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!

Wordless Videos, isthisprime, and Fan Chung Graham

Welcome to this week’s Math Munch! For the final Thursday of May, we’ll be looking back at some of this month’s posts from our Facebook page. We’ll see some wordless videos by The Global Math Project, look at a prime number quiz game, and meet Fan Chung Graham, one of the world’s leading mathematicians.

I don’t know much about The Global Math Project, but I know James Tanton is involved, and that is always a good thing. (Remember his Exploding Dots?) Well, they’ve posted a couple of wonderful videos featuring Tanton’s “math without words.” Need I say more? See for yourself.

If you like those, here are some more math without words from Tanton’s website.

Up next is a neat little thing by Christian Lawson-Perfect from The Aperiodical. Christian bought isthisprime.com and set up a little quiz game. Click over and see for yourself how it goes… I’ll wait… click below…

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It’s good practice for divisibility tests and getting your prime recognition up, but I suppose it’s not all that mathematical, is it? But Christian did something interesting. He recorded data from all the games played, and he wrote a summary of the results. I love all the charts and graphs in there. The one below shows how likely a number is to be missed by players.

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Finally, I hadn’t heard of Fan Chung-Graham until I found an interview of her posted on Facebook. She is one of the world’s leading mathematicians in several fields, and though she recently retired, she still conducts some research. The interview is a little academic, but it’s still nice to get to know such a talented mathematician.

fan1

Well that wraps up the week and month. I hope you’ve found some tasty math.  Have a great week and bon appetite!

Forest Fires, Scrubbing Calculator, and Bongard Problems

Welcome to this week’s Math Munch!

Last summer where I live, in California, there were a lot of forest fires. We’re having a big drought, and that made fires started for lots accidental of reasons– lightning, downed power lines, things like that– get much bigger than usual. I thought I’d learn a little about forest fires so that I can be a more responsible resident of my state.

And I found this great website with an awesome computer simulation that you can manipulate to experiment with the factors that lead to forest fires!

Forest fire.png

My forest fire, just starting to burn. Click on this picture to play with the simulation! (Note: It doesn’t work as well in certain browsers. I recommend Firefox.)

 

This site was built by Nicky Case, who studies things that mathematicians and scientists call complex systems. Basically, a complex system is some phenomenon that has a simple set of causes but unpredictable results. An environment with forest fires is a good example– a simple lightning strike in a drought-ridden forest can cause a wildfire to spread in patterns that firefighters struggle to predict. It turns out that simulations are perfect for modeling complex systems. With just a few simple program rules we can create a huge number of situations to study. Even better, we can change the rules to see what will happen if, say, the weather changes or people are more careful about where and when they set fires.

Forest fire what if

What do you think? Click the picture to see Nicky’s simulation.

You can use Nicky’s program to change the probability that trees grow, a burning tree sets its neighbors on fire, and many other factors. You can even invent your own and model them with emojis. What if there were two kinds of trees and one was more flammable than the other? What if trees grew quickly but lightning was common? As Nicky shows, simulations are useful for exploring what-if questions in complex systems. Use your imagination and explore!

Next up, have you ever heard of a scrubbing calculator? No, it’s not a calculator that doubles as a sponge. It’s a calculator that helps you solve for unknowns in equations by “scrubbing,” or approximating, the answer until you find a number that works.

Scrubbing calculator equation

Here’s how it works: Say you’re trying to solve for an unknown, like the x in the equation above (maybe for some practical reason or just because you’re doing your homework). You could do some pretty complicated algebraic manipulations so that the x alone equals some number. But what if you could make a guess and change it until the equation worked?

Scrubbing calculator

Click on this image to see a scrubbing calculator in action!

If your guess was too big, you’d know because the expression wouldn’t equal 768 anymore– it would equal something larger. And if you had a calculator that instantly told you the solution based on your guess, you could do this guessing and checking pretty quickly.

Well, lucky for you I found a scrubbing calculator that you can use online! It’s very simple– just type in your equation and your guess, and click on the number you want to change (most likely the guess) to make it larger or smaller. It’s useful for solving equations, like I said. But I actually find it most interesting to watch how the whole expression changes as you change one of the numbers in it. For instance, check out the Pythagorean triple calculator I built. What do you notice as you gradually change one of the numbers in the expression?

https://www.cruncher.io/?/embed/xiY1IjwUJS

Finally, I’m excited to share with you one of my favorite kinds of puzzles– Bongard problems!
Bongard 1A Bongard problem has two sets of pictures, with six pictures in each set. All of the pictures on the left have something in common that the pictures on the right do not. The challenge is to figure out what distinguishes the two groups of pictures.

