| |
 |
|
|
Science Forum Index » Fractals Science Forum » Good Math, Bad Math : The Mandelbrot Set
Page 1 of 1
|
| Author |
Message |
| Roger Bagula |
 Posted: Wed Jul 11, 2007 11:23 am |
|
|
|
Guest
|
http://homepage.mac.com/markcc/mandelbrot.m3u
http://scienceblogs.com/goodmath/2007/07/the_mandelbrot_set_1.php
The Mandelbrot Set
Category: goodmath > topology > Fractals
Posted on: July 11, 2007 10:18 AM, by Mark C. Chu-Carroll
800px-Mandelbrot_set_with_coloured_environment.png 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.)
So what is the Mandelbrot set?
overall-mandelbrot.gif Take the set of functions fC(x)=x2+C where for
each fC, C is a particular complex constant. That gives an infinite set
of simple functions over the complex numbers. For each possible complex
number C, you look at the recurrence relation generated by repeatedly
appling f, starting with x=0:
* m(0,C)=fC(0)
* m(i+1,C)=fC(m(i,C))
If m(i,C) doesn't diverge (escape) towards infinity as i gets larger,
then the complex number C is a member of the Mandelbrot set. That's it -
that simple definition - repeatedly apply f(x)=x2+C for complex numbers
- produces the astonishing complexity of the Mandelbrot set.
If we use that definition of the Mandelbrot set, and draw the members of
the set in black, we get an image like the one above. That's nice, but
it's probably not what you expected. We're all used to the beautiful
colored bands and auras around that basic pointy black blob. Those
colored regions are not really part of the set.
Mandelbrot1.png The way we get the colored bands is by considering how
long it takes for the points to start to diverge. Each color band is an
escape interval - that is, some measure of how many iterations it takes
for the repeated application of f(x) to diverge. Images like the ones to
the right and below are generated using various variants of
escape-interval colorings.
images-1.jpg
images-2.jpg
images.jpg
My personal favorite rendering of the Mandelbrot set is an image called
the Buddhabrot. In the Buddhabrot, what you do is look at values of C
which aren't in the mandebrot set. For each point m(i,C) before it
escapes, plot a point. That gives you the escape path for the value C.
If you take a large number of escape paths for randomly selected values
of C, and you plot them so that the brightness of a pixel is determined
by the number of escape paths that cross that pixel, you get the
Budddhabrot. It's fascinating because it reveals the structure in a
particularly amazing way. If you look at a simple unzoomed image of the
madelbrot set, what you see is a spiky black blob; the actually
complexity of the structure isn't obvious until you spend some time
looking at it. The Buddhabrot is more obvious - you can see the
astonishing complexity much more easily. |
|
|
| Back to top |
|
| |
|
Page 1 of 1
All times are GMT - 5 Hours
The time now is Fri Nov 21, 2008 8:20 pm
|
|