Monthly Archives: June 2016

Temari, Function Families, and Clapping Music

We have a rare 5 Thursdays this month, so we get an extra rerun post. This one features a Q&A with mathematical artist Carolyn Yackel and much more beautiful stuff. Enjoy!

Math Munch

Welcome to this week’s Math Munch!

Carolyn Yackel Carolyn Yackel

As Justin mentioned last week, the Math Munch team had a blast at the MOVES conference last week.  I met so many lovely mathematicians and learned a whole lot of cool math. Let me introduce you to Carolyn Yackel. She’s a math professor at Mercer University in Georgia, and she’s also a mathematical fiber artist who specializes in the beautiful Temari balls you can see below or by clicking the link. Carolyn has exhibited at the Bridges conference, naturally, and her 2012 Bridges page contains an artist statement and some explanation of her art.

temari15 temari3 temari16

Icosidodecahedron Icosidodecahedron

Truncated Dodecahedron Truncated Dodecahedron

Cuboctahedron Cuboctahedron

Temari is an ancient form of japanese folk art. These embroidered balls feature various spherical symmetries, and part of Carolyn’s work has been figure out how to create and exploit these symmetries on the sphere.  I mean how do you actually make it…

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Combinatorial Games, Redistricting Game, and Graph Music

Welcome to this week’s Math Munch!

Have you ever played tic-tac-toe? If so, maybe you’ve noticed that unless you or your opponent makes a bad move, the game always ends in a tie! (Oops– spoiler alert!) Why is that? And what makes tic-tac-toe different from other games that have unpredictable outcomes, like Monopoly or the card game War?

strategy-for-x

We wrote about tic-tac-toe in this post! Click to learn more.

Tic-tac-toe is similar to other kinds of game that mathematicians call combinatorial games— or games where there is no chance involved in the outcome and neither player has information that the other one doesn’t. This means that depending on who starts, where they go, and where each player decides to go next, the outcome is completely predictable and everyone playing could know what it is before it even happens. No surprises!

Now, this might also sound like NO FUN to you (why play the game at all if everyone knows what’s going to happen?) but I think it introduces a new kind of fun– figuring out what the outcomes could be! One of my favorite combinatorial games is the game NIM.

NIM 1

Here’s an example of a starting NIM board. If you go first, can you win? (Assuming your opponent never makes a mistake.)

NIM is a two-player game. You start with several piles or rows of objects (here they’re matches). On each turn, a player removes some objects from a pile– any number they want. BUT the player who’s forced to remove the last match loses!

 

There’s no chance in NIM– no dice determining how many matches you can remove, for example. Also both players know the rules and how many matches are in the piles at all times. That means that if you thought about it for a while, you could figure out who should win or lose any game of NIM. Maybe playing the game NIM isn’t super fun– but thinking about it like a puzzle is!

More versions of online NIM can be found here and here. And to read about combinatorial games we’ve written about in the past, check out this interview with mathematician Elwyn Berlekamp!

Next up, it’s presidential election time here again in the U.S.! Did you know that there’s a lot of mathematics behind what makes elections work? Four years ago, before the last presidential election, we shared a great series of YouTube videos about the math of elections.

redistricting game

A map in the redistricting game.

A big way that math gets involved in elections is through how politicians decide to draw districts, or regions of states that get to elect their own representative to the House of Representatives and elector to the Electoral College. The math behind drawing districts ranges from simple arithmetic to graph theory, or the field of math that deals with how parts of a shape or diagram are connected. To learn more about drawing election districts and the math behind it, check out the Re-Districting Game! In this game, you play the part of a map maker who works with the Congress, governor of your state, and courts to make a district map that meets everyone’s needs.

Finally, I recently ran across a series of graph music videos! What’s that? Videos in which a graph (made on Desmos) dances along to music, much like people would in a regular music video. Here’s one of my favorites:

The equations on the left-hand side of the screen create the images you see and the rhythm of the animation. Want to make your own graph music video? Share it with us!

Bon appetit!

Halving-Fun, Self-Tiling Tile Sets, and Doodal

This post comes to us all the way from June of 2014! Enjoy this blast from the past!

Welcome to this week’s Math Munch!

Print out two copies of this pattern, cut them out, and fold each along the dotted lines, making two identical solids. Then fit these two pieces together to make a regular tetrahedron.

Print out two copies of this pattern, cut them out, and fold each along the dotted lines, making two identical solids. Then fit these two pieces together to make a regular tetrahedron.

Our first bit of fun comes from a blog called Futility Closet (previously featured). It’s a neat little cut-and-fold puzzle. The shape to the right can be folded up to make a solid with 5 sides. Two of them can be combined to make a solid with only 4 sides, the regular tetrahedron. If you’d like, you can use our printable version, which has two copies on one sheet.

What do you know, I also found our second item on Futility Closet! Check out the cool family of tiles below. What do you notice?

A family of self-tiling tiles

A family of self-tiling tiles

Did you notice that the four shapes in the middle are the same as the four larger shapes on the outside? The four tiles in the middle can combine to create larger versions of themselves! They can make any and all of the original four!!

Lee Sallows

Recreational Mathematician, Lee Sallows

Naturally, I was reminded of the geomagic squares we featured a while back (more at geomagicsquares.com), and then I came to realize they were designed by the same person, the incredible Lee Sallows! (For another amazing one of Lee Sallows creations, give this incredible sentence a read.) You can also visit his website, leesallows.com.

reptile3

A family of 6 self-tiling tiles

For more self-tiling tiles (and there are many more amazing sets) click here. I have to point out one more in particular. It’s like a geomagic square, but not quite. It’s just wonderful. Maybe it ought to be called a “self-tiling latin square.”

And for a final item this week, we have a powerful drawing tool. It’s a website that reminds me a lot of recursive drawing, but it’s got a different feel and some excellent features. It’s called Doodal. Basically, whatever you draw inside of the big orange frame will be copied into the blue frames.  So if there’s a blue frame inside of an orange frame, that blue frame gets copied inside of itself… and then that copy gets copied… and then that copy…!!!

To start, why don’t you check out this amazing video showing off some examples of what you can create. They go fast, so it’s not really a tutorial, but it made me want to figure more things out about the program.

I like to use the “delete frame” button to start off with just one frame. It’s easier for me to understand if its simpler. You can also find instructions on the bottom. Oh, and try using the shift key when you move the blue frames. If you make something you like, save it, email it to us, and we’ll add it to our readers’ gallery.

Start doodaling!

Make something you love. Bon appetit!

A fractal Math Munch Doodal

A fractal Math Munch Doodal

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.)

ThereAreNoSetsHere

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:

tumblr_o0k7mkhNSN1uk13a5o1_500

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.

suntriangle

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

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.)

Mondrian

Mondrainverted.

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!