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Author Topic: 3d Fractal Type that works for z^2  (Read 3308 times)
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M Benesi
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« on: February 13, 2010, 05:09:44 AM »

  Hi all.  I posted it in the "Search for the holy grail" thread, but I think it's worth starting an independent thread.  Maybe some of you will start exploring it...  It seems to have great variety within it.  Potential for crazy fractals similar to the variety of 2d Mandelbrots.  At low iterations in the higher n sets (z^n n>6), you find great variety (although it follows the same basic pattern, it just is slightly different, just like 2d mandelbrots).  

  You could find a z^2 image like Gold Top:

 A z^2 like The Fortress of Solitude (this one needs to be rendered at a higher res for full effect):

 or the z^5  Diamond Skull (simple one, and lucky on the clipping) :

  or the z^13 Wampum Predatorum (clipping not lucky this time, needs re-rendered, recolored, relighted... re-done):

  The formula is pretty simple: a rotation of the linear x axis component against the planar y&z component, rotation of the y axis component vs. the x&z planar component, and rotation of the z axis component vs. the x&y planar component.

pixel_x = x component of pixel
pixel_y = y component of pixel
pixel_z = z component of pixel

plane_yz = y&z planar component =  sqrt(pixel_y^2 + pixel_z^2)

plane_xz = x&z planar component  =  sqrt(pixel_x^2 + pixel_z^2)

plane_xy = x&y planar component =   sqrt(pixel_x^2 + pixel_y^2)

 If you desire to enter a seed value, you should only set 3 seed values, and use similar equations to the above to create your other 3 seed value components:

seed_x, seed_y, seed_z  <-- self referential

seed_yz = sqrt (seed_y^2 + seed_z^2)
seed_xz = sqrt (seed_x^2 + seed_z^2)  
seed_xy = sqrt (seed_x^2 + seed_y^2)

We set x,y, and z to their corresponding components, and the planar variables yz, xz, xy to their corresponding components (for Julias, the first set would be the pixel values, for Mandelbrot type the first set would be seed values).

  After our variables are assigned, we do out iteration loop:

Type R2 D2:     //  All Images in thread are this type, which apparently isn't as "fractal" after further tests  
r= sqrt[ (x^2 + y^2 + z^2) ^ n + (yz^2 + xz^2 +xy^2)^n ]
Type D2:
r= (x^2+y^2+z^2+yz^2+xz^2+xy^2)^(n/2)
    Comparison Shots Follow... Maybe not as varied, but definitely more fractal and symmetric

whiskey= atan2 (x + i*yz)    // angle assignment.  The names you use for your variables are generally very important.      
tango =  atan2 (y + i*xz)     // or not.
foxtrot = atan2 (z + i*xy)

x= r * cos (whiskey * n)     // r is already calculated above
y= r * cos (tango * n)
z= r * cos (foxtrot * n)

yz=r* sin (whiskey * n)
xz=r* sin (tango    * n)
xy=r* sin (foxtrot  * n)

  Then add in either the corresponding pixel or seed components, depending on if you are doing Julia type or Mandelbrot type.

For Mandelbrot:                           For Julia:

x=x+pixel_x                                x=x+seed_x
y=y+pixel_y                               y=y+seed_y

dot dot dot dot dot dot........

xy=xy+plane_xy                         xy=xy+seed_xy


After all that rigamorale check your bailout before your next iteration.

 I simply bail on |x| + |y| + |z| < 10^30 so I get a smooooooth fractal, although you do need to do a few more iterations for details.  Which might explain a thing or 2, perhaps I don't need to do 8 whole iterations to get an insanely detailed 7th order fractal... anyways.


  Anyways, there are several other starting magnitudes that I use to generate different fractal types.  Some for pyramidal fractals (that I call "Type D3") some for hybrid fractals (fractals that are a cross between d2 and d3), some for Mandala types (which generate really cool patterns at different angular rotations/magnitude settings, but aren't actually fractal), and some for other types, such as this one which I call R2 D2 because it is the 2 radius (magnitude) formula for D2 that produces the most awesome fractals (R2= 2 magnitude).

  I'm including this set of z^7 images, because I just found a neat location at the "top" (rotated 45 degrees) of the fractal, which might correspond to the bottom of a quadrant of the fractal... or not.  I should check.

  The third image below is a zoom into the alter like thing in the center of the second image (and the second is a zoom into basically somewhat below the center of the first).  The first is I don't know how many iterations, the second is 9, the third and 4th are 10, and the 5th is the same location as the 4th but 11 iterations.  



 Calculating one more small iteration increase at the same location before I switch to the search in z^2 (now that I know cool locations are at the "top and bottom" of 45 degree rotated fractal.

Here is the 12 iteration one:


I will upload a few z^2 images later, been exploring z^7 and my z^2s aren't that interesting as I haven't gone that deep in the z^2s in this new magnitude formula (which appears really cool, like all the other ones).  I'll also attach a text file that can be used in ChaosPro: it is the set of my formulas, including the one that makes this image "Type R2 D2".  I'll get together a help file for it eventually...  To use it the FormulaPack in ChaosPro, you just rename it from "Benesi.cfm.txt" to "benesi.cfm" then copy it into the "compiler" directory in the "formula" directory of your chaospro directory (which is in your program files directory, or wherever you installed it).  

