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 Author Topic: Mandelbulb any-power expressions  (Read 40689 times) Description: 0 Members and 1 Guest are viewing this topic.
Paolo Bonzini
Guest
 « Reply #105 on: January 21, 2010, 10:18:40 AM »

Okay, so it *is* associative, since you're keeping it in spherical coordinates and only using addition/multiplication.
Only using multiplication/division, actually.  Addition requires cartesian coordinates, so it is associative but not distributive.

Also, the division/multiplication inverse *does* work, agreed? I can understand, though, why the distributive property doesn't work. I'm curious as to how exactly exp(a) is defined - you could use taylor series, or the exponentiation formula I suggested with A=2.71..., or exp(x)=e^x * (cos x,sin y,0) * (cos z, 0, sin z) [or WHATEVER that was on the first page].... it seems there is an agreed upon definition for Mandelbulb purposes, can someone please tell me what it is?
I have no idea.  The whatever was on the first page probably does not agree with the Taylor series, which is a problem.  Maybe quaternions could help, I don't have much time now.

Forgive me if I'm being a pain with having all this explained, but why doesn't (z*z)*z=z^3?
When you do z*z*z*z*z in cartesian form, every multiplication includes an implicit conversion to and from spherical coordinates.  It's a bit hard to visualize, but the conversion loses information because of the different range of the elevation vs. the y-axis rotation term.  Instead, you have to use specially crafted formulas for each exponent(*) that include exactly one conversion to spherical coordinates and one from, which is what is called z^n.

(*)I had a generic exponentiation formula expressed using Chebyshev polynomials, but I haven't looked at it for a while and I'm not sure I had no calculation errors since I did it on paper.
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soler
Forums Newbie

Posts: 2

 « Reply #106 on: September 19, 2010, 04:15:35 AM »

Bugman has defined a triplex polar form using a particular matrix product. I have extended this idea to give 48 different polar forms. If you are interested then please have a look at: http://soler7.com/Fractals/Matrices%20to%20Triplex.pdf
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M Benesi
Fractal Schemer

Posts: 1049

 « Reply #107 on: September 20, 2010, 05:42:37 AM »

Something a hell of a lot easier is.. the complex triplex (from another post of mine elsewhere in the forums).  It really makes the mathematical relationship to the original 2d set apparent:

Quote from:  M Benesi
You don't need anything too complex** to do triplex "algebra".  You simply require:

A) a square root function to calculate a magnitude:
r1= sqrt(y^2+z^2)

B) a complex power function:
complex_1= (x + i r1)^n
complex_2= (y + i z)^n

C) a real power function:
r3=r1^-n       (you are applying the magnitude of y and z two times (once in each complex number), so need to divide it out once)

D) the ability to directly access the real and imaginary components of the complex numbers:
new x = real part of complex 1   + x pixel value  OR  x Julia seed   (for Julias use the seed, Mandys use the pixel value)
new y = imaginary part of complex_1 * real part of complex 2 * r3  + y pixel value  OR  y Julia seed
new z = imaginary part of complex_1 * imaginary part of complex 2 * r3  + z pixel value  OR  z Julia seed

It's easily extended to higher dimensions... and is faster than the trig version in certain compilers (although I haven't tried them all).

** pun was and is still intended.... 2 complex..  although I didn't mention that it was intentional in the original post, as I felt it was a bit heavy handed to point out the pun.  This, however, has changed.
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soler
Forums Newbie

Posts: 2

 « Reply #108 on: September 22, 2010, 02:33:45 AM »

Here is an image from one of the 48 variations:

Many more are at
http://soler7.com/Fractals/3D0.html
All were made using Visins of Chaos.
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Fractal Molossus

Posts: 673

 « Reply #109 on: September 22, 2010, 03:11:26 AM »

"I have extended this idea to give 48 different polar forms"
I think you could say there are an infinity of polar forms, and I think it is right to say there is one for each rotation, which is equivalent to rotating the point by this rotation each iteration.
 Mandelbulb.gif (75.09 KB, 70x70 - viewed 1195 times.) Logged
Softology
Conqueror

Posts: 101

 « Reply #110 on: September 26, 2010, 07:25:52 AM »

Have a look at Soler's PDF.  It does show variations that do cover 48 possible variations.  Some of them do match the orginal +SIN -SIN COS etc variations, but the rest are new and do give unique bulb types.  I have included Phase, Theta and Phi scaling/shifting in all the existing bulb formulas and these 48 do give new/unique variations.

Anyway, I hope the next 3D fractal that gets as much publicity as the Mandelbulb comes from these forums.  Keep going guys.  The Mandelbulb and Kaleidoscopic IFS were great examples of someone thinking "what if" outside the scientific community.

Jason.
 « Last Edit: September 26, 2010, 07:36:20 AM by Softology » Logged
Nahee_Enterprises
World Renowned
Fractal Senior

Posts: 2250

use email to contact

 « Reply #111 on: September 27, 2010, 02:23:11 PM »

Bugman has defined a triplex polar form using a particular matrix product.
I have extended this idea to give 48 different polar forms.  If you are
interested then please have a look at:
http://soler7.com/Fractals/Matrices%20to%20Triplex.pdf

Here is an image from one of the 48 variations:
//soler7.com/Fractals/3D17/Mandelbulb1585.JPG
Many more are at:  http://soler7.com/Fractals/3D0.html
All were made using Visins of Chaos.

Greetings, and Welcome to this particular Forum !!!

Some nice work you have there.  Looking forward to your future contributions.

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KRAFTWERK
Global Moderator
Fractal Senior

Posts: 1418

Virtual Surreality

 « Reply #112 on: November 23, 2010, 10:45:23 AM »

Here is an image from one of the 48 variations:
<Quoted Image Removed>
Many more are at
http://soler7.com/Fractals/3D0.html
All were made using Visins of Chaos.

Wow, nice image! Very interesting!
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Softology
Conqueror

Posts: 101

 « Reply #113 on: July 23, 2011, 07:05:41 AM »

Here are another 20 new variations.

http://softologyblog.wordpress.com/2011/07/21/new-mandelbulb-variations/

A few sample images...

Jason.
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Softology
Conqueror

Posts: 101

 « Reply #114 on: July 27, 2011, 12:00:11 AM »

27 more new variations
http://softologyblog.wordpress.com/2011/07/27/more-new-mandelbulb-variations/

Jason.
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jehovajah
Global Moderator
Fractal Senior

Posts: 2674

May a trochoid in the void bring you peace

 « Reply #115 on: July 27, 2011, 01:57:16 PM »

Very nice! Paticularly like the plant analogue.
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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!
Softology
Conqueror

Posts: 101

 « Reply #116 on: August 01, 2011, 02:59:45 AM »

And yet another 16 new varieties.
http://softologyblog.wordpress.com/2011/08/01/even-more-new-mandelbulb-variations/

Jason.
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