# Functionized Photos, Projective Games, and Traffic

Welcome to this week’s Math Munch!

Have you ever looked in a distorted mirror– one that stretched and squeezed your face so that you looked very, very silly? If you like that, check out this program called the Function Explorer that distorts your picture according to different functions!

My cat under the “fraction” function

To use the program, you’ll have to turn on your webcam. Then, select one of the functions listed– maybe similarity, log, or fraction. Then, watch as the image in front of your webcam twists, expands, and repeats as the function distorts the picture!

What’s going on here? The program treats your picture like it’s on something called the complex plane— which is kind of like the regular two-dimensional plane we’re used to, except that some of the numbers multiply strangely. One of the dimensions on the complex plane is made of regular, normal numbers– which, in this situation, are called the “real numbers”– while the other dimension is made of different numbers, called “imaginary numbers.” These are the numbers that do weird things when you multiply them together. Maybe you’ve heard that you can’t take the square-root of a negative number. Well, on the complex plane you can. And when you do, you get an imaginary number!

Windows, under 1/z

If you’re curious about these crazy creatures called imaginary numbers and how they work to make images go wild on the complex plane, I recommend you check out this site. It gives a great interactive explanation of imaginary numbers (and teaches you about fractals, too!). But I also wouldn’t blame you if you wanted to spend a few hours holding things in front of your webcam and seeing what happens to them under different function transformations!

Gummy bears! Which function did this?

Meet Donna

Next up, I’d like to share a fun collection of games with you. They’re all made by mathematician Donna Dietz, and they all have to do with a particular kind of math that I find very interesting– projective geometry! You can still enjoy the games even if you know nothing about projective geometry (and you might learn something at the same time).

The rules are pretty simple: Donna gives you a bunch of cards with symbols on them. For example, in the version shown here, you get 13 cards with 4 symbols on them each. There are a bunch of different symbols. Your task is to pick four cards to discard and arrange the remaining nine so that the cards in each row, column, and diagonal share exactly one symbol.

Donna’s projective geometry games page has links to lots more games (if you think the game with cards in three rows and columns is too easy, try one with five) and information about them.

“What does this have to do with geometry?” you might be wondering. These games show a very important property of points and lines in projective geometry. In regular geometry (which you could also call Euclidean geometry), you can have two lines that don’t share any points– meaning that they’d be parallel. But this isn’t possible in projective geometry. All pairs of lines share exactly one point. How is this related to Donna’s games? If lines are rows, columns, and diagonals of cards, and points the symbols on them…

Finally, I’ve been driving a lot lately. I live in the Bay Area, and there is SO MUCH TRAFFIC AAAAAAAA!!! I went searching for solutions, and I came across this great video by our friend CGP Grey (who also made these great videos about voting theory). There’s a lot of math going on here, even if it isn’t immediately apparent. Can you find the math? (Oh, and can you stop causing traffic jams? Thanks.)

Don’t Math Munch and drive, and bon appetit!

# Squricangle, Magic Angle Sculpture, and …

Welcome to this week’s Math Munch!

There’s a neat old problem/puzzle that goes like this: make a 3-D shape that could fit snugly through each of three holes—one a square, one a circle, and one a triangle. To make a shape that works for just two holes isn’t so tricky. For example, a cylinder that is just as tall as it is across would fit snugly through a circle hole and a square hole. Can you think of what would work for each of the other two shape combos? What about all three?

Three holes, three shapes…and what’s that over in the corner??

If you’re curious about the answer, you might enjoy this post by Kit Wallace or this page by George Hart or—believe it or not—roundsquaretriangle.com. I don’t know the origin of this puzzle and would love to. I haven’t found any info about it after to poking around the internet for a while. So if you locate any information about the backstory of the squircangle—which is not its real name, just one that I made up—please let us know!

Even though I knew about the square-circle-triangle problem, I was not at all prepared to encounter the solution to the jet-butterfly-dragon problem!

Dragon Butterfly Jet is just one of several “magic angle sculptures” created by artist, chemist, and PhD, and high school dropout John V. Muntean. John writes the following in his Artist Statement:

As a scientist and artist, I am interested in the how perception influences our theory of the universe. … Every 120º of rotation, the amorphous shadows evolve into independent forms. Our scientific interpretation of nature often depends upon our point of view. Perspective matters.

