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Author Topic: True 3D mandelbrot fractal (search for the holy grail continues)  (Read 47608 times)
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illi
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« on: November 20, 2009, 03:59:12 PM »

Hi all.
I followed the original thread since before the /. linking, and first of all you guys are amazing.

So since the original thread is now locked I'm making a new one to continue the quest for the true 3D/4D equivalent of the mandelbrot set.

The "perfect" z^2+c 3D Mandelbrot would be made up of a single cardioid (a simple rotation of the 2D one around the real axis) then many near-spherical objects (bulbs) analagous to the near circles in the 2D version. For the circles in the 2D version that are centred on the real axis then the analogous bulbs would again simply be a rotation of the near-circle around the real axis. For the other near-circles that make up the rest of the inside i.e. those that exist symmetrically above and below the real axis then there would be analagous bulbs in the appropriate positions not only with respect to the i axis but also with respect to the j axis (or in the appropriate rotated positions around parent bulbs in both i and j orientations).
The attractors of the orbits in the bulbs would have the same period as the attractors of the orbits in the analogous circles[...]


>This isn't a mandelbulb discussion<
« Last Edit: November 23, 2009, 01:24:54 PM by illi » Logged
illi
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« Reply #1 on: November 25, 2009, 12:13:44 PM »

I guess no one wants to continue this?
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« Reply #2 on: November 25, 2009, 12:21:46 PM »

I guess no one wants to continue this?

When I have time I'll post all the links to the other objects discovered while we were searching smiley
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twinbee
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« Reply #3 on: November 25, 2009, 01:30:32 PM »

Welcome to the forum. Take a look in the Mandelbulb forum - there are quite a few discussions there to find the real McCoy.

Aka, the Mandelisk wink
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« Reply #4 on: November 27, 2009, 12:07:59 AM »

Most attempts so far are displayed on Paul Nylanders pages including links to their origins and various renders:

http://www.bugman123.com/Hypercomplex
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« Reply #5 on: November 28, 2009, 01:07:19 AM »

I guess this goes here, though the idea is based on the Mandelbulb method, just using different angles:

UF iteration code (magn precalculated):

            magn = sqrt(magn)^@mpwr
            th = @mpwr*atan2(zri)                       ;az
            ph = @mpwr*atan2(imag(zri) + flip(zj))  ;ax
            sz = sin(th)
            zri = magn*(cos(th) + flip(cos(ph)*sz)) + cri
            zj = cj - magn*sz*sin(ph)
            magn = |zri| + sqr(zj)

Reslts in these:

z^2+c:



z^8+c:


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« Reply #6 on: November 28, 2009, 11:25:08 AM »

Hi all, I think I found *it*, or at least maybe at least a better bet than the Mandelbulb, first animation (z^2+c):

<a href="http://www.youtube.com/v/9Oy4VLzndto&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/9Oy4VLzndto&rel=1&fs=1&hd=1</a>

More later....
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« Reply #7 on: November 28, 2009, 03:39:35 PM »

Hi all, I think I found *it*, or at least maybe at least a better bet than the Mandelbulb, first animation (z^2+c):
More later....

OK, maybe there's still something better to find, this version has "correct" cross-sections in both XZ and XY planes for even powers>=2 but the odd ones are slightly off.
Anyway here's the iteration loop calculation as programmed in UF (all variables real except zri and cri which are complex, magn precalculated on loop entry):

            magn = sqrt(magn)^@mpwr
            th = @mpwr*atan2(zri)                      ;az
            ph = @mpwr*atan2(imag(zri) + flip(zj)) ;ax
            r = @mpwr*atan2(real(zri) + flip(zj))    ;ay
            sx = sin(ph)
            sy = sin(r)
            cy = cos(r)
            sz = sin(th)
            cz = cos(th)
            zri = magn*(cz*cy - sx*sy*sz + flip(cos(ph)*sz)) + cri
            zj = magn*(sx*cy*sz + sy*cz) + cj
            magn = |zri| + sqr(zj)

Note that |zri|+sqr(zj) is the square of the magnitude.

