Tag Archives: tessellation

Nautilus, The Riddler, and Brain Pickings

Welcome to this week’s Math Munch!

Sometimes math pops up in places when you aren’t even looking for it. This week I’d like to share three websites that I enjoy. What they have in common is that they all cover a wide range of subjects—astronomy, politics, pop culture—but also host some great math if you know where to look for it.

nautilusFirst up is a site called Nautilus. In their own words, “We are here to tell you about science and its endless connections to our lives.” Each month they publish articles around a theme. This month’s theme is “Heroes.” Included in Nautilus’s mission is discussing mathematics, and you can find their math articles on this page. Here are a few articles to get you started. Read about how Penrose tiles have made the leap from nonrepeating abstraction to the real world—including to kitchen items. Learn about one of math’s beautiful monsters and how it shook the foundations of calculus. Or you might be interested in learning about how a mathematician is using computers to change the way we write proofs.

riddler_4x3_defaultNext, you might think that, since the presidential election is now over, you won’t be heading to Nate Silver’s FiveThirtyEight quite as often. But do you know about the site’s column called The Riddler? Each week Oliver Roeder shares two puzzles, the newer Riddler Express and the Riddler Classic. Readers can send in their solutions, and some get featured on the website—that could be you! Here are a couple of puzzles to get you started, and you can also check out the full archive. The Puzzle of the Lonesome King asks about the chances that someone will win a prince-or-princess-for-a-day competition. Can You Win This Hot New Game Show? asks you to come up with a winning strategy for a round of Highest Number Wins. And Solve The Puzzle, Stop The Alien Invasion is just what is says on the tin.


The third site I’d like to point you to is Brain Pickings. It’s a wide-ranging buffet of short articles on all kinds of topics, written and curated by Maria Popova. If you search Brain Pickings for math, all kinds of great stuff will pop up. You can read about John ConwayPaul Erdős, Margaret WertheimBlaise Pascal, and more. You’ll find book recommendations, videos, history, and artwork galore. I particularly want to highlight Maria’s article about the trailblazing African American women who helped to put a man on the moon. Their story is told in the book Hidden Figures by Margot Lee Shetterly, and the feature film by the same name is coming soon to a theater near you!

I hope you find lots to dig into on these sites. Bon appetit!

SET, Ptolemy, and Malin Christersson

Welcome to this week’s Math Munch!

To set up the punchline: if you haven’t played the card game SET before, do yourself a favor and go try it out now!

(Or if you prefer, here’s a video tutorial.)


Are there any sets to be found here?

(And even if you have played before, go ahead and indulge yourself with a round. You deserve a SET break. 🙂 )

Now, we’ve shared about SET before, but recently there has been some very big SET-related news. Although things have been quieter around Georgia Tech since summer has started, there has been a buzz both here and around the internet about a big breakthrough by Vsevolod Lev, Péter Pál Pach, and Georgia Tech professor Ernie Croot. Together they have discovered a new approach to estimate how big a SET-less collection of SET cards can be.

In SET there are a total of 81 cards, since each card expresses one combination of four different characteristics (shape, color, filling, number) for which there are three possibilities each. That makes 3^4=81 combinations of characteristics. Of these 81 cards, what do you think is the most cards we could lay out without a SET appearing? This is not an easy problem, but it turns out the answer is 20. An even harder problem, though, is asking the same question but for bigger decks where there are five or ten or seventy characteristics—and so 3^5 or 3^10 or 3^70 cards. Finding the exact answer to these larger problems would be very, very hard, and so it would be nice if we could at least estimate how big of a collection of SET-less cards we could make in each case. This is called the cap set problem, and Vsevolod, Péter, and Ernie found a much, much better way to estimate the answers than what was previously known.

To find out more on the background of the cap set problem, check out this “low threshold, high ceiling” article by Michigan grad student Charlotte Chan. And I definitely encourage you to check out this article by Erica Klarreich in Quanta Magazine for more details about the breakthrough and for reactions from the mathematical community. Here’s a choice quote:

Now, however, mathematicians have solved the cap set problem using an entirely different method — and in only a few pages of fairly elementary mathematics. “One of the delightful aspects of the whole story to me is that I could just sit down, and in half an hour I had understood the proof,” Gowers said.

(For further wonderful math articles, you’ll want to visit Erica’s website.)

 Vsevolod  Peter  Ernie
 Charlotte  Erica  Marsha

These are photos of Vsevolod, Péter, Ernie, Charlotte, Erica, and the creator of SET, geneticist Marsha Jean Falco.

