My 2001 calendar has fractals as the theme, and each month has a different equation graphed.
However, I’ve never understood how an equation is graphed. For example, this one:
z = z[sup]2[/sup] + c.
(Chosen because it was the simplest.)
The result is an interesting bubble fractal. But no numbers that I plug into it get anything close. And what’s with the colors. I don’t see that in the equations, either.
aw crud, I should know this. I just read a book on chaos theory
http://www.amazon.com/exec/obidos/ASIN/0060916966/o/qid=979071736/sr=8-1/ref=aps_sr_b_1_1/107-3041726-7371715
Anyway, plot z on one axis and z^2+c on the other. Start plugging in every number you can think of for z (1, 2, 2.1, 2.01, 2.001, 2.000000012324, etc.). Get a computer to do this for you with millions/billions of entries for z. Notice how the points fall in a pattern and that a seemingly random thing actually has an amazing structure to it.
The black regions are where no points lie. I don’t know about the colors. Maybe it’s indicative of the density of points in that region.
but I’m no mathematician, so maybe I’m wrong.
Anyway, it become fractal because the result (z) is self-referencing (=z^2+c).
That particular fractal is the Mandelbrot set. Here’s how you get it:
Pick a complex number c. Square it and then add c.
Square that result, then add c again.
Square that result, then add c again.
Square that result, then add c again.
And so on. Depending on your original choice for c, one of two different things may happen.
The results get larger and larger without bound. (For a complex number a+bi, this means a[sup]2[/sup]+b[sup]2[/sup] (the (square of the) distance a+bi is from 0 in the complex plane) gets larger and larger without bound).
The results jump around, but have don’t get any larger than a certain limit. These points are what we call the Mandelbrot set (the interior of the “bubble”).
The different colors you see along the border are basically measures of how fast the points from case 1 are going to infinity.
A really good, free, fractal generating program is Fractint.
Fractal graphs, like the Mandelbrot set, are actually graphing which values make the iterating equation diverge and which are finite. Colors often indicate how fast a particular point diverged. For instance, red might indicate that the result left a particular envelope after only five iterations, while yellow might show a spot where it diverged slower, staying within the envelope for ten steps.
In the case of your example, you keep feeding the result of z[sup]2[/sup]+c into itself as a new z, and see whether it goes infinite on you.
Cabbage, I wish I’d said that. Here I thought I finally had something to add, and I typed it too slow.
I used to do non-linear dynamics for a living (not a very good one). IIRC, the mapping you are looking at is the one that generates the Mandelbrot set. Again, IIRC, here is what you do:
Define a set of mappings in the complex plane (z=x+iy) that map z[sub]n[/sub] to z[sub]n+1[/sub]=z[sub]n[/sub][sup]2[/sup] + c. Start with z[sub]0[/sub]=0+0i, and iterate the map indefinitely. The set of all complex numbers, c, for which z[sub]infinity[/sub] remains bounded (that is, does not end up at infinity) is the Mandelbrot set, which looks vaguely (Valentine) heart shaped with lots of little bubbles and projections on it. The Mandelbrot set itself is typically colored black, with the surrounding parts of the plane colored to reflect how fast the mapping iterates out to infinity.
There are related sets in the complex plane called Julia sets. Instead of looking at what different values of c do to the mapping of the origin, look at all z[sub]0[/sub] for a fixed value of c. One can show that for all z[sub]0[/sub], there are three possibilities: you eventually map to the origin, you eventually map to infinity, or you stay on some compact set. The third possiblility, which is the boundary between the two other cases, is the Julia set for c. For all c within the Mandelbrot set, the Julia set is a closed, connected curve (for c=0, it’s just the circle of radius one centered at the origin), that may look pretty crinkled and complicated, but encloses a finite area of the complex plane. For all c on the boundary of the Mandelbrot set, the Julia set is connected, but encloses no area (it’s a fractal curve). For all c outside the Mandelbrot set, the Julia set is essentially dust–it’s no longer connected, but is comprised of an infinite number of pieces at all length scales (cf. the Cantor set).
OK, that was probably too much information.
(Cabbage and Saltire were more concise and typed faster than me, but I’ll post anyway for all of you Julia set fans.)
Rick
:o
thanks for correcting me, dudes.