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Author Topic: True 3D mandelbrot type fractal  (Read 262809 times)
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David Makin
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« Reply #360 on: November 03, 2009, 12:56:44 AM »

Hi all, you may remember I had a 3D based suggestion for a "true 3D" Mandy using the following:

 *  |    r    i    j
-----------------
  r  |    r    i    j
  i  |    i   -r  -j
  j  |   j   -j   -r

Which gives a square of (x,y,z):

new x = x^2 - y^2 - z^2
new y = 2*x*y
new z = 2*z*(x-y)

I finally got around to investigating further and realised why no-one had commented on it, here's a rotation of the top of the Mandy:

http://www.fractalgallery.co.uk/FlatMandy.mov

However it wasn't a dead loss because here's the main minibrot:

<a href="http://www.youtube.com/v/rIMqmDR30ks&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/rIMqmDR30ks&rel=1&fs=1&hd=1</a>
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cKleinhuis
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« Reply #361 on: November 03, 2009, 03:38:20 AM »

i rendered an animation with your formula dave, 1000 frames of julia pertubation, rendered in ~35 hours on a quad core 2,2 ghz in uf

<a href="http://www.youtube.com/v/ZNIQ50SuIMM&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/ZNIQ50SuIMM&rel=1&fs=1&hd=1</a>

 afro afro afro
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« Reply #362 on: November 03, 2009, 06:22:44 AM »

Hi

I wrote completely new program for rendering 3D fractals. I resigned with rendering slices and fixed grid. Now fractal is calculated directly in 3D space using scanline algorithm. DE is not implemented yet but for accurate searching of fractal boundary I use binary search algorithm. I included all shading algorithms which I used in old program: global illumination, hard shadows and normal vector calculation (angle of incidence of light). Without volumetric fog (opacity proportional to number of iterations) rendering is much more faster than in old program and quality is higher. Details are sharper because I didn't have to use any interpolation algorithms. This image was rendered 25 minutes in 2560x2560 resolution and max. 20 iterations (rendered on Intel Core 2 Duo Quad 8200)


Wow, 24 minutes without any distance estimation? That's quite impressive for such quality.
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twinbee
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« Reply #363 on: November 05, 2009, 12:58:46 PM »

The article I'm working on is almost finished. I want to leave some things as a surprise, but couldn't resist these.....  grin Hope you enjoy, the second Cave piece is available at aprx. 4000x2000 resolution here (downscaled from 8000x4000!).

"Ice Cream From Uranus"




"Cave of Lost Secrets"
« Last Edit: November 14, 2009, 09:31:13 AM by twinbee » Logged
jehovajah
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« Reply #364 on: November 05, 2009, 01:37:22 PM »

Hi all, you may remember I had a 3D based suggestion for a "true 3D" Mandy using the following:

 *  |    r    i    j
-----------------
  r  |    r    i    j
  i  |    i   -r  -j
  j  |   j   -j   -r

Which gives a square of (x,y,z):

new x = x^2 - y^2 - z^2
new y = 2*x*y
new z = 2*z*(x-y)

I finally got around to investigating further and realised why no-one had commented on it, here's a rotation of the top of the Mandy:

http://www.fractalgallery.co.uk/FlatMandy.mov




I would like to see the results of the table you discarded as uninteresting where j x i transforms to -i. If you have time could you render that please.

« Last Edit: November 05, 2009, 02:24:06 PM by jehovajah, Reason: remove you tube ref. » Logged

May a trochoid of 去逸 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!
jehovajah
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« Reply #365 on: November 05, 2009, 03:18:11 PM »

  : r   i   j 
...................
r : rr ri rj
i :  ir  ii  ij
j :  jr  ji  jj
.......................



Define rr as r2   , ii as i2   = -1 , jj as j2  = -1 all of which are real values.

Define ri  =  ir  which is an i value  and rj = jr  which is a j value.

