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Author Topic: 3d Mandelbrot attempt  (Read 8757 times)
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M Benesi
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« on: February 24, 2011, 06:52:50 AM »

  So basically, I dug out an old idea, and apparently found a realllllllly good way to apply it.  

  The thing was, for a long time, I had this gut feeling that it was the sign assignment that was holding us away from fractally goodness for z^2.  I attempted many different methods, but not this one (and this one can be applied to the other formulas as well, one in particular comes to mind, the one with the perfect 2d cross section).  Anyways...

  I did my standard trick- double complex triplex (use complex numbers instead of trig- it runs about 2 times as fast on my old comp), using my favorite formula:  x vs. [y vs. z]   (the other formula that comes to mind is the [x vs. y] vs. z formula).  

  This time, I constrained the x variable to absolute value, but allowed sign variation on the y and z.  So basically, the new x value (before adding in pixel component) is |new x value|.

  Anyways, here are a few images.  As you will see, it has a few different types of forms... well, a lot.  There is variety... weird shapes and the like.  There are seahorse spins.  There are... other things we've caught glimpses of in other attempts to hit the true deal....  Anyways, nothing that great yet, but I wanted to put it out there.  Haven't even tried the other z^n yet.

  Here's a Night Mare:

  Here's a dragon totem thingy:

Here's some twisty stuff:

Here's in the rings around the stalk.  You gotta jack up the iterations and zoom in to find the variety.  I bet it gets weirder at higher iterations.  This is one of the few formulas (besides the one with the towers from a while back) that generates SOLID fractallyness around the stalk area, instead of fuzzy spiky crap).

And here is a branchy area.  I've found these in other formula variations, but I'm not sure if it was as good.

Here is a different branchy area:
« Last Edit: February 26, 2011, 02:24:54 AM by M Benesi » Logged

KRAFTWERK
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« Reply #1 on: February 24, 2011, 09:02:08 AM »

Wowza, a "solid" Power 2!

Do not fully understand what you did but I like the result! Nice images Benesi!

Waiting to see the P8...  afro
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miner49er
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« Reply #2 on: February 24, 2011, 02:53:14 PM »

Looks great and I have not produced anything better but it does still have the 'taffy' areas.
I would like to see the entire object as well.
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M Benesi
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« Reply #3 on: February 25, 2011, 12:48:53 AM »

Nice images Benesi!
Thanks KRAFTWERK.  I'm sure Jesse will implement this when he notices the thread.  cheesy  

 I constrained the new x values (before adding in pixels) to their absolute value (to avoid sign (+/-) variation).  

Waiting to see the P8...  afro
 I messed with it a bit, but the lower powers are so detailed... It seemed to add too much detail..  hrmm... just gotta explore it.  

Looks great and I have not produced anything better but it does still have the 'taffy' areas.
 There are a few areas of cancellation (smooth/taffy).  I tend to lean towards the idea that certain areas (until VERY high iteration) will remain smooth because the one variable (x for example) cancels out the (rotational) effects of the other 2 (y and z).  The idea may be incorrect, but it seems like a logical assumption to me, despite lack of mathematical proof.  
I would like to see the entire object as well.
 I'll post a few views.  It doesn't have the 2d cross section, and although I've made one with the other formula, the [x vs. y] vs. z variety, that does have the cross section, it doesn't have all the cool sections (at least at first glance, I've missed things before, including this particular fractal because I was looking for something else).  

"I'll post a few views."
  Sorry about that.  Moved on to other formulas, as I'm trying to find the one that I like the most.  Rest be assured, the one above is closely related to the burning ship fractal (the name escapes me at the moment, although I know I've seen it before) and it isn't all that special (simply being another type of 3d fractal, albeit a z^2, which I am pursuing at the moment).  The other formulas below rely on something other than absolute value, rather they use sign assignment methods (although absolute value is used for specific parts of several of the equations, rather they don't completely rely upon it so as to... be truer to traditional brots).
« Last Edit: February 26, 2011, 04:49:52 AM by M Benesi » Logged

M Benesi
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« Reply #4 on: February 26, 2011, 02:33:20 AM »

  Alright, I've further refined the formula.  Taking an idea from an older formula (sign assignment with certain dependencies) I applied it to the other part of the formula (instead of applying it to y vs. z, I applied it to x vs. [y & z]).

  There are 3 comparison shots.  I thing this works better than using plain old absolute value, and.. well, the one type has a sphere with fractallyness!!  A z^2 nonetheless...

