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Thanks gerrit, it helps a bit, but the expression for \( f(x, y) \) is quite hairy, with part of it being implicit:

Given complex \( C \) in an atom domain of period \( P \), find \( Z_0 = F_C^P(Z_0) \) where \( F_c(z)=z^2+c \).  Now, if \( \left|\frac{\partial}{\partial z}F_C^P(Z_0)\right| \le 1 \) then \( C \) is interior to a component of period \( P \).  Now define the surface height
\[ h = f\left(\Re(C), \Im(C)\right) = \frac{ \sqrt{1 - \left| \frac{\partial}{\partial z} \right|^2} }{\left|\frac{\partial}{\partial{c}}\frac{\partial}{\partial{z}} + \frac{\left(\frac{\partial}{\partial{z}}\frac{\partial}{\partial{z}}\right) \left(\frac{\partial}{\partial{c}}\right)} {1-\frac{\partial}{\partial{z}}}\right|} \]
with all the derivatives in that expression being evaluated at \( F_C^P(Z_0) \).

Finding the derivatives of \( f \) seems to be a big challenge, I was wondering if maybe there is a simpler way using properties of conformal mapping etc?
OK, note that h^2 is the sum of two absolute values of analytic functions, and of course \( \nabla h = \nabla h^2 /(2h) \).
Section 2 of http://persianney.com/fractal/fractalNotes.pdf shows how to go back and forth between \( R^2 \) and \( C \), both for variables and gradient/derivatives.

For analytic function f() consider grad of |f|. In the notes I show that norm of grad is just abs of derivative of f. Similarly you can work out that
\( (\partial_x - i \partial_y)|f| = \partial_z|f| = \bar{f}f^{'}/(2|f|) \). So all you have to work out is \( \partial_c  h^2 \) which is only a bit unpleasant.
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Image Threads / Re: Lyapunov fractals
« Last post by ThunderboltPagoda on Today at 01:26:57 AM »
The b*sinē(r+x) formula creates much more symmetry than the logistic equation.
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Image Threads / Re: Lyapunov fractals
« Last post by ThunderboltPagoda on Today at 01:21:04 AM »
More Emmental cheese. Or is it an exploding Borg ship? In the second image, the exploded ship has been abandoned ...
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Fragmentarium / Re: Attempting to port Riemann to Fragmentarium
« Last post by mclarekin on Today at 12:28:34 AM »
It amazes me what you guys can produce in FragM, especially when the analytic DE is bad.
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Collaborations & Jobs / Re: Help- what self-similar shapes do you know of?
« Last post by TGlad on Today at 12:10:26 AM »
Thanks! great suggestions, I didn't know that about the Sierpinski tetrahedron.

The simplest examples are just recursive 'filled' shapes, like how the Pythagoras tree is made of filled squares. But there are surprisingly few of these that have names.


Is there a name for one which is a tree of disks? or spheres?
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The last 24 hours to upload your Art in  "The image of the month contest"     :thumbs:

I've seen a lot of new members, so it would be nice if you will also participate in this friendly contest.  For FF honor and Glory  :toast:
Tomorrow 'at Midnight'  the topic will be locked. After that the voting period will begin

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Fractal Mathematics And New Theories / Solid on (3D+) orbit trapping
« Last post by FractalDave on Yesterday at 11:46:37 PM »
Just out of interest does anyone know of any maths done to find an analytical method for getting a decent distance estimation for solid based on orbit trapping (at least say for "nearest" and at least for the default trap value i.e. the origin) ?
I have done code previously for solid based on orbit trapping but it was just a brute force method and either a bit inaccurate or a bit slow ! My own formal pure math isn't up to the task :(
If not an analytical one - any other ideas ?
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Fragmentarium / Re: Attempting to port Riemann to Fragmentarium
« Last post by timemit on Yesterday at 10:09:33 PM »
This looks great, quite playable with ,, thanks mclarekin : )  no stability problems yet.  and less messy.
Sabine , :yes: I might avoid your one atm or change the settings a little and see if I can stop it freezing me   :yes: ..
Thanks for new toyz : ) even if de can't be completely  fixed this still makes nice things
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Image Threads / Re: zm49 (Stars)
« Last post by CFJH on Yesterday at 09:30:14 PM »
128Stars (at E26329)
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btw this is my work in progress with bad normals (simply the direction to the nucleus of the component)
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