END OF AN ERA, FRACTALFORUMS.COM IS CONTINUED ON FRACTALFORUMS.ORG

I think we may have something...



Sorry about the low resolution - even this took around 30 hours to render!! Formula and full story coming when I tidy up the code...

2D renders of 3D slices from the 4 dimensional "Julibrot" space are possible from Fractint and some formulas in Ultrafractal but they aren't exactly what is being sought

Here's one view of a Julibrot:

http://makinmagic.deviantart.com/art/Classic-3D-Julibrot-43168532

Wow--it's not often that something I wrote 15 years ago comes back to life!  :-)

I think that the search for a 3D Mandelbrot analog depends on a suitable definition of a 3D number system.  As has probably been pointed out, the more useful extensions of the real numbers have 2^n components, so there isn't a standard 3D system.  Thus, you're free to define your own and see how it works with the Mandelbrot set.

(Lloyd) Kerry Mitchell

interesting screenshot, unfortunately the program crashes at startup

edit: it was crashing because of the included opengl dlls, deleted them and it works. unfortunately it still crashes when making high resolution meshes, but i was able to render these with resolution 960:

Oh, my deeply sleeping thread!
I revive Thee from the dead... 


Ok, seriously, after staying away from FractalForums for a while because of RealLife(TM), I came back and (obviously) found this thread. I had a little exchange of mails with twinbee about the formula involved and he has asked me to post the results here. So, here it comes:


twinbee defined (a "few" posts back)

double r    = sqrt(x*x + y*y + z*z );   
double yang = atan2(sqrt(x*x + y*y) , z  ) // that would be theta in std polar coordinates
double zang = atan2(y , x);                      // that would be phi  in std polar coordinates

so I would suppose he has implicitly

x = r*sin(yang)*cos(zang)
y = r*sin(yang)*sin(zang)
z = r*cos(yang)

and I would have expected (doubling the angles!)

newx = (r*r) * sin(yang*2)*cos(zang*2)
newy = (r*r) * sin(yang*2)*sin(zang*2)
newz = (r*r) * cos(yang*2)

but he defines

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 );

which can be simplified by taking into account the symmetries of sin() and cos() to

newx = - (r*r) * cos(yang*2) * cos(zang*2)
newy = - (r*r) * cos(yang*2) * sin(zang*2)
newz = - (r*r) * sin(yang*2)

which is not exactly equal to doubling the angles.
... but it leads to interesting pictures.
One DOES NOT need atan2(), sin() and cos() to implement these formulae,
because they can be simplified a lot by using the following identities for any
angle a :
cos(a)*cos(a)+sin(a)*sin(a)=1  (well known, I suppose)
cos(2*a) = cos(a)*cos(a)-sin(a)*sin(a)   (less well known, it seems )
sin(2*a) = 2*cos(a)*sin(a)

I'll spare you the details, but you end on:

newx = ( x*x + y*y - z*z )*( x*x - y*y) / ( x*x + y*y )
newy = 2 * ( x*x + y*y - z*z )*x*y / ( x*x + y*y )
newz = - 2 * z * sqrt( x*x + y*y )

no trigonometric functions at all, just additions, multiplications, divisions and a squareroot.
There is NO pole on the z-axis, BUT there might be numerical problems because of the
denominator, solvable e.g. by taking

if( abs(y) < really_small_value )
newx = x*x-z*z
newy = 0
newz = -2*z*sqrt(x*x)
else (view above)

(end of simplification)

Interesting side effect: when I first read about "doubling both angles" I wanted
to try that for myself. When doing geometry instead of math, I prefer measuring
the angle theta not against the z-axis, but against the x-y-plane, so in that case

phi = atan2( y, x ) // just like before
theta = atan2( z, sqrt(x*x + y*y) ) // exchange of arguments, angle is positive above, negative below x-y-Plane

then
x = r*cos(theta)*cos(phi)
y = r*cos(theta)*sin(phi)
z = r*sin(theta)

and simply doubling the angles

newx = r*r*cos(2*theta)*cos(2*phi)
newy = r*r*cos(2*theta)*sin(2*phi)
newz = r*r*sin(2*theta)

simplifying this analogous to above gives

newx = ( x*x + y*y - z*z )*( x*x - y*y) / ( x*x + y*y )
newy = 2 * ( x*x + y*y - z*z )*x*y / ( x*x + y*y )
newz = 2 * z * sqrt( x*x + y*y )

IDENTICAL to twinbee's stuff except for the sign in z !

Since I'm sitting at an OLD Mac (350MHz) and try to use POV-Ray to produce pictures I can't show any yet; that takes TIME! But the first little "thumbnails" I produced show that this change of sign dramatically changes the resulting set!

Forgive my ranting - I hope somebody might find these "simplifications" useful - twinbee thought so, at least!

Happy iterating...
Karl

whoa cool, with ~100x speedup it should be possible to compute nice images of this thing, finally and at last i suspected there was a nice simplification to be found instead of the constant cartesian<->polar conversions...

i'm busy rendering another (absolutely insane) 3d fractal, once i've got those cycles freed up i'll try do some more 3d mandelbrot pictures 

ps. twinbee no idea mate, sorry; it could be a huge number of things (camera pointed wrong way, light intensity set to zero, forgot to comment out this and that, ...)

Hi twinbee,

sorry, haven't been here for a while - my new part time job puts quite a strain on my capabilities to look at a computer screen 
I'm quite happy that someone made use of my mathematical gymnastics - finally, I might add.


OK - that's not THAT much, but it's SOMETHING. I would seriously love a speedup like that, but I'm looking forward to the next pay check from aforementioned new job; that just MIGHT become a faster computer (crosses fingers)

My pictures came up very fast (and empty!) too, before I worked around that non-singularity with that "really_small_value"-trick...

Happy iterating
Karl

Hi twinbee, I'm a brand new member here. I stumbled upon your page (http://www.skytopia.com/project/fractal/mandelbrot.html) on my quest to understand how to render the "whipped cream" Julia quaternions. I've always liked how they look... but there's something missing... I was so glad to find your page documenting your attempt to find this object. Ever since I wrote my first crappy Mandelbrot program (probably ten years ago), I've had the same question you do, about a "true" 3D Mandelbrot fractal solid. I just wanted to post to let you know that I understand EXACTLY what you're looking for, and why its so enticing. I briefly searched for this thing years ago, but gave up pretty quickly. Some of the images I've seen here and on your page make me think that it does exist though.

Also, while looking over the description of your 3D multiplication, I wondered why you doubled both angles - that has kind of a weird effect. Wouldn't it be better to double the angle between your number, and the number 1 (for simplicity, say 1=(0,0,1))? This is similar to your approach, but not exactly the same thing. It makes the multiplication operation a little more uniform, in that all points on the circle of constant r and phi, but varying theta, will end up on another circle of constant, but different, r and phi. Did you try this at all? (sorry if that's hard to follow - I can clarify with a diagram if necessary)

lyc, it seems that most of your renders attached to your older posts are gone, are those still available anywhere?

I use my TI89 frequently (it uses a program called Derive). There's also Mathematica, and Maple. Matlab has a symbolic manipulation package that just uses Maple.

[sorry for reviving a dying thread, but I was totally astounded to find another person in the world searching for this]