Ready for more? Earlier this week, I ran across this animation:


It shows two ways of modeling the motions of the sun and the planets in the sky. On the left is a heliocentric model, which means the sun is at the center. On the right is a geocentric model, which means the earth is at the center.


Around 250 BC, Aristarchus calculated the size of the sun, and decided it was too big to revolve around the earth!

Now, I’m sure you’ve heard that the sun is at the center of the solar system, and that the earth and the planets revolve around the sun. (After all, we call it a “solar system”, don’t we?) But it took a long time for human beings to decide that this is so.

I have to confess: I have a soft spot for the geocentric model. I ran across the animation in a Facebook group of some graduates of St. John’s College, where I studied as an undergrad. We spent a semester or so reading Ptolemy’s Almagest—literally, the “Great Work”—on the geocentric model of the heavens. It is an incredible work of mathematics and of natural science. Ptolemy calculated the most accurate table of chords—a variation on a table of the sine function—that existed in his time and also proved intricate facts about circular motion. For example, here’s a video that shows that the eccentric and epicyclic models of solar motion are equivalent. What’s really remarkable is that not only does Ptolemy’s system account for the motions of the heavenly bodies, it actually gave better predictions of the locations of the planets than Copernicus’s heliocentric system when the latter first debuted in the 1500s. Not bad for something that was “wrong”!

Here are Ptolemy and Copernicus’s ways of explaining how Mars appears to move in the sky:

ptolemy Copernicus_Mars

Maybe you would like to learn more about the history of models of the cosmos? Or maybe you would like tinker with a world-system of your own? You might notice that the circles-on-circles of Ptolemy’s model are just like a spirograph or a roulette. I wonder what would happen if we made the orbit circles in much different proportions?


Malin, tiled hyperbolically.

Now, I was very glad to take this stroll down memory lane back to my college studies, but little did I know that I was taking a second stroll as well: the person who created this great animation, I had run across several other pieces of her work before! Her name is Malin Christersson and she’s a PhD student in math education in Sweden. She is also a computer scientist who previously taught high school and also teaches many people about creating math in GeoGebra. You can try out her many GeoGebra applets here. Malin also has a Tumblr where she posts gifs from the applets she creates.

About a year ago I happened across an applet that lets you create art in the style of artist (and superellipse creator) Piet Mondrian. But it also inverts your art—reflects it across a circle—so that you can view your own work from a totally different perspective. Then just a few months later I delighted in finding another applet where you can tile the hyperbolic plane with an image of your choice. (I used one tiling I produced as my Twitter photo for a while.)



tiling (4)

Me, tiled hyperbolically.

And now come to find out these were both made by Malin, just like the astronomy animation above! And Malin doesn’t stop there, no, no. You should see her fractal applets depicting Julia sets. And her Rolling Hypocycloids and Epicycloids are can’t-miss. (Echoes of Ptolemy there, yes?!)

And please don’t miss out on Malin’s porfolio of applets made in the programming language Processing.

It’s a good feeling to finally put the pieces together and to have a new mathematician, artist, and teacher who inspires me!

I hope you’ll find some inspiration, too. Bon appetit!

Continents, Math Explorers’ Club, and “I use math for…”

Welcome to this week’s Math Munch!


Steven Strogatz.

All of our munches this week come from the recent tweets of mathematician, author, and friend of the blog Steven Strogatz. Steve works at Cornell University as an applied mathematician, tackling questions like “If people shared taxis with strangers, how much money could be saved?” and “What caused London’s Millennium Bridge to wobble on its opening day?”

On top of his research, Steve is great at sharing math with others. (This week I learned one great piece of math from him, and then another, and suddenly there was a very clear theme to my post!) Steve has written for the New York Times and was recently awarded the Lewis Thomas Prize as someone “whose voice and vision can tell us about science’s aesthetic and philosophical dimensions, providing not merely new information but cause for reflection, even revelation.”

NMFLogo_Horiz_RGB_300DPI2This Saturday, Steve will be presenting at the first-ever National Math Festival. The free and fun main event is at the Smithsonian in Washington, DC, and there are related math events all around the country this weekend. Check and see if there’s one near you!

Here are a few pieces of math that Steve liked recently. I liked them as well, and I hope you will, too.

First up, check out this lovely image:

tesselation1-blog480It appeared on Numberplay and was created by Hamid Naderi Yeganeh, a student at University of Qom in Iran. Look at the way the smaller and smaller tiles fit together to make the design. It’s sort of like a rep-tile, or this scaly spiral. And do those shapes look familiar? Hamid was inspired by the shapes of the continents of Africa and South America (if you catch my continental drift). Maybe you can create your own Pangaea-inspired tiling.

If you think that’s cool, you should definitely check out Numberplay, where there’s a new math puzzle to enjoy each week!

