1. **Marillion, "If My Heart Were a Ball It Would Roll Downhill"**: Very neat track from

one of my favorite neo-progressive bands. Catchy, but with lots of layers.

2. **Mandelbrot Set, "Constellation of Rings"**: math-geek postrock. What's not to love?

3. **The Police, ;Every Breath You Take"**: I've always been a fan of the Police. But

what I like most about this song is how often it's been used by clueless people. I've

heard this at multiple weddings, where the couple thought it was a beautiful romantic

song. If you listen to it, it's anything but romantic. It's actually a rather evil

little song about a stalker: "Every breath you take, every vow you break,

every smile you fake, I'll be watching you... Oh can't you see, you belong to me?"

How can anyone miss that?

4. **Naftule's Dream, "Speed Klez"**: John Zorn-influenced klezmer mixed with

a bit of thrash. Insane, but very very cool. Thrash with a trombone line!

5. **Jonathan Coulton, "Todd the T1000"**: Sci-fi geek pop. It's a catchy little pop

song about trading in your old robot for a new one which turns out to be a

psychopath.

6. **Hamster Theater, "Reddy"**: A short track from a great band. Hamster Theater

is a sort-of spin-off from Thought Plague. It's a bit more traditional than

what you'd hear from TP; still very dissonant, sometimes atonal, but more often

closer to traditional tonality and song structure. This track is a short instrumental

featuring an accordion solo.

7. **Transatlantic, "Mystery Train"**: great little song. It's a track by one of

those so-called supergroups; Transatlantic is a side-project formed by members of

Marillion (bassist Pete Travawas), Dream Theater (drummer Mike Portnoy), Spock's Beard (singer Neil Morse), and the Flower Kings (guitarist Royne Stolt). In general, these

supergroups have a sort of shaky sound. These guys are *great* together; it sounds

like they've been playing together for years: they're sharp, there's a great interplay

between the different instruments, it's all incredibly precise. I've heard that the

music was written in advance mainly by Morse, but even with polished music pre-written,

it's got a great sound, and you can here the distinctive musical voices of each of the

musicians.

8. **Godspeed You! Black Emperor, "Antennas To Heaven"**: It's Godspeed - which means

that it's brilliant post-rock. This starts off with a very rough recording of a very

old-timey folkey tune, and uses it as a springboard into a very typical God-speed

texture.

9. **The Flower Kings, "Devil's Playground"**: more neo-progressive stuff. This is an

incredibly long piece (25 minutes), very typical of Roine Stolt's writing. It's not

the sort of way-out-there kind of thing that you'd hear from, say, King Crimson; it's

very structured, very melodic, but put together more in the structure of a symphony

(theme, development, restatement) than the typical ABACAB structure of a rock song.

10. **Porcupine Tree, "Sleep Together"**: a brilliant song by yet another neo-prog

band. Very odd... a strange electronic pulse drives the entire song; but it starts

off as a very quiet song with this electronic pulse giving it a tense feel. Then

the percussion comes in, and shifts your sense of the rhythm... And then it gets

to the chorus, which is big and loud, and features a full string section. Strange,

but wonderful.

## Archive for: July, 2007

## Friday Random Ten, July 13th

## Fractal Borders

Part of what makes fractals so fascinating is that in addition to being beautiful,

they also describe real things - they're genuinely useful and important for helping us to

describe and understand the world around us.

A great example of this is maps and measurement.

## Fractal Woo: Video TransCommunication

This is a short one, but after mentioning this morning how woo-meisters constantly invoke

fractals to justify their gibberish, I was reading an article at the 2% company

about Allison DuBois, the supposed psychic who the TV show "Medium" is based on. And that

led me to a perfect example of how supposed fractals are used to justify some of the

most ridiculous woo you can imagine.

## The Mandelbrot Set

The most well-known of the fractals is the infamous Mandelbrot set. It's one of the first

things that was really studied *as a fractal*. It was discovered by Benoit Mandelbrot during his early study of fractals in the context of the complex dynamics of quadratic polynomials the 1980s, and studied in greater detail by Douady and Hubbard in the early to mid-80s.

