Inverse Fractals

[cye]

Hi,

I'm new to fractals and to this board, though not new to complex mathematics and programming. One quickly gets the impression that everything has been done already! So, in seeking to do something different, I set out to test the idea of 'inverse' fractals (not to be confused with 'inverse iteration'). By inverse fractal I mean that if a fractal is defined by some f(z0), then the inverse is given by f(1/z0). I've seen some examples of this for the Mandelbrot fractal. I've developed a number of inverse fractals and created animations that can be found at Pinterest (Fractals Animated) (https://www.pinterest.com/cyewaldman/animated-fractals/). These include the phoenix fractal (whose inverse I call the scorpion fractal), as well as the self-squared dragon, the Glynn- Julia, and various trigonometric Julia sets, including sin(), cos(), csc(), csc()*csch(), csc()/csch(). All of these have the argument z^n. These have animations for various values of n. The others have animated zooms on specific regions.

So, that's what I'm doing. If you have a chance to take a look, I'd certainly welcome any comments, suggestions, and yes, even criticisms.

FYI, I'm a retired engineer with 50 years of experience in mathematical modeling. My recreational mathematical endeavors in plane curves, tessellations, fractal geometry, and 3-D modeling can be found at the National Curve Bank (http://curvebank.calstatela.edu/home/home.htm). All that work is done in complex mathematics.

Thanks for reading.

Cye [cye@att.net]

[3dickulus]

after looking, I think you would enjoy Fragmentarium or Synthclipse

[SamTiba]

hey cye and welcome to the forum!
nice to see, that you show so much interest in julia-sets

I dont particulary do a lot with your 'inverse-fractals' but julia-fractals in general.
especially the variations of exp(z) (so all trigonometric functions) are tricky to handle but can be really beautiful.
Have a short look in three examples I have in my personal gallery here.

What method do you use to compute the Julia-sets? Did you program the application for that by yourself?
I'm looking forward to share some knowledge and method with you

[cye]

Hi,

All of my work is in codes that I developed by myself, not to say that I haven't learned and appropriated things from many others. My work is facilitated by programming in Matlab, insofar as it handles both complex variables and matrices seamlessly.

There has been so much done with the Julia sets that I was discouraged from trying to do something new. So I turned to (1) inverse fractals and (2) more complex (i.e., more complicated) trigonometric functions. For example, try programming csc(z)/csch(z) in Cartesian coordinates. Matlab may not be the fastest kid on the block, but the development time is extremely quick.

I'm interested to see your personal gallery, but you did not provide a link.

Keep in touch.

Cye

[SamTiba]

Oh cool!

I am also doing a lot in MatLab, but for fractals just mostly correction or tests, the rest is programmed in java due to computation speed. But as you said I need way more time to develop my code. Spent hours and hours for it.

I would be able to compute the julia-set for your example quite fast, built myself some tools to handle with a lot of different complex functions.

How much time do you need for an image to render in average?

You can find those few images on my profile, I don't have uploaded that much anywhere else, but I can show you some of my work.

[cye]

The question about how long it takes to render is very difficult to answer insofar as it is highly variable. Mostly it depends on how much black space there is (i.e., never escaped, reached max iteration). Thus, for example, Mandelbrot takes much loner than most Julia sets. But throw in a Julia with a lot of black space and the time goes way up. The Glynn-Julia set, as far as I've seen, has no black space and computes fiercely quick. Also, it depends upon the size of the output. I usually run ~4000 square array with a max iter of 2048 and escape set at 512. These can take minutes to hours. Some sets, like the dragon are very time consuming. I always note the time of computation. Often, I'll run a set at 1000-square to check how it looks and gauge the time for the full set. Okay, to try to attempt a direct answer, when I make the initial run of a
reimagined fractal, using the original computed z0, it takes seconds to minutes (say, 3-5) to compute and plot.

In the end, all I can say is, patience.

And I'm curious about something. Does Java support complex variables and operations?

As for showing your work, you email them to me personally [algorithmicart@att.net], or do as I do, and start a Pinterest page (I've got three going presently).

[SamTiba]

jep, patience is all that it needs
I always start with a 128x128 array and then increase it to 1024x1024 for some more settings and then for images compute the final version on 4096x4096
Some calculations take up to 10 hours, depending mostly on the complexity of the formula.
Normal time for 1024x1024 would be around 5-20 mins.

Java itself does not support complex numbers normally. I built something to handle that by myself, but it was quite easy. With oop in java you can easily implement that. Furthermore I also want to include Quaternion numbers for higher dimensional fractals in the future.

I think I'm going to open an account on pinterest to show some of my work, I'll let you know then.

[superheal]

The question about how long it takes to render is very difficult to answer insofar as it is highly variable. Mostly it depends on how much black space there is (i.e., never escaped, reached max iteration). Thus, for example, Mandelbrot takes much loner than most Julia sets. But throw in a Julia with a lot of black space and the time goes way up. The Glynn-Julia set, as far as I've seen, has no black space and computes fiercely quick. Also, it depends upon the size of the output. I usually run ~4000 square array with a max iter of 2048 and escape set at 512. These can take minutes to hours. Some sets, like the dragon are very time consuming. I always note the time of computation. Often, I'll run a set at 1000-square to check how it looks and gauge the time for the full set. Okay, to try to attempt a direct answer, when I make the initial run of a
reimagined fractal, using the original computed z0, it takes seconds to minutes (say, 3-5) to compute and plot.

In the end, all I can say is, patience.

And I'm curious about something. Does Java support complex variables and operations?

As for showing your work, you email them to me personally [algorithmicart@att.net], or do as I do, and start a Pinterest page (I've got three going presently).

No java does not have a built-in complex class, if you are interrested on developing something in java here is my complex implementation (https://github.com/hrkalona/Fractal-Zoomer/blob/master/src/fractalzoomer/core/Complex.java). It has most of the trig functions.