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Author Topic: Summary of 3D Mandelbrot Set Formulas  (Read 17400 times)
Description: Summary of many different 3D Mandelbrot set formulas
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bugman
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« on: November 25, 2009, 11:39:46 PM »

Here is a summary of some different 3D Mandelbrot set formulas for your convience. I have tried to include a second order and 8th order rendering for each formula presented here. Additional formulas are welcome.


* Definitions.gif (3.62 KB, 491x175 - viewed 6221 times.)

* 01. White.jpg (137.84 KB, 844x304 - viewed 2924 times.)

* 02. WhiteZ.jpg (127.51 KB, 844x304 - viewed 2144 times.)

* 03. Cosine.jpg (141.62 KB, 844x304 - viewed 1785 times.)
« Last Edit: December 18, 2009, 08:53:31 AM by bugman » Logged
bugman
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« Reply #1 on: November 25, 2009, 11:40:36 PM »

.


* 04. Directional.jpg (128.85 KB, 844x304 - viewed 1704 times.)

* 05. Tricorn.jpg (119.5 KB, 844x304 - viewed 1641 times.)

* 06. Rucker.jpg (112.77 KB, 844x304 - viewed 1670 times.)

* 07. Reversed.jpg (128.38 KB, 844x304 - viewed 1556 times.)
« Last Edit: December 18, 2009, 08:54:34 AM by bugman » Logged
bugman
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« Reply #2 on: November 25, 2009, 11:41:02 PM »

.


* 08. ReversedZ.jpg (115.41 KB, 844x304 - viewed 1545 times.)

* 09. 2Stage.jpg (117.83 KB, 844x304 - viewed 1524 times.)

* 10. 2StageZ.jpg (89.16 KB, 844x304 - viewed 1472 times.)

* 11. 8Stage.jpg (104.53 KB, 844x304 - viewed 1521 times.)
« Last Edit: December 18, 2009, 08:55:36 AM by bugman » Logged
bugman
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« Reply #3 on: November 25, 2009, 11:41:18 PM »

.


* 13. Combined.jpg (113.45 KB, 844x304 - viewed 1611 times.)
« Last Edit: December 18, 2009, 08:56:03 AM by bugman » Logged
David Makin
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« Reply #4 on: November 26, 2009, 04:34:07 AM »

Hi Paul, you might want to try normalising that last one to 1 before multiplying by the magnitude - if the magnitude of the trig is zero then use (1,0,0) multiplied by the magnitude.
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ZsquaredplusC
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« Reply #5 on: November 26, 2009, 05:56:37 AM »

Thanks for all these.  That is the sort of summary I requested back in the original thread.  This will really help others looking for formula info.
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dougfractal
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« Reply #6 on: November 27, 2009, 01:22:56 AM »

This is the method that I used to create my 3D versions

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TedWalther
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« Reply #7 on: November 27, 2009, 03:17:03 AM »

This is the method that I used to create my 3D versions

<Quoted Image Removed>

Doug, I don't know enough math to follow that; could you turn it into a formula so I can see how new x,y,z values are created?

Ted
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dougfractal
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« Reply #8 on: November 27, 2009, 03:31:22 AM »

Here's a link to that question that I've all ready put into a post

http://www.fractalforums.com/theory/has-anyone-tried-this-formula/msg9198/#msg9198
« Last Edit: November 27, 2009, 03:43:15 AM by David Makin » Logged
Tglad
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« Reply #9 on: November 27, 2009, 10:43:07 AM »



These top two aren't a different formula, they're using the second method down in the posted list, but I don't think even powers are appropriate for this type as it makes the iteration discontinuous in space. Power 3 is my favourite, it would be interesting to get some more detailed shots of this. I'm fairly sure it is fully connected, and shows a lot of variety.


These bottom two from the third formula down in the table. The iteration should be continuous in space at odd and even powers, its certainly a bizarre and interesting power 2 shape.


* mandelcousinPower3.jpg (78.56 KB, 634x635 - viewed 1459 times.)

* mandelcousinPower3top.jpg (120.21 KB, 624x622 - viewed 1445 times.)

* mandelPower2.jpg (80.67 KB, 638x627 - viewed 1444 times.)

* mandelPower2side.jpg (64.78 KB, 617x619 - viewed 1462 times.)
« Last Edit: November 27, 2009, 10:49:34 AM by Tglad, Reason: not reading thread properly » Logged

BradC
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« Reply #10 on: December 18, 2009, 04:21:49 AM »

Possible typo in the definitions at the top of this thread (image http://www.fractalforums.com/theory/summary-of-3d-mandelbrot-set-formulas/?action=dlattach;attach=524;image): r is defined as the nth power of the original vector length, and then the definition of phi uses r in the denominator. I think this value in the denominator should actually be the original vector length, not its nth power.
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bugman
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« Reply #11 on: December 18, 2009, 08:56:58 AM »

Thanks for pointing that out. I made the correction.
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cbuchner1
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« Reply #12 on: December 18, 2009, 03:39:02 PM »

I am getting confused as to why I cannot reproduce the exact same shapes of 3D mandelbulbs when I plug above formulas into a voxel generator. I do apply the exact same iterative formulas in a spherical coordinates, yet the voxel representations of the bulbs look distinctly different (especially apparent for power 2).

I am beginning to think that maybe a 3D raymarching approach based on Hubbard Duady does not really give the "correct" result. Could anyone verify this hypothesis or prove it wrong?
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jehovajah
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« Reply #13 on: December 19, 2009, 03:22:24 PM »

Couple more formulae for you paul.

(x,y,z)2 = (x^2 - y^2 - z^2 - 2yz, 2xy, 2xz)  left hand version.

(x,y,z)2 = (x^2 - y^2 - z^2 + 2yz, 2xy, 2xz)  right hand version.

Do you want the correct vx versions as well?
« Last Edit: December 19, 2009, 03:45:05 PM by jehovajah » Logged

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M Benesi
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« Reply #14 on: February 06, 2010, 09:54:09 PM »

  I've added 4 new formulas, starting in post 114 of this thread:
http://www.fractalforums.com/3d-fractal-generation/truerer-true-3d-mandelbrot-fractal-(search-for-the-holy-grail-continues)/msg12604/#msg12604

  They definitely don't strictly stick to the accepted Mandelbulb format, but they are all 3d fractals (not simply "regions inside level sets for 3d fractals" (quoted from an unnamed source, haven't asked permission to use their name)  like the previous formulas I posted in the thread).

  Images are in the thread.  As it is, some of them are pretty cool and have really neat fractal symmetry (I like Type B (not Mandelbulb B) the most).  Zooming into the point (front) of type B higher order Z^8++ is pretty fun.

  If you think they are appropriate for this thread, go ahead an import them.  May have a mod move them over here and put a link to this thread from that one.... or whatever.

  matt
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