GASTON JULIA HAD SETS TOO...

There is only one Mandelbrot set and every number in the complex plane is either

in or out of the set. Associated with every point in the complex plane is a set

somewhat similar to the Mandelbrot set called a "Julia" set after the

mathematician Gaston Julia. In developing the Mandelbrot set we examined points

in the complex plane to see if z2+c diverged or not. Remember that in this

function c represents the coordinates of the pixel location being examined. When

we shift to a different point we use that point's coordinates in the function as

c.

If we examine points in the complex plane to see if z2+k diverges, where k is a

fixed complex number, we get the Julia set associated with the point k. Instead

of changing the constant in the function as we move from pixel to pixel, we hold

the value of k fixed as we scan the screen. Julia sets and the Mandelbrot set

are close relatives. The Julia set boundary will be illuminated by the escape

time algorithm as was the case for the Mandelbrot set. Julia sets take several

different forms depending on the location in the plane of the fixed point k.

Now that you have had a description of how Julia sets are generated we will give

you a guided tour of some of the possibilities. Each of the next several

displays shows an entire Julia set. Note the coordinates of the starting point

in the label above the drawing area. Each Julia set is contained in the same

region of the complex plane as is the Mandelbrot set. In fact the Mandelbrot set

has been called a catalog of the Julia sets. Many similar structures are seen in

both sets. The boundaries of either might be considered the ultimate fractal

object. Run the following series of Julia set displays to see the sets develop.

I have included below each display title an image of the final results.

Julia Set 1 - Near the Real Axis

 

Julia Set 2 - Near the Origin

 

Julia Set 3 - Map of an Asteroid

 

Julia Set 4 - Three Cylinder Wankle Head Gasket

 

Next we will pick a particular Julia set and zoom in on it as we did the

Mandelbrot set. You will see that at higher magnifications the similarity

between Julia and Mandelbrot increases. If we jumped from one to the other at

high magnification you would have trouble telling them apart. In the research

mode of this section you will be able to proceed from the Mandelbrot set at high

magnification to the corresponding Julia set. Keep in mind that the instructions

for all this detail is contained in a few lines of computer code. Run this group

of Julia Set displays.

 

Julia Set 5 - Some Holes Full Scale

 

Julia Set 5 - Some Holes Once Zoomed

 

Julia Set 5 - Some Holes Twice Zoomed

 

Next you may explore the Julia sets by marking points on the screen and using

the Action button to generate the corresponding Julia set. You will find that

there are several types of Julia sets depending on where the reference point is

located. Run the Exploring the Julia Sets display.

In the next display you will be able to select points for the Julia set with the

Mandelbrot set outline on the screen. Run the Mandelbrot Set/Julia Sets

Relationship display.

The next display in this section allows considerable flexibility in exploring

the Mandelbrot and Julia sets. Run the Complex Sets Research display.