
Classic Mandelbrot set fractal.

formulas: mandel


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 xmfract, 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 computer'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 value of Z is less than 2 after 150 iterations (150 is the default
setting, changeable via the <X> options screen or "maxiter=" parameter).
All the surrounding "contours" of other colors represent points for which Z
exceeds 2 after 149 iterations (the contour closest to the M-set itself),
148 iterations, (the next one out), and so on.

We actually don't test for Z exceeding 2 - we test Z squared against 4
instead because it is easier. This value (FOUR usually) is known as the
"bailout" value for the calculation, because we stop iterating for the
point when it is reached. The bailout value can be changed on the <Z>
options screen but the default is usually best.

Some features of interest:

1. Use the <X> options screen to increase the maximum number of iterations.
Notice that the boundary of the M-set becomes more and more convoluted
(the technical terms are "wiggly," "squiggly," and "utterly bizarre")
as the Z-values for points that were still within the set after 150
iterations turn out to exceed 2 after 200, 500, or 1200. In fact, it can
be proven that the true boundary is infinitely long: detail without limit.

2. Although there appear to be isolated "islands" of blue, zoom in -- that
is, plot for a smaller range of coordinates to show more detail -- and
you'll see that there are fine "causeways" of blue connecting them to the
main set. As you zoomed, smaller islands became visible; the same is true
for them. In fact, there are no isolated points in the M-set: it is
"connected" in a strict mathematical sense.

3. The upper and lower halves of the first image are symmetric (a fact
that xmfract makes use of here and in some other fractal types to speed
plotting). But notice that the same general features -- lobed discs,
spirals, starbursts -- tend to repeat themselves (although never exactly)
at smaller and smaller scales, so that it can be impossible to judge by
eye the scale of a given image.

4. In a sense, the contour colors are window-dressing: mathematically, it
is the properties of the M-set itself that are interesting, and no
information about it would be lost if all points outside the set were
assigned the same color. If you're a serious, no-nonsense type, you may
want to cycle the colors just once to see the kind of silliness that other
people enjoy, and then never do it again.

Go ahead. Just once, now. We trust you.

