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Author Topic: A new paradigm for fractal graphics  (Read 1373 times)
Description: Ths model allows an arbitrary initial iterant
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cye
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« on: January 26, 2017, 08:22:43 PM »

I've developed a new paradigm for fractal graphics. The idea is that the initial iterant, call it z0, is not limited to a uniform array in the complex plane. In the new model, z0 can be derived from any curve in the complex plane. Beyond that, it behaves in the usual way of the iteration.

I've prepared a manuscript explaining this in detail with many images of results. The paper has been posted at the National Curve Bank and this Web page is a portal to the PDF file: (Fractals Reimagined)(http://web.calstatela.edu/curvebank/waldman20/waldman20.htm). In addition, I've posted a large number of animations on Pinterest to demonstrate how the fractals change with variations in the generating curves; they can be seen here: (Fractals Reimagined)(https://www.pinterest.com/cyewaldman/fractals-reimagined/).

I'll be happy to answer of your questions (or dodge any of your barbs). I certainly welcome any suggestions or criticisms.

Have a look!

Cye [cye@att.net]
« Last Edit: January 31, 2017, 01:00:25 AM by cye » Logged
SamTiba
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« Reply #1 on: January 27, 2017, 12:46:49 AM »

so basically all you do is to take a function z0 = f(z) and use this in the initation to compute sets?
like folding the complex plane and then using the normal mandelbrot- or julia-method?

sounds interesting, some examples look really good! But a bunch of them just seem to be very stretched standard-sets.

I thought about this idea for a while now but never had the time to implement and play around with it somehow.
Busy life, why can't we just do fractal programming all day long?
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Some of my images: Pinterest
cye
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« Reply #2 on: January 27, 2017, 02:35:22 AM »

It's true that many of them look like useless messes. I allowed that on purpose, that is, to show the good, the bad, and the ugly. However, there are always surprising results when you drill into them. Most of the images are such details. My intent is to demonstrate the great variety that can be achieved with this method. I worked with generating curves that are both strictly mathematical as well as completely free-form (can only be read as a vector of complex numbers). At any rate, I hope this will inspire you to try some things on your own. Please keep in touch.

I hope you also had a chance to look at the animations as well. Even a small change in the generating curve can produce substantial changes in the final result.

Thanks for taking the time to write. I appreciate the feedback.

Cye
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SamTiba
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« Reply #3 on: January 27, 2017, 08:46:16 AM »

is there a general form in how you achieve those curves?
or are there some functions where you say 'damn they look good' ?
tell me some more about that smiley
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Some of my images: Pinterest
cye
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« Reply #4 on: January 27, 2017, 05:57:25 PM »

When you ask about my curves, I assume you mean what I call the 'generating curves.' That is, the curves used to derive the array z0. The fact is, you can any curve f(z). The ones I've used in the PDF come from four principal sources: (1) plane curves of my own design, such as the 'apple of my i' and superconics and spirolaterals, (2) spirals, (3) tessellating tiles that I developed, and (4) original and more common glyphs that I have used in fractal tiling. In this last instance, I wrote a program that can take an outline figure (say, a png file) and output a complex vector of points that gives the shape.

The results usually produce some interesting areas in the final fractal. I've never started with a particular generating curve and rejected it. There is always something interesting to be found. Remember, you can drill down into the fractals just as you do with the Julia sets.

Cye
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SamTiba
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« Reply #5 on: January 28, 2017, 01:32:57 PM »

wow that sounds so cool!
I really have to test that by myself!

What method are you then using to compute the fractals?
I never had a look into inverse function systems so I'm quite curious to see how you handle it.
I mainly use a converging method and it's working pretty fine.
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cye
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« Reply #6 on: January 28, 2017, 07:48:02 PM »

There are two parts to the answer (since I'm not sure if you are asking about the inverse fractals or the reimagined fractals, or both). First, for the reimagined fractals, assuming you are starting with a curve f(z) in the complex plane, getting the initial iterant z0 is described in the PDF manuscript. Then, for performing the iterations, let's say you have a function cout=f(z0, c, maxiter, escape, etc) for either Mandelbrot or Julia sets. This applies to the reimagined fractals as is. For the inverse fractals you merely use cout=f(1/z0, c, ...). It's really that simple. Clearly, there can be singularities, but I've never had a problem with them, per se.
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SamTiba
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« Reply #7 on: January 29, 2017, 05:31:16 PM »

are you using the typical bail-out-algorithm?
with abs(z) < escape ?

