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>

Wow, 24 minutes without any distance estimation? That's quite impressive for such quality.

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.

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.

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!

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?
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/

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        (±jz)2 = -z2      yxi = iyx

-yzij = -ijyz      (±iy)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.