  It has a bunch of my formulas, and I am working on higher dimensional extensions for them that will be included later.  Already have a 5d that works, but decided 6d would be better (and haven't totally finished the 5d, so it is not released with these).  I decided 6d would be better because I like to set the "time" dimensions to vary with the space dimensions (a little bit, perhaps have arbitrary offset (or starting) values, then vary them by (space dimension)/(some value), one for each space dimension: thus 6 dimensions total).  

* Benesi Fractals.cfm.txt (26.86 KB - downloaded 226 times.)
« Last Edit: February 14, 2010, 08:26:52 AM by M Benesi » Logged

M Benesi
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« Reply #1 on: February 13, 2010, 07:18:54 AM »

Here are a couple of z^2 images.  I might have found the 3d Mandelbrot "seahorse valley".  Lots of variety in a little location: only 22 iterations.

First image is the boring hole set:


Second is Frowning Man Peak


Third is a zoom in above Frowning Man Cave (no more iterations necessary):


Fourth is a zoom into Frowning Man Cave, one branch of the Mandelbrot Caverns:


  More on the way... eventually.  I decided not to correct the spelling in this... for whatever reasons. 
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M Benesi
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« Reply #2 on: February 14, 2010, 02:56:27 AM »

These are in corresponding locations of the fractals.  The types have very similar structures, similar replication, but different overall feel and complexity.  Perhaps one of them is "less" fractal, or at least more chaotic (R2D2).

  So which one is the "correct" or "better" formula?  I really don't know.  R2D2 generates very dirty- organic looking images.  I haven't explored D2 as much, but it makes more ordered (like Mandelbulbs) images, although it has variety as well.  

  Here are a couple shots of z^8:

First is d2 (1 more iteration, forget if it was 13 vs.12 or something else...):

Second is R2D2:


Here are a couple shots of z^2, both 18 iterations:

First is d2:                                                                 Second is R2D2:


More z^2, found a good location to compare, although the second 2 shots (not the first 2) are mismatched: same vicinity, different zoom in..

First is D2, 19 iterations:

second is R2 D2 19 iterations:


Second set, remember, I think I screwed up matching the location, so I think THIS IS NOT MATCHED UP CORRECTLY

First is D2, 22 iterations:


Second is R2D2, 22 iterations (I like this one, very dendritic appearing, I like how it is all connected, not that the D2 isn't but..):


The next one is an UNMATCHED 27 iteration D2... just a screw up, but here it is anyways:


This is a 30 iteration D2 to match the second image, a 26 iteration R2D2 (might be off a bit... or not):


I went down in the hole of the 30 iteration D2 above, and either took a right or a left after hitting the bottom, then cranked the iterations up to 36 and zoomed towards the end of the canyon a bit (it was pretty boring prior to the increase in iterations)... maybe still boring:
« Last Edit: February 14, 2010, 06:46:35 AM by M Benesi » Logged

jehovajah
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« Reply #3 on: February 14, 2010, 08:26:10 AM »

I like your guided tours! however the colour palette is flat, so does that make it boring? Great to see that the detail is there in the z^2.
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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!
M Benesi
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« Reply #4 on: February 14, 2010, 08:30:30 AM »

  Thanks jehovajah,

  I'll start messing with colors a bit more.  Thinking of some coloring algorithms.  For now, colored lighting is nice, lit up the top 2 images that I just added to the first post in the thread.

  matt
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kram1032
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« Reply #5 on: February 14, 2010, 10:45:57 AM »

A lot of really nice images cheesy

R2D2, despite being more chaotic, looks better in most cases for me smiley
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M Benesi
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« Reply #6 on: February 14, 2010, 09:10:58 PM »

  I felt the same way, until I started exploring the D2s a bit more yesterday, and now I'm just not sure (although perhaps the R2D2s will prove to have far greater variety when zoomed into, but I just don't know at this time: in addition it's hard for me to pick one over the other, because both have their "good points").  I suppose we need to check the R2D2 "Gold Crown" and "Fortress of Solitude" (those were D2s at the top of the thread).


  The "Wampum Predatorum" z^13 image is an R2D2, and arguably, if rendered correctly would produce an AWESOME fractal (perhaps a nice 1280x800 for my desktop, with colored (RGB) lighting, drab or multicolored object).  I suppose it's time to find the corresponding D2 location, although that might be a bit of work (hoping the zoom out is "self explanatory" so that I can match zoom ins to the same location in both fractals).

  Well, off to do some stuff... after I look for matching locations on the Wampum....

