Julia Explorer Java Applet

JAVA DISABLED

Instructions

  • Requirements:
    This applet requires a browser with Java 1.1 support. I have seen it work only with Internet Explorer 5 in Windows and Netscape Communicator 4.5 in Linux and Windows.
  • Reset:
    Reset the applet to the initial configuration.
  • Color Choice:
    Use the color choice pull-down menus to change the color scheme.
  • Zoom mode:
    In zoom mode, click on a point in a fractal to zoom in toward that point.
  • Julia mode:
    In Julia mode, click on a point in the Mandelbrot fractal (on the left) to specify a new Julia fractal.
  • Auto-zoom:
    This relatively lame feature choses a point of interest and zooms in. Points closer to the image center bordering points of different colors are more likely to be chosen.

Details

Program History

This applet was adapted from a stand-alone Java 1.1 program I originally wrote for CS 3 at Caltech. I added many extra features to make it more spiffy. For example, points in the fractals are painted at increasing resolution so you can get a rough idea of what the fractal looks like very quickly. Also, the color scheme can be changed to any one of the preset color schemes. The main strength of this program (and where it gets the name "Julia Explorer") lies in the ability to change the Julia fractal by choosing a point in the Mandelbrot fractal. Why this is interesting is discussed below.

Code

The java source files:

Fractals

Fractals are generated by iterating a function from the complex plane to the complex plane.
A point is in the Mandelbrot (or Julia) set if the function does not eventually diverge to infinity.
The mandelbrot set uses the function z[i+1] = z[i]^2 + z[0]
There is only one mandelbrot set.
Julia sets use the function z[i+1] = z[i]^2 + c
Where c is some fixed point in the complex plane. So, for every c, there is an associated Julia set.

Any point z0 with absolute value greater than 2 eventually diverges under iteration of both Julia and Mandelbrot functions. This will be important.

Colorful fractal images can be generated by defining the color of a point z0 in the complex plane as follows:

  • Define a list of colors.
  • Iterate the function once for each color in the list starting at z0 until either z has an absolute value greater than 2 or until the end of the list has been reached.
  • When the iteration stops, the color is found.
Points that take longer to diverge to greater than 2 will be colored with colors closer to the end of the list. If a point is colored with the last color from the list, it is either in the Mandelbrot (or Julia) set or, if not, then we have reached the limit of our approximation. We can only approximate these sets, but by allowing the function to iterate more, we can improve the approximation.

Finally, note that by definition, a Julia set is connected if and only if the point c is in the mandelbrot set. So, to find the most interesting Julia fractals, select points near the edge of the Mandelbrot set (the black in the left fractal) either inside of it or near it.

by B. Oran Switzer

Home | Resume | Samples | Fractals | oran@its.caltech.edu