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Author Topic: Simplest formula for producing arbitrarily complex fractal?  (Read 2146 times)
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DarkBeam
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« Reply #15 on: January 16, 2017, 10:17:24 PM »

What looks simple in glsl might be complicated in javascript, java, basic, fortran etc.
The very word simple is meaningless in the world of fractals, raytracing and in math where nothing is really simple!
 smiley
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No sweat, guardian of wisdom!
kram1032
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« Reply #16 on: January 16, 2017, 10:35:36 PM »

what are you talking about, there is nothing in math that ISN'T simple grin
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PieMan597
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« Reply #17 on: January 16, 2017, 10:51:19 PM »

what are you talking about, there is nothing in math that ISN'T simple grin
Except needlessly confusing notation.
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claude
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« Reply #18 on: January 16, 2017, 11:42:08 PM »

make a program that generates formulas from every possible combination

That's a nice idea, I might try it sometime.  The number of expressions gets big quickly though, with n slots in each formula there are (4^n + 4^(n - 1))^2 possible formula (you can reduce this quite a bit using symmetries, though I suspect it will remain O(16^n)).  But as you are interested in small n I think it'll be tractable.

EDIT: got the formula wrong: needs squaring instead of doubling because the choices for the second formula are independent of the choices for the first formula
« Last Edit: January 17, 2017, 02:18:59 AM by claude, Reason: bad maths » Logged
kram1032
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« Reply #19 on: January 17, 2017, 12:01:58 AM »

Math notation, oddly enough, has for the longest of times not gotten the same scrutiny as math itself. I think that might start changing right now. But that's a very different topic smiley
I'm curious what that survey of formulae will yield. With the proposed three occurrences per variable we'd get 160 distinct fractals. Though none of them would be, say, the normal Mandelbrot Set, since you wouldn't allow the 2 in y \to 2xy+b.
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quick yellow whale
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« Reply #20 on: January 17, 2017, 10:49:40 PM »

I tried all combination of formulas that use exactly three variables. Some of the images were interesting, but none had a complexity similar to Mandelbrot. This is one of the more interesting ones and is a fractal, but doesn't have a complexity similar to Mandelbrot which I'm looking for.

Z^x_{n+1} = Z^y_n - Z^x_n + C_y
Z^y_{n+1} = Z^x_n*Z^x_n - C_x

Mandelbrot itself has 5 variables in the X equation and 4 in the Y, if you count the '2', so I'm going to have to optimize my program to narrow down the number of combinations before I go that high.

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quick yellow whale
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« Reply #21 on: January 23, 2017, 12:09:24 AM »

After producing images for all formulas that use between 3 and 4 variables I got 196,365 images after removing lots of duplicates. There's too many images to go through manually, but sorting by image size allows viewing the more complex images. None of the interesting I found so far had the shape-stacking feature of the Mandelbrot set. There were also no interesting ones that didn't seem distorted or stretched in some way.

This is the formula for the most interesting one I've found so far:

X_{n+1} = C_x*C_x - X_n - Y_n
Y_{n+1} = X_n*X_n - Y_n + C_y



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kram1032
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« Reply #22 on: January 23, 2017, 12:48:10 AM »

With the kinds of operations you allow you aren't going to get any "interesting ones that don't seem distorted or stretched" except for very specific parameters. It's partially due to how these formulae generate their images in the first place: You keep folding \mathbb{R}^2 onto itself in very specific ways that happen to stretch and distort the space. - The 2 in the Mandelbrot formula is actually pretty vital in that regard. It in part makes sure the distortion happens in a fairly regular manner.
I realize this will explore complexity even more but I'd suggest  perhaps also enumerating, for a small range of integers or perhaps fractions, pre-factors for each term. Something like any fraction between \frac{1}{3} and \frac{3}{1} that you can build with the numbers between 1 and 3, just to give you some idea.
And if you are smart about it, you can reduce quite a bit of the complexity that is caused by some of what you do. For instance, changing around X and Y will just rotate your set by 90░. Multiplying X*Y is the same as multiplying Y*X.... stuff like that.

