The colors in these pictures represent how "close" you are to
the Mandelbrot set. If you're not
already familiar with what that means, here
is how it's done. For purposes of computation,
the Mandelbrot set M is defined
as the set of all points p in the
plane for which a certain sequence
of points (associated with the point
p and which depends on p
in a simple nonlinear way) remains bounded. In order to
compute an approximation to M one has to give
computable meaning to the phrase "remains bounded". Then
one can plot the set of points in the picture that satisfy this
condition to produce an approximate drawing of M.
The simplest way of doing this is to specify a
"large" constant R > 0 and a "large" integer N > 0.
One then considers a point p to be in M if all of
the first N terms of the sequence associated with p
belong to the disc of radius R about the origin. In practice, R
doesn't have to be large for good results
(R = 2.0 works nicely) and the value N = 100
produces a fairly good picture of M (though
these images were generated using much larger
values than N=100).
Thus every pixel p of the picture you see has a number N(p) associated with it: N(p) = N if all of the first N terms of the sequence associated with the location of p remain in the disc of radius R, or else N(p) is the first n < N for which the next term of the sequence is outside the central disc of radius R. If N(p) = N then we consider p to be a pixel belonging to M and we may color it black. Otherwise N(p) < N, we consider that p does not belong to M, and we may color it white. This produces a black-and-white picture of M.
On the other hand, we are also free to use the value of N(p) to choose more interesting colors than white for a pixel p when N(p) < N...and that is what this program does. For example, a pixel p is colored red if N(p) is very close to N, and for smaller values of N(p) the colors vary "smoothly" with the value of N(p). Most of the pixels with colors other than black are not in M, and most of the black pixels are in M. But because R and N are finite there can be small scale misrepresentations in a picture. For example, a few isolated black pixels in a tiny field of 100 or so red pixels can't be accurate, because M is a connected set.
Highly magnified views of any point near the boundary of a Julia set are not very different from an unmagnified view of the entire set. This interesting fact is a reflection of the self-similarity of Julia sets under changes of scale; it is in strong contrast to the Mandelbrot set, whose general "look" changes very dramatically under high magnification at various locations (see views 1, 2, 3 above).
You can see a clear demonstration of the self-similarity
of Julia sets in views 6 and 8. If you're handy with
your browser, you should open a second window and
view them simultaneously. View 6 is a standard
Julia set, composed of fattened black "S" shapes and
colored spirals. View 7 is a somewhat closer look at
one of the spirals. View 8 is an extreme blowup of
the upper tip of the picture shown in view 6. The
similarity of this enormously magnified bit to the
total Julia set shown in view 6 is quite apparent.
The magnification in view 8 is about
5,000,000,000,000,000 times that of view 6 (whatever
this number is, Webster's calls it five quadrillion).
(Last updated: August, 1999)
Here is the Vaserely. Enjoy.