Tag Archives: Fibonacci

Dots-and-Boxes, Choppy Waves, and Psi Day

Welcome to this week’s Math Munch!

And happy Psi Day! But more on that later.

dots

Click to play Dots-and-Boxes!

Recently I got to thinking about the game Dots-and-Boxes. You may already know how to play; when I was growing up, I can only remember tic-tac-toe and hangman as being more common paper and pencil games. If you know how to play, maybe you’d like to try a quick game against a computer opponent? Or maybe you could play a low-tech round with a friend? If you don’t know how to play or need a refresher, here’s a quick video lesson:

In 1946, a first grader in Ohio learned these very same rules. His name was Elwyn Berlekamp, and he went on to become a mathematician and an expert about Dots-and-Boxes. He’s now retired from being a professor at UC Berkeley, but he continues to be very active in mathematical endeavors, as I learned this week when I interviewed him.

Elwyn Berlekamp

Elwyn Berlekamp

In his book The Dots and Boxes Game: Sophisticated Child’s Play, Elwyn shares: “Ever since [I learned Dots-and-Boxes], I have enjoyed recurrent spurts of fascination with this game. During several of these burst of interest, my playing proficiency broke through to a new and higher plateau. This phenomenon seems to be common among humans trying to master any of a wide variety of skills. In Dots-and-Boxes, however, each advance can be associated with a new mathematical insight!”

Elwyn's booklet about Dots-and-Boxes

Elwyn’s booklet about
Dots-and-Boxes

In his career, Elywen has studied many mathematical games, as well as ideas in coding. He has worked in finance and has been involved in mathematical outreach and community building, including involvement with Gathering for Gardner (previously).

Elywn generously took the time to answer some questions about Dots-and-Boxes and about his career as a mathematician. Thanks, Elywn! Again, you should totally check out our Q&A session. I especially enjoyed hearing about Elwyn’s mathematical heros and his closing recommendations to young people.

As I poked around the web for Dots-and-Boxes resources, I enjoyed listening to the commentary of Phil Carmody (aka “FatPhil”) on this high-level game of Dots-and-Boxes. It was a part of a tournament held on a great games website called Little Golem where mathematical game enthusiasts from around the world can challenge each other in tournaments.

What's the best move?A Sam Loyd Dots-and-Boxes Puzzle

What’s the best move?
A Dots-and-Boxes puzzle by Sam Loyd.

And before I move on, here are two Dots-and-Boxes puzzles for you to try out. The first asks you to use the fewest lines to saturate or “max out” a Dots-and-Boxes board without making any boxes. The second is by the famous puzzler Sam Loyd (previously). Can you help find the winning move in The Boxer’s Puzzle?

Next up, check out these fantastic “waves” traced out by “circling” these shapes:

Click the picture to see the animation!

Lucas Vieira—who goes by LucasVB—is 27 years old and is from Brazil. He makes some amazing mathematical illustrations, many of them to illustrate articles on Wikipedia. He’s been sharing them on his Tumblr for just over a month. I’ll let his images and animations speak for themselves—here are a few to get you started!

A colored-by-arc-length Archimedean spiral.

A colored-by-arc-length Archimedean spiral.

File:Sphere-like_degenerate_torus

A sphere-like degenerate torus.

A Koch cube.

A Koch cube.

There’s a great write-up about Lucas over at The Daily Dot, which includes this choice quote from him: “I think this sort of animated illustration should be mandatory in every math class. Hopefully, they will be some day.” I couldn’t agree more. Also, Lucas mentioned to me that one of his big influences in making mathematical imagery has always been Paul Nylander. More on Paul in a future post!

Psi is the 23rd letter in the Greek alphabet.

Psi is the 23rd letter in the Greek alphabet.

Finally, today—March 11—is Psi Day! Psi is an irrational number that begins 3.35988… And since March is the 3rd month and today is .35988… of the way through it–11 out of 31 days—it’s the perfect day to celebrate this wonderful number!

What’s psi you ask? It’s the Reciprocal Fibonacci Constant. If you take the reciprocals of the Fibonnaci numbers and add them add up—all infinity of them—psi is what you get.

psisum

Psi was proven irrational not too long ago—in 1989! The ancient irrational number phi—the golden ratio—is about 1.61, so maybe Phi Day should be January 6. Or perhaps the 8th of May—8/5—for our European readers. And e Day—after Euler’s number—is of course celebrated on February 7.

That seems like a pretty good list at the moment, but maybe you can think of other irrational constants that would be fun to have a “Day” for!

And finally, I’m sure I’m not the only one who’d love to see a psi or Fibonacci-themed “Gangham Style” video. Get it?

Bon appetit!

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EDIT (3/14/13): Today is Pi Day! I sure wish I had thought of that when I was making my list of irrational number Days…

Sandpiles, Prime Pages, and Six Dimensions of Color

Welcome to this week’s Math Munch!

Four million grains of sand dropped onto an infinite grid. The colors represent how many grains are at each vertex. From this gallery.

We got our first snowfall of the year this past week, but my most recent mathematical find makes me think of summertime instead. The picture to the right is of a sandpile—or, more formally, an Abelian sandpile model.

If you pour a bucket of sand into a pile a little at a time, it’ll build up for a while. But if it gets too tall, an avalanche will happen and some of the sand will tumble away from the peak. You can check out an applet that models this kind of sand action here.