Next, up check out the Math Explorers’ Club, a collection of great math activities for people of all ages. The Club is a project of Cornell University’s math department, where Steve teaches.

The first item every sold on the auction site eBay. Click through for the story!

The first item every sold on the auction site eBay. Click through for the story!

One of the bits of math that jumped out to me was this page about auctions. There’s so much strategy and scheming that’s involved in auctions! I remember being blown away when I first learned about Vickrey auctions, where the winner pays not what they bid but what the second-highest bidder did!

If auctions aren’t your thing, there’s lots more great math to browse at the Math Explorer’s Club—everything from chaos and fractals to error correcting codes. Even Ehrenfeucht-Fraïssé games, which are brand-new to me!

And finally this week: have you ever wondered “What will I ever use math for?” Well, SIAM—the Society for Industrial and Applied Mathematics—has just the video for you. They asked people attending one of their meetings to finish the sentence, “I use math for…”. Here are 32 of their answers in just 60 seconds.

Thanks for sharing all this great math, Steve! And bon appetit, everyone!

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!

Spheres, Gears, and Souvenirs

92GearSphere-20-24-16Welcome to this week’s Math Munch!

Whoa. What is that?

Is that even possible?

This gear sphere and many others are the creations of Paul Nylander. There are 92 gears in this gear sphere. Can you figure out how many there are of each color? Do you notice any familiar shapes in the gears’ layout?

What’s especially neat are the sizes of the gears—how many teeth each gear color has. You can see the ratios in the upper left corner. Paul describes some of the steps he took to find gears sizes that would work together. He wrote a computer program to do some searching. Then he did some precise calculating and some trial and error. And finally he made some choices about which possibilities he liked best. Sounds like doing math to me!

Along the way Paul figured out that the blue gears must have a number of teeth that is a multiple of five, while the yellow ones must have a multiple of three. I think that makes sense, looking at the number of red gears around each one. So much swirly symmetry!

Spiral shadows!

Spiral shadows!

Be sure to check out some of Paul’s other math art while you’re on his site. Plus, you can read about a related gear sphere in this post by mathematician John Baez.

I figured there had to be a good math game that involves gears. I didn’t find quite what I expected, but I did find something I like. It’s a game that’s called—surprise, surprise—Gears! It isn’t an online game, but it’s easy to download.

Can you find the moves to make all the gears point downwards?

Can you find the moves to make all the gears point downwards?

This Wuzzit is in trouble!

Wuzzit Trouble!

And if you’re in the mood for some more gear gaming and you have access to a tablet or smartphone, you should check out Wuzzit Trouble. It’s another free download game, brought to you by “The Math Guy” Keith Devlin. Keith discusses the math ideas behind Wuzzit Trouble in this article on his blog and in this video.


Last up this week, I’d like to share with you some souvenirs. If you went on a math vacation or a math tour, where would you go? One of the great things about math is that you can do it anywhere at all. Still, there are some mathy places in the world that would be especially neat to visit. And I don’t mean a place like the Hilbert Hotel (previously)—although you can get a t-shirt or coffee mug from there if you’d like! The mathematician David Hilbert actually spent much of his career in Göttingen, a town and university in Germany. It’s a place I’d love to visit one day. Carl Gauss lived in Göttingen, and so did Felix Klein and Emmy Noether—and lots more, too. A real math destination!

Lots of math has been inspired by or associated with particular places around the world. Just check out this fascinating list on Wikipedia.

Arctic Circle Theorem

The Arctic Circle Theorem

The Warsaw Circle

The Warsaw Circle

Cairo Pentagonal Tiling

The Cairo Pentagonal Tiling

Did you know that our word souvenir comes from the French word for “memory”? One thing that I like about math is that I don’t have to memorize very much—I can just work things out! But every once in a while, there is something totally arbitrary that I just have to remember. Here’s one memory-helper that has stuck with me for a long time.

May you, like our alligator friend, find some good math to munch on. Bon appetit!

Squaring, Water Calculator, and Snap the Turtle

Welcome to this week’s Math Munch!

I’ve been really into squares lately. Maybe it’s because I recently ran across a new puzzle involving squares– something called Mrs. Perkin’s quilt.

Mrs. Perkin's quilt 1

69 by 69 Mrs. Perkin’s quilt.

The original version of the puzzle was published way back in 1907, and it went like this: “For Christmas, Mrs. Potipher Perkins received a very pretty patchwork quilt constructed of 169 square pieces of silk material. The puzzle is to find the smallest number of square portions of which the quilt could be composed and show how they might be joined together. Or, to put it the reverse way, divide the quilt into as few square portions as possible by merely cutting the stitches.”