.. guess I will have to take a closer look on your pdf, did not have the time to inspect it properly. It should answer those questions, right?
« Last Edit: January 29, 2017, 05:42:40 PM by SamTiba » Logged

Some of my images: Pinterest
cye
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« Reply #8 on: January 30, 2017, 06:58:12 PM »

Yup. Just the standard routine. All of the new material takes place before the fractal calculation begins. That's why I say that you can use 95-99% of your existing codes.

I saw your Pinterest page. You do some really nice work. Mostly things I haven't seen before, so kudos to you.

I've never followed anyone on Pinterest before, so I'm curious if they'll notify me when you make changes to the page.

In the PDF file I give a brief list of the generating curves and fractal families that I use. I used an old book called 'Fractal Programming in C' by Stevens to get the basics of the Phoenix and (self-squared) Dragon fractals (which I then wrote in complex notation for Matlab).
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SamTiba
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« Reply #9 on: January 30, 2017, 07:56:40 PM »

thanks for your kind words!

I built most of it mainly by myself understanding parts of codes from others.
Fractals are a fun way to learn programming!
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SamTiba
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« Reply #10 on: January 31, 2017, 05:38:08 PM »

I finally tested it with an easy distortion term.
The initial value in my examples is z0 + sin(abs(z0))*(1+i) (so the z i use is a+sin(abs(z)) + (b+sin(abs(z)))i)

For some examples you can easily see the sine-distortion:

Especially for single-value-algorithms only depending on itself we would often get the look of folding the complex plane, however, if we have algorithms depending on each other we can get different shapes that vary from the 'normal' ones. Especially for 3D-fractals it would be quite interesting to distort the surfaces, an example for that here:
(Where the small Julia-set gets distorted to a different looking shape, just imagine that in 3D)

Images created using a convergent method to compute Julia-sets with quite complex formulas. Red is the Fatou- and blue the Julia-set.
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cye
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« Reply #11 on: January 31, 2017, 06:44:29 PM »

Fantastic. You've created a new variation on my idea. To wit, my idea was to take a line in the complex plane, say f(z) and create an array from real(f(z)) and imag(f(z)). You have started with a regular array, z0 and transformed it. Thus creating a totally new form. The reminds me of the posting (How do Complex Numbers warp the plane?)(http://www.fractalforums.com/general-discussion-b77/how-do-complex-numbers-warp-the-plane/msg40217/#msg40217). Take a look at it.

I have used similar transformations in another context. See (The Paradox of Greater and Lesser Infinities)(http://web.calstatela.edu/curvebank/waldman11/waldman11.htm) where I developed transformations for warping the complex plane.
« Last Edit: January 31, 2017, 06:54:00 PM by cye » Logged
jehovajah
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« Reply #12 on: February 24, 2017, 10:15:58 AM »

Did you look into any of Hermanns Grassmanns ideas when conceiving this general line as an array ?
In addition, in fluid dynamics the dynamic boundary condition could be represented by some general curve in a surface( space-time surface) , have you looked at that in relation to the Finite Element Method( Claes et. al.) ?

I will look at your PDF when I get time
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May a trochoid of ¥h¶h iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
cye
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« Reply #13 on: February 24, 2017, 11:17:16 PM »

Well, you certainly posed some interesting questions. And I was surprised that I was unfamiliar with both of them, especially in light of the fact that I have a PhD in Aerospace Engineering (almost 50 years ago!) and that about half of my course work was in mathematics (albeit in the usual calculus, vectors, matrices, complex variables, approximate methods, ordinary and partial DEs, perturbation and asymptotic analysis, and so on).

So the answer to both of your questions is--no. Specifically, I don't think I have ever come across Grassmann (even when I self-taught quaternions for a particular problem). And dynamic BCs never came up my in particular studies of boundary layers, combustion, and propulsion or in my later working years in ocean electromagnetics.

The idea arose, in my retirement, as a consequence of application of complex variables to plane curves, tessellations, fractal tilings, and, more recently, fractals, as you see here. Much of my other work can be found at the National Curve Bank, and a few things at the Google Tiling List.

I will be interested to see what you do if you follow up on these ideas. Please let me know.
« Last Edit: February 24, 2017, 11:40:54 PM by cye » Logged
jehovajah
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« Reply #14 on: March 01, 2017, 10:06:45 AM »

Very interesting background and life's work!
Ocean electromagnetics sounds kind of cool .
At the moment I am only committing to reading your PDF .  I have so much on my plate that I have to discriminate.
I will try to keep you informed as I progress xxx
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May a trochoid of ¥h¶h iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
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