### documentation about MANDELBROT-fractals from fractint.exe (19.2)

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This set is the classic: the only one implemented in many plotting
programs, and the source of most of the printed fractal images published
in recent years. Like most of the other types in Fractint, it is simply
a graph: the x (horizontal) and y (vertical) coordinate axes represent
ranges of two independent quantities, with various colors used to
symbolize levels of a third quantity which depends on the first two. So
far, so good: basic analytic geometry.

Now things get a bit hairier. The x axis is ordinary, vanilla real
numbers. The y axis is an imaginary number, i.e. a real number times i,
where i is the square root of -1. Every point on the plane -- in this
case, your PC's display screen -- represents a complex number of the
form:

x-coordinate + i * y-coordinate

If your math training stopped before you got to imaginary and complex
numbers, this is not the place to catch up. Suffice it to say that they
are just as "real" as the numbers you count fingers with (they're used
every day by electrical engineers) and they can undergo the same kinds
of algebraic operations.

OK, now pick any complex number -- any point on the complex plane -- and
call it C, a constant. Pick another, this time one which can vary, and
call it Z. Starting with Z=0 (i.e., at the origin, where the real and
imaginary axes cross), calculate the value of the expression

Z^2 + C

Take the result, make it the new value of the variable Z, and calculate
again. Take that result, make it Z, and do it again, and so on: in
mathematical terms, iterate the function Z(n+1) = Z(n)^2 + C. For
certain values of C, the result "levels off" after a while. For all
others, it grows without limit. The Mandelbrot set you see at the start
-- the solid-colored lake (blue by default), the blue circles sprouting
from it, and indeed every point of that color -- is the set of all
points C for which the magnitude of Z is less than 2 after 150
iterations (150 is the default setting, changeable via the options
screen or "maxiter=" parameter). All the surrounding "contours" of
other colors represent points for which the magnitude of Z exceeds 2
after 149 iterations (the contour closest to the M-set itself), 148
iterations, (the next one out), and so on.

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