Bongard 2

This problem was made by Douglas Hofstadter, who introduced Bongard problems to the U.S. I haven’t figured it out yet. Can you?

I got the Bongard problems shown above from a collection of problems put together by cognitive scientist Harry Foundalis. He has almost 300 of them, some made by Mikhail Bongard himself, who developed these problems while studying how to train computers to recognize patterns.

Harry also has guidelines for how to develop your own Bongard problems. He encourages people to send their problems to him and says he might even put them up on his site!

Bon appetit!

Slides and Twists, Life in Life, and Star Art

Happy second Thursday, and get your engines star-ted! We hope you’ll enjoy this throwback post from May 2012. Bon appetit!

Math Munch

Welcome to this week’s Math Munch!

I ran across the most wonderful compendium of slidey and twisty puzzles this past week when sharing the famous 15-puzzle with one of my classes.  It’s called Jaap’s Puzzle Page and it’s run by a software engineer from the Netherlands named Jaap Scherphuis. Jaap has been running his Puzzle Page since 1999.


Jaap Scherphuis
and some of his many puzzles

Jaap first encountered hands-on mathematical puzzles when he was given a Rubik’s Cube as a present when he was 8 or 9. He now owns over 700 different puzzles!

Jaap’s catalogue of slidey and twisty puzzles is immense and diverse. Each puzzle is accompanied by a picture, a description, a mathematical analysis, and–SPOILER ALERT–an algorithm that you can use to solve it!

On top of this, all of the puzzles in Jaap’s list with asterisks (*) next to them have playable Java applets on…

View original post 365 more words

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.

Solomon Golomb.

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

Here at Math Munch we are big fans of pentominoes and polyominoes—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” in 1953, investigated them, wrote about them—including this book—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 and 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 led to such technologies as radio telescopes. You can hear him talk about the applications that came from his work and more 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 biography of Solomon written by Elwyn Berlekamp. And how about a tutorial on a 16-bit Fibonacci linear feedback shift register—which Solomon mentions as the work he’s most proud of—in Minecraft!

Another kind of mathematical object that Solomon invented is a 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.

An optimal Golomb ruler of order 4.

Portrait of Solomon by Ken Knowlton.

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. 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. 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 a Tetris or Blokus party—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 trick—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. 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. Generalizing—it’s what mathematicians do!

 img_3975  measuringtapes1

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.

A seam allowance curve ruler.

David Cohen

David Cohen

Speaking of Gathering for Gardner: it was announced recently that G4G 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” of some occupation, like an architect or a physician. Here are a couple of examples:

MedicinePuzzle
 ArchitectPuzzle

Notice that both 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 site—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 A

Maze B

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 1/4 shaded. They were designed by David Butler, who teaches 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.

Six ways to quarter the cross pentomino. 94 more await you!

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

Bon appetit!

Near Miss, Curiosa Mathematica, and Poincaré

Welcome to this week’s Math Munch!

For this last Thursday of April, we’ll be taking a look at some recent posts from our facebook page. Craig Kaplan writes about “near miss” polyhedra, a Pythagorean gif takes us to an curious math blog, and we find a beautiful portrait of a great mathematician.  Let’s go!

Craig Kaplan

Craig Kaplan

First is an article from a wonderful mathematician and mathematical artist by the name of Craig Kaplan. His name has popped up on Math Munch before (1, 2 ,3), in case it sounds familiar. You can check out Craig’s stuff on his website, Isohedral, or download his really great game, “Good Fences,” which I have on my iPhone.

near missWhat I really wanted to share, however was Craig’s writing on “A New Near Miss.” This is a polyhedron that almost is… but just isn’t. It looks pretty good, but it can’t be. You’ll have to read to see what I mean.

