In the post about Koch curves, I talked about how a grammar-rewrite system could be used to describe fractals. There’s a bit more to the grammar idea that I originally suggested. There’s something called an L-system (short for Lindenmayer system, after Aristid Lindenmayer, who invented it for describing the growth patterns of plants), which is a variant of the Thue grammar, which is extremely useful for generating a wide range of interesting fractals for describing plant growth, turbulence patterns, and lots of other things.
While reading Mandelbrot’s text on fractals, I found something that surprised me: a relationship
between Shannon’s information theory and fractals. Thinking about it a bit, it’s not really that suprising;
in fact, it’s more surprising that I’ve managed to read so much about information theory without
encountering the fractal nature of noise in a more than cursory way. But noise in a communication channel
is fractal – and relates to one of the earliest pathological fractal sets: Cantor’s set, which Mandelbrot
elegantly terms “Cantor’s dust”. Since I find that a wonderfully description, almost poetic way of describing it, I’ll adopt Mandelbrot’s terminology.
One of the strangest things in fractals, at least to me, is the idea of space filling curves. A space filling curve is a curve constructed using a Koch-like replacement method, but instead of being
self-avoiding, it eventually contacts itself at every point.
What’s so strange about these things is that they start out as a non-self-contacting curve. Through
further steps in the construction process, they get closer and closer to self-contacting, without touching. But in the limit, when the construction process is complete, you have a filled square.
Why is that so odd? Because you’ve basically taken a one-dimensional thing – a line – with no width at all – and by bending it enough times, you’ve wound up with a two-dimensional figure. This isn’t
just odd to me – this was considered a crisis by many mathematicians – it seems to break some of
the fundamental assumptions of geometry: how did we get width from something with no width? It’s nonsensical!
Often, we use graphs as a structured representation of some kind of information. Many interesting problems involving the things we’re representing can then by described in terms of operations over
graphs. Today, I’m going to talk about some of the basic operations that we can do on graphs, and in later posts, we’ll see how those operations are used to solve graph-based problems.
After seeing PZs comments on Stuart Pivar’s new version of his book, titled “Lifecode: From egg to embryo by self-organization”, I thought I would try taking a look. I’ve long thought that much of the stuff that I’ve read in biology is missing something when it comes to math. Looking at things, it often seems like there are mathematical ideas that might have important applications, but due to the fact that biology programs rarely (if ever) require students to study any advanced math, they don’t recognize the way that math could help them. So, hearing about Pivar’s book, which claims to propose a theory of structural development based on the math describing structural distortions of an expanding figure in a constrained space – well, naturally, I was interested.
So I wrote to the publisher of his book, to see if I could get a review copy. I wanted to try writing a review from the perspective of a mathematician. To my immense surprise, a courier arrived at my door two hours later with a copy of the book! It’s a lucky thing I was working from home that day! So I started reading it monday afternoon. I didn’t have a lot of time to read this week, so I didn’t finish the main text until thursday, despite the fact that it’s really quite short.
I got hit by a mutant meme; I don’t remember who tagged me. I’m not terribly into these
meme things, but I don’t pass up excuses to post recipes. So below the fold are four recipes that I’ve created: seared duck breast with ancho chile sauce; saffron fish stew; smoked salmon hash; and
spicy collard greens.
I just finally got my copy of Mandelbrot’s book on fractals. In his discussion of curve fractals (that is, fractals formed from an unbroken line, isomorphic to the interval (0,1)), he describes them in terms of shorelines rather than borders. I’ve got to admit
that his metaphor is better than mine, and I’ll adopt it for this post.
In my last post, I discussed the idea of how a border (or, better, a shoreline) has
a kind of fractal structure. It’s jagged, and the jags themselves have jagged edges, and *those* jags have jagged edges, and so on. Today, I’m going to show a bit of how to
generate curve fractals with that kind of structure.
One application of graph theory that’s particularly interesting to me
is called Ramsey theory. It’s particularly interesting as someone who
debunks a lot of creationist nonsense, because Ramsey theory is in
direct opposition to some of the basic ideas used by bozos to
purportedly refute evolution. What Ramsey theory studies is when some
kind of ordered structure *must* appear, even in a pathologically
chaotic process. Ramsey theory is focused largely on *structures* that
exhibit particular properties, and those structures are usually
represented as graphs.
Today, I’m going to talk a bit about two closely related problems in graph theory: the maximal clique detection problem, and the maximal common subgraph problem. The two problems are interesting both on their own as easy-to-understand but hard-to-compute problems; and also because they have so many applications. In particular, the two are used extensively
in bioinformatics and pharmacology for the analysis of complex molecular structure.
Yesterday’s Wall Street Journal has a *spectacular* example of really bad math.
The WSJ is, in general, an excellent paper with really high quality coverage of economic
issues. But their editorials page has long been a haven for some of the most idiotic
reactionary conservative nonsense this side of Fox News. But this latest piece takes the
cake. They claim that this figure is an accurately derived Laffer curve describing the relationship
between tax rates and tax revenues for different countries; and that the US has the highest corporate tax
rates in the world.