Edit: Apologies I overestimated how good the 2D cross-sections where - I thought some deviations from the "correct" outlines where errors in my clipping routine but it turns out that the cross-sections are close to being correct, but slightly disturbed in one way or another smiley


« Last Edit: November 29, 2009, 01:49:49 PM by David Makin » Logged

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« Reply #8 on: November 28, 2009, 11:29:05 PM »

Hi all, I think I found *it*, or at least maybe at least a better bet than the Mandelbulb, first animation (z^2+c):
More later....

OK, maybe there's still something better to find, this version has "correct" cross-sections in both XZ and XY planes for even powers>=2 but the odd ones are slightly off.
Anyway here's the iteration loop calculation as programmed in UF (all variables real except zri and cri which are complex, magn precalculated on loop entry):

            magn = sqrt(magn)^@mpwr
            th = @mpwr*atan2(zri)                      ;az
            ph = @mpwr*atan2(imag(zri) + flip(zj)) ;ax
            r = @mpwr*atan2(real(zri) + flip(zj))    ;ay
            sx = sin(ph)
            sy = sin(r)
            cy = cos(r)
            sz = sin(th)
            cz = cos(th)
            zri = magn*(cz*cy - sx*sy*sz + flip(cos(ph)*sz)) + cri
            zj = magn*(sx*cy*sz + sy*cz) + cj
            magn = |zri| + sqr(zj)

Note that |zri|+sqr(zj) is the square of the magnitude.



Here's the "non-trig" version using complex and real - this can be used to get the "non-trig" all-reals version for the integer powers (zri,ztemp,zjk and cri are complex and the rest are real):

            magn = sqrt(magn)^@mpwr
            ztemp = (real(zri) + flip(zj))^@mpwr
            zjk = (imag(zri) + flip(zj))^@mpwr
            zri = zri^@mpwr
            ztemp = ztemp/cabs(ztemp) ; cy, sy
            zjk = zjk/cabs(zjk) ; cx, sx
            zri = zri/cabs(zri) ; cz, sz
            zj = magn*(imag(zjk)*real(ztemp)*imag(zri)+imag(ztemp)*real(zri)) \
                 + cj
            zri = magn*(real(zri)*real(ztemp)-imag(zjk)*imag(ztemp)*imag(zri) \
                        + flip(real(zjk)*imag(zri))) + cri
            magn = |zri| + sqr(zj)
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« Reply #9 on: November 29, 2009, 01:30:53 AM »

Here's how I came up with the "Mandelview":

The usual way of transforming a scene for viewing in 3D is using the rotation matrices for example:
 
| cz -sz  0 |
| sz  cz  0 |
|  0    0   1 |
 
and
 
|  cy   0   sy |
|   0    1   0  |
|  -sy  0   cy |
 
and
 
|  1   0     0  |
|  0  cx  -sx |
|  0   sx  cx |
 
The reversal of the sign of the sines in the Y angle rotation being the "correct" method.
Now we can transform space using a combination of these matrices in any order, we want the order where the z is first, so either z*y*x or z*x*y.
I tried z*x*y first, i.e.
 
| cz -sz  0 ||  1   0     0  | = | cz   -sz.cx  sz.sx |
| sz  cz  0 ||  0  cx  -sx |    | sz    cz.cx  -cz.sx |
|  0    0   1 ||  0   sx  cx |    | 0        sx       cx   |
 
| cz   -sz.cx   sz.sx ||  cy   0   sy |  = | cz.cy-sz.sx.sy   -sz.cx   cz.sy+sz.sx.cy |
| sz    cz.cx  -cz.sx ||   0    1   0  |
| 0        sx        cx   ||  -sy  0   cy |
 
For the equivalent of the "Mandelbulb" method we only need that first row assuming you multiply like this (as I was taught):
 
|x y z|| a b c | = | x.a+y.d+z.g  x.b+y.e+z.h  x.c+y.f+z.i |
         | d e f |
         | g h i |
 