There’s much more to see on John’s website. And you can check out Dragon Butterfly Jet in action in the video below, along with Knight Mermaid Pirate-Ship. I also recommend this video made by John where he demonstrates how his sculpture works himself. It also includes a stop-frame animation of the sculpture being built! So cool.

No, not ellipses…

And finally, what you’ve all been waiting for…

…!

That’s right! My final share of the week is that most outspoken of punctuation marks, the ellipsis. Because often what you don’t say says a whole lot! That’s true when writing a story or some dialogue, and it’s also true in mathematics. Watch: 1+2+3+…+100. See? Pretty neat! Those three dots sure say a mouthful…

The ellipsis is probably my second favorite punctuation mark—after the em dash, of course. But don’t take my word for it. Instead, check out this article about the history and uses—mathematical and otherwise—of the humble ellipsis. Author Cameron Hunt McNabb writes:

Thus the ellipsis has been used to indicate anything from the erroneous to the irrational, and its intrigue lies in resistance to meaning. As long as we have things to say, we will have things to omit.

The very first equals sign, in 1557.

I could go on and on about the ellipsis, just like pi does: 3.1415… But anyway, while we’re on the subject of punctuation, let me point you to one of my favorite sites on the mathematical internet: the Earliest Uses of Various Mathematical Symbols page, maintained by Jeff Miller. Jeff teaches high school math in Florida and also has some other great pages, too, including this one about mathematicians featured on stamps.

Bon…

A nice visualization of the squircangle by Matt Henderson

…appetit!

# Lucea, Fiber Bundles, and Hamilton

Welcome to this week’s Math Munch!

The Summer Olympics are underway in Brazil. I have loved the Olympics since I was a kid. The opening ceremony is one of my favorite parts—the celebration of the host country’s history and culture, the athletes proudly marching in and representing their homeland. And the big moment when the Olympic cauldron is lit! This year I was just so delighted by the sculpture that acted as the cauldron’s backdrop.

Isn’t that amazing! The title of this enormous metal sculpture is Lucea, and it was created by American sculptor Anthony Howe. You can read about Anthony and how he came to make Lucea for the Olympics in this article. Here’s one quote from Anthony:

“I hope what people take away from the cauldron, the Opening Ceremonies, and the Rio Games themselves is that there are no limits to what a human being can accomplish.”

Here’s another view of Lucea from Anthony’s website:

Lucea is certainly hypnotizing in its own right, but I think it jumped out at me in part because I’ve been thinking a lot about fiber bundles recently. A fiber bundle is a “twist” on a simpler kind of object called a product space. You are familiar with some examples of products spaces. A square is a line “times” a line. A cylinder is a line “times” a circle. And a torus is a circle “times” a circle.

Square, cylinder, and torus.

So, what does it mean to introduce a “twist” to a product space? Well, it means that while every little patch of your object will look like a product, the whole thing gets glued up in some fancy way. So, instead of a cylinder that goes around all normal, we can let the line factor do a flip as it goes around the circle and voila—a Mobius strip!

Now, check out this image:

It’s two Mobius strips stuck together! Does this remind you of Lucea?! Instead of a line “times” a circle that’s been twisted, we have an X shape “times” a circle.

Do you think you could fill up all of space with an infinity of circles? You might try your hand at it. One answer to this puzzle is a wonderful example of a fiber bundle called the Hopf fibration. Just as you can think about a circle as a line plus one extra point to close it up, and a sphere as a plane with one extra point to close it up, the three-sphere is usual three-dimenional space plus one extra point. The Hopf fibration shows that the three-sphere is a twisted product of a sphere “times” a circle. For a really lovely visualization of this fact, check out this video:

That is some tough but also gorgeous mathematics. Since you’ve made it this far in the post, I definitely think you deserve to indulge and maybe rock out a little. And what’s the hottest ticket on Broadway this summer? I hope you’ll enjoy this superb music video about Hamilton!

William Rowan Hamilton, that is. The inventor of quaternions, explorer of Hamiltonian circuits, and reformulator of physics. Brilliant.

