2.2 Julia Sets
(type=julia)
These sets were named for mathematician Gaston Julia, and can be
generated by a simple change in the iteration process described for the
Mandelbrot Set (p. 43). Start with a specified value of C, "C-real + i
* C-imaginary"; use as the initial value of Z "x-coordinate + i * y-
coordinate"; and repeat the same iteration, Z(n+1) = Z(n)^2 + C.
There is a Julia set corresponding to every point on the complex plane
-- an infinite number of Julia sets. But the most visually interesting
tend to be found for the same C values where the M-set image is busiest,
i.e. points just outside the boundary. Go too far inside, and the
corresponding Julia set is a circle; go too far outside, and it breaks
up into scattered points. In fact, all Julia sets for C within the M-set
share the "connected" property of the M-set, and all those for C outside
lack it.
Fractint's spacebar toggle lets you "flip" between any view of the M-set
and the Julia set for the point C at the center of that screen. You can
then toggle back, or zoom your way into the Julia set for a while and
then return to the M-set. So if the infinite complexity of the M-set
palls, remember: each of its infinite points opens up a whole new Julia
set.
Historically, the Julia sets came first: it was while looking at the M-
set as an "index" of all the Julia sets' origins that Mandelbrot noticed
its properties.
The relationship between the Mandelbrot (p. 43) set and Julia set can
hold between other sets as well. Many of Fractint's types are
"Mandelbrot/Julia" pairs (sometimes called "M-sets" or "J-sets". All
these are generated by equations that are of the form z(k+1) =
f(z(k),c), where the function orbit is the sequence z(0), z(1), ..., and
the variable c is a complex parameter of the equation. The value c is
fixed for "Julia" sets and is equal to the first two parameters entered
with the "params=Creal/Cimag" command. The initial orbit value z(0) is
the complex number corresponding to the screen pixel. For Mandelbrot
sets, the parameter c is the complex number corresponding to the screen
pixel. The value z(0) is c plus a perturbation equal to the values of
the first two parameters. See the discussion of Mandellambda Sets
(p. 49). This approach may or may not be the "standard" way to create
"Mandelbrot" sets out of "Julia" sets.
Some equations have additional parameters. These values are entered as
the third or fourth params= value for both Julia and Mandelbrot sets.
The variables x and y refer to the real and imaginary parts of z;
similarly, cx and cy are the real and imaginary parts of the parameter c
and fx(z) and fy(z) are the real and imaginary parts of f(z). The
variable c is sometimes called lambda for historical reasons.
NOTE: if you use the "PARAMS=" argument to warp the M-set by starting
with an initial value of Z other than 0, the M-set/J-sets correspondence
breaks down and the spacebar toggle no longer works.