Tag Archives: partitions

Rectangles, Explosions, and Surreals

Welcome to this week’s Math Munch!

What is 3 x 4?   3 x 4 is 12.

Well, yes. That’s true. But something that’s wonderful about mathematics is that seemingly simple objects and problems can contain immense and surprising wonders.

How many squares can you find in this diagram?

As I’ve mentioned before, the part of mathematics that works on counting problems is called combinatorics. Here are a few examples for you to chew on: How many ways can you scramble up the letters of SILENT? (LISTEN?) How many ways can you place two rooks on a chessboard so that they don’t attack each other? And how many squares can you count in a 3×4 grid?

Here’s one combinatorics problem that I ran across a while ago that results in some wonderful images. Instead of asking about squares in a 3×4 grid, a team at the Dubberly Design Office in San Francisco investigated the question: how many of ways can a 3×4 grid can be partitioned—or broken up—into rectangles? Here are a few examples:

How many different ways to do this do you think there are? Here’s the poster that they designed to show the answer that they found! You can also check out this video of their solution.

In their explanation of their project, the team states that “Design tools are becoming more computation-based; designers are working more closely with programmers; and designers are taking up programming.” Designing the layout of a magazine or website requires both structural and creative thinking. It’s useful to have an idea of what all the possible layouts are so that you can pick just the right one—and math can help you to do it!

If you’d like to try creating a few 3×4 rectangle partitions of your own, you can check out www.3x4grid.com. [Sadly, this page no longer works. See an archive of it here. -JL, 10/2016]

Next up, explosions! I could tell you about the math of the game Minesweeper (you can play it here), or about exploding dice. But the kind of explosion I want to share with you today is what’s called a “combinatorial explosion.” Sometimes a problem that appears to be an only slightly harder variation of an easy problem turns out to be way, way harder. Just how BIG and complicated even simple combinatorics problems can get is the subject of this compelling and also somewhat haunting video.

Donald Knuth

Finally, all of this counting got me thinking about big numbers. Previously we’ve linked to Math Cats, and Wendy has a page where you can learn how to say some really big numbers. But thinking about counting also made me remember an experience I had in middle school where I found out just how big numbers could be! I was in seventh grade when I read this article from the December 1995 issue of Discover Magazine. It’s called “Infinity Plus One, and Other Surreal Numbers” and was written by Polly Shulman. I remember my mind being blown by all of the talk of infinitely-spined aliens and up-arrow notation for naming numbers. Here’s an excerpt:

Mathematicians and precocious five-year-olds have long been fascinated by the endlessness of numbers, and they’ve named the endlessness infinity. Infinity isn’t a number like 1, 2, or 3; it’s hard to say what it is, exactly. It’s even harder to imagine what would happen if you tried to manipulate it using the arithmetic operations that work on numbers. For example, what if you divide it in half? What if you multiply it by 2? Is 1 plus infinity greater than, less than, or the same size as infinity plus 1? What happens if you subtract 1 from it?

After I read this article, John Conway and Donald Knuth became heros of mine. (In college, I had the amazing fortune to have breakfast with Conway one day when he was visiting to give a lecture!) Knuth has a book about surreals that’s the friendliest introduction to the surreal numbers that I know of, and in this video, Vi Hart briefly touches on surreal numbers in discussing proofs that .9 = 1. Boy, would I love to see a great video or online resource that simply and beautifully lays out the surreal numbers in all their glory!

It was fun for me to remember that Discover article. I hope that you, too, run across some mathematics that leaves a seventeen-year impression on you!

Bon appetit!

Visualizations, Inspirations, and the Super Ultimate Graphing Challenge

Welcome to this week’s Math Munch!

Jason Davies

Meet Jason Davies, a freelance mathematician living in the UK. Growing up in Wales (one of the 4 countries of the United Kingdom) his classes were taught in Welsh. This makes Jason one of only about 611,000 people that speak the language, only 21.7% of the population of Wales! Imagine if only 1/5 of France spoke French!! These statistics are from a 2004 study, so the numbers may have changed a bit, but they still say something interesting don’t they?

Prime Seive

Jason is all about what numbers and pictures can tell us.  Since graduating from Cambridge, he’s been doing all sorts of data visualization and computer science on his own for various companies and IT firms. I originally found Jason through a link to his Prime Seive visualization, but take a look at his gallery and you’re bound to find something beautiful, interesting, interactive, and cool. I’ve linked to some of my favorites below.

Interactive Apollonian Gasket

Rhodonea Curves

Set Partitions

I asked Jason a few questions about his interest in data visualization and math in general. Here’s a tasty little excerpt:

MM: What’s the most important trait for a mathematician to have? Is there one?

JD: Persistance is always useful in maths! I think the stereotype is to be analytical and logical, but in fact there are many other traits that are highly important, for instance communication skills. Mathematics is passed on from person to person, after all, so being able to communicate ideas effectively is dynamite.

MM: Do you have a message you’d like to give to young mathematicians?

JD: The world needs you!

Read the rest in our Q&A with Jason Davies, and you can see all of our interviews on the Q&A page we’ve just created.

Up next, a beautiful and inspiring video from Spain. The video is actually called Insprations, and it comes to us from Etérea Studios, the online home of animator Cristóbal Vila. In the intro he says, “I looked into that enormous and inexhaustible source of inspiration that is Escher and tried to imagine how it could be his workplace, what things would surround an artist like him, so deeply interested in science in general and mathematics in particular.”

I’d die to have an office like this!

It gets better.  Cristóbal added a page explaining all of the wonderful maths in the video. Click to read about Platonic solids, tilings, tangrams, and various works of art by M.C. Escher.

Finally, a nifty new game that explores the relationship between graphs and different kinds of motion. Super Ultimate Graphing Challenge is a game developed by Physics teacher Matthew Blackman to help his students understand the physics and mathematics of motion. You might not understand it all when you start, but keep playing and see what you can make of it. If you need a bit of help or have something to say, post it in our comments, and we’ll happily reply.

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!