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TITLE: Fractals - The Effects of Changes in the Complex
Number C In Julia Set Fractals.
STUDENT RESEARCHER: Iain Hunt
SCHOOL ADDRESS: Del Norte High School
Del Norte, Colorado, 81132
GRADE: 11
TEACHER: Laura Stuemky
I. STATEMENT OF PURPOSE AND HYPOTHESIS:
In my experiment, I explored the effects of changing the real
part of c in Julia Set fractals representing the equation
f(z)=z^2+c. I predicted that fractals with close values for the
real part of the complex number c would have similar shapes and
color patterns.
II. METHODOLOGY:
In my experiment, I originally intended to test a longer series
of values that would have ranged from -1 to 1. I planned on
using numbers that were exact only to one decimal place. In
conducting this experiment I found that the numbers were too far
apart to produce any recognizable pattern. I then chose two
imaginary values for c and two sets of real number values. The
values I chose were suggested on the website. I tested two
different sets as a way of repeating my experiment, in order to
reduce error in the interpretation of the images.
Other than the value for c, everything in experiment remained
constant. All the fractals were generated by the same means:
the program on the Internet, at the site
library.advanced.org/3288/myojulia.html. I tested only Julia
Set fractals representing the equation f(z)=z^2+c. The starting
value for z was always 0+0i. I always used the red/green/blue
color palette, and the magnification factor always remained at
1.
Procedure
1. Get on the Internet and go to
library.advanced.org/3288/myojulia.html.
2. Enter .097i as the value for the imaginary part of c.
3. Enter -.750 as the value for the real part of c. Click the
finish button.
4. Save the picture of the fractal for future analysis.
5. Repeat steps 3 and 4 with the same value for the imaginary
part of c and -.749, -.748, -.747, -.746, -.745, -.744, -.743, -
.742, and -.741.
6. Repeat the experiment with .043i as the imaginary part of c
and .320, .321, .322, .323, .324, .325, .326, .327, .328, and
.329 as the values for the real part of c.
7. Analyze the fractals. Look for patterns in the shapes and
colors of the fractals.
III. ANALYSIS OF DATA:
The only data in my experiment is included on a separate page.
These are the resultant fractal images from my investigation.
IV. SUMMARY AND CONCLUSION:
In my experimentation I found that all the fractals with close c
values had similar shapes. The basic shapes of these fractals
were nearly identical. This supports the first part of my
hypothesis, where I predicted that fractals with close c values
would have similar shapes. I also found that the color patterns
weren't near as similar. With a difference of .001 in the real
part of the c value, the color patterns were similar. However,
with a difference of .009, the color patterns seemed to have no
similarity whatsoever. From this I conclude that there is
greater and more rapid change in the color patterns of fractals
than in the shapes of fractals. The second part of my
hypothesis, where I predicted that fractals with close c values
would have similar color patterns, was somewhat correct,
although not correct to the extent of the first one.
V. APPLICATION:
The science of fractals is still a young one, so at this point
applications for fractals haven't been fully developed, although
there is potential in fields such as fluid dynamics. Fractals
can be used to model complex systems, such as the stock market,
population trends, or weather patterns. These models could
possibly lead to a greater understanding of the stock market and
more accurate weather forecasting. There is potential for
fractals in most, if not all non-linear systems.