Because our vector to be transformed is |1 0 0| (pr more accurately |magnitude 0 0 | smiley
 
Anyway I tried: magn*(cz.cy-sz.sx.sy, -sz.cx, cz.sy+sz.sx.cy) which gave a combination of Mandelbrot and Mandelbar cross-sections if I remember correctly.
Now a Mandelbar is just a Mandelbrot with a sign change so I just played around with the signs of the terms with sines until both cross-sections were Mandelbrots and ended up with: magn*(cz.cy-sz.sx.sy, sz.cx, cz.sy+sz.sx.cy) which is totally incorrect if applying "normal" rules - hence my comment on the animation regarding a secondary "imaginary" level since the "incorrectness" involves a sign change.

Afterwards I tried the z*y*x order and got a modified version of that which produced the correct cross-sections for the z^2+c version but still had the problem with the odd powers and in addition only one of the cross-sections was correct for each of the even powers above 2.

Edit: I have since found that some of the cross-sections are only close to the "correct" versions - I thought the difference was a problem with my clipping smiley

« Last Edit: November 29, 2009, 02:22:10 PM by David Makin » Logged

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« Reply #10 on: November 29, 2009, 02:28:50 AM »

I think the "Mandelview" gives the most interesting "power morph" for a 3D Mandelbrot so far, here's 2 to 10 and back:

<a href="http://www.youtube.com/v/xgHr-aVUV_4&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/xgHr-aVUV_4&rel=1&fs=1&hd=1</a>

Apologies for it being a bit fast, I didn't want to spend too much CPU time on rendering animations as I'm still trying to find a "perfect" 3D Mandy smiley
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« Reply #11 on: November 29, 2009, 03:06:13 AM »

Another alternative method of getting a true 3D Mandelbrot I've been considering is a 3D version of trigonometry using an idea that I've never actually seen described or used but I think is sound (I know I'm not rigorous enough for a proof):

Consider a right-angled tetrahedron (which is an object that can by defined simply by (x,0,0),(0,y,0) and (0,0,z) it's fairly obvious that any two such objects are similar in the same way as triangles if the ratios of the *areas* of the sides stay unchanged.
Considering this I thought that maybe we could define a specific right-angled tetrahedron by any two known ratios of areas - for example if we define the areas of the sides as xy, xz and yz for the small sides and xyz as the base (=="hypoteneuse") then if we know xy/xyz and either xz/xyz or yz/xyz then we have defined a set of self-similar tetrahedrons.
Is this correct ? If so it probably gives an alternative method of defining rotation in 3D and squaring, multplying etc. once we have a version of Pythagoras for tetrahedrons relatng the areas.

Edit: OK I know - make that the ratios of the square roots of the areas smiley
« Last Edit: November 29, 2009, 03:26:41 AM by David Makin » Logged

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« Reply #12 on: November 29, 2009, 05:31:19 AM »

Here's a full 360 of the z^2+c "Mandelview":

<a href="http://www.youtube.com/v/V2c2SInnDNw&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/V2c2SInnDNw&rel=1&fs=1&hd=1</a>
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David Makin
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« Reply #13 on: November 29, 2009, 02:23:44 PM »

The "Mandelview" cross sections are not as close to being "correct" as I at first thought, I initially thought the difference was a problem with my clipping algorithm but have since found that some of the cross-section are only similar to the "correct" versions.
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« Reply #14 on: November 29, 2009, 04:55:18 PM »

The rotation matrices of the mandleview are analogous to the transform i have designed so it is interesting to see that someone else has thought along similar lines.

Your idea of a solid pythagoras formula is also very interesting.

I have to say that since Bellini described them these operators have inspired so much of modern utilitarian math as well as pure inventiveness, but underlying all of this is the fundamental power of iteration.  am exploring a fundamental revision of math that recognises this fundamental in all math and beyond maths into perception.
« Last Edit: December 07, 2009, 04:09:53 AM by jehovajah » Logged

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