Print Page - Rozuvan Circles Fractal

Title: Rozuvan Circles Fractal
Post by: valera_rozuvan on July 21, 2016, 12:24:20 PM

Hi there folks!

I have been tinkering with fractals for a long time now, trying to come up with something new. Basic idea is to take the Mandelbrot set, generalize it, and see if something cool turns up. So, I think I am on to something :) I will report my formula, code, and more. Need to sort things out, clean up some code... For now, an image of my new creation. I call it the Rozuvan Circles Fractal!

(http://nocache-nocookies.digitalgott.com/gallery/19/14000_21_07_16_12_16_04.png)
source: http://www.fractalforums.com/index.php?action=gallery;sa=view;id=19439

Title: Re: Rozuvan Circles Fractal
Post by: valera_rozuvan on July 21, 2016, 10:38:37 PM

OK folks! I decided to do a quick Jupyter Notebook to demonstrate the algorithm I use to render the RCF (Rozuvan Circles Fractal). You can see a static output from the notebook at https://valera-rozuvan.github.io/rozuvan-jupyter-notebooks/rozuvan_circles_fractal.html (https://valera-rozuvan.github.io/rozuvan-jupyter-notebooks/rozuvan_circles_fractal.html). On the page, there are a few more renderings of the fractal.

The mathematical formula, and a bit of theory will come next. In the meantime, what do you think about the uniqueness of the RCF? Have you seen anything like this before?

Title: Re: Rozuvan Circles Fractal
Post by: valera_rozuvan on July 22, 2016, 09:43:24 PM

A Generalized Mandelbrot Set (GMS), is a set of complex numbers $c$ for which a defined function $M(z, \bar{w})$ does not diverge when iterated from $z = 0$ , i.e. for which the sequence $M(0, \bar{w}), M(M(0, \bar{w}), \bar{w})$ , etc., remains bounded in absolute value. The generating function $M$ must fulfill one condition - for at least one set $\bar{w}$ , the function should produce the original Mandelbrot Set. I.e. it should be possible for the function to take on the form $M(z, \bar{w})=z^2+c$ .

Some explanations as to the notation above. The set $\bar{w}$ represents all of the constant parameters to the function $M$ . A parameter $w_i$ might (or might not) depend on the complex number $c$ . So:

$\bar{w} = \{w_1, w_2, w_3, ...\}$
$w_{i} = W_{i}(c, \bar{a_{i}})$
$\bar{a_{i}} = \{a_{i1}, a_{i2}, a_{i3}, ...\}$
$c,a_{ij}\in\mathbb{C}$

Let's look at an example of a GMS generating function $M$ :

$M(z, \bar{w})=(w_1+z)^{w_{2}}+w_3$

where:

$w_1=W_{1}(c, \bar{a_1})=a_{11}+a_{12}\times{c}$
$w_2=W_{2}(c, \bar{a_2})=a_{21}$
$w_3=W_{3}(c, \bar{a_3})=a_{31}\times{c}$

Combining all of the above together, we can write out fully our recursive function as follows:

$z_{n+1} = (a_{11}+a_{12}\times{c} + z_n)^{a_{21}} + a_{31}\times{c}$

The above function $M$ is a GMS generating function because it can produce the original Mandelbrot Set. If we use the following:

$w_1=0$
$w_2=2$
$w_3=c$

we get the recursive function:

$z_{n+1} = z_{n}^2 + c$

which generates us the original Mandelbrot Set.

In the next post I will go over the mathematical formula for the RCF. In the meantime, any questions so far?

PS: Another image of the RCF for you below :)

(http://nocache-nocookies.digitalgott.com/gallery/19/14000_22_07_16_9_38_20.png)
source: http://www.fractalforums.com/index.php?action=gallery;sa=view;id=19444

Title: Re: Rozuvan Circles Fractal
Post by: TheRedshiftRider on July 22, 2016, 10:00:55 PM

I'm curious, is it zoomable and does it contain interesting figures that are or can be made visible? Does it have julia-variants?

Title: Re: Rozuvan Circles Fractal
Post by: valera_rozuvan on July 23, 2016, 09:42:51 PM

Quote from: TheRedshiftRider on July 22, 2016, 10:00:55 PM

I'm curious, is it zoomable and does it contain interesting figures that are or can be made visible? Does it have julia-variants?

What do you mean by "Does it have julia-variants"? Is there some method to determine this? Why is this important?

As for zooming the RCF, I believe you can do it. Though, I haven't tried any deep zooms yet. Here is a small example of details that arise after zooming in a couple of times:

(http://nocache-nocookies.digitalgott.com/gallery/19/14000_23_07_16_9_39_10.png)
source: http://www.fractalforums.com/index.php?action=gallery;sa=view;id=19447

Title: Re: Rozuvan Circles Fractal
Post by: TheRedshiftRider on July 23, 2016, 09:52:37 PM

Ah. Those details look like the negative (f(z)=z^-1+c) version of the m-set.

