Julia Explorer v3

Oran's Julia Explorer

It started as a standalone program written for CS3 at Caltech (the original applet version can be found here). It have played with various parts since then, and essentially nothing is left of the original design, although the functionality is similar. One major visible difference is that it now shows both fractals on the same canvas with (optional) transparency multiplied into each layer and/or intrinsic to a color(s) in the colorscheme. I have made the Rainbow3* colorscheme to have points in the Mandelbrot set clear and points outside of the set opaque. The Trig2*+ colorscheme has its color components oscillating, including the transparancy. I used an AColor class for this (I couldn't get Color to do it with my 1.1 jdk -- and I wanted to keep it in 1.1). The other major difference is that you can now play with the equations that produce the fractal, and there are severa presets. Here's what you need to know:

Instructions

Alt
To set the Julia fractal's c parameter, hold down the Alt key and click on the canvas. The value of the complex c parameter depends on which point in the Mandelbrot fractal you clicked on (This matters because the Julia and Mandelbrot zoom independently).
Shift
Clicking on an image while holding Shift will zoom out by a factor of 2. Click without holding Shift to zoom in by a factor of 2. Aside from zooming, the new center point will be where you clicked.
Ctrl
In order to zoom in or out on the Julia fractal, you must hold down the Control key. Without the control key, you are clicking on the Mandelbrot fractal.
Set Args and Render
Sets the fractal parameters. Changes you make (other than clicking on the canvas) will not take effect until you click this.
Zoom Standard
Zoom to: { x + iy: -2 < x < 2, -2 < y < 2 }
Color Schemes
enjoy
Top layer
Usae the checkboxes to set whether the Julia or Mandelbrot is the top layer
Opacity
User the sliders to set the opacity of the layer. Use 0% to make a layer invisible and 100% to make it completely opaque (unless the colorscheme has intrinsic transparency)
Function
Set the equation that produces the fractal (as described below). The two lists allow you to pick one or two functions to add (or multiply) with the complex parameter "c" (although for multiplication mode, it should be called "lambda"). The operation is determined by the checkbox, and the equation is shown in the text field.
Bailout
This tells the algorithm when to decide that the starting point is definitely not in the fractal set. For some fractals, you may need to adjust this, and you can also play with the appearance of the fractal this way, but for suitable bailout tests, the underlying set is the same.
Presets
Select a preset to change all parameters to the preset configuration.
Hints and Tips
- Click then quickly Ctrl+Click to zoom in on both
- Find an interesting design on the Mandelbrot and Alt+Click there. This should produce an interesting Julia fractal
- Turn one layer completely off (by setting opacity to 0%) to more quickly explore the other fractal
- The bailout test can extremely important or extremely unimportant. For the classic Mandelbrot and Julia, mod >= 4 is used. The bailout test can significantly change the underlying fractal set that is being approximated since points that pass the bailout test on the first iteration will be excluded from the set even if there is an interesting set that does include this point... To see what the bailout tests look like, zoom out and use a colorscheme with lots of contrast like Trig3 where the first color is orange and the second is more red. Points that pass the bailout test without applying the function are colored with the first color.

Fractals

This type of fractal is generated by iterating a function that maps complex numbers to complex numbers. The basic idea is that there are interesting sets for which applying the function to any point in the set produces another point in the set. This is not enough to make it interesting, it should also have "lots" of periodic points (points that that will repeat as you iterate the function) in any region. Also, there is a sensitivity to arbitrarily small changes, and any two arbitrarily small regions of the set have points whose orbits eventually reach the other region. So, there are places where you could, in theory though not in a computer approximation, zoom in forever and always have an interesting image.

Many such sets are known to have bailout test such that any point not in the set will eventually pass the bailout test if the function is iterated long enough. It is known, for example, that points more than 2 units from the origin are not in the classic Mandelbrot set and that any other point not in the mandelbrot set will "eventually" reach outside of that region upon iteration of the function (in that case, f(z) = z^2 + c, where c is the first point).

These images are generated as follows:

start with a [long] list of colors
for each point, iterate the function (f(z) = g(z) + c OR f(z) = g(z)*c) starting at that point
when a point is reached that passes the bailout test, stop
when the number of iterations reaches the number of colors, stop
paint the point based on the number of iterations it took to pass
the points colored with the last color (points whose orbits never pass the bailout test) represent our approximation of the set.
The difference between the algorithm for the Mandelbrot and Julia images is that in the Mandelbrot, the point c is always the point to be colored, and in the Julia, c is fixed for the entire image

The chaotic set itself is all colored the same color, so it might have an interesting shape, but it does not account for the detail of the image. The colors are added as a way of visualizing the rate at which the orbits of a point diverge (or pass the bailout test). The most interesting regions in the image are those near the edge of the set whose orbits eventually diverge, but approximate some of the behavior described above in the early iterations. The properties of the chaotic set also ensure a degree of self-similarity -- the patterns repeat as you zoom in.

Finally, you might recognize immediately that if you zoom in very far on a point in the Mandelbrot fractal, the Julia fractal associated with that point, zoomed into that point, should approximate the same image in the Mandelbrot. This is illustrated somewhat in the "Classic Julia/Mandelbrot (zoomed)" preset. It is also known that a Julia set is connected if and only if the point c is in the Mandelbrot set. If c is a point near the edge of the Mandelbrot set, it may be difficult to tell whether the Julia set is connected. These properties lead to the strategy for producing interesting Julia images by setting c to an interesting point on the Mandelbrot image.