In the case of commutativity  ij = ji

so define ij = -i or ij = -j   

in the case where the operators are non commutative ij / ji

define ij = -j and ji = -i

or    ij = -i  and  ji=  -j   

If you have explored all these alternatives i would love to see the results.
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May a trochoid of 去逸 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!
bugman
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« Reply #366 on: November 05, 2009, 05:26:10 PM »

The article I'm working on is almost finished. I want to leave some things as a surprise, but couldn't resist these.....  grin Hope you enjoy, the second Cave piece is available at aprx. 4000x2000 resolution here (downscaled from 8000x4000!).

Twinbee, your deep zoom renderings are still my favorite. I suspect that your "Ice Cream From Uranus" rendering is hiding some beautiful spirals in it, but we cannot see them because the iteration depth is not deep enough. I wonder if it is possible to make iteration depth a function of the cumulative derivative or Cauchy method in such a way that we could render the fine detail in the spirals without going into too much detail in other regions.
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jehovajah
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« Reply #367 on: November 05, 2009, 05:42:13 PM »

Just in passing  ; it occurs that using the non commutative forms model left handed and righr handed forms in 3D so called. Combining the forms in this way may produce relevant results to natural forms.
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May a trochoid of 去逸 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!
David Makin
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« Reply #368 on: November 06, 2009, 02:18:52 AM »

Twinbee, your deep zoom renderings are still my favorite. I suspect that your "Ice Cream From Uranus" rendering is hiding some beautiful spirals in it, but we cannot see them because the iteration depth is not deep enough. I wonder if it is possible to make iteration depth a function of the cumulative derivative or Cauchy method in such a way that we could render the fine detail in the spirals without going into too much detail in other regions.

Just solid based solely on distance estimate threshold would probably achieve the desired result in terms of detail i.e. use distance estimation with maxiter set higher than is ever used in the distance estimation.
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The meaning and purpose of life is to give life purpose and meaning.

http://www.fractalgallery.co.uk/
"Makin' Magic Music" on Jango
twinbee
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« Reply #369 on: November 06, 2009, 04:05:32 AM »

Spirals!! Oh if only... y'see this exactly why I don't think we've found nearly anything like the real McCoy (tm) yet. So far, I haven't seen anything really very spiral yet, although yes, the "Ice Cream from Uranus" pic does indeed hint that spirals may exist.

I'll see if I can get methods (such as Cauchy) working with the 2D mandelbrot first before I attempt it with the 3D bulb. Any particularly good links with example pictures?

In the meantime, I'll render the Icecream picture with higher resolution and more iterations. And you never know;  that curl may itself develop into a fully blown spiral. If so we're in for a treat I'm sure...
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twinbee
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« Reply #370 on: November 06, 2009, 10:54:04 PM »

Right here we go. At this level of magnification (150x150 to 300x300), and with this area of the brot, things are rendering really slow. But here are some results. For full size, save the pic, and reopen in an image viewer because the forum has shrunk it a bit.



Any more iterations, and the spiral is 'covered up' by that overhanging subspiral to the right of the main spiral's eye. Another great thing which I was hoping for; notice how the subpsirals are branching off in oblique directions!

And yes, I will be rendering high resolution of 40 iterations or so. Even though the main spiral will be covered up, one of the subspirals is surely bound not to be.

I should have been more optimistic about the potential of spirals. Thanks to Paul for pushing me to try this!
« Last Edit: November 14, 2009, 09:33:20 AM by twinbee » Logged
xenodreambuie
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« Reply #371 on: November 07, 2009, 09:14:34 AM »

Beautiful renders, Dan. It seems to me that in 2D the relative size of spirals for z^n+c is about inversely proportional to abs(power)-1. For the triplex the same is true in certain planes but the spiral structures get obscured by the 3D structures, especially in powers greater than about 3. In that case, taking a slice should show spirals with the right parameters. Negative powers tend to have larger spirals and also invert them towards the outside, so power -2 should be best. It turns out that the standard (sine) triplex handles these spirals surprisingly well. Here is a sample of power -2, although the straight on view doesn't show how much depth there is.


