and we’ll post it in the Readers’ Gallery!

Bon appetit!

George Washington, Tessellation Kit, and Langton’s Ant

Welcome to this week’s Math Munch!

002What will you do with your math notebook at the end of the school year? Keep it as a reference for the future? Save it as a keepsake? Toss it out? Turn it into confetti? Find your favorite math bits and doodles and make a collage?

Lucky for us, our first president kept his math notebooks from when he was a young teenager. And though it’s passed through many hands over the years—including those of Chief Justice John Marshall and the State Department—it has survived to this day. That’s right. You can check out math problems and definitions copied out by George Washington over 250 years ago. They’re all available online at the Library of Congress website.

Or at least most of them. They seem to be out of order, with a few pages missing!

Fred Rickey

That’s what mathematician and math history detective Fred Rickey has figured out. Fred has long been a fan of math history. Since he retired from the US Military Academy in 2011, Fred has been able to pursue his historical interests more actively. Fred is currently studying the Washington cypher books to help prepare a biography about Washington’s boyhood years. You can see two papers that Fred has co-authored about Washington’s mathematics here.

Fred writes:

Washington valued his cyphering books and kept them as a ready source of reference for the rest of his life. This would seem to be particularly true of his surveying studies.

Surveying played a big role in Washington’s career, and math is important for today’s surveyors, too.

Do you have a question for Fred about the math that George Washington learned? Send it to us and we’ll try to include it in our upcoming Q&A with Fred!

A tessellation, by me!

A tessellation, by me!

Next up, check out this Tessellation Kit. It was made by Nico Disseldorp, who also made the geometry construction game we featured recently. The kit is a lot of fun to play with!

One thing I like about this Tessellation Kit is how it’s discrete—it deals with large chunks of the screen at a time. This restriction make me want to explore, because it give me the feeling that there are only so many possible combinations.

I’m also curious about the URL for this applet—the web address for it. Notice how it changes whenever you make a change in your tessellation? What happens when you change some of those letters and numbers—like bababaaaa to bababcccc? Interesting…

For another fun applet, check out this doodling ant:

Langton's Ant.

Langton’s Ant.

Langton’s Ant is following a simple set of rules. In a white square? Turn right. In a black square? Turn left. And switch the color of the square that you leave. This ant is an example of a cellular automaton, and we’ve seen several of these here on Math Munch before. This one is different from others because it changes just one square at a time, and not the whole screen at once.

Breaking out of chaos.

Breaking out of chaos.

There’s a lot that is unknown about Langton’s ant, and it has some mysterious behavior. For example, after thousands of steps of seeming randomness, the ant goes into a steady pattern, paving a highway out to infinity. What gives? Well, you can try out some patterns of your own in the applets on the Serendip website. (previously). And you can read some amusing tales—ant-ecdotes?—about Langton’s ant in this lovely article.

DSC03509I learned about Langton’s Ant from Richard Evan Schwartz in our new Q&A. In the interview, Rich shares his thoughts about computers, art, what to pursue in life, and of course: Really Big Numbers.

Check it out, and bon appetit!