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Welcome to Fractal Forums > Fractal Math, Chaos Theory & Research > Mandelbrot & Julia Set > The 3D Mandelbulb > A new 3d mandelbrot variant - Mandelcup
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Author Topic: A new 3d mandelbrot variant - Mandelcup  (Read 5719 times)
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pupukuusikko
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« on: November 12, 2015, 06:19:20 PM »
Hello,

While tinkering with Tglad's formulas from thread http://www.fractalforums.com/the-3d-mandelbulb/2d-conformal-formula-suggestion/ ,
replacing x here and y there, I came upon an interesting variant which I'd call a mandelcup.
It is simpler than the originals, with just two complex operations:

Code:
void mandelcup(inout vec3 p) {
        // c1 and c2 are adjustable constants, with default values c1=0. c2=2.
vec2 z = complexDivision(p.xy,vec2(p.z,c1));
z = complexMult(z,z)*c2;
p=vec3(2.*z.x, 2.*z.y, dot(z,z)-1.)/(dot(z,z)+1.);
}

Because no insight was involved due to my lack of mathematical understanding, let's move straight to images.
Sideview from mandelbrot mode, notice the overall grailish shape  A Beer Cup




The top contains embedded mandelbrot, there's one in the bottom as well.




At glance the julias seem disappointing, having rather featureless 2d julias at the ends of the tube,
with stretched bands in the middle.




But wait, the tube is hollow, and it contains layers of beautiful 3d julias of the same theme shocked



Attached is the fragmentarium .frag file with presets for the above images. Be careful with screen buffer size
in inside rendering, it can be quite slow.

Any comments, corrections or modifications of the formula would be greatly appreciated from the well educated folks around here.  wink
* mandelcup.frag (6.42 KB - downloaded 234 times.)
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3dickulus
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« Reply #1 on: November 13, 2015, 04:09:46 AM »
Beautiful! Nice work, I like the embedded mandelbrot and the "Grail" shape, may I include this in the Fragmentarium Experimental folder?
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KRAFTWERK
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« Reply #2 on: November 13, 2015, 07:20:03 AM »
WOW, love that last image. I like your layman approach too, kind of the same as mine...  afro
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3dickulus
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« Reply #3 on: November 13, 2015, 07:58:05 AM »
Yikes! it is slow for the inside, maybe some of the larger math brains can help speed this up a bit.
@KRAFTWERK yours was using Mandelbulb 3D?

this pic is rendered with DE-Kn2, some nice structural features smiley

* MandelCup1.jpg (82.58 KB, 640x360 - viewed 641 times.)
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DarkBeam
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« Reply #4 on: November 13, 2015, 11:25:19 AM »
This is incredibly interesting. A Beer Cup
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pupukuusikko
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« Reply #5 on: November 13, 2015, 11:58:22 AM »
Thanks everyone for your interest, it's nice to be able to contribute something here. wink

Quote from: KRAFTWERK on November 13, 2015, 07:20:03 AM
WOW, love that last image. I like your layman approach too, kind of the same as mine...  afro

I wish I had other approaches in my arsenal.. I have a bit more polished version of that scene in
http://pupukuusikko.deviantart.com/art/Towards-the-Light-571646559, check out also the new tetra works  wink

Quote from: 3dickulus on November 13, 2015, 07:58:05 AM
Yikes! it is slow for the inside, maybe some of the larger math brains can help speed this up a bit.
this pic is rendered with DE-Kn2, some nice structural features smiley

Nice pic, although I find using more fog is useful for separating the foreground, otherwise inside renderers tend
to result a bit messy. Perhaps wait a bit on including this on the Fragmentarium package, maybe some improvements
will come up concerning inside DE or the formula. At the moment, play with number of iterations, detail and DE adjustment to
get acceptable speed in navigation.


Quote from: DarkBeam on November 13, 2015, 11:25:19 AM
This is incredibly interesting. A Beer Cup

Thanks Luca, should not be impossible to code for MB3d? cheesy Maybe a bit later though,
there are issues of which constants are good for users, also some variations might come up.
 
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DarkBeam
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Fragments of the fractal -like the tip of it
« Reply #6 on: November 13, 2015, 01:07:19 PM »
More ideas?

Code:
void mandelcup(inout vec3 p) {
        // c1 and c2 are adjustable constants, with default values c1=0. c2=2.
vec2 z = complexDivision(p.xy,vec2(p.z,c1));
z = complexMult(z,z)*c2;
p=vec3(2.*z.x, 2.*z.y, dot(z,z)-1.)/(dot(z,z)+1.);
}

First idea that comes to mind; is c1 truly useful? smiley
What variation pops out when you do this instead...
It is identical to your formula if c1 is still zero.

