Archive for the 'Fractals' category

L-System Fractals

Jul 30 2007 Published by under Fractals

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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.

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15 responses so far

Fractal Dust and Noise

Jul 27 2007 Published by under Fractals, information theory

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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.

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45 responses so far

Fractal Pathology: Peano's Space Filling Curve

Jul 25 2007 Published by under Fractals

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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!

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35 responses so far

Fractal Curves and Coastlines

Jul 17 2007 Published by under Fractals

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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.

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12 responses so far

Fractal Borders

Jul 12 2007 Published by under Fractals

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.

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57 responses so far

The Mandelbrot Set

Jul 11 2007 Published by under Fractals

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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.)

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23 responses so far

An Introduction to Fractals

Jul 09 2007 Published by under Fractals

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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?

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35 responses so far

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