The flipped root cosine triplex also has some good regions for power -2, but it takes some searching.

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http://xenodream.com
jehovajah
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« Reply #372 on: November 08, 2009, 01:45:47 AM »

Hope the exam went okay. Here's the way to create the object. I'm not sure how Cayley tables would fit in this context, but here's the multiplication function:

double pi=3.14159265;
double r    = sqrt(x*x + y*y + z*z );
double yang = atan2(sqrt(x*x + y*y) , z  )  ;
double zang = atan2(y , x);
newx = (r*r) * sin( yang*2 + 0.5*pi ) * cos(zang*2 +pi);
newy = (r*r) * sin( yang*2 + 0.5*pi ) * sin(zang*2 +pi);
newz = (r*r) * cos( yang*2 + 0.5*pi );

Since in the standard Mandelbrot forumla, the only multiplication is the number multiplied by itself (the square), that's why there's only x, y and z in the above function, and not x2, y2, and z2 aswell. Hope that makes sense.

The addition function for these 3D numbers is as expected:
newx = x1 + x2;
newy = y1 + y2;
newz = z1 + z2;

Do you mind explaining what you mean here? The method you have arrived at seems to be the right stuff but it does not seem to follow from any standard math. Karl rogers called it doubling you call it 3d numbers and in both cases the description is non standard. This is good in my book as the whole area of complex numbers is not rigorous enough to determine what we are constructing.

The simple question is : when you calculate newx etc you put in an extra r factor. Why is this?
Quote
The main Mandelbrot formula stays the same ("a" and "point" are 3D numbers of course - X, Y and Z):
add(    multiply(a,a)   , point).

Now what do you mean by this? What specifically is the a? What form do these 3d numbers have? Then following that what definition of multiplication are you using for these 3d numbers?

If Karl has discussed this with you before then i do not mind if he wants to email me his points. However it is not clear whether his "doubling" and your 3d multiplication are one and the same or not.

I am about to post some analysis for David's alternative method but just wondered if your spherical coordinate approach had not already covered the alternatives more elegantly.

Whatever i think that your method has illustrated the point that calling these operators numbers is not helpful, at least not as helpful as vector /tensor type denotation.  Thanks for your time and maybe i need to congratulate you on your fantastic discovery of the 3d mandelbrot method. There is just one thing that needs to be clarified and that is that you have not found a 3d juliabrot that has the mandelbrot form

http://www.fractalforums.com/mandelbrot-and-julia-set/julia-set-that-looks-like-a-mandelbrot-set/

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May a trochoid of 去逸 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!
twinbee
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« Reply #373 on: November 08, 2009, 07:53:44 AM »

Hi Jehova,

Quote
The simple question is : when you calculate newx etc you put in an extra r factor. Why is this?

Oh right, it's for the same reason that one would calculate the radius squared in the standard 2D mandelbrot. Instead of approaching/repelling way from the centre of a circle though, it'll do the same for a sphere.

Paul Nylander printed a neater representation of the formula, which you might find clearer:

{x,y,z}^n = r^n { cos(n*theta)cos(n*phi)  ,  sin(n*theta)cos(n*phi)  ,  -sin(n*phi) }
where r = sqrt(x^2+y^2+z^2),   theta = atan2(y,x),   phi = atan2(z,sqrt(x^2+y^2))

So that's the equivalent of the squaring part of the 2D mandelbrot formula (except of course it's for 3D, and for a higher power).

Also, to get a better idea, see my old tuturial found here.


Quote
Quote
The main Mandelbrot formula stays the same ("a" and "point" are 3D numbers of course - X, Y and Z):
add(    multiply(a,a)   , point).

Now what do you mean by this? What specifically is the a? What form do these 3d numbers have?

