# 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!

# Solomon Golomb, Rulers, and 52 Master Pieces

Welcome to this week’s Math Munch.

I was saddened to learn this week of the passing of Solomon Golomb.

Solomon Golomb.

Can you imagine the world without Tetris? What about the world without GPS or cell phones?

Here at Math Munch weÂ are big fans of pentominoes and polyominoesâ€”we’ve written about them often and enjoy sharing them and tinkering with them. While collections of glued-together squares have been around since ancient times, Solomon invented the term “polyominoes”Â in 1953, investigated them, wrote about themâ€”including this bookâ€”and popularized them with puzzle enthusiasts. But one of Solomon’s outstanding qualities as a mathematician is that he pursued a range of projects that blurred the easy andÂ often-used distinction between “pure” and “applied” mathematics. While polyominoes might seem like just a cute plaything, Solomon’s work with discrete structures helped to pave the way for our digital world. Solomon compiled the first book on digital communications and his work ledÂ to such technologies as radio telescopes. You can hear him talk about the applications that came from his work and moreÂ in this video:

Here is another video, one that surveys Solomon’s work and life. It’s fast-paced and charming and features Solomon in a USC Trojan football uniform! Here is a wonderful short biographyÂ of Solomon written by Elwyn Berlekamp.Â And how about aÂ tutorial on a 16-bit Fibonacci linear feedback shift registerâ€”which Solomon mentions as the work he’s most proud ofâ€”in Minecraft!

Another kind of mathematical object that Solomon invented is aÂ Golomb ruler. If you think about it, an ordinary 12-inch ruler is kind of inefficient. I mean, do we really need all of those markings? It seems like we could just do away with the 7″ mark, since if we wanted to measure something 7 inches long, we could just measure from the 1″ mark to the 8″ mark. (Or from 2″ to 9″.) So what would happen if we got rid of redundancies of this kind? How many marks do you actually need in order to measure every length from 1″ to 12″?

An optimal Golomb ruler of order 4.

Portrait of Solomon by Ken Knowlton.

I was pleased to find that there’s actually a distributed computing project at distributed.net to help find new Golomb Rulers, just like the GIMPS project to find new Mersenne primes. It’s called OGR for “Optimal Golomb Ruler.” Maybe signing up to participate would be a nice way to honor Solomon’s memory.Â It’s hard to know what to do when someone passionate and talented and inspiring dies. Impossible, even. We can hope, though, to keep a great person’s memory and spirit alive and to help continue their good work.Â Maybe this week you’ll share a pentomino puzzle with a friend, or check out the sequences on the OEIS that have Solomon’s name attached to them, or host aÂ Tetris or Blokus partyâ€”whatever you’re moved to do.

Thinking about Golomb rulers got me to wondering about what other kinds of nifty rulers might exist. Not long ago, at Gathering for Gardner, Matt Parker spoke about a kind of ruler that foresters use to measure the diameter of tree. Now, that sounds like quite the trickâ€”seeing how the diameter is inside of the tree! But the ruler has a clever work-around: marking things off in multiples of pi! You can read more about this kind of ruler in a blog post by Dave Richeson.Â I love how Dave got inspired and took this “roundabout ruler” idea to the next level to make rulers that can measure area and volume as well. Generalizingâ€”it’s what mathematicians do!

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I was also intrigued by an image that popped up as I was poking around for interesting rulers. It’s called a seam allowance curve ruler. Some patterns for clothing don’t have a little extra material planned out around the edges so that the clothes can be sewn up. (Bummer, right?) To pad the edges of the pattern is easy along straight parts, but what about curved parts like armholes? Wouldn’t it be nice to have a curved ruler? Ta-da!

A seam allowance curve ruler.

David Cohen

Speaking of Gathering for Gardner: it was announced recentlyÂ that G4GÂ is helping to sponsor an online puzzle challenge called 52 Master Pieces. It’s an “armchair puzzle hunt” created by David Cohen, a physician in Atlanta. It will all happen online and it’s free to participate. There will be lots of puzzle to solve, and each one is built around the theme of a “master”Â of some occupation, like an architect or a physician.Â Here are a couple of examples:

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Notice that bothÂ of these puzzles involve pentominoes!

The official start date to the contest hasn’t been announced yet, but you can get a sneak peek of the siteâ€”for a price! What’s the price, you ask? You have to solve a puzzle, of course! Actually, you have your choice of two, and each one is a maze. Which one will you pick to solve? Head on over and give it a go!

 Maze A Maze B

And one last thing before I go: if you’re intrigued by that medicine puzzle, you might really like checking out 100 different ways this shape can be 1/4Â shaded. They were designed by David Butler, who teachesÂ in the Maths Learning Centre at the University of Adelaide. Which one do you like best? Can you figure out why each one is a quarter shaded? It’s like art and a puzzle all at once! Can you come up with some quarter-shaded creations of your own? If you do, send them our way! We’d love to see them.

EightÂ ways to quarter the cross pentomino. 92 more await you!

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!