# 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. đź™‚ )

Now, we’ve shared about SET before, but recently there has been some very big SET-related news. Although things have beenÂ quieter around Georgia Tech sinceÂ summer has started, there has been a buzz both here and around the internet about aÂ big breakthrough byÂ Vsevolod Lev, PĂ©ter PĂˇl Pach, and Georgia Tech professorÂ Ernie Croot. Together they have discovered a new approachÂ 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 characteristicsâ€”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 ofÂ SET-less cards we could make in each case.Â This is called the cap set problem, and Vsevolod,Â PĂ©ter, 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 byÂ 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 â€” and in only a few pages of fairly elementary mathematics. â€śOne 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,â€ť Gowers said.

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

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These are photos of Vsevolod,Â PĂ©ter, Ernie, Charlotte, Erica, and the creator of SET, geneticist Marsha Jean Falco.

Ready for more? Earlier this week, I ran acrossÂ 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 rightÂ 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 toÂ 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 Almagestâ€”literally, the “Great Work”â€”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 chordsâ€”a variation on a table of the sine functionâ€”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,Â 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 somethingÂ 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 toÂ 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 happenedÂ across an applet that lets you create art in the style of artist (and superellipse creator) Piet Mondrian. But it also inverts your artâ€”reflects it across a circleâ€”so that you can view your own work from a totally different perspective.Â Then just a few months later I delighted in finding anotherÂ 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 herÂ Rolling Hypocycloids and EpicycloidsÂ are can’t-miss. (Echoes of Ptolemy there, yes?!)

And please don’t miss out on Malin’s porfolioÂ 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 hopeÂ 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 munchesÂ this week come from theÂ recent tweetsÂ 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 Londonâ€™s Millennium Bridge to wobble on its opening day?”

On top of his research,Â Steve is great at sharing math with others. (This week I learnedÂ one great piece of mathÂ from him, and then another, and suddenly there was a very clear theme to my post!) SteveÂ has written for the New York Times and was recently awarded the Lewis Thomas Prize as someone “whoseÂ voice and vision can tell us about scienceâ€™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 isÂ at the SmithsonianÂ inÂ Washington, DC, andÂ there are relatedÂ 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 enjoyÂ 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 aboutÂ 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 Clubâ€”everything from chaos and fractals to error correcting codes. EvenÂ Ehrenfeucht-FraĂŻssĂ© games, which are brand-new to me!

And finally this week: have you ever wondered “What will I ever use math for?” Well, SIAMâ€”theÂ Society for Industrial and Applied Mathematicsâ€”has just the video for you.Â 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!