It's a beautiful

example of what makes fractals so attractive to us: it's got an extremely simple definition; an incredibly complex structure; and it's a rich source of amazing, beautiful images. It's also been glommed onto by an amazing number of woo-meisters, who babble on about how it represents "fractal energies" - "fractal" has become a woo-term almost as prevalent as "quantum", and every woo-site

that babbles about fractals invariably uses an image of the Mandelbrot set. It's

also become a magnet for artists - the beauty of its structure, coming from a simple bit of math captures the interest of quite a lot of folks. Two musical examples are Jonathon Coulton and the post-rock band "Mandelbrot Set". (If you like post-rock, I definitely recommend checking out MS; and a player for brilliant Mandelbrot set song is embedded below.)

## An Unsolved Simple Graph Problem

One of the things that I find fascinating about graph theory is that it's so simple, and

yet, it's got so much depth. Even when we're dealing with the simplest form of a graph - undirected graphs with no 1-cycles, there are questions that *seem* like that should be obvious, but which we don't know the answer to.

For example, there's something called the *reconstruction theorem*. We strongly suspect that it's really a theorem, but it remains unproven. What it says is a very precise formal version of the idea that a graph is really fully defined by a canonical collection of its subgraphs.

## An Introduction to Fractals

I thought in addition to the graph theory (which I'm enjoying writing, but doesn't seem

to be all that popular), I'd also try doing some writing about fractals. I know pretty

much *nothing* about fractals, but I've wanted to learn about them for a while, and one

of the advantages of having this blog is that it gives me an excuse to learn about things that that interest me so that I can write about them.

Fractals are amazing things. They can be beautiful: everyone has seen beautiful fractal images - like the ones posted by my fellow SBer Karmen. And they're also useful: there are a lot of phenomena in nature that seem to involve fractal structures.

But what is a fractal?

## Graph Contraction and Minors

Another useful concept in simple graph theory is *contraction* and its result, *minors*

of graphs. The idea is that there are several ways of simplifying a graph in order to study

its properties: cutting edges, removing vertices, and decomposing a graph are all methods we've seen before. Contraction is a different technique that works by *merging* vertices, rather than removing them.

## Graph Decomposition and Turning Cycles

One thing that we often want to do is break a graph into pieces in a way that preserves

the structural relations between the vertices in any part. Doing that is called

*decomposing* the graph. Decomposition is a useful technique because many ways

of studying the structure of a graph, and many algorithms over graphs can work by

decomposing the graph, studying the elements of the decomposition, and then combining

the results.

To be formal: a graph G can be decomposed into a set of subgraphs {G_{1}, G_{2}, G_{3}, ...}, where the edges of each of the G_{i}s are

*disjoint* subsets of the edges of G. So in a decomposition of G, *vertices* can be shared between elements of the decomposition, but *edges* cannot.

## Edge Coloring and Graph Turning

In addition to doing vertex and face colorings of a graph, you can also do edge colorings. In an edge coloring, no two edges which are incident on the same vertex can share the same color. In general, edge coloring doesn't get as much attention as vertex coloring or face coloring, but it can be an interesting subject. Today I'm going to show you an example of a really clever visual proof technique called *graph turning* to prove a statement about the edge colorings of complete graphs.

Just like a graph has a chromatic index for its vertex coloring, it's got a chromatic

index for its edge coloring. The edge chromatic index of a graph G is the minimum number of colors in any edge-coloring of G. The theorem that I'm going to prove for you is about the edge chromatic index of complete graphs with 2n vertices for some integer n:

**The edge-chromatic index of a complete graph K_{2n} = 2n-1.**

## A Taxonomy of Some Basic Graphs

Naming Some Special Graphs

When we talk about graph theory - particularly when we get to some of the

interesting theorems - we end up referencing certain common graphs or type of graphs

by name. In my last post, I had to work in the definition of snark, and struggle around

to avoid mentioning another one, so it seems like as good a time as any to run through

some of the basics. This won't be an exciting post, but you've got to do the definitions sometime. And there's a bunch of pretty pictures, and even an interesting simple proof or two.