Monthly Archives: October 2012

30

Oct. ’12

Pentago, Geometry Daily, and The OEIS

Welcome to this week’s Math Munch!

Pentago Board

Hurricane Sandy is currently slamming the East coast, but the Math Munch Team is safe and sound, so the math must go on.  First up, if you’ve visited our games page lately, you may have noticed a recent addition.  Pentago is a 2-player strategy with simple rules and an enticing twist.

  • Rules: Take turns playing stones.  The first person to get 5 in a row wins.  (5 is the “pent” part.)
  • Twist: After you place a stone you must spin one of the 4 blocks.  This makes things very interesting.

Why don’t you play a few games before you read on?  You can play the computer on their website, play with a friend by email, or download the Pentago iPhone app.  But if you’re ready, let’s dig into some Pentago strategy and analysis.

Mindtwister CEO, Monica Lucas

Mindtwister (the company that sells Pentago) put out a free strategy guide that names 4 different kinds of winning lines and rates their relative strengths.  The weakest strategy is called Monica’s Five, and it’s named after Mindtwister CEO and Pentago lover, Monica Lucas.  You can read our Q&A for more expert game strategies and insights.  We also had a chance to speak with Tomas Floden, the inventor of Pentago, so it’s a double Q&A week.

As you play, you start to build your own strategy guide, so let me share three basic rules from mine.  I call them the first 3 Pentago Theorems.  (A theorem is a proven math fact.)

  1. If you have a move to win, take it!  This one is obvious, but you’ll see why I include it.
  2. If your opponent is only missing one stone from a line of 5 you must play there.  It seems like you could play somewhere else and spin the line apart, but your opponent can play the stone and spin back!  The only exception to this rule is rule 1.  If you can win, just do that!
  3. 4 in a row, with both ends open will (almost always) win.  This is a classic double trap.  Either end will finish the winning line, so by rule 2 both must be filled, but this is impossible.  The exceptions of course will come when your opponent is able to win right away, so you still have to pay close attention.

Up next, check out the beautiful math art of Geometry Daily.

#288 Fundamental

#132 Eight Squares

#259 Dudeney’s Dissection

#296 Downpour

#236 Nova

#124 Cuboctahedron

#136 Tesseract

#26 Pentaflower

#92 Circular Spring

The site is the playground for the geometrical ideas of Tilman Zitzmann, a German designer and teacher, who’s been creating a new image every day for almost a year now!  He also took some time to write about his creative process, so if you’re interested, have a read.  Visit the Geometry Daily archives to view all the images.

Finally, an amazing resource – the On-Line Encyclopedia of Integer Sequences.  What’s the pattern here?  1, 3, 6, 10, 15, 21, …  Any idea?  Do you know what the 50th number would be?  Well if you type this sequence into the OEIS, it’ll tell you every known sequence that matches.  Here’s what you get in this case.  These are the “triangular numbers,” also the number of edges in a complete graph.  It also tells you formula for the sequence:

  • a(n) = C(n+1,2) = n(n+1)/2 = 0+1+2+…+n.

If you make n=1, then you get 1.  If n=2, then you get 3.  If n=5, you get the 5th number, so to get the 50th number in the sequence, we just make n=50 in the formula.  n(n+1)/2 becomes 50(50+1)/2 = 1275.  Nifty.  Who’s got a pattern that needs investigating?

Have a great week, and bon appetit!

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22

Oct. ’12

Pixel Art, Gothic Circle Patterns, and First Past the Post

Welcome to this week’s Math Munch!

Guess what? Today is Math Munch’s one-year anniversary!

We’re so grateful to everyone who has made this year so much fun: our students and readers; everyone who has spread the word about Math Munch; and especially all the people who do and make the cool mathy things that we so love to find and share.

Speaking of which…

Mathematicians have studied the popular puzzle called Sudoku in numerous ways. They’ve counted the number of solutions. They’ve investigated how few given numbers are required to force a unique solution. But Tiffany C. Inglis came at this puzzle craze from another angle—as a way to encode pixel art!

Tiffany studies computer graphics at the University of Waterloo in Ontario, Canada. She’s a PhD candidate at the Computer Graphics Lab (which seems like an amazing place to work and study—would you check out these mazes!?)

Tiffany C. Inglis, hoisting a buckyball

Tiffany tried to find shading schemes for Sudoku puzzles so that pictures would emerge—like the classic mushroom pictured above. Sudoku puzzles are a pretty restrictive structure, but Tiffany and her collaborators had some success—and even more when they loosened the rules a bit. You can read about (and see!) some of their results on this rad poster and in their paper.

Thinking about making pictures with Sudoku puzzles got Tiffany interested in pixel art more generally. “I did some research on how to create pixel art from generic images such as photographs and realized that it’s an unexplored area of research, which was very exciting!” Soon she started building computer programs—algorithms—to automatically convert smooth line art into blockier pixel art without losing the flavor of the original. You can read more about Tiffany’s pixelization research on this page of her website. You should definitely check out another incredible poster Tiffany made about this research!

To read more of my interview with Tiffany, you can click here.

Cartoon Tiffany explains what makes a good pixelization. Check out the full comic!

I met Tiffany this past summer at Bridges, where she both exhibited her artwork and gave an awesome talk about circle patterns in Gothic architecture. You may be familiar with Apollonian gaskets; Gothic circle patterns have a similar circle-packing feel to them, but they have some different restrictions. Circles don’t just squeeze in one at a time, but come in rings. It’s especially nice when all of the tangencies—the places where the circles touch—coincide throughout the different layers of the pattern. Tiffany worked on the problem of when this happens and discovered that only a small family has this property. Even so, the less regular circle patterns can still produce pleasing effects. She wrote about this and more in her paper on Gothic circle patterns.

I’m really inspired by how Tiffany finds new ideas in so many place, and how she pursues them and then shares them in amazing ways. I hope you’re inspired, too!

A rose window at the Milan Cathedral, with circle designs highlighted.

A mathematical model similar to the window, which Tiffany created.

An original design by Tiffany. All of these images are from her paper.

Here’s another of Tiffany’s designs. Now try making one of your own!

Using the Mathematica code that Tiffany wrote to build her diagrams, I made an applet where you can try making some circle designs of your own. Check it out! If you make one you really like—and maybe color it in—we’d love to see it! You can send it to us at MathMunchTeam@gmail.com.

(You’ll may have to download a plug-in to view the applet; it’s the same plug-in required to use the Wolfram Demonstrations Project.)

Finally, with Election Day right around the corner, how about a dose of the mathematics of voting?

I’m a fan of this series of videos about voting theory by C.G.P. Grey. Who could resist the charm of learning about the alternative vote from a wallaby, or about gerrymandering from a weasel? Below you’ll find the first video in his series, entitled “The Problems with First Past the Post Voting Explained.” Majority rule isn’t as simple of a concept as you might think, and math can help to explain why. As can jungle animals, of course.

Thanks again for being a part of our Math Munch fun this past year. Here’s to a great second course! Bon appetit!


PS I linked to a bunch of papers in this post. After all, that’s the traditional first anniversary gift!

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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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