PythagorasPerigalP.gifUp next, I found this little gif on our facebook page, and I absolutely loved it. It demonstrates the Pythagorean Theorem which says that as long as that’s a right triangle there, the big square on bottom is exactly as big as the two smaller squares combined. The animation shows you how to chop up the middle-sized square and recombine it with the small one to make the big one. I knew there were demonstrations/proofs like this, but this one opened my eyes to something I didn’t quite know before.

This gif sent me off on a journey through the internet to track down the source, and it led me to a site called Curiosa Mathematica. It’s a math blog featuring lots of random math goodies. There’s lots to see and get into (much like Math Munch). Here’s a quote I found there.  I hope you find something you like too.

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Finally, I was really taken by this piece of art (below). It’s a portrait of French mathematician Henri Poincaré, and it was drawn by Bill Sanderson. I can’t find much info on Bill, but WOW the piece is so cool. I love how he’s surrounded by his mathematical creations. I was hoping he had done more, and I did find a couple more (below), but not all I had hoped for.

 

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French Mathematician Henri Poincaré

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Alan Turing

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Isaac Newton

Have some illustrative talent? I’d love to see your mathematician’s portrait. Feel free to send us something… anything.

I hope you enjoy your weekend and find something tasty out there in the mathematical interwebs. Bon appetit!

Mobiles, Mathematical Objects, and Math Magazine

Welcome to this week’s Math Munch!

Before we start, a little business. You may have noticed that posts have been few and far between lately. Those of you who know us, the members of the Math Munch Team, know that we’ve all made a lot of life changes in the past year or two. We started out together teaching in the same school in New York– but now we live on far corners of the country and spend our time doing very different things. In case you’re curious, here are some pictures of the things we’ve been up to!

stack

Justin’s genus 19, rotationally symmetric surface

But even though we’ve moved apart physically, we’ve decided that we really want to keep the Math Munch Team together. We LOVE sharing our love of math with you– and we love hearing from you about the amazing things you make and do with math, too.

So, we’ve decided to revamp our posting process and came up with a schedule for when you can expect posts. There will be a new post every Thursday. (Though if Anna is posting from the West Coast, it might come out in the wee hours of Friday morning for some of you!) And here’s the monthly schedule of Thursday posts:

  • The first Thursday of the month will be a post from Justin
  • The second Thursday of the month will be a rerun!! Did you know we have over 150 posts on this site?? And we’ve been posting for almost five years??
  • The third Thursday of the month will be a post from Anna
  • The last Thursday of the month will be a post from Paul

And for those mysterious months with five Thursdays (ooh, when will that be, I wonder?)… There will be a surprise!

And now… for some math!

Screen Shot 2016-04-22 at 1.44.50 AMFirst up is a little game called SolveMe Mobiles! This game is full of little puzzles in which you have to figure out what each of the different shapes in a mobile weighs. You’re given different clues in different puzzles. So, for instance, in the puzzle to the left, you’re given the weight of the red circle and you have to figure out how much a blue triangle is. But you’re not given the weight of the whole mobile… Hmmm…

Screen Shot 2016-04-22 at 1.53.04 AMAnd this one, to the right, gives you the weight of one of the shapes and of the whole mobile– but now there are three shapes! Tricky!

Even better, you can build your own mobile puzzle for others to solve! I made this one, shown below– like my use of a mobile within a mobile?

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Next up, I found a beautiful Tumblr account that I’d like to share with you full of pictures of found mathematical objects. It’s called… Mathematical Objects! (How clever.) The author of the site writes that the aim of the blog is to “show that mathematics, aside from its practicality, is also culturally significant. In other words, mathematics not only makes the trains run on time but also fundamentally influences the way we view the world.

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“Counting to One Hundred with my Four-Color Pen”

tumblr_n07pvisVvm1rnwsgbo2_1280Some of the images are mathematical art, like the one above; others are more “practical,” such as plans for buildings or images drawn from science.

Do you ever see an interesting mathematical object in the wild and feel the urge to take a picture of it? If so, go ahead and send it to us! We’d love to see what you find.

I’m very excited to share this last find with you all. It was sent to me by a wonderful math teacher, Mark Dittmer, and his math students. This year, they were inspired by Math Munch to make their own fun online math sites! I think what they made is super awesome– and I want to share it with you. I’ll be featuring some of their work in my next few posts, one thing at a time to give each its own day in the sun. First up is this adorable adventure story about the residents of Number Land. I hope you enjoy it!

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Bon appétit!