  These shots are of the "same" location, to demonstrate various methods implemented.  

  These first 2 aren't as "good" in my opinion... They are off a bit, don't have the pure fractallyness of the 3rd. not to mention the third applies the change uniformly to the x and [y & z] components, rather than not applying it to one component, or applying abs to one instead of the dependency sign assignment:


  This one is the kicker.  If you look, you will see a sphere of fractally twists:


  The problem is that the last formula, while it has awesome fractally areas (such as the self twisting sphere above), is not continuous (it has breaks in it- that I've seen with other misapplied sign changes in the past).  So, there is some component that needs to be changed in order for it to work correctly.

  The first image has twisty areas, the kind I like (I used to love to find them in old, complicated 2d formulas).  It also has the advantage of having gaps between the twists, rather than being as solid as the 2nd:

  Same as first type:
« Last Edit: February 28, 2011, 08:28:48 AM by M Benesi » Logged

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« Reply #5 on: February 26, 2011, 02:49:53 AM »

I like that 3rd image!

It almost has little ...  Bulbs!
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M Benesi
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« Reply #6 on: February 26, 2011, 03:37:04 AM »

  I'm quite disappointed that it (the third type) isn't a solid fractal (it has large gaps sharply chopped out of it).  

  I've stumbled upon solutions in the past, however, so maybe an idea will pop into my head.  For now, I am reduced to the absolute value method of the first image, which doesn't have that awesome sphere of fractallyness (which we all expect to see in z^2 3d brots).  

edit:   Or...  This is related to the above, but I switched the signs on the sign assignment (which resulted in a different fractal because I used the abs method for the one assignment):


edit again:  Also, noticed the one variety, with a simple sign assignment (no absolute value!) has interesting areas all over the fractal.  I even found one (an area) similar to the absolute value variety above, but this version has greater variety (at least at first glance...   cheesy).    edit to add:  the part below is from a regular xmas tree variety z^2, which shares parts with the sign assignment variety.  The sign assignment variety has additional areas of fractally goodness (more so than the standard xmas tree variety), but it isn't anything way to special.  
« Last Edit: February 28, 2011, 08:29:32 AM by M Benesi » Logged

M Benesi
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« Reply #7 on: February 28, 2011, 08:26:47 AM »

  All right.  I've done a little bit more work, narrowed down the formula to what I consider to be the best of the bunch.  

  So, I posted a few <images> of this formula in the images forum, will duplicate those here, along with other demo pics of this fractal type.  It has great variety, although it is a single formula.  I've noticed that it has sections that resemble other z^2 attempts I've worked on, but those other attempts generally only had a few areas of interest, and didn't have the variety of "types" of fractal behavior that this single formula possesses (it combines the best parts of several earlier attempts).

  So.. here are some more images (some can be clicked to view a larger image in its own tab, recommended for the first especially).  

  The first 4 are shots of the fractal, rotated around horizontally.  Didn't do the other 2 views... probably should someday...

  The other shots are demos of various areas of the fractal.  


I'll get more up later, it's slow rendering the one area.... lots of empty background, and gotta do a standard check so I don't miss the filaments (you'll see what I mean by filaments).  

  The first one is a "bulb" with bulbs on it.  The next one is a cookie cutter section (where the fractal gets really thin- there are some missed pixels near the edge, but it's just a render to demonstrate the variety of this fractal type).  Above, if you clicked on the first image, you got to see one twisty section of the fractal.  There are different types of twisty sections, chaotic ones, circular ones, threaded ones...



  Last one for the night.  The "radio tower" at the one end... or whatever it ends up getting called.  


  There are a few more sections with slightly different properties... Suppose there is some fractal spelunking in order...  as this is an intricate fractal.
« Last Edit: February 28, 2011, 09:53:00 AM by M Benesi » Logged

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« Reply #8 on: February 28, 2011, 10:56:42 AM »

That last "Tower" image looks really nice!!!
Thank you for keeping up the good work Benesi!  A Beer Cup
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miner49er
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« Reply #9 on: February 28, 2011, 11:12:19 AM »

I love it! Can't wait to start exploring this beauty!

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ker2x
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« Reply #10 on: February 28, 2011, 04:47:40 PM »

Very very good. i love it  shocked
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Jesse
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« Reply #11 on: February 28, 2011, 08:34:14 PM »

Thanks KRAFTWERK.  I'm sure Jesse will implement this when he notices the thread.  cheesy  

Cool stuff, Benesi!

