
Julia Sets

formulas: 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.
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.

Xmfract'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 set and Julia set can hold between
other sets as well. Many of xmfract'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.

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 for 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.