I should have used just one a, and put 'square' instead of 'multiply' really. I was just trying to be more general to account for any 'triplex' multiplication.
Let's rewrite that:

add( square(a), point)

Okay so that corresponds to the original 2D mandelbrot. Using more standard math we can say:
a -> a^2 + point

...which of course is the same as Wikipedia's definition:
z -> z^2 + c

(I of course use 'a' instead of 'z', and 'point' instead of 'c').

Bear in mind of course that you replace the square function for higher powers if you wanted them.

Hope that clears things up a bit.

Quote
If Karl has discussed this with you before then i do not mind if he wants to email me his points. However it is not clear whether his "doubling" and your 3d multiplication are one and the same or not.

I think so yes. He made a quicker version which cuts out the time consuming trig stuff, but other than it's the same.

Garth, nice and unusual spirally renders - I'll have to play around with power -2 as well...
« Last Edit: November 08, 2009, 11:24:45 AM by twinbee » Logged
jehovajah
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« Reply #374 on: November 08, 2009, 07:54:27 AM »

Hi all, you may remember I had a 3D based suggestion for a "true 3D" Mandy using the following:

 *  |    r    i    j
-----------------
  r  |    r    i    j
  i  |    i   -r  -j
  j  |   j   -j   -r

Which gives a square of (x,y,z):

new x = x^2 - y^2 - z^2
new y = 2*x*y
new z = 2*z*(x-y)


I attempt to be explicit and thorough.

 *  x       iy     jz     -iy    -jz
------------------------------------
x:     x2   xiy   xjz   -xiy   -xjz
iy:   iyx   -y2   iyjz  -iyiy  -iyjz
jz:    jzx   jziy   -z2  -jziy   -jzjz
-iy:   -iyx  -iyiy -iyjz  -y2   iy(-jz)
-jz:   -jzx  -jziy -jzjz  -jz(-iy)  -z2


The rules are:
     i2 = j2 = (-j)2 = (-i)2 = -1
  
     -ii = -jj = +1

The manipulations are:

 yix = yxi = xyi = xiy = ixy = iyx

 zjx = zxj = xzj = xjz = jxz = jzx

-yix = -yxi = -xyi = -xiy = -ixy = -iyx

-zjx = -zxj = -xzj = -xjz = -jxz = -jzx


double operator manipulations non commutative :

ziyj =zyij = yzij = yizj = iyzj = [iyjz = -iy(-jz)] = ijyz = ijzy= izjy

+y2 = -y2i2 = -yiyi = -iy2i = [-iyiy = iy(-iy)] = -i2y2 = +y2


-ziyj = -zyij = -yzij = -iyzj = [-iyjz = iy(-jz)] = -ijyz = -ijzy = -izjy

yjzi = yzji = zyji = zjyi = jzyi = [jziy = -jz(-iy)] = jizy = jiyz = jyiz

-yjzi = -yzji = -zyji = -zjyi = -jzyi = [-jziy = jz(-iy)] = -jizy = -jiyz = -jyiz

+z2 = -z2j2 = -zjzj = -jz2j = [-jzjz = jz(-jz)] = -j2z2 = +z2


summary of main manipulations:

yzij  = ijyz        (強z)2 = -z2      yxi = iyx
-yzij = -ijyz      (張y)2 = -y2     zxj = jzx
yzji  = jiyx     -(iy)2 = +y2     -yxi = -iyx
yzji  = jiyz     -(jz)2  = +z2     -zxj = -jzx
-yzji   = -jiyz


So (x + iy +jz)2 gives the following parts

x2 + (iy)2 + (jz)2


i(xy + yx)

j(xz + zx)

yz(ij + ji)

So for the function the last term which i am calling the handedness term is decided by the programmer.

For the geometrical space mandelbrot iteration

newx =x2 - y2 - z2

newy = 2xy

newz = 2xz

handedness term affects either x or y or z or 2 out of the 3 and has magnitude yz so:
if ij / ji

newyz =y.z.


for the handedness term
 if ij = ji
newyz = 2yz

I will suggest in another post how the handedness term might be applied but i think there is enough here for you to play with.
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May a trochoid of 去逸 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!
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