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Author Topic: Rozuvan Circles Fractal  (Read 1728 times)
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valera_rozuvan
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valera.rozuvan valerarozuvan
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« 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  smiley 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!


source: http://www.fractalforums.com/index.php?action=gallery;sa=view;id=19439
« Last Edit: July 26, 2016, 11:53:43 PM by valera_rozuvan » Logged

valera_rozuvan
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a test in time


valera.rozuvan valerarozuvan
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« Reply #1 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. 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?
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valera_rozuvan
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valera.rozuvan valerarozuvan
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« Reply #2 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  smiley


source: http://www.fractalforums.com/index.php?action=gallery;sa=view;id=19444
« Last Edit: July 26, 2016, 11:54:20 PM by valera_rozuvan » Logged

TheRedshiftRider
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« Reply #3 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?
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valera_rozuvan
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a test in time


valera.rozuvan valerarozuvan
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« Reply #4 on: July 23, 2016, 09:42:51 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:


source: http://www.fractalforums.com/index.php?action=gallery;sa=view;id=19447
« Last Edit: July 26, 2016, 11:54:40 PM by valera_rozuvan » Logged

TheRedshiftRider
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« Reply #5 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 co÷rdinate.
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
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Motivation is like a salt, once it has been dissolved it can react with things it comes into contact with to form something interesting. nerd
valera_rozuvan
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valera.rozuvan valerarozuvan
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« Reply #6 on: July 23, 2016, 11:02:41 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).
« Last Edit: July 23, 2016, 11:21:49 PM by valera_rozuvan » Logged

TheRedshiftRider
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« Reply #7 on: July 24, 2016, 10:00:23 AM »

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).
Yes, I forgot to use the right term for it. Thanks for pointing that out. smiley
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Motivation is like a salt, once it has been dissolved it can react with things it comes into contact with to form something interesting. nerd
valera_rozuvan
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a test in time


valera.rozuvan valerarozuvan
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« Reply #8 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:


source: http://www.fractalforums.com/index.php?action=gallery;sa=view;id=19450
« Last Edit: July 26, 2016, 11:55:00 PM by valera_rozuvan » Logged

valera_rozuvan
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Posts: 80


a test in time


valera.rozuvan valerarozuvan
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« Reply #9 on: July 26, 2016, 03:13:03 AM »

Announcing interactive version of my RCF. Done in JavaScript, it can be accessed here. Any modern web browser should be able to handle it  smiley

Here is another interesting rendering:


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.
« Last Edit: July 27, 2016, 12:17:26 AM by valera_rozuvan » Logged

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