You got a formula like thing for an old(feeling) man to implement it in other program?  wink
« Last Edit: February 28, 2011, 08:59:59 PM by Jesse » Logged
M Benesi
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« Reply #12 on: February 28, 2011, 09:32:57 PM »

  Thanks all!

 @ Jesse-   hahaha... I was waiting for you to ask.  cheesy

  So, basically, we can do it 2 ways.  The formula works for z^2,6,10   (every 4th z), but the best is the z^2 (6 or 10... become a lot like a z^6 regular mandelbulb... and we've seen those a bunch)- which has a very simple write up.

  I'll just cover the z^2 for now, as it avoids trig and complex numbers  (I'll post a trig version after dinner, got a bunch of stuff to do though):
Code:
r1= y^2 + z^2;

if [x < sqrt (r1)  &&  pixelr<0 ]     //   IMPORTANT Forgot to check the the pixel part before..   pixelr is the x axis pixel component
    {new_x = x^2-r1}                //  makes the one side very similar to a xmas tree fractal.. just... maybe better.  I dunno.
else                                       //  try it with and without the pixelr (x axis pixel component) check..   I think it's better with it.. :dink:
    {new_x=  -x^2 + r1}

r1=- r1^-.5 * 2 * |x|;                     //yeah, it's negative r1 to the negative 1/2 times 2  times the absolute value of x (original x, not new_x)

new_y= r1 * (y^2 - z^2);
new_z= r1 * 2 * y * z;

  For coloring, I really like trapping the z component (although the others are interesting as well- the z is the coolest).  I generally just get a magnitude.  At the end of every iteration, after pixels are added in, I grab the z component.  

First iteration:
old_z_component= current_z_component;  //current_z_component is the one you grab at the end of the iteration

After one iteration:
old_z component= sqrt(old_z_component^2 + current_z_component^2);  //I like grabbing the magnitude...

  But regular averaging works as well, giving slightly different color patterns (sign variation in the z component adds a bit of variation).

  I generally grab the z components for all <iterations> except the last 2 (1 to 3) iterations.  If you do all the iterations, it's way more complicated (although you can slow it down with a low multiplier like .0001 or something).  I've been using a pretty standard palette/index method.  

  Here's a cookie cutter with filaments- it looks like it might origami itself off to the one side.. who knows... click to biggify
« Last Edit: March 01, 2011, 10:01:53 AM by M Benesi » Logged

Jesse
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« Reply #13 on: February 28, 2011, 09:52:11 PM »


Many thanks to you, i will try the pow2!

Some coloring options are available, they must work for now.  wink
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M Benesi
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« Reply #14 on: March 01, 2011, 01:38:02 AM »

  UPDATE    Corrected the formulas.  Sorry Jesse, realized we should add an x-pixel component part to the sign check- it really improves the quality of the fractal on the -x axis side, in my opinion.  Might need to look a bit closer.. but ehh....


Cool..  So, the trig version.  Keep in mind, it doesn't make a continuous fractal for powers that aren't z^ (2 +4*n), haven't checked -2,-6... yet... hrmmm... heheheh.  Will do so after this write up.  I didn't check this code, so... just doing it on the fly.  Might forget something as trig versions aren't my preferred method.

Code:
r=(x^2+y^2+z^2) ^ (n/2);
r2=sqrt (y^2+z^2);

theta= atan2 [x + i r2];   //  atan2 is equivalent of arg... I think... or something like that....
phi=    atan2  [y+ i z];

newx= r * cos [theta*n];

if (x>r2  && pixelr>0) {newx=-newx;}              //  you'll notice the > sign is pointed the other way... eliminate an else statement... ehh?
                                    //  [b] IMPORTANT [/b]  Forgot to check the pixel part before..   pixelr is the x axis pixel component
                                   // makes the negative x-axis side.. better in my opinion.
r=-r* |sin [theta*n] |;      // if you want to try something different, you can leave out the absolute value.
                                     //  I don't think the fractal is nearly as cool, but... heh..  it's continuous through all n... I think.
newy= r * cos [phi*n];
newz= r * sin [phi*n];

.... so now add in yur pixel components, or maybe yer julia components, or do whatever.  Enjoy.  

  Now off to try a -2...  hrmm.. boring.  Maybe turn it inside out.  That could work.
« Last Edit: March 01, 2011, 10:04:37 AM by M Benesi » Logged

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