Is there any shape that can not be found in the Mandelbrot?

[Levi]

I've successfully built a program to find grids of unique escape times. It's not very smart - it just repeatedly picks a random location and a random zoom factor then checks to see if the grid created is unique.

I can confirm that unique grids of 15x15 can be found - however it takes a little while. For an iteration cap of 10000 (the fastest number I've tried so far), it averages about 1 unique grid for every 551021 grids tested. If this is going to be feasible, then I'm going to have to give my program a little bit of intuition so that it spends more time searching in areas that are more likely to be successful.

[Dinkydau]

Nice work, Levi

[Alef]

A question for the mathematicians. Is there any shape that is not present, at least approximately, in the Mandelbrot set?

Here is one of them:


Here is another that don't appears in mandelbrot:

[Levi]

Actually, Alef, I may just have to prove you wrong when I get this program finished. The problem is getting it to run in a feasible time frame.

[Alef]

Actually, Alef, I may just have to prove you wrong when I get this program finished. The problem is getting it to run in a feasible time frame.

Would love to see it.
 I was thinking that using 3D CAD approach to combine fractals could allow to make fractal chair, but then it woun't be just generated.

[grobblewobble]

I'm sure you could find such images where the escape time of every pixel is unique (assuming that all visible points are escaping), and from there you could adjust your coloration rules in such a way that it would create any image you want.

Actually, I wonder now. What if the image has multiple pixels with the exact same color? Of course you can always approximate such an image by choosing a different color that is arbitrarily close to it, but..

edit: and the answer is: then you choose a color scheme that assigns the same color to several different escape times. Doh. 

[Levi]

@Alef: Read the posts on the previous page. The OP meant "Is there any image that cannot be generated within the Mandelbrot using the right configurations", not "Is there any shape that does not occur in the Mandelbrot set itself". Which changes the question completely. I should be able to have a demo for very very small images in a few days, but it's going to require some more research into interpolation techniques which I haven't worked with much.

@grobblewobble: For a large image size, almost every possible grid will have two pixels of the exact same escape time if you're using non-normalized iteration counts. However, unique grids of every size should exist - they are just hard to find. The largest ones I have found successfully are 15*15, but I found about 100 of those. Large sizes will require ridiculous computation times (and luck, since these are nondeterministic computations) but they can, in theory, be found.

[Alef]

IMHO some are irrational and trying to see just everything in mandelbrot set;)

[Levi]

Udate: have found unique grids of 64x64, still without using normalized iteration count. Started working on an interpolation program but got distracted learning SDL
But I will resume this project soon!!