# 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!

# Harriss Spiral, Math Snacks, and SET

Happy New Year, and welcome to this week’s Math Munch!

The Harriss Spiral

Exciting news, folks! The Golden Ratio curve– that beautiful spiral that everyone adores– has evolved. And not into a freak of nature, either! Into something– dare I say it?– evenÂ more beautiful…

Meet the Harriss spiral. It was discovered/invented by mathematician and artist Edmund Harriss (featured before here and here) when he began playing around with golden rectangles. A golden rectangle is a very special rectangle, whose sides are in a particular proportion. You can read more about them here— but what’s most important to this new discovery is what you can do with them. If you make a square inside a golden rectangle you get another golden rectangle– and continuing to make squares and new golden rectangles inside of ever-shrinking golden rectangles, and drawing arcs through the squares, is one way to make the beautiful golden ratio spiral.

Edmund Harriss decided to get creative. What would happen, he wondered, if he cut the golden rectangle into two similar rectangles (same shape, just one is a scaled-down version of the other) and a square? And then what if he did the same thing to the new rectangles, again and again to make a fractal? Edmund’s new golden rectangle fractal makes this pattern, and when you draw a spiral through it, you get a lovely branching shape.

But don’t take my word for it. Math journalist Alex Bellos broke the news just this week in his article inÂ The Guardian.Â His article explains much, much more than I can here– check it out to learn many more wonderful things about the Harriss spiral (and other spirals that Harriss has created…)!

(Bonus: Here’s a GeoGebra demonstration created by John Golden that builds the Harriss Spiral. It’s awesome!)

Next up is a site that sounds quite a lot like Math Munch. But it’s all games and cartoons, all the time. (Maybe that means you’ll like it better…) Check out Math Snacks, a site developed by a group of math educators atÂ New Mexico State University. They worked hard to create games and animations that are both fun and full of interesting math.

One of my favorite games on Math Snacks is calledÂ Game Over Gopher.In this game, you have to save your carrot from an army of gophers by placing little machines that feed the gophers. Where’s the math, you may wonder? Placing the gopher-feeders and the other equipment that can help you save the carrot requires you to think carefully about geometry and coordinates.

Finally, speaking of games, here’s one of my favorites. I love to play SET, and I recently found a way to play online– either against a friend or against the computer. Click on this link to start your own game!

To play SET, you deal 12 cards. Then, you try to find a group of 3 cards that all share and all don’t share the same characteristics. For example, in the picture to the right– do you see the cards with the empty red ovals? They’re all the same shape, shading, and color (oval, empty, red), but they’re all different numbers (1, 2, and 3). Can you find any other sets in the picture? (Hint: One involves purple.)

Want to hone your SET skills without competing? Here’s a daily SET puzzle to challenge you.

Enjoy the games (and maybe invent a spiral of your own) and bon appetit!