True 3D mandelbrot fractal (search for the holy grail continues)

[jehovajah]

The wizards staff! A julia of arcane power. He who wields it has power to subdue dragons!

[jehovajah]

These sculptures arise out of an exploration of the sin and cos functions and the principle range.

[jehovajah]

The road to the holy grail is long and winding. Why not fly or go by boat?

[jehovajah]

Lets stop for breakfast and fry an egg! Wait what appears in the distance......?

[jehovajah]

[x=real(z)
y=imag(z)
s=imaj(z)
a=real(c)
b=imag(c)
d=imaj(c)
r2=1*(|c|+d*d)+0*(|z|+s*s)
t0=1*sqrt(|c|)+0*sqrt(|z|)
|z|<10
|c|<10
s2t=sin(2*t0)
c2t=cos(2*t0)
z=1*r2*(s2t*cos(2*b)+s2t*sin(b)*i+c2t*j)+0*r2*(s2t*cos(2*y)+s2t*sin(y)*i+c2t*j)+0*c+0*z

Have fun with the formula. Maybe you might have an interesting adventure on the way to your holy grail! Yom Tov!

[jehovajah]

Grail mandy settles down to lay a mandy egg. But she becomes egg bound right at the crucial dropping point! Argh!!!

[jehovajah]

First some variations based on the following grailmandy with the approximations to theta and phi incorrectly applied in the tan and cos versions.

y=sin(imag(z))
s=imaj(z)
b=sin(imag(c))
d=imaj(c)
r2=1*(|c|+d*d)+0*(|z|+s*s)
t0=1*sin(sqrt(|c|))+0*sin(sqrt(|z|))
|z|<10
|c|<10
s2t=sin(2*t0)
t2c=cos(2*t0)
z=1*r2*(s2t*cos(2*b)+s2t*sin(2*b)*i+t2c*j)+0*r2*(s2t*cos(2*y)+s2t*sin(2*y)*i+t2c*j)+0*c+0*z

wherey,b,t0 are approximations to theta and phi.

[jehovajah]

ctd

[jehovajah]

And now the intended versions for cos and tan.

I deliberately altered the colour palette in the last one. Not so boring n'est ce pas?

[jehovajah]

y=sin(z)
s=imaj(z)
b=sin(c)
d=imaj(c)
r2=1*(|c|+d*d)+0*(|z|+s*s)
t0=1*sin(sqrt(|c|))+0*sin(sqrt(|z|))
|z|<10
|c|<10
s2t=sin(2*t0)
t2c=cos(2*t0)
z=1*r2*(s2t*cos(2*b)+s2t*sin(2*b)*i+t2c*j)+0*r2*(s2t*cos(2*y)+s2t*sin(2*y)*i+t2c*j)+0*c+0*z

A set based on this variation of grailmandy now follows.

[jehovajah]

a cos and a closeup of a sinz. Remember these functions are approximating theta and phi, so other functions that have a link to theta and phi may produce interesting results: for example e^iz.

[David Makin]

Thought that it's relevant to this thread too, so here's a cross-link for anyone interested:

http://www.fractalforums.com/theory/summary-of-3d-mandelbrot-set-formulas/msg14463/#msg14463

[jehovajah]

Just to finish this part of the exploration, using cosz as an approximation to theta and arranging the formula for a traditional mandy thusly:
 y=cos(z)
s=imaj(z)
b=cos(c)
d=imaj(c)
r2=0*(|c|+d*d)+1*(|z|+s*s)
t0=0*cos(sqrt(|c|))+1*cos(sqrt(|z|))
|z|<10
|c|<10
s2t=sin(2*t0)
t2c=cos(2*t0)
z=0*r2*(s2t*cos(2*b)+s2t*sin(2*b)*i+t2c*j)+1*r2*(s2t*cos(2*y)+s2t*sin(2*y)*i+t2c*j)+1*c+0*z

[jehovajah]

to finish with grailmandy i have set the formula to reflect mandlebulb formulae, and used sin and tan as approximations for theta, phi.

[David Makin]

Just had a really crazy idea and tried it, I didn't expect it would actually "work" and it didn't - you just get a sort of slightly plumped out 2D Mandelbrot, but I like the Julias:

UF code:

            magn = sqrt((m=|zri|)+(s=sqr(zj)))
            if magn>0.0
              zri = sqr(zri)
              magn = sqr(magn)/sqrt(sqr(m)+s)
              zri = magn*zri + cri
              zj = magn*zj + cj
            else
              zri = cri
              zj = cj
            endif

zri/cri are complex, magn/zj/m/s are real and in UF |zri| means the magnitude of zri squared.