  R2D2 "Wampum" looks better, and less messy even though it is one more iteration than the D2 fractal: this is the opposite of what was happening for the z^2s earlier.
  r2D2 10 iterations:

For this D2, had to lower iterations to 9 or it had too many fine details (got fuzzy):


  I checked out the little "bud" right above the water like area in the fractals, an image of the R2D2:

and of the D2's bud (one less iteration, once again):
« Last Edit: February 15, 2010, 11:01:00 PM by M Benesi » Logged

jehovajah
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« Reply #7 on: February 17, 2010, 06:44:20 AM »

Whiskey foxtrot tango! Realised a few days ago you are using 3 angle reference scheme i was outlining in other thread! So to me this shows that the problem with twinbees formula is that is has some rotational elements missing. Did you correct that in your original post in the other thread, the really flat mandy?

http://www.fractalforums.com/3d-fractal-generation/truerer-true-3d-mandelbrot-fractal-(search-for-the-holy-grail-continues)/msg12084/#msg12084
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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!
M Benesi
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« Reply #8 on: February 17, 2010, 08:25:47 AM »

  No.  I think that really flat one was completely off.  It's possible that your (just edited today?) post prior to my "really flat mandy" post triggered the thoughts that lead to the next formula in the thread (which is a form of this one).  The terminology you use in that post is not familiar to me, so I don't completely understand it.  I wouldn't hesitate to credit you with the idea if I thought you both wanted and deserved credit.  As it is, I am not entirely sure what triggered it: a lifetime of experiences?  I'll tell you my line of thinking:  

  I remember thinking (about the Mandelbulb formulas): They are rotating x and y around eachother, and z around x and y: this is unbalanced.  That's the first thing that stands out when looking at the original formulas:  they are not balanced correctly!  How can you rotate x vs y, then z vs both of them?  I thought that these were more like "double 2d" fractals, rather than true 3d fractals in which all variables influence others equally.

  So I came up with the formula to balance the system:  if you have 3 variables, you have to rotate each individual around the other 2 to maintain balance.  I thought it wasn't a fractal formula for a bit, just a cool (non-fractal) mandala generator with r=sqrt(x^2+y^2+z^2).  I was going to abandon it, messed around with a couple of other formulas, but something made me go back to it.

  I remember thinking (maybe on re-reading one of your posts, which have helped me along, even if I don't understand some of the terminology you use) that balancing variable interactions has to work: this has to be the key to true 3d fractals.  It also could have been rereading the "Is this... the real 3d Mandelbrot then?" on Daniel White's web page.

  Here are a few more z^2:

First is a 20 iteration image of the 3x120 degree symmetric quadrant (I've got ideas about this symmetry):


Here is a 31 iteration zoom in off to the top side of the symmetry (went fairly "deep" in some hills to the side of the triple symmetry):


Here is a 36 iteration zoom into the cave under the front hill in the above image (looks like a binded demon):

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jehovajah
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« Reply #9 on: February 17, 2010, 10:51:06 AM »

Yes the next formula you announce uses phi tango theta. Your train of thought is yours and your formulation is yours so i am not seeking your credit or even any for myself. In fact strictly speaking i only propose a three angle system way after you have been using it for 2 pages of posts! Any way my strange language is an attempt to give me a headache so i don't go assuming things which ain't necessarily so  cheesy. Take 3d space for example you are referencing it with a so called non quaternionic 4d vector, except as it turns out quaternions are everywhere, its 3d space that is a misnomer, hence i use geometrical space.

Anyway enough of my language, Well done. See if you can rewrite twinbees formulation using your 3 angles without the symmetry.
« Last Edit: February 20, 2010, 02:47:39 AM by jehovajah » Logged

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!
M Benesi
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Posts: 1075



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« Reply #10 on: February 17, 2010, 09:42:23 PM »

  I'm not entirely sure what you mean by "rewrite twinbees formulation using your 3 angles without the symmetry." 

  I believe Twinbee's formula was of the standard type with one variable against the other 2, and those 2 against one another.  It's really a different type of formula: one which (sort of) plays 2 2d Mandelbrot sets off one another, instead of generating a single 3d Mandelbrot set as my formula does.

  In any of the original Mandelbulbs, you have a situation in which:

  1. you calculate x vs. y like it is a 2d Mandelbrot iteration
  2. you calculate z vs. the magnitude of x&y like it is a 2d Mandelbrot iteration
  3. you add in the x,y,z values (which changes the magnitude of x&y and has other effects on the outcome)

  So this is going to give us a "complex intersection" of the 2 2d Mandelbrot sets, which happens to be our Mandelbulbs.

  However, I don't think that the "true 3d" Mandelbrot is one in which all 3 variables lack equal influence upon one another: the original 2d Mandelbrot was 1 variable vs. another: a balanced system.  I think the system I am using in this thread is balanced, perhaps not perfectly yet (I've still got some work to do over in the triplex algebra thread I started), so is a better candidate for "true 3d" Mandelbrot status. 

  Like I've thought (and probably said) in the past:  a system in which 3 variables influence one another is not going to follow or generate the same basic pattern that a system with 2 variables influencing one another follows or generates.  It's going to generate something different (like these fractals), likely to be more complex and varied than its 2 dimensional analog. 

 
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