Möbius music

Hofstadter’s Gödel, Escher, Bach is one of my all-time favorite books. As the name implies, there are many references to Bach’s music, particularly his fugues and canons. When I was reading the book back in high school, it was hard to track down and listen to all the music that came up in the book, let alone their musical scores. The world is different now, though. Take Bach’s Crab Canon, for example. Now you can find sites aplenty that describe it, show you the music, and play it. It’s called a crab because it is played against itself forward and backwards simultaneously. Don’t believe me? Look at this MIDI roll visualization (it looks like a crab!), and then listen to the audio file played backwards. You can’t do that to too many pieces of music and still have something worth hearing.

But wait! Why not look at how Bach’s canons resemble functions, and our friend the crab is g(t) = f(18-t). And if you print the piece out on a Möbius strip, you and a friend can play it together, assuming you’re on the same differentiable manifold (ha ha! you knew that). But don’t take my word for it. Watch the video.

Curiously, when it comes to Möbius music, Bach is not the only game in town. I was thoroughly charmed by this video of Vi Hart playing her comparatively recent composition, the Harry Potter Septet on a Möbiola. I like how the variable crank speed is part of the performance.

PLEASE NOTE: I don’t know if it’s really called Möbiola, but that’s what I would call it if I were king.

Journey to the bottom of the Mandelbrot set

If you want to explore the Mandelbrot set, or fractals in general, you have endless options, but you should definitely look at the Xaos Fractal Browser. It’s been super-streamlined for zooming around quickly. Lots of people use it and upload their pictures to Flickr, and Flickr, in turn, makes it easy for me to embed this dandy slideshow.


Zooming around in Mandelbrot space got me thinking about the problem of mathematical exploration in general. I did a little googling and found this news item. (Note: actually I made it up)

WASHINGTON, DC (May 1, 2009)

Government officials were scrambling this morning to put the finishing touches on what’s being touted as the “most significant exploration initiative since Apollo.” The program, called Fractal One, has been established to plant an American flag on the bottom of the Mandelbrot set. Lead mathematician for the program Curt Canneford explained: “The Mandelbrot set is a mathematical object, a fractal set of fantastic richness. The deeper you delve into it, the more mysteries you find. It’s unconscionable that, since its discovery more than twenty years ago, we still don’t know where it stops.”

Bipartisan support has been building for the initiative. As Representative Malcolm Blakey (R, Missouri) explained at a press conference yesterday, “When we heard that the French were funding an expedition to set foot on the bottom of the Mandelbrot set, we realized this was a matter of national pride and competitiveness. Why, in an age when Everest has been climbed, the Marianas trench plumbed, and the moon itself claimed for this great republic, is the Mandelbrot set still hiding secrets? This math resource should be probed and exploited. We might find oil, mineral wealth, or lots of cool pictures for American students to put up in their dorm rooms.”

Late last night, underscoring the urgency of the effort, came word of an imminent Chinese expedition, and there was an unsubstantiated claim that a Russian team was “already there.”

What will it look like at the bottom? It’s hard to say. Lead Mandelnaut Irving Bell was sanguine about the dangers. “We’re talking about extremely small numbers… the linear dimensions alone will measure less than 10 to the minus 128. The computational pressures will be enormous. The iteration counts near the bottom could pin us down for weeks, and a divergent blowout could happen at any time.”

It was a stirring scene as he and his team, against the backdrop of a large American flag, were sealed into the stainless steel Fractal One compute pod and lowered into the complex set.

Binary marble adding machine

I like this hand-crafted wooden adding machine modeled on the digital logic found in microchips. It’s a very simple wooden computer. Still, as simple as it is, it really is showing you how to add any two positive integers that sum to less than 64. Any further addition is just more of the same.

You look at it and you think, whatever those computer circuits are doing on silicon, it just can’t be this simple. But of course it really is that simple. You just need a heaping great crapload of adders to do anything interesting. Which really isn’t a problem when you’ve got a few billion transistors floating around. “Crapload” factors a billion a good many times.

Enigma and the Danish school teacher

Via the Mathematical Tourist I came across this article on the Enigma machine. The short version of the story is that during World War Jr. the Germans were convinced that their code machine, known as Enigma, kept their military secrets safe. It didn’t, partly through the efforts of three Polish mathematicians and partly because of the British crypto-analysts working at Bletchley Park in England (any story that involves a machine called Enigma and a place called Bletchley Park is already off to a really good start). The story is often told as if the Polish mathematicians had spirited away a physical copy of the machine, but in fact what they had was a reasonable mathematical model of how the machine worked. This they gave to the British. Goodness ensued and Hitler was vanquished.

It’s an often told story, and I mention it here because I was so impressed with this page maintained by a Danish high school teacher named Erik Vestergaard. I consider this to be the best kind of amateur reporting, and the kind of thing that really distinguishes the Age of the Web. It’s got beautiful illustrations, lovely old photos, and deep colorful piles of mathematical detail that really tell you what happened. Vestergaard figured out how Enigma worked, and now he’s telling you. No editor, not even of a specialist book with a limited print run, would let you include all that detail. “Cut out all that boring permutation crap,” says the Editor, Mr. Book, “You’re killing your market!” But Mr. Web, the imperturbable Not-An-Editor, says “Leave it in!”

Mr. Web wins again.