Trip... formulas

[fractality]

Hey there,

once again I am a little overwhelmed by fractal mathematics and therefore I wanted to ask you here for some help.
I´m working with the 3D fractals in XenoDream, so mostly with the different "Trip..." Metamorphs. I understand, that they (all?) are based on the Juliabulb, which seems to be an alteration of the common Mandelbulb. The Discription just doesn´t tell me enough to really grasp the formula.

For a poster and booklet I just wanted to know the complete formulas for those 3D fractals (if not all, then at least Triplex3T). Is there anyone, who could tell me those or how to find them?

Many thanks,
fractality.

[Garth Thornton]

The Juliabulb formulas are the same as the standard Mandelbulbs, z^n+c, where z and c are triplex numbers (a version of 3D complex number defined in spherical coordinates). The Triplex series transforms to the polar axis using sin(phi), while the Tripco series uses cos(phi). For Triplex3T, n is 3. As usual, the Mandelbulb iterates from a fixed z and varies c, while the Juliabulbs have a fixed c and vary z. XD uses inverse iteration towards the attractor, which works for Julia sets but not Mandelbrot sets.

[fractality]

Thanks for explaning this topic so clearly to a layman like me.

Still I don´t understand, if these triplex numbers can be expressed in another way, like in this wikipedia article for example.

So I wonder, if this formula from the same article is correct for the Triplex3T.

I hope you understand my issue ^^"
Many thanks for helping me out,
fractality.

[Garth Thornton]

Yes, the alternative formulas should give the same results. In general, some operations are easiest to think about in terms of x,y,z and others are easiest to think about (or compute) in other coordinate systems, such as spherical (radius, longitude and latitude). For complex numbers (of any dimension) it's much easier to calculate arbitrary powers in the polar, spherical or equivalent higher dimensional system. The trigonometry just specifies the transformation from x,y,z to r,theta,phi; then you do the power operation, transform back to x,y,z and add C.

If XD wasn't limited to 3 parameters for metamorphs there would only be one TriplexT metamorph with a parameter for power.