I like the one you found there. It has quite an interesting structure going on.
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quick yellow whale
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« Reply #23 on: January 23, 2017, 04:29:12 AM »

With the kinds of operations you allow you aren't going to get any "interesting ones that don't seem distorted or stretched" except for very specific parameters. It's partially due to how these formulae generate their images in the first place: You keep folding <Quoted Image Removed> onto itself in very specific ways that happen to stretch and distort the space. - The 2 in the Mandelbrot formula is actually pretty vital in that regard. It in part makes sure the distortion happens in a fairly regular manner.
I realize this will explore complexity even more but I'd suggest  perhaps also enumerating, for a small range of integers or perhaps fractions, pre-factors for each term. Something like any fraction between <Quoted Image Removed> and <Quoted Image Removed> that you can build with the numbers between <Quoted Image Removed> and <Quoted Image Removed>, just to give you some idea.
And if you are smart about it, you can reduce quite a bit of the complexity that is caused by some of what you do. For instance, changing around X and Y will just rotate your set by 90░. Multiplying X*Y is the same as multiplying Y*X.... stuff like that.

I like the one you found there. It has quite an interesting structure going on.

I underestimated how many different combinations there are for two equations with 4 variables each, and 3 different operations between the variables. Manually looking at thousands of images is far too tedious. What I could try is create images with different combinations of constant coefficients in front of the variables for the most interesting equation I found to see if one combination removes the distortion and reveals shape stacking in deeper areas.
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Tglad
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« Reply #24 on: January 23, 2017, 09:31:22 AM »

Surely you want to only search conformal transforms.?.. they are the only ones that will produce detail without stretching in one direction.
If you're only using multiplies and additions then you've only really got 2D rotations, squaring, cubing etc, and translations. But since multibrots are universal any combination is equivalent to a standard multibrot.

If you allow more primitives like complex sin, cos, exp etc then you can do more cool (and conformal) patterns, but then you have to decide on the relative complexity of each function.
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kram1032
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« Reply #25 on: January 23, 2017, 09:58:33 AM »

That's actually part of the reason why complex numbers are a good choice: All smooth functions over the complex numbers are conformal.
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quick yellow whale
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« Reply #26 on: January 23, 2017, 10:41:30 PM »

Surely you want to only search conformal transforms.?.. they are the only ones that will produce detail without stretching in one direction.
If you're only using multiplies and additions then you've only really got 2D rotations, squaring, cubing etc, and translations. But since multibrots are universal any combination is equivalent to a standard multibrot.

If you allow more primitives like complex sin, cos, exp etc then you can do more cool (and conformal) patterns, but then you have to decide on the relative complexity of each function.

Is the "standard multibrot" the Mandelbrot set? I'm wondering if there exists a formula that is more complicated than the Mandelbrot one in complex terms, but is simpler than the two Mandelbrot formulas when reduced to the real plane.

Also, can every conformal function when represented as two equations in the real plane be converted into one equation in complex terms?
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Tglad
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« Reply #27 on: January 24, 2017, 03:18:13 AM »

Without thinking too thoroughly... yes and yes
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kram1032
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« Reply #28 on: January 24, 2017, 11:16:30 AM »

you could simply take the split-complex numbers for an example of the first. You can emulate them in complex numbers with conditionals. (Something you can always do in principle) But it's awkward and probably no longer holomorphic (differentiable in the sense of the complex numbers) and as such probably also no longer conformal.
And I'm pretty sure all 2D conformal maps are indeed convertible to complex numbers.
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quick yellow whale
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« Reply #29 on: January 28, 2017, 01:46:57 AM »

After thinking about it some more, it would probably be easier to analytically check if a transform is conformal rather than generate an image for the transform and manually check if the image is distorted.
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