**GASTON JULIA HAD SETS TOO...
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**There is only one Mandelbrot set and every number in the
complex plane is either
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**in or out of the set. Associated with every point in the
complex plane is a set
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**somewhat similar to the Mandelbrot set called a "Julia"
set after the
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**mathematician Gaston Julia. In developing the Mandelbrot
set we examined points
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**in the complex plane to see if z2+c diverged or not.
Remember that in this
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**function c represents the coordinates of the pixel
location being examined. When
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**we shift to a different point we use that point's
coordinates in the function as
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**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
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**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
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**time algorithm as was the case for the Mandelbrot set.
Julia sets take several
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**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
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**displays shows an entire Julia set. Note the coordinates
of the starting point
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**in the label above the drawing area. Each Julia set is
contained in the same
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**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
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**both sets. The boundaries of either might be considered
the ultimate fractal
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**object. Run the following series of Julia set displays
to see the sets develop.
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**I have included below each display title an image of the
final results.
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**Julia Set 1 - Near the Real Axis
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**
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**Julia Set 2 - Near the Origin
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**
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**Julia Set 3 - Map of an Asteroid
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**
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**Julia Set 4 - Three Cylinder Wankle Head Gasket
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**
**

**Next we will pick a particular Julia set and zoom in on
it as we did the
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**Mandelbrot set. You will see that at higher
magnifications the similarity
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**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
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**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
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**for all this detail is contained in a few lines of
computer code. Run this group
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**of Julia Set displays.
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**
**

**Julia Set 5 - Some Holes Full Scale
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**
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**Julia Set 5 - Some Holes Once Zoomed
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**
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**Julia Set 5 - Some Holes Twice Zoomed
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**
**

**Next you may explore the Julia sets by marking points on
the screen and using
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**the Action button to generate the corresponding Julia
set. You will find that
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**there are several types of Julia sets depending on where
the reference point is
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**located. Run the Exploring the Julia Sets display.
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**In the next display you will be able to select points
for the Julia set with the
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**Mandelbrot set outline on the screen. Run the Mandelbrot
Set/Julia Sets
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**Relationship display.
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**The next display in this section allows considerable
flexibility in exploring
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**the Mandelbrot and Julia sets. Run the Complex Sets
Research display.
**