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Author Topic: True 3D Mandelbrot ?  (Read 12805 times)
Description: Yet another try at the "true 3D" Mandelbrot
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Alef
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« Reply #60 on: August 20, 2014, 06:59:39 PM »

Didn't read thorougly as I prefare to read hard texts when sitting in cousy chair and drinking a tea but bicomplex numbers just produce square like mandelbrot.


Everything I posted is made by Chaos Pro -> update Formulas -> MalinovskyFract -> DavidsGrail. Very easy to write 3D formulas throught slightly outdated renderer, and not so easy to get nice images as Mandelbulb3D. And Ultra Fractal you alsou have to write your own 3D raytracer (most of the authors are individualists so no plugins;)  ).

Code:
     r     i    j     k
r    r     i    j     k
i    i    -r    k    -j
j    j     k    0     0
k    k    -j    0     0

I think bottom right zeros in multiplication matrix should not be inherently wrong. There are dual numbers where epsilon represented unit with property e*e=0.
http://en.wikipedia.org/wiki/Dual_number
More strange is that both k*k and j*j is 0. But 4th dimension is absent thus j is like k and J*J = k*k =0. Thus in 3rd and 4th dimensions this fractal must look identical (like quaternion in i, j, k). So this could be extruded (linear) or revolved between 3dr and 4th dimensions. Thus it practicaly is 3 dimensional and not 4 dimensional, exept julia set when z,w aren't 0.

Maybe there could be multiplication matrix with 3 (or 1) zeroes only for k and j*j=r.
But then if k allways =0 there are just r, i, j parts and r*r, i*i, j*j are perfectly different r, -r ,0. Here r*r + i*i + j*j =0 as r*r + i*i =0 of complex, not shure is this important.

Twinbees 3D mandelbrot site have this: On October 13th, 2006, Marco Vernaglione put out the question and challenge to the world with this memorable document. Looks very much like sincos version.


http://www.renderosity.com/mod/gallery/index.php?image_id=1308487&member

Notice simmilarity.

« Last Edit: August 20, 2014, 07:17:40 PM by Alef » Logged

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Alef
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« Reply #61 on: August 20, 2014, 07:01:11 PM »

Rendered 4 zero matrix mandelbrot from unusual perspective.

There are some strange angles and strands on the top but alsou there are bunch of mini mandelbrots (universality?) near the object. Hadn't seen these small repeated minibulbs on the mandelbulb fractal.


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KRAFTWERK
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« Reply #62 on: August 20, 2014, 07:30:17 PM »

Very beautiful and VERY INTERESTING render Alef!!!

And you are right, looks very much like Marcos drawing, I remember it from Daniels site!
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Alef
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« Reply #63 on: August 20, 2014, 08:33:19 PM »

Thanks KRAFTWERK  cheesy
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« Reply #64 on: August 20, 2014, 09:52:07 PM »

wow, this all looks better than the mandelbulb (in my opinion).
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« Reply #65 on: August 21, 2014, 04:49:39 PM »

Yep, because the original mandelbulb is ugly anyway. Never really understood why that power was chosen for the bulb.
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Alef
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« Reply #66 on: August 22, 2014, 05:28:49 PM »

I think pow8 was chosen becouse of its simmetry but probably alsou becouse square mandelbrot have strange ungrailish things on its top and pow8 hides them embarrass

Indeed, 4 zeros vector multiplication table of cos formula is the same as a dual complex numbers. Throught what happens could differ, probably so far only the author understands what is going on wink

These strange angled stalks on top have something to do with bailout value, on minimal bailout they aren't visible. Probably a slow escape.

Looked at 3D slices of 4D, or in plain language rendered it with normaly absent 4th component of pixel beeing w=-0.11 and w=0.11. This curved fractal to the side and all the smaller bulbs are gone. I think w and z could be identical, there are no smaller bulbs on z axis and probably in 4th dimension its somewhere linear or revolved huh?

pixel w=-0.11


pixel w=0.11


p.s.
Sincos looks like spacecraft. I was watching news about another political ^&^#% and they showed spacecraft who looked just like sincos bulb.
« Last Edit: August 22, 2014, 05:50:42 PM by Alef » Logged

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« Reply #67 on: August 24, 2014, 04:56:40 PM »

