Tag Archives: video

14

Oct. ’12

Harmonious Sum, Continuous Life, and Pumpkins

Welcome to this week’s Math Munch!

We’ve posted a lot about pi on Math Munch – because it’s such a mathematically fascinating little number.  But here’s something remarkable about pi that we haven’t yet talked about. Did you know that pi is equal to four times this? Yup.  If you were to add and subtract fractions like this, for ever and ever, you’d get pi divided by 4.  This remarkable fact was uncovered by the great mathematician Gottfried Wilhelm Leibniz, who is most famous for developing the calculus.  Check out this interactive demonstration from the Wolfram Demonstrations Project to see how adding more and more terms moves the sum closer to pi divided by four.  (We’ve written about Wolfram before.)

I think this is amazing for a couple of reasons.  First of all, how can an infinite number of numbers add together to make something that isn’t infinite???  Infinitely long sums, or series, that add to a finite number have a special name in mathematics: convergent series.  Another famous convergent series is this one:

The second reason why I think this sum is amazing is that it adds to pi divided by four.  Pi is an irrational number – meaning it cannot be written as a fraction, with whole numbers in the numerator and denominator.  And yet, it’s the sum of an infinite number of rational numbers.

In this video, mathematician Keith Devlin talks about this amazing series and a group of mathematical musicians (or mathemusicians) puts the mathematics to music.

This video is part of a larger work called Harmonious Equations written by Keith and the vocal group Zambra.  Watch the rest of them, if you have the chance – they’re both interesting and beautiful.

Next up, Conway’s Game of Life is a cellular automaton created by mathematician John Conway.  (It’s pretty fun: check out this to download the game, and this Munch where we introduce it.)  It’s discrete – each little unit of life is represented by a tiny square.  What if the rules that determine whether a new cell is formed or the cell dies were applied to a continuous domain?  Then, it would look like this:

Looks like a bunch of cells under a microscope, doesn’t it?  Well, it’s also a cellular automaton, devised by mathematician Stephan Rafler from Nurnberg, Germany.  In this paper, Stephan describes the mathematics behind the model.  If you’re curious about how it works, check out these slides that compare the new continuous version to Conway’s model.

Finally, I just got a pumpkin.  What should I carve in it?  I spent some time browsing the web for great mathematical pumpkin carvings.  Here’s what I found.

A pumpkin carved with a portion of Escher’s Circle Limit.

A pumpkin tiled with a portion of Penrose tiling.

A dodecapumpkin from Vi Hart.

I’d love to hear any suggestions you have for how I should make my own mathematical pumpkin carving!  And, if you carve a pumpkin in a cool math-y way, send a picture over to MathMunchTeam@gmail.com!

Bon appetit!

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07

Oct. ’12

Martin Gardner, G4G, and Many More Flexagons

Welcome to this week’s Math Munch!

Meet the incredible Martin Gardner. If you’re a mathematician then chances are good you already have, but any reader of this blog has certainly felt the ripples of his influence. Nearly everything we share on Math Munch can be traced back to the recreational mathematics of Martin Gardner. For 25 years he wrote the “Mathematical Games” column for Scientific American, in which he shared wonderful puzzles, riddles, and games that have inspired generations of mathematicians. We’re trying to do the same here at Math Munch, so Gardner’s a sort of hero for us.

Maybe the best way to connect with this math legend would be to take on a few of his puzzles. (I’ve linked to some of my favorites below.) Many of the puzzles have a “Print ‘n Play” option you may want to take advantage of. His columns and other writings now live in the more than 100 books he wrote. Pick one up at your local library and dig in!

Martin Gardner’s Puzzles

Heavy Weight

Crazy Cut

Two Sipirals

Cube Dates

Scott Kim’s ambigram logo for G4G5

In this video you can hear Scott Kim (remember his ambigrams?) talk about his own connection to Martin Gardner, and here he talks about his involvement with the Gathering 4 Gardner. These are events that take place across the world to celebrate Gardner’s work and legacy. If you’re inspired by some of this stuff, maybe you’ll get a few friends together and share it with them.

Gardner’s very first article for Scientific American was about the hexaflexagon, so as part of this year’s Gathering 4 Gardner, people are making flexagons of all sorts. Here’s a video of Martin himself talking about them, which is part of full-length video about Gardner called “The Nature of Things.” Justin wrote about flexagons here, and Vi Hart followed suit with a pair of fantastic videos telling the true story of their discovery.

And here’s part 2.

That’s not all! There’s a whole world of flexagons to build and play with. To see another kind, check out the cyclic hexa-tetraflexagon as shown off by James Grime in this video.

A Flexagon Bestiary

Here’s one last flexagon resource with instructions you might prefer.  There are so many videos to watch, puzzles to solve, and flexagons to flex. Hopefully you’ll have a very mathy week in honor of Martin Gardner.

Bon appetit!

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01

Oct. ’12

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!

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