Mrs. Perkin's quilt 18

18 by 18 Mrs. Perkin’s quilt

Said in another way, if you have a 13 by 13 square, how can you divide it up into the smallest number of smaller squares? Don’t worry, you get to solve it yourself– I’m not including a picture of the solution to that version of the puzzle because there are so many beautiful pictures of solutions to the puzzle when you start with larger and smaller squares. Some are definitely more interesting than others. If you want to start simple, try the 4 by 4 version. I particularly like the look of the solution to the 18 by 18 version.

Mrs. Perkin's quilt 152

152 by 152 Mrs. Perkin’s quilt

Maybe you’re wondering where I got all these great pictures of Mrs. Perkin’s quits. And– wait a second– is that the solution to the 152 by 152 version? It sure is– and I got it from one of my favorite math websites, the Wolfram Demonstrations Project. The site is full of awesome visualizations of all kinds of things, from math problems to scans of the human brain. The Mrs. Perkin’s quilts demonstration solves the puzzle for up to a 1,098 by 1,098 square!

Next up, we here at Math Munch are big fans of unusual calculators. Marble calculators, domino calculators… what will we turn up next? Well, here for your strange calculator enjoyment is a water calculator! Check out this video to see how it works:

I might not want to rely on this calculator to do my homework, but it certainly is interesting!

Snap the TurtleFinally, meet Snap the Turtle! This cute little guy is here to teach you how to make beautiful math art stars using computer programming.

On the website Tynker, Snap can show you how to design a program to make intricate line drawings– and learn something about computer programming at the same time. Tynker’s goal is to teach kids to be programming “literate.” Combine computer programming with a little math and art (and a turtle)– what could be better?

I hope something grabbed your interest this week! Bon appetit!


Zentangle, Graph Paper, and Pancake Art

My recent doodling.

Some recent doodling, by me.

Welcome to this week’s Math Munch!

As you start a new school year, you might be looking for some new mathy doodle games to play in the margins of your notebooks. Doodling helps me to listen sometimes, and I love making neat patterns. I especially like seeing what new shapes I can make.

This summer I was very happy to run across Zentangle®—”an easy-to-learn, relaxing, and fun way to create beautiful images by drawing structured patterns.” I’ve learned a lot about Zentangle from a blog called Tangle Bucket by Sandy Hunter. She shares how to doodle snircles, snafoozles, and oodles. There’s a whole dictionary of zentangle shapes over at tanglepatterns.com.

My favorite idea in Zentangle is trying to combine two kinds of designs. Sometimes this is described as one pattern “versus” another one. For instance, check out these:


Maybe you’ll pick some tangle patterns to combine with each other. If you try some, maybe you’ll share them in our Readers’ Gallery.

Sandy writes:

It’s so true that the more I tangle, the more I see the potential in patterns all around me. I catch myself mentally deconstructing them (whether I want to or not) to figure out if they can be broken down into simple steps without too much effort. That’s the trademark of a good tangle pattern.

Try some of Sandy’s weekly challenges, or check out Tiffany Lovering’s time-lapse videos—here’s one with music and one with an interview. Can you learn the names of any of the shapes she creates? I spy a Rick’s Paradox. There are lots of ways to begin zentangling—I hope you enjoy giving it a try.

Squares and dots and crosses, oh my!

Squares & dots & crosses, oh my!

If zentangling is too freeform for your doodling tastes, then let me share with you one of my longtime favorite websites. I’ve used it for years to help me to do math and to teach math, and it’s great for math doodling, too. I might even call it a trusty friend, one that I met one day through the simple online search: “free online graph paper”.

That’s right, it’s Free Online Graph Paper.

Something I love about the site is that it lets you design different aspects of your graph paper. Then you can print it out. First you get to decide what kind of grid you would like: square? triangular? circular? Then you get to tinker with lots of variables, like how big the grid cells are, how dark the lines are, and what color they are. And more!

Free Online Graph Paper was created by Kevin MacLeod, who composes music and shares it for free. That way other people can use it for creative projects. That’s really awesome! I enjoyed listening to Kevin’s “Winner Winner“. It’s always good to be reminded that everything you use or enjoy was almost certainly made by a person—including custom graph paper websites!

A 7/3 star spirocake.

A 7/3 star spirocake.

Last up this week is some doodly math that you can really munch on. Everyone knows that breakfast is the most important meal of the day and that the most important food group is roulette curves.

To get your daily recommended allowance of groovy math, look no further than the edible doodles of Nathan Shields and his family over at Saipancakes.

I can wait until the Shields family tackles the cissoid of Diocles.

Bon appetit!