Regarding fractals it is easy. You see there are only 4 people in the entire
world that really understand the fractal fomulae and 1 of them is dead. What
happened is that these 4 people discovered these cool fomulae and showed
them to other people. Those other people then started playing with them and
tweaked a few values or whatever and then said they'd invented a new
fractal. Take for example the Mandelbrot fractal. There are about 2
gazillion variations on that one fractal. People try to make it 3d or 4d or
5d or whatever and they all claim they invented a new fractal. Not only is
that not true but no one really understand the Mandelbrot in the first
place, not even Mandelbrot himself. Here is how it happened:

One day Benoit Mandelbrot was playing with numbers on a sheet of paper and
started with this:

1 = 1

He knew this was boring so changed it to:

x = 1

This wasn't much better so how about:

x = 1 + 1

but since that is the same as x = 2 he livened it up with:

x = 1 + y

but since that was just a static equation and really didn't go anywhere he
decided some reiteration would be nice:

x = x + y

still this wasn't too interesting, specifically it was too predictable so he
changed it to:

x = x^2 + y

the problem with this was that it would grow too large too quickly for any
normal number, but what about abnormal numbers, specifically imaginary
numbers? so Benoit said that x and y would be imaginary numbers, he also
changed them to z and c. C is for constant and z if for the letter after y
since it is usually plotted on a x-y grid. So that gave him:

z = z^2 + c where z and c are imaginary numbers

Finally in order to make the plot he said that the x coordinate of the grid
would be the real part of c and the y coordinate would be the imaginary part
of c and that z starts out as zero. How do you determine which color to plot
each pixel? Simple, just count how many times you iterate the equation and
as soon as Z goes off to infinity use the number of iterations up to the
point as the value (color, height, transparency, or whatever) for that
pixel. Of course in a typical M-set plot the value of 4 is used for infinity
during this check (4 == infinity? where the heck'd that come from?).

Now, do you really think Benoit actually understood what he was doing? OF
COURSE NOT, it was all doodles on a peice of paper that happened to come out
looking interesting. You never hear about another mathematician named
Charles Brottlemand, that's because his doodles just turned out as a big
dark ugly brown circle, no one was intersted, it wasn't colorful like the
M-set.


Sorry for the digression. What I said above was about 1/2 serious. Most
people that do stuff with fractals don't really understand them, they just
copy and play with the formulas. The best way to learn is to play. Take a
fractal formula and plot it. Then change some values and plot it again. With
POV you can do some cool things by changing values as you progress along an
axis and then you get a 3D fractal. Likewise you can make a 4d fractal
(i.e., animation) by rendering lots of scenes with parameters changing a bit
each frame.


--Rainer


P.S. For more info on the M-set check out
http://www.cygnus-software.com/theory/theory.htm

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