Furan's corner

[Furan]

My latest work. Unfortunately my coloring scheme was not very good and I had to decolor/recolor it, so it is monotone.
And I should have added a noise function.

Available in 1920x1920 and HQ 3840x3840 (4x4AA), 11MB
http://furan.sweb.cz/

I will try to zoom in as much as double precision allows me.
The concept of visualization with a glowing aura seems legit.

[Furan]

MandelDonut Fractal.
Or maybe not a fractal after all, I'm not sure if it has stable orbits of any kind.
10 iterations, 1920x1440 (5x5 AA)
http://furan.sweb.cz/Wall2012/MandelDonutHD.jpg

Code:

rho2  = sqrt(x**2+y**2)
rho8 = ((rho2-2)**2+z**2)**4
phi    = 8*ATAN2(y,x)
theta = 8*ATAN2(z,rho2-2.)
x = (rho8*COS(theta)+2) * COS(phi) + x_0
y = (rho8*COS(theta)+2) * SIN(phi) + y_0
z = rho8*SIN(theta)                 + z_0

[Furan]

Same fractal in black smoke.

http://furan.sweb.cz/Wall2012/MandelDonutBHD.jpg 1920x1440
http://furan.sweb.cz/Wall2012/MandelDonutB3840.jpg 3840x2880 (6x6AA), 5MB

[Furan]

Ok, now the true MandelDonut:

http://furan.sweb.cz/ (1920x1440)
HiDef 7680x5760 is in progress, I will also try some zoom in.
I had this idea about an external region of space that would save some escaping trajectories and immediately thought about the toroid, as the new iteration of rho would go across the center and land safely inside again. I will try some more elaborate structures, like knots etc.

Code:

R=2
ni=0
RHO = sqrt(x^2+y^2)
rho = sqrt((RHO-R)^2+z^2)
phi    = ATAN2(y,x)
theta = ATAN2(z,RHO-R)
Cx   = rho*COS(theta) * COS(phi)  //This Cx,Cy,Cz adjustment is new. Now the fractal set has a volume, a well defined orbit trap.
Cy   = rho*COS(theta) * SIN(phi)
Cz   = rho*SIN(theta)

Iterate, while rho<2R+2
------------------
RHO = sqrt(x^2+y^2)
rho = ((RHO-R)^2+z^2)^4
phi   = 8.*ATAN2(y,x)
theta = 8.*ATAN2(z,RHO-R)
x = (rho*COS(theta)+R) * COS(phi) + Cx
y = (rho*COS(theta)+R) * SIN(phi) + Cy
z =  rho*SIN(theta)               + Cz
ni++
------------------
I didn't manage to derivate this, so I used:
step=min(max(1E-8; 0.05/ni/ni*0.5**ni); 0.05)
There hasn't been done much tweaking of this formula, so it's probably pretty inefficient.

[cKleinhuis]

nice, hello and welcome to the forums btw

what do you use for rendering ? fragmentarium ?!

[Furan]

I'm rendering in my own Fortran 77 program (I'm working on implementation of material models to FE-systems. Now only in F77, so it's a sort of an excercise.)

[Furan]

After about 10 days on 6 cores, this is the result: 7680x5760 (5x5 AA)
http://furan.sweb.cz/Wall2013/MandelDonutVegas7680.jpg 35MB

[cKleinhuis]

you have been featured on orbittrap.ca
http://orbittrap.ca/?p=4480

[Furan]

Beautifully written. Thank you for the memo. I will definitely continue my work on this piece.

[Furan]

Math background in http://www.fractalforums.com/theory/a-generalization-of-triplex-zpplusc/
Behold the Trifold MandelKnot:

http://furan.sweb.cz/Wall2013/TrifoldMandelKnot.jpg 1920x1920 (5x5 AA)
It's still work in progress. It looks biological, so I was wondering about coloring it as nerve bundle or an artery ripped out from a beating heart 
My initial attempts were not successful.

[Pauldelbrot]

I just had a thought about 3d Mandelbrots myself.

The points on a non-singular elliptic curve in the plane form an additive group under the operation "draw a line through two points on the curve (the tangent there if they're not distinct) and find the third intersection (or if it's tangent at one of the two points, that's the third)".

I wonder if there's some higher-dimensional analogue with three degrees of freedom and a reasonable multiplication operation that can be defined as well?

[Furan]

The coloring is a bit too much. Watch your eyes
Trifold Julia Knot

[Dinkydau]

Nice work