When rendering a mandelbrot set you make use of both the z and the c variables. When generating a juliaset you use a fixed complex number for the c coördinate.
https://en.m.wikipedia.org/wiki/Julia_set

The mandelbrot set is made up out of multiple juliasets.

https://m.youtube.com/watch?v=JuyPwwG-Cr8

Title: Re: Rozuvan Circles Fractal
Post by: valera_rozuvan on July 23, 2016, 11:02:41 PM

Quote from: TheRedshiftRider on July 23, 2016, 09:52:37 PM

The mandelbrot set is made up out of multiple juliasets.

https://m.youtube.com/watch?v=JuyPwwG-Cr8

Initially I understood your answer wrong. I see now that your answer doesn't imply that the Mandelbrot set is a subset (in the mathematical meaning of the word) of the Julia set (or vice versa). Actually, there is a famous paper on the similarity of the 2 sets "Similarity Between the Mandelbrot Set and Julia Sets" Tan Lei, 1989 (http://www.math.univ-angers.fr/~tanlei/papers/similarityMJ.pdf (http://www.math.univ-angers.fr/~tanlei/papers/similarityMJ.pdf)).

Title: Re: Rozuvan Circles Fractal
Post by: TheRedshiftRider on July 24, 2016, 10:00:23 AM

Quote from: valera_rozuvan on July 23, 2016, 11:02:41 PM

Yes, I forgot to use the right term for it. Thanks for pointing that out. :)

Title: Re: Rozuvan Circles Fractal
Post by: valera_rozuvan on July 25, 2016, 01:18:25 AM

In this post, I will outline the mathematical formula that creates the Rozuvan Circles Fractal. The RCF is one of infinitely many possible fractals that come from a Generalized Mandelbrot Set defined by the following equation:

$M(z, \bar{w}) = ((w_1 + z)^{w_2} + w_3)^{w_4}$

where:

$w_1 = W_{1}(c, \bar{a_1}) = a_{11}\times{|c|} + a_{12}$
$w_2 = W_{2}(c, \bar{a_2}) = a_{21}\times{|c|} + a_{22}$
$w_3 = W_{3}(c, \bar{a_3}) = (a_{31}\times{|c|} + a_{32})^{a_{33}\times{c} + a_{34}} + (a_{35}\times{|c|} + a_{36})\times{c}$
$w_4 = W_{4}(c, \bar{a_4}) = a_{41}\times{|c|} + a_{42}$

In the above equations, $|c|$ is the absolute value of the complex number $c$ . If $c = a + b\times{i}$ , then $|c| = \sqrt{a^2 + b^2}$ .

To get the RCF, one needs to use the following $\bar{a}$ :

$a_{11} = 1$
$a_{12} = -10 \times{i}$

$a_{21} = 1$
$a_{22} = 10 \times{i}$

$a_{31} = -2 \times{i}$
$a_{32} = -100$
$a_{33} = 1$
$a_{34} = 0$
$a_{35} = -1 \times{i}$
$a_{36} = -200$

$a_{41} = 1 \times{i}$
$a_{42} = 0$

It can be easily shown, that our $M$ is indeed a GMS. It produces the orinal Mandelbrot fractal with the following $\bar{a}$ :

$a_{11} = 0$
$a_{12} = 0$

$a_{21} = 0$
$a_{22} = 2$

$a_{31} = 0$
$a_{32} = 0$
$a_{33} = 0$
$a_{34} = 1$
$a_{35} = 0$
$a_{36} = 1$

$a_{41} = 0$
$a_{42} = 1$

I created a Jupyter Notebook that demonstrates the above equation for $M$ and 3 different parameter sets $\bar{a}$ (two from above, and a bonus one). Check it out at https://valera-rozuvan.github.io/rozuvan-jupyter-notebooks/rozuvan_circles_fractal_part_2.html .

For the impatient ones, here is a collage of the 3 fractals:

(http://nocache-nocookies.digitalgott.com/gallery/19/14000_25_07_16_1_16_23.png)
source: http://www.fractalforums.com/index.php?action=gallery;sa=view;id=19450

Title: Re: Rozuvan Circles Fractal
Post by: valera_rozuvan on July 26, 2016, 03:13:03 AM

Announcing interactive version of my RCF. Done in JavaScript, it can be accessed here (https://valera-rozuvan.github.io/FractalViewer/FractalViewer.html#maxI=50$centerX=4.4$centerY=-0.6$viewWidth=370$cyclePeriod=15$cyclePhase=0.35$fractalType=rozuvan-circles-fractal). Any modern web browser should be able to handle it :)

Here is another interesting rendering:

(http://nocache-nocookies.digitalgott.com/gallery/19/14000_26_07_16_3_07_32.png)
source http://www.fractalforums.com/index.php?action=gallery;sa=view;id=19455

PS: The above rendering can be accessed in the interactive viewer here (https://valera-rozuvan.github.io/FractalViewer/FractalViewer.html#maxI=500$centerX=1.704258147354928$centerY=4.951237100860878$viewWidth=0.000004500473582474214$cyclePeriod=15$cyclePhase=0.35$fractalType=rozuvan-circles-fractal).

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Fractal Math, Chaos Theory & Research => Other types => Topic started by: valera_rozuvan on July 21, 2016, 12:24:20 PM