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Deadstar
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Let A be the following set (NOTE: The part within the red square is NOT part of the original question, I have added that in and it will be explained)
[PLAIN]http://img695.imageshack.us/img695/1037/unleddo.png
After choosing coordinates, construct the iterated function system whose fixed point it A.
Now I find the whole iterated function system a bit odd as it is but I can usually construct ones for most fractals (Koch snowflakes and the like...) But this one is a bit different since it looks like the bottom of the fractal has a different system to the rest of it. I have included an example of the iterated function of the Koch snowflake at the end of this post to give you an idea of what I'm trying to do.
The bottom of the fractal seems like it's missing some parts and the part within the red square is what I think should be added on to the fractal to make any iterated function system work. I'm finding it difficult to explain what I think the pattern is in terms of an actual function system as I keep involving too many parts at once.
I would like to select coordinates such that the 0 occurs at the bottom left of the entire fractal and the length of the largest line is one (which bring problems as the I think the largest line should be the one I added on!).
Letting Phi be the iterated function.
[tex]\Phi_1(z) = z e^{i\pi/2}[/tex]
[tex]\Phi_2(z) = z e^{i \pi/2} + \frac{1}{2}[/tex]
[tex]\Phi_3(z) = \frac{z}{2} e^{i \pi/2} + \frac{1}{4}[/tex]
[tex]\Phi_4(z) = \frac{z}{4} e^{i \pi/2} + \frac{1}{8}[/tex]
This (in my mind at least) creates the following...
[PLAIN]http://img202.imageshack.us/img202/6851/unledxgi.png
So is this correct? Repeated iterations of this will give the fractal I'm after..?
Below is the iterated function system for the Koch snowflake.
[PLAIN]http://img860.imageshack.us/img860/6989/unledpjt.png
Thanks in advance for any answers!
[PLAIN]http://img695.imageshack.us/img695/1037/unleddo.png
After choosing coordinates, construct the iterated function system whose fixed point it A.
Now I find the whole iterated function system a bit odd as it is but I can usually construct ones for most fractals (Koch snowflakes and the like...) But this one is a bit different since it looks like the bottom of the fractal has a different system to the rest of it. I have included an example of the iterated function of the Koch snowflake at the end of this post to give you an idea of what I'm trying to do.
The bottom of the fractal seems like it's missing some parts and the part within the red square is what I think should be added on to the fractal to make any iterated function system work. I'm finding it difficult to explain what I think the pattern is in terms of an actual function system as I keep involving too many parts at once.
I would like to select coordinates such that the 0 occurs at the bottom left of the entire fractal and the length of the largest line is one (which bring problems as the I think the largest line should be the one I added on!).
Letting Phi be the iterated function.
[tex]\Phi_1(z) = z e^{i\pi/2}[/tex]
[tex]\Phi_2(z) = z e^{i \pi/2} + \frac{1}{2}[/tex]
[tex]\Phi_3(z) = \frac{z}{2} e^{i \pi/2} + \frac{1}{4}[/tex]
[tex]\Phi_4(z) = \frac{z}{4} e^{i \pi/2} + \frac{1}{8}[/tex]
This (in my mind at least) creates the following...
[PLAIN]http://img202.imageshack.us/img202/6851/unledxgi.png
So is this correct? Repeated iterations of this will give the fractal I'm after..?
Below is the iterated function system for the Koch snowflake.
[PLAIN]http://img860.imageshack.us/img860/6989/unledpjt.png
Thanks in advance for any answers!
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