Here are a couple of pages of Hamiltonian circuit puzzles. The goal is to visit every dot exactly once as you draw one continuous path. Try them out! Rio, where the Olympics is happening, pops up as a dot in the first one. You might even try your hand at making some Hamiltonian puzzles of your own.

Happy puzzling, and bon appetit!

# Wild Maths, Ambiguous Cylinders, and 228 Women

Welcome to this week’s Math Munch!

You should definitely take some time to explore Wild Maths, a site dedicated to the creative aspects of mathematics. Wild Maths is produced by the Millennium Mathematics Project, which also makes NRICH and Plus.

I won!

One fun things you’ll find on Wild Maths is a game called Square It! You can play it with a friend or against the computer. The goal is to color dots on a square grid so that you are the first to make a square in your color. It is quite challenging! To the left you’ll find my first victory against the computer after losing the first several matches.

You’ll find lots more on Wild Maths, including an equal averages challenge, a number grid journey, and some video interviews with mathematicians Katie Steckles and Nira Chamberlain. Wild Maths also has a Showcase of work that has been submitted by their readers, much like our own Readers’ Gallery. (We love hearing from you and seeing your creations!)

Next up is a video of an amazing illusion:

Now, I am as big of a fan of squircles as anyone, but this video really threw me for a loop. The illusion just gets crazier and crazier! The illusion was designed by Kokichi Sugihara of Meiji University in Japan. It recently won second place in the Best Illusion of the Year Contest.

We are fortunate that Dave Richeson has hit it out of the park again, this time sharing both an explanation of the mathematics behind the illusion and a paper template you can use to make your own ambiguous cylinder!

Finally this week, I’d like to share a fascinating document with you. It is a supplement to a book called Pioneering Women in American Mathematics: The Pre-1940s PhD’s by Judy Green and Jeanne LaDuke.

The supplement gives biographies of all 228 American women who earned their PhD’s in mathematics during the first four decades of the 20th century. You might enjoy checking out this page from the National Museum of American History, which describes some about the origin of the book project.

Judy Green, Jeanne LaDuke, and fifteen women who received their PhD’s in math before 1940.

I hope you will find both pleasure and inspiration in reading the stories of these pioneers in American mathematics. I have found them to be a lot of fun to read.

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.)

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:

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.)

Mondrainverted.

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!

# Web Applets, Space Fillers, and Sisters

Welcome to this week’s Math Munch!

Recently I’ve been running across tons of neat, slick math applets. I feel like they all go together. What do they have in common? Maybe you’ll be able to tell me.

First up, you can tinker with some planetary gears. Then try out these chorded polygons. And then how about some threaded lines?

Ready for some more? Because with these sorts of visualizations, Dan Anderson has been on fire lately. Dan is a high school math teacher in New York state. He and his students had fifteen minutes of fame last year when they investigated whether or not Double Stuf Oreos really have double the stuf.

Here is Dan’s page on OpenProcessing. (Processing is the computer language in which Dan programs his applets.) And check out the images and gifs on Dan’s Tumblr. Here’s a sampling!

Dan also coordinates Daily Desmos, which we’ve feature previously. Check out the latest periodic and “obfuscation” challenges!

That’s a chunk of math to chew on already, but we’re just getting started! Next up, check out the space-filling artwork of John Shier.

John’s artwork places onto the canvas shapes of smaller and smaller sizes. Notice that the circles below fill in gaps, but they don’t touch each other, they way circles do in an Apollonian gasket.

You can learn more about John’s space-filling shapes on this page and find further details in this paper.

Thanks for making us this sweet banner, John!

Last up this week, head to this site to watch an awesome trailer of a film about Julia Robinson. The short clip focuses on Julia’s work on Hilbert’s tenth problem. It includes interviews with a number of people who knew Julia, including her sister Constance Reid. Constance wrote extensively about mathematics and mathematicians. I’ve read her biography of Hilbert and can highly recommend it. You can read more about Julia and Constance here and here.

 Julia Robinson Julia’s sister, Constance Reid

Julia and Constance as young girls.

You might enjoy visiting the site of the Julia Robinson Mathematics Festival. Check to see if a festival will be hosted in your area sometime soon, or find out how you can run one yourself!

With May wrapped up and June getting started, I hope you have a lot of math to look forward to this summer. 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.”

This 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:

It 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!

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!