More detail of the sin/cos combi version for z^2+c - solid at 1e-8

Top of the whole thing showing many minis above:



http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16571

The Minibrot M1 from the above:



http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16572

The Minibrot M2 from the above:



http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16573


That's the good bit - the bad bit is that all 3 forms, cos, sin and cos/sin combined fail as mathematical fields because although unlike quaternions they are commutative and unlike bi-complex they have a full division algebra, they are neither distributive nor associative sad
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« Reply #68 on: August 24, 2014, 05:23:01 PM »

Who cares?  afro Those minibulbs looks scaringly Grailish...  A Beer Cup
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« Reply #69 on: August 24, 2014, 06:49:06 PM »

congrats, indeed a very good candidate wink
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« Reply #70 on: August 25, 2014, 04:40:02 AM »

I mentioned the field thing because I believe if a true R4+ field is found then we'll have the perfect Grail wink

Also an interest point I thought I'd add is that neither the plain cos nor sin versions of this form have minibrots at the "top", they only appear in the combined version.
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Alef
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« Reply #71 on: August 26, 2014, 04:58:13 PM »

This sounds as almoust an universality, most of "grails" didn't had such nice self repeating and mini mandelbrots. Could the author publish sincos formula for pow2 for those who have no idea how to turn ^-1 into ^2? Unless of corse this is going to arxive or there are another reason for not sharing it.

Looking at pow4 zooms gave the idea that without DE it have too much details. But a walk throught 4th dimension was more promising.
Julia sets should have very elaborated 4D geometry as when you render it with non zero 4th dimension pixel value you sometimes can see something appearing, features changing and spirals rotating. In Mandelbrot set strange areas with disconnected lines and angles are just in the same regions where with changing 4th dimension pixel value solids appears (in animation at right corner). So it could be that these distorted lines are remnants of some 4th dimensional objects. So I was wrong, it's fully 4 dimensional.
Hadn't tested the fractal with switched dimensions throught.

Cos Julia set (-0.5; -0.3; 0.15; -0.25)  rotating and moving throught 4th dimension with pixel value from -0.3 to 0.



smaller


And a cos mandelbrot set going thought 4th dimension from -1.24 to +1.24.



smaller, 534 kb


Quote
nor distributive nor associative
I had forgotten what it means. At least there are division, so it's more practical than bi-complex.
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Kalles Fraktaler
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« Reply #72 on: September 01, 2014, 04:36:32 PM »

The images you posted here are interesting, but unfortunately they don't look much like the picture from Marco Vernaglione.

I have an idea about this, but I am not able to realize it myself (at least not for a long time).
Because the roots of the expanded Mandelbrot formula are indeed located where the bulbs are expected.

First iteration, x, has root in 0. Thats the main bulb, so it can be rotated around the real axis.
Second iteration, x^2+x, has a new root in
X2=-1
And around this points there is the second big bulb. So, just rotate them around the real axis.

Second iteration, (x^2+x)^2+x, has (new) roots in
X2=(-0.1225611668766536+i0.7448617666197443)
X3=(-0.1225611668766536-i0.7448617666197443)
X4=-1.7548776662466927
So, you could put the 2D bulbs there, rotate around the axis of the angle from (0,i0) and put additional bulbs rotated 90 degrees.

Third iteration, ((x^2+x)^2+x)^2+x, has (new) roots in
X2=-1.310702641336833
X3=(0.28227139076691393+i0.5300606175785253)
X4=(0.28227139076691393-i0.5300606175785253)
X6=(-0.156520166833755-i1.032247108922832)
X7=(-0.156520166833755+i1.032247108922832)
X8=-1.9407998065294843
So, you could put new spherical bulbs there, and put additional bulbs rotated 45 degrees.

etc etc
Would that be possible?
Would it give a good result?
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« Reply #73 on: September 07, 2014, 07:05:18 PM »

Like this?
 cheesy

Edit:
Made a fun movie of it as well:
<a href="https://www.youtube.com/v/vMfbp2xmAQk&rel=1&fs=1&hd=1" target="_blank">https://www.youtube.com/v/vMfbp2xmAQk&rel=1&fs=1&hd=1</a>


* mandel3d.png (63.94 KB, 1058x368 - viewed 312 times.)
« Last Edit: September 07, 2014, 08:23:18 PM by Kalles Fraktaler » Logged

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David Makin
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« Reply #74 on: September 09, 2014, 03:59:51 AM »

Yes I think that's a way to visualise what we're aiming for - but we want it from a straightforward z^2+c formula rather than a manipulation that produces what we're after wink
We're getting close enough now for me to be convinced the required form does exist, probably as a 4D system rather than a 3D one though wink
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