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 Author Topic: Mandelbrot on real numbers (new post)  (Read 28741 times) Description: Linear combination of two 1D Mandelbrots sets 0 Members and 1 Guest are viewing this topic.
lkmitch
Fractal Lover

Posts: 238

 « Reply #15 on: January 24, 2011, 05:56:41 PM »

I like the notion, but tweaked it a bit, using two different 1D Julia sets (using the same set on each axis was just too symmetric for me).  Here are a couple of examples using my Rose Range Lite coloring in Ultra Fractal.

(Sorry if this shows up twice--my images were too large so I had to try again.)
 jan24-b.jpg (96.59 KB, 600x450 - viewed 415 times.)  jan24-a.jpg (81.04 KB, 600x600 - viewed 359 times.) Logged
trafassel
Fractal Bachius

Posts: 531

 « Reply #16 on: January 24, 2011, 08:46:01 PM »

Sorry, the last 3d pictures uses |x|+|y|+|z| instead of |x|+|y|-|z| for the bailout value.

Here is the correct 3d fractal.
 Data537pic10633a.jpg (242.59 KB, 1200x1200 - viewed 362 times.) Logged
trafassel
Fractal Bachius

Posts: 531

 « Reply #17 on: January 24, 2011, 09:08:48 PM »

Another start value
 Data537pic10669small.jpg (243.67 KB, 1200x1200 - viewed 466 times.) Logged
Kali
Fractal Supremo

Posts: 1138

 « Reply #18 on: January 24, 2011, 09:19:30 PM »

Nice pics, @lkmitch, I have also tried the Julia version with differents values for each function, it works nice.

---

Good work, @traffassel, The first looks like some kind of 3d cantor sets, and the second is a nice architectural piece
Did you try using abs function, like abs(x-y-z) or abs(x+y+z)?
And perhaps inverting the bailout setting for values that tend to zero? (I mean setting a small bailout value and exiting the iteration loop if the values get smaller than this)

P.S.: What do you use to render? I'd love to have a 3D renderer that let's me edit the formula and the drawing method as I want... I'm planning to code one myself.

---

@Ken, any new results?

---

As for me, I just discovered this:

Using the formula/method I first described, but evaluating abs(f1+f2) and setting the bailout to values getting close to 0, I got this kind of bifurcation map (I had to fine-tune the bailout and max iteration values to get a nice view)

And this is evaluating abs(f1-f2)

I used just plain inside painting (paint only the points inside the set, the same color)

Then, after trying several things, I got this strange drawing:

I looked closer and it's a waveform!

I got this by evaluating f1*f1-f2*f2, with a bailout value of .0000000001 (remember that I look for values that get smaller than this) and max iterations above 1000

It definitely looks like a plot of a sound wave to me, don't you think? it has different frequencies and amplitudes, I wonder how it sounds.
I'll try to get this waveform somehow by programming a VB.NET code and perhaps I can make it sound!

I don't have any conclusion as for now, I leave it for you.
( Can´t think too much as I' having a bad headache )

 « Last Edit: January 24, 2011, 09:52:27 PM by Kali » Logged

bib
Global Moderator
Fractal Senior

Posts: 2070

At the borders...

 « Reply #19 on: January 24, 2011, 09:24:29 PM »

Fantastic pictures Kali, the results of your experiences are so beautiful, and the first two pictures above are very impressive! I wonder what your fractal is trying to tell us with its strange waveform. That's pure fractal sound. Anyone to translate it to .wav?
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Between order and disorder reigns a delicious moment. (Paul Valéry)
trafassel
Fractal Bachius

Posts: 531

 « Reply #20 on: January 24, 2011, 10:26:44 PM »

You wrote:
>Did you try using abs function, like abs(x-y-z) or abs(x+y+z)?
> And perhaps inverting the bailout setting for values that tend to
> zero? (I mean setting a small bailout value and exiting the iteration loop
> if the values get smaller than this)

No, I use abs(x)-abs(y)+abs(z)

I used Gestaltlupe (http://www.boitels.de/Gestaltlupe/) for 3d drawing of arbitrary formulas.

