Tag Archives: paper

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!

Fields Medal, Favorite Numbers, and The Grapes of Math

Welcome to this week’s Math Munch! And, if you’re a student or teacher, welcome to a new school year!

fieldsOne of the most exciting events in the world of math happened this August– the awarding of the Fields Medal! This award honors young mathematicians who have already done awesome mathematical work and who show great promise for the future. It also only happens every four years, at the beginning of an important math conference called the International Congress of Mathematicians, so it’s a very special occasion when it does!


Maryam Mirzakhani, first woman ever to win a Fields Medal

Maryam Mirzakhani, first woman ever to win a Fields Medal

This year’s award was even more special than usual, though. Not only were there four winners (more than the usual two or three), but one of the winners was a woman!

Now, if you’re like me, you probably heard about the Fields Medal and thought, “There’s no way I’ll understand the math that these Field Medalists do.” But this couldn’t be more wrong! Thanks to these great articles from Quanta Magazine, you can learn a lot about the super-interesting math that the Fields Medalists study– and why they study it.


Manjul Bhargava

One thing you’ll immediately notice is that each Fields Medalist has non-math interests that inspire their mathematical work. Take Manjul, for instance. When he was a kid, his grandfather introduced him to Sanskrit poetry. He was fascinated by the patterns in the rhythms of the poems, and the number patterns that he found inspired him to study the mathematics of number patterns– number theory!

But, don’t just take my word for it– you can read all about Manjul and the others in these great articles! And did I mention that they come with videos about each mathematician? 

Want to read more about this year’s Fields Medallists? Check out Alex Bellos’s article in The Guardian. Which brings me to…

download… What’s your favorite number? Is it 7? If it is, then you’re in good company! Alex polled more than 30,000 people about their favorite number, and the most popular was 7. But why? What’s so special about 7? Here’s why Alex thinks 7 is such a favorite:

grapes-of-mathWhy do you like your favorite number? People gave Alex all kinds of different reasons. One woman said about 3, her favorite number, “3 wishes. On the count of 3. 3 little pigs… great triumvirates!” Alex made these questions the topic of the first chapter of his new book, The Grapes of Math. (Get the reference?) In this book, Alex shares many curious ways that math appears in our world. Did you know that a weird pattern in numbers can be used to catch criminals? Or that the Game of Life, a simple computer program, shares surprisingly many characteristics with real life? These are only a few of the hundreds of topics Alex covers in his book. Whether you’re a math whiz or a newbie, you’ll learn something new on every page.

Alex currently writes about math for The Guardian in a blog called, “Alex’s Adventures in Numberland”— but he also loves and writes about soccer (or futbol, as it’s called in his native Brazil)! He even wrote a few articles for his blog about math and soccer. 

Do you have any questions for Alex? (About math, soccer, or their intersection?) Write them here and you might find them featured in our interview with Alex!

Good writing about math is hard to find. If you’ve ever picked up a standard math textbook, you’ll know what I mean. But reading something fascinating, that grabs your interest from the first page and leads you through the most complex ideas like they’re as natural as anything you’ve observed, is a great way to learn. The Grapes of Math and “Alex’s Adventures in Numberland” do just that. Give them a go!

Bon appetit!


Byrne’s Euclid, Helen Friel, and PolygonJazz

Welcome to this week’s Math Munch! We’ve got geometry galore, starting with a series of historical math diagrams and a color update to Euclid’s Elements. Then it’s onto modern day paper artist Helen Friel, and finally a nifty new app that makes music from polygons. Let’s get into it.

Euclid’s “Elements” was written around 300BC. It was the first great compilation of geometric knowledge, broken into 13 books, and it is one of the most influential books of all time. Euclid’s proof of the Pythagorean Theorem may be his most famous proof from the book (and all of mathematics for that matter), and in the pictures below you can see three diagrams of the proof, spanning seven centuries.

Nasir al-Din al-Tusi's 13th century arabic translation of Euclid's proof.

Persian mathematician Nasir al-Din al-Tusi‘s 13th century arabic translation of Euclid’s proof.

Late 14th century English manuscript

A late 14th century English manuscript of Euclid’s “Elements.”

The idea in each picture is that the area of the top two squares adds up exactly to the area of the bottom square. In the picture below, we see the big square broken up into blue and yellow pieces, whose areas are the same as the squares above them.

Oliver Byrne's 1871 color edition

Oliver Byrne’s 1847 color edition.  Click the image for the full proof of the Pythagorean Theorem as presented by Oliver Byrne in 1847.

This color version comes from Oliver Byrne’s 1847 edition, “The First Six Books of the Elements of Euclid, with Coloured Diagrams and Symbols.” (completely available online). I find the diagrams really beautiful and charming. There’s something extremely modern about them (see De Stijl) though they’re more than 150 years old now. See if you can follow his Oliver Byrne’s version of Euclid’s proof. It’s quite short.

Paper Engineer Helen Friel
Paper Engineer Helen Friel


“They’re an absolutely beautiful piece of work and far ahead of their time,” said paper engineer Helen Friel. Helen lives in London, and and as part of a charity project, she designed paper sculptures of Oliver Byrne’s diagrams.