Code:
void mandelcup(inout vec3 p) {
        // c1 and c2 are adjustable constants, with default values c1=0. c2=2.
vec2 z = (p.xy / p.z);
z = complexMult(z,z)*c2;
        float mag1 = dot(z,z)-1.; float mag2 = mag1+2.;
        z.xy = complexDivision(z.xy,vec2(mag2,c1));
p=vec3(2.*z.x, 2.*z.y, mag1/mag2);
}
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pupukuusikko
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« Reply #7 on: November 13, 2015, 06:14:55 PM »
Quote
First idea that comes to mind; is c1 truly useful? smiley

Yes it is, I use it to fatten the julia cups to make more room inside. Azn

However, in your version, if we replace
Code:
//vec2 z = (p.xy / p.z);
vec2 z = (p.xy * p.z);

we get a different and at least as interesting fractal  shocked




Now, the only significant  flaw with both these variants is the stretching around the equator, seen
clearly in the previous julia pic, yet always present. Could something simple be done to alleviate that without
hitting the theoretical conformality limit? (Layman approach does not help here)
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pupukuusikko
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« Reply #8 on: November 13, 2015, 10:56:07 PM »
Oh my Tglad, the brot insides are also navigable! nerd
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mclarekin
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« Reply #9 on: November 13, 2015, 11:28:03 PM »
Looks great.  Will try it soon afro afro afro
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3dickulus
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« Reply #10 on: November 14, 2015, 02:21:42 AM »
I've done a few things like this with other peoples fragments but have never really published much more than pictures because of my "layman's" approach, thinking the real mathematicians would probably get a chuckle out of my silly little hacks.
@pupukuusikko I find this very encouraging and may include a few more things in the Experimental folder wink
Once this gets polished a bit I would be happy to include it too.
hmmm... the "other" side of mandelbrot and julia sets smiley
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quaz0r
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« Reply #11 on: November 14, 2015, 03:53:32 AM »
Quote from: pupukuusikko on November 12, 2015, 06:19:20 PM
Because no insight was involved

 grin  cheesy  grin  A Beer Cup

really nice!

it looks like a magnificent glass bowl for a bong  angel
should name it "mandelbowl" instead
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pupukuusikko
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« Reply #12 on: November 14, 2015, 05:10:35 PM »
Thanks quazor and mclarekin for support.   Azn

Quote from: 3dickulus on November 14, 2015, 02:21:42 AM
I've done a few things like this with other peoples fragments but have never really published much more than pictures because of my "layman's" approach, thinking the real mathematicians would probably get a chuckle out of my silly little hacks.
@pupukuusikko I find this very encouraging and may include a few more things in the Experimental folder wink

Nice that you find my example therapeutic. I think there is a well established tradition in fractal community on
more or less random tinkering, seldom punctuated by mathematicians bringing something totally new to table.

Luca, here's the current version with some added variation and c1 moved to a more interesting role.
If you could serve this to the mb3d community, I'd be a happy camper. Yes !!

Code:
void mandelcup(inout vec3 p) {
        // c1 and c2 are adjustable constants, with default values c1=0. c2=2.

        // squaring z appears to increase conformality and adds intesting
        // variations on outside julias
        if (squareZ)
                p.z*=p.z;

        // c1 can be used to modify the overall shape in nice ways and
        //  turn surface to aestethically pleasing quarternion-like smoothnes
        p.z+=c1;

        vec2 z;
        // inversion here produces another nice fractal with similar properties
        if (multiply)
                z = p.xy * p.z;
        else
                z = p.xy/p.z;

        z = complexMult(z,z)*c2;
        float mag1 = dot(z,z)-1.; float mag2 = mag1+2.;
        z.xy = complexDivision(z.xy,(vec2(mag2,0.)));
        p=vec3(2.*z.x, 2.*z.y, mag1/mag2);
}
 
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matsoljare
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« Reply #13 on: November 15, 2015, 12:43:14 AM »
But will you color it red for Christmas?
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pupukuusikko
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« Reply #14 on: November 15, 2015, 07:43:41 AM »
Quote from: matsoljare on November 15, 2015, 12:43:14 AM
But will you color it red for Christmas?

Nah, I will paint it in gold and send it to fine folks in FIFA as a christmas present.
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