Fractals:
As pointed out by a commenter, there are some really surprising places where fractal patterns can appear. For example, there was a recent post on the Wolfram mathematica blog by the engineer who writes the unlimited precision integer arithmetic...
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Posted on September 29, 2007 3:18 PM • 18 Comments •
In the course of the series of posts I've been writing on fractals, several people have either emailed or commented, saying something along the lines of "Yeah, that fractal stuff is cool - but what is it good for?...
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Posted on September 25, 2007 9:44 PM • 13 Comments •
When you mention fractals, one of the things that immediately comes to mind for most people is fractal landscapes. We've all seen amazing images of mountain ranges, planets, lakes, and things of that sort that were generated by fractals....
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Posted on September 6, 2007 9:50 PM • 9 Comments •
Aside from the Mandelbrot set, the most famous fractals are the Julia sets. You've almost definitely seen images of the Julias (like the ones scattered through this post), but what you might not have realized is just how closely...
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Posted on August 22, 2007 9:45 PM • 8 Comments •
One of the most fundamental properties of fractals that we've mostly avoided so far is the idea of dimension. I mentioned that one of the basic properties of fractals is that their Hausdorff dimension is larger than their simple...
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Posted on August 8, 2007 11:41 AM • 5 Comments •
So, in my last post, I promised to explain how the chaos game is is an attractor for the Sierpinski triangle. It's actually pretty simple. First, though, we'll introduce the idea of an affine transformation. Affine transformations aren't strictly...
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Posted on August 6, 2007 4:59 PM • 10 Comments •
Most of the fractals that I've written about so far - including all of the L-system fractals - are examples of something called iterated function systems. Speaking informally, an iterated function system is one where you have a transformation...
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Posted on August 3, 2007 3:15 PM • 17 Comments •
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...
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Posted on July 27, 2007 2:31 PM • 44 Comments •
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...
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Posted on July 25, 2007 9:33 AM • 35 Comments •
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...
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Posted on July 17, 2007 1:17 PM • 12 Comments •
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...
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Posted on July 12, 2007 8:08 PM • 57 Comments •
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...
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Posted on July 11, 2007 10:18 AM • 19 Comments •
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...
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Posted on July 9, 2007 7:40 PM • 35 Comments •
