Jason's Fractals

Each of the images was made from C++ and Mathematica. C++ produced a data file that Mathematica converted to an image.


fractal image of f(z)=(1+0.4i)Sin(z)

The first image we see if from the function f(z)=(1+0.4i)Sin(z) where z=x+iy

Here is how the image was created: Let z0=x+iy be a point in the plane that we want colored.
Then we iterate like so: zn=fn(z0)
Finally we stop iterating when |zn-zn-1| < error bound

Here are some specifics of the image: 
x min: 0
x max: 6.28
y min: -6.28
y max: 6.28
Number of pixels for the width (x-axis): 1256
Number of pixels for the length (y-axis): 2512
max number of iterations to check: 35
error bound: 0.01
Please click on the image for a more detailed jpeg image. (482x965 pixels 164 KB)

fractal image of exponential function This next fractal was created the same way as above but with a different function.
f(x+iy)=u(x,y)+iv(x,y) where
u(x,y)=ex(e-1cos(y)+0.01cos(y))
v(x,y)=ex(e-1sin(y)+0.01sin(y)) 
x min: -2
x max: 8
y min: -3
y max: 3
The number of pixels for the width (x-axis): 3000
The number of pixels for the length (y-axis): 1800
The max number of iterations to check: 35
The error bound: 0.01
Please click on the image for a more detailed jpeg image. (1154x692 pixels 265 KB)


fractal image of Newton's Method of a cubic polynomial

Again, we use the same method mentioned above. This fractal is interesting since it is made from Newton's Method on the the function: z3-1. Thus, the function used in this fractal is: f(z)=z-(z3-1)/(3z2) 
x min: -1
x max: 1
y min: -1
y max: 1
The number of pixels for the width (x-axis): 2000
The number of pixels for the length (y-axis): 2000
The max number of iterations to check: 35
The error bound: 0.01
Please click on the image for a more detailed jpeg image. (1019x1019 pixels 174 KB)
fractal image of f(z)=sin(z)


This fractal was made by the same method as before. The function is: f(z)=Sin(z). Notice how similiar this fractal looks to the first fractal! 
x min: 0
x max: 3.14
y min: -6.28
y max: 6.28
The number of pixels for the width (x-axis): 600
The number of pixels for the length (y-axis): 2400
The max number of iterations to check: 34
The error bound: 0.01
Please click on the image for a more detailed jpeg image. (662x2651 pixels 255 KB)













When I first saw last image and the first image, I began to wonder what a movie might look like if I animated ft(z)=(1+it)Sin(z) where t changes from 0 to 0.4. In other words, how does the fractal to the right change to the fractal at the top of the page? Below, the movie experiment is available for viewing.

Fractal Movie of f(z)=(1+it)sin(z). (Where t goes from 0 to 0.4)



(It may take awhile to load. This GIF file is about 6 mb.)


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Jason Harrington
mathguy@math.ufl.edu