Euclid 2 Euclid 4 Euclid 3 Euclid 1

In an interview, she explained, “It’s a more visual and intriguing way to describe the geometry. I love anything that simplifies. I find it very appealing!” In the interview, Helen also talks a little about her attraction to math. “There’s order in straight lines and geometry. Although my job is creative, I use as much logical progression as possible in my work.”

It’s also cool to see Helen’s work side by side with Oliver Byrne‘s, so click for that.

Screen Shot 2014-02-05 at 11.46.00 PM

Click to send us a pic.  Yes, that is a paper camera Helen made.

Downloadable model

Downloadable model

Perhaps the best part in all of this, though, is that you can download Helen’s Pythagaorean Theorem model and make your own! There are plain white version as well as color. If you end up making one, definitely email us a picture, and we’ll show it off here on Math Munch.

Oh, and here’s a quick video documenting the many versions Helen decided not to use.  So cool.

Now, on to our final bite.
PolygonJazz Recently, John Miller sent me an email showing off his new iPad app called PolygonJazz. In the app, you control the starting direction for a ball inside a polygon. Once you start it moving, the ball bounces off the walls, making a sound every time it hits a side. Check out the video below. I noticed something about the speed of the ball. Can you spot it? (PolygonJazz is available for $0.99 on the iTunes store.)

Speaking of bouncing around, here‘s a previous Math Munch featuring some billiards, and here‘s another bouncy post that features one of my favorite juggling routines. Michael Moschen built a gigantic equilateral triangle and juggles silicon balls inside and off of it. As with the app, Michael is utilizing the sound and geometry of the collisions to make something beautiful. It’s quite mesmerizing.

Have a bouncy week, and bon appetit!

Celebration of Mind, Cutouts, and the Problem of the Week

Welcome to this week’s Math Munch!  We’re going to revisit the work of Martin Gardner, look at some beautiful mathematical art, and see if we can dig into a college’s “problem of the week” program.

Martin Gardner

Martin Gardner

Last October, I wrote about Martin Gardner. He is one of the great popularizers of mathematics, known for his puzzles, columns in Scientific American, and over 100 books. Around the time of his birthday, October 21st, each year, people around the world participate in a global “Gathering4Gardner” — a so-called Celebration of Mind.

Two Sipirals

One of Martin Gardner’s many puzzles

These are gatherings of two or more people taking time to dig into the kinds of mathematics that Martin Gardner loved so much. Below you can find lots of ways to participate and share with family, friends, or strangers.

First, If you want to learn more about Gardner himself, here’s a very detailed interview. You can also try solving some of Gardner’s great puzzles. We featured both of these last year, but I recently found a whole new page of resources and activities for the Celebration of Mind.

In the video on the left you can see a geometric vanish like those we’ve previously featured (Get off the Earth, and Chocolate). The second is a surprising play on the Möbius Strip which we’ve also featured before (Art and Videos + Möbius Hearts). I hope you’ll find some time this week to celebrate Martin Gardner’s love of math and help grow your own. (Though, I guess if you’re reading this, you already are!)

Up next, check out the work of artist Elena Mir. This video shows a series of artworks she created over the last four years. They feature stacks of cut paper to form geometric shapes, and they make me wonder what I could make out of cut paper. If you make something, please let us now.

It reminds me of the work of Matt Shlian that we featured in our very first post.  You can watch Matt’s TED Talk or visit his website to see all sorts of cutouts and other paper sculptures, plus incredible videos like the one below. It might be my favorite video I’ve ever posted on Math Munch.

Finally, Macalester College in St. Paul, Minnesota has a weekly problem that they offer to their students, and the problem archives can be found online. These are for college students, so some of them are advanced or phrased in technical language, but I think we can find some that all of us can dig in to. Give these a try:

Have a mathematical week, and let us know if you do anything for the Celebration of Mind. Bon appetit!

The Fractal Foundation, Schoolhouse Rock, and More

Welcome to this week’s Math Munch!

Triangle Cutout Fractal

Up first, check out the Fractal Foundation.  They’re mission is simple: “We use the beauty of fractals to inspire interest in Science, Math and Art.”  If you played around with recursive drawing a few weeks ago, then perhaps you were as inspired by fractals as they hope you’ll be.  If you’re not really sure what fractals actually are, here’s a great one-page explanation from the Fractal Foundation website.  They also have an excellent page of “fractivities,” including instructions for the beautiful paper cutout fractal pictured on the right.  If you want to have your mind blow, check out their fantastic page of fractal videos.  Just amazing.

Next up, have you ever heard of Schoolhouse Rock?  It’s a series of rocking animated music videos that originally ran on TV from 1973 to 1985.  Vintage math goodness!  They cover all kinds of educational stuff like grammar and history, but I totally love the math videos, and a few of them are on YouTube!  Down below you can watch two of my favorites, and you can find the others here.  if you poke around YouTube, you could probably find a few more as well.

Finally, a few additions to our resource pages.  For the Math Games page, we’re adding Linebounder.  You and the computer battle to draw a line towards your goal.  I had a really hard time with this at first, but there are certain strategies that the computer simply cannot beat.  You just have to find them.  Also new is Shift, another fun game that plays with the relationship between figure and ground.  For the new Math Art Tools page, we’re adding Tessellate!  It’s an interactive applet that lets you make custom tiles to cover the plane.  Here’s a few examples I just made.

Bon appetit!