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 Author Topic: IFS fractals from Mobius transforms  (Read 13111 times) Description: 0 Members and 1 Guest are viewing this topic.
Thunderwave
Guest
 « Reply #15 on: July 15, 2010, 05:06:54 AM »

A 3D variant would be from just adding another variable...

I wonder about a mixed variation:

(a*x+b*y[+c*z]+d)/(e*x+f*y[+g*z])+h)

Or even further extended... You could do that from a polynomial of any degree, basically...

<Quoted Image Removed>

That would be a mix of a Möbius transform and a general 3-cone section.

You could extend that even further by allowing permutations....

x*x=x²     y*x=x*y     z*x=x*z
x*y         y*y=y²       z*y=y*z
x*z         y*z            z*z=z²

So... a*x²,b*y²,c*z²,d*2xy,e*2xz,f*2yz

Could we be talking dimensions beyond the usual, or am I stupid about this?  I mean a fractal dimension above 3, like 3.23 etc.  Is that even possible?  Maybe I'm not really understanding this.  Sorry if I sound stupid.  I keep trying to learn more.
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paxinum
Guest
 « Reply #16 on: July 15, 2010, 09:46:06 AM »

A 3D variant would be from just adding another variable...

I wonder about a mixed variation:

(a*x+b*y[+c*z]+d)/(e*x+f*y[+g*z])+h)

Or even further extended... You could do that from a polynomial of any degree, basically...

<Quoted Image Removed>

That would be a mix of a Möbius transform and a general 3-cone section.

You could extend that even further by allowing permutations....

x*x=x²     y*x=x*y     z*x=x*z
x*y         y*y=y²       z*y=y*z
x*z         y*z            z*z=z²

So... a*x²,b*y²,c*z²,d*2xy,e*2xz,f*2yz

Could we be talking dimensions beyond the usual, or am I stupid about this?  I mean a fractal dimension above 3, like 3.23 etc.  Is that even possible?  Maybe I'm not really understanding this.  Sorry if I sound stupid.  I keep trying to learn more.

You can have any positive number as fractal dimension. We mathematicians deals with n-dimensional (integer dimensional) spaces on a daily basis, but they are quite complicated to visualize.
You can easily generalize the sierpinski trianlge to any dimension, as an example.
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