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.
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 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 . 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.
The spacebar toggle has been enhanced for the classic Mandelbrot and Julia types. When viewing the Mandelbrot, the spacebar turns on a window mode that displays the Inverse Julia corresponding to the cursor position in a window. Pressing the spacebar then causes the regular Julia escape time fractal corresponding to the cursor position to be generated. The following keys take effect in Inverse Julia mode.
[Space] Generate the escape-time Julia Set corresponding to the cursor position. Only works if fractal is a "Mandelbrot" type.
[n] Numbers toggle - shows coordinates of the cursor on the screen. Press [n] again to turn off numbers.
[p] Enter new pixel coordinates directly
[h] Hide fractal toggle. Works only if View Windows is turned on and set for a small window (such as the default size.) Hides the fractal, allowing the orbit to take up the whole screen. Press [h] again to uncover the fractal.
[s] Saves the fractal, cursor, orbits, and numbers.
[<] or [,] Zoom inverse julia image smaller.
[>] or [.] Zoom inverse julia image larger.
[z] Restore default zoom.
The Julia Inverse window is only implemented for the classic Mandelbrot (type=mandel). For other "Mandelbrot" types [space] turns on the cursor without the Julia window, and allows you to select coordinates of the matching Julia set in a way similar to the use of the zoom box with the Mandelbrot/Julia toggle in previous Fractint versions.