A mathematical sandpile formalizes this idea. First, take any graph—a small one, a medium sided one, or an infinite grid. Grains of sand will go at each vertex, but we’ll set a maximum amount that each one can contain—the number of edges that connect to the vertex. (Notice that this is four for every vertex of an infinite square grid). If too many grains end up on a given vertex, then one grain avalanches down each edge to a neighboring vertex. This might be the end of the story, but it’s possible that a chain reaction will occur—that the extra grain at a neighboring vertex might cause it to spill over, and so on. For many more technical details, you might check out this article from the AMS Notices.

This video walks through the steps of a sandpile slowly, and it shows with numbers how many grains are in each spot.

A sandpile I made with Sergei’s applet

You can make some really cool images—both still and animated—by tinkering around with sandpiles. Sergei Maslov, who works at Brookhaven National Laboratory in New York, has a great applet on his website where you can make sandpiles of your own.

David Perkinson, a professor at Reed College, maintains a whole website about sandpiles. It contains a gallery of sandpile images and a more advanced sandpile applet.

Hexplode is a game based on sandpiles.

I have a feeling that you might also enjoy playing the sandpile-inspired game Hexplode!

Next up: we’ve shared links about Fibonnaci numbers and prime numbers before—they’re some of our favorite numbers! Here’s an amazing fact that I just found out this week. Some Fibonnaci numbers are prime—like 3, 5, and 13—but no one knows if there are infinitely many Fibonnaci primes, or only finitely many.

A great place to find out more amazing and fun facts like this one is at The Prime Pages. It has a list of the largest known prime numbers, as well as information about the continuing search for bigger ones—and how you can help out! It also has a short list of open questions about prime numbers, including Goldbach’s conjecture.

Be sure to peek at the “Prime Curios” page. It contains intriguing facts about prime numbers both large and small. For instance, did you know that 773 is both the only three-digit iccanobiF prime and the largest three-digit unholey prime? I sure didn’t.

Last but not least, I ran across this article about how a software company has come up with a new solution for mixing colors on a computer screen by using six dimensions rather than the usual three.

Dimensions of colors, you ask?

The arithmetic of colors!

Well, there are actually several ways that computers store colors. Each of them encodes colors using three numbers. For instance, one method builds colors by giving one number each to the primary colors yellow, red, and blue. Another systems assigns a number to each of hue, saturation, and brightness. More on these systems here. In any of these systems, you can picture a given color as sitting within a three-dimensional color cube, based on its three numbers.

A color cube, based on the RGB (red, green, blue) system.

If you numerically average two colors in these systems, you don’t actually end up with the color that you’d get by mixing paint of those two colors. Now, both scientists and artists think about combining colors in two ways—combining colored lights and combining colored pigments, or paints. These are called additive and subtractive color models—more on that here. The breakthrough that the folks at the software company FiftyThree made was to assign six numbers to each color—that is, to use both additive and subtractive ideas at the same time. The six numbers assigned to a given number can be thought of as plotting a point in a six-dimensional space—or inside of a hyper-hyper-hypercube.

I think it’s amazing that using math in this creative way helps to solve a nagging artistic problem. To get a feel for why mixing colors using the usual three-coordinate system is such a problem, you might try your hand at this color matching game. For even more info about the math of color, there’s some interesting stuff on this webpage.

Bon appetit!

Partitions, Riddles, and Escher Videos

Welcome to this week’s Math Munch!

Meet James Tanton, one of my very favorite mathematicians. According to his bio, James is “deeply interested in bridging the gap between the mathematics experienced by school students and the creative mathematics practiced and explored by mathematicians.” Me too! Dr. Tanton is an author and math teacher, but I know him best through his internet videos. Some of them cover some pretty advanced mathematics, but this video on partitions and the Fibonacci numbers is very clear and WAY COOL!

o o oo ooo ooooo

Up next, check out Steve Miller’s Math Riddles, a website full of fantastic (you guessed it) math riddles collected by Steve Miller. Steve’s a math professor at Williams College, and according to him, these riddles, “have two very desirable properties: they have an elegant solution, and that solution doesn’t involve advanced mathematics… What you do need is some patience, and a willingness to explore. Don’t be afraid to try something — see where it leads!”

With that in mind, why not give some a try? You can sort the riddles by topic or difficulty, but here a few possible starters:

There are fifteen sticks. Remove six sticks and be left with ten.

Finally, some relaxing videos I’ve found to showcase once again the fantastic artwork of Dutch graphic artist, M.C. Escher. We’ve featured his work before, but I can never get enough.

3 Spheres II by M.C. Escher

“Mathematicians know their subject is beautiful. Escher shows us that it’s beautiful.” That’s a lovely little quote from mathematician Ian Stewart in this short little clip called, The Mathematical Art of M.C. Escher. If you’re up for something more substantial, here’s an hour-long documentary called Metamorphose, which features video of Escher himself hard at work, something I had never seen before! If you end up watching, leave us a comment and let us know what you think.

We’ve also put together a YouTube playlist of every video ever featured on Math Munch, which we will continue to update. If you want to find the coolest math vids on the internet, I’d say that’s a good place to start.

Bon appetit!