 Data537pic10688small.jpg (255.75 KB, 1200x1200 - viewed 421 times.) Logged
matsoljare
Fractal Lover

Posts: 215

 « Reply #21 on: January 24, 2011, 11:16:34 PM »

Isn't this related to the Lyapunov function?
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Fractal Ken
Fractal Lover

Posts: 246

Proud to be 2D

 « Reply #22 on: January 24, 2011, 11:36:17 PM »

It definitely looks like a plot of a sound wave to me, don't you think? it has different frequencies and amplitudes, I wonder how it sounds.
I'll try to get this waveform somehow by programming a VB.NET code and perhaps I can make it sound!

Cool picture and a great idea! This could be a really neat approach to making fractal music.

@Ken, any new results?

Here's an image which uses Kali's initial method, but with bailout based on sqrt(x2 + y2) instead of |x - y|.

I see what look like barcodes within the picture.
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Fortran will rise again
Kali
Fractal Supremo

Posts: 1138

 « Reply #23 on: January 25, 2011, 12:38:30 AM »

Thanks @bib for your words (I love your artwork) and @trafassel for the info, I just downloaded the program, I'm about to try it. I realy like the images you posted.

@matsoljare, don't know about Lyapunov function, but I think it's related somehow to the logistic map, if you see the bifurcations graph I posted (I already know standard Mandelbrot is also related).

What I see, in all diferent variants, it's that the are distortions in the shapes (stretchings) that are always present within the patterns. I'm pretty sure  it's related to the waveform I found. Maybe the distortions obbey or encodes somehow the waveform frequency/amplitude.

I'm still looking for a way to reproduce the waveform, but also we must find how to decode the barcode Ken mentioned

 « Last Edit: January 25, 2011, 10:23:28 AM by Kali » Logged

Fractal Ken
Fractal Lover

Posts: 246

Proud to be 2D

 « Reply #24 on: January 25, 2011, 06:56:36 AM »

Here's a variant based on the formula (x, y) = (x3/y, y2) + c.

I think it's a surreal teapot.
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Fortran will rise again
jehovajah
Global Moderator
Fractal Senior

Posts: 2749

May a trochoid in the void bring you peace

 « Reply #25 on: January 25, 2011, 08:23:46 AM »

I will look into it Kali and get back to you, but for now i think that the fundamental basis of all these patterns is ratios of equivalence. We live in a marvellous age, where we can now visualise our relational patterns.

I think rusty ken! has nailed it for you in terms of generalising your algorithm to the (x,y) plane, but it may be to symbolic for you, so i will look for some natural process for you to think on.

Finally, i would be interested if you could explore outputs for prime number coefficients or constraints as far as possible.
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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!
KRAFTWERK
Global Moderator
Fractal Senior

Posts: 1439

Virtual Surreality

 « Reply #26 on: January 25, 2011, 11:37:48 AM »

Wow, this is so interesting Kali, even for a novice, cant give you any real input, like the others here, just had to tell you this.
Your images looks very interesting, so does Trafassels 3D versions.... and...
I WANT TO HEAR THAT SOUND!!!!

Regards
Johan
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bib
Global Moderator
Fractal Senior

Posts: 2070

At the borders...

 « Reply #27 on: January 25, 2011, 12:25:31 PM »

PSSSSCCCCHHHHHHHHHHHHHHH

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Between order and disorder reigns a delicious moment. (Paul Valéry)
Sockratease
Global Moderator
Fractal Senior

Posts: 3181

 « Reply #28 on: January 25, 2011, 06:17:51 PM »

I WANT TO HEAR THAT SOUND!!!!

<Quoted Image Removed>

Me too!

So I did something about it

And - OF COURSE my desktop wallpaper is Lissa!!  I just couldn't bring myself to crop Her out

That's part one of a couple stages.  I'll post more over the next day or 2.

I began by using a jpg to wav program that makes music out of pictures (found here: http://rekkerd.org/virtual-ans-image-to-sound-converter/ )

Next I'll show how to convert the wav to midi voices and compile them into something a bit more Musical
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Fractal Ken
Fractal Lover

Posts: 246

Proud to be 2D

 « Reply #29 on: January 25, 2011, 06:37:34 PM »

Sockratease, the conversion you did is super cool!

Kali, don't quit your day job to compose music quite yet.
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