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nice
though , grade 16 already is very spherical... could you do a lower grade one? z^8 seems to work quite well in many cases...
also, how does the z² look in this case?

hey, cool
Very nice experiments.
What's the modification in this case?
(And, if it's more interesting, how's it in the following one? )

OK, here's the alternative version of "Walnut":



These formulas are based on the rotation idea as in the Mandelbulb but (not worrying about "correct" maths) I wondered what would happen if you simply rotate by multiplying the original 2D angles i.e. simultaeneously rotate by the multiples of the angles around the axes of the original value.

So here are the formulas that produced the images in this thread, note that zri and cri are complex, r, magn, ph, zj, th and @mpwr are real and in Ultra Fractal |zri| is x^2+y^2 where zri = (x+i*y).

First I tried adding multiplication/rotation around the x (real) axis in exactly the same way that complex numbers rotate around the non-existant z axis:

              r = (magn=sqrt(magn))^@mpwr
              ph = @mpwr*atan2(imag(zri) + flip(zj))
              th = @mpwr*atan2(zri)
              zj = r*sin(ph) (edit: there should be +cj here)
              zri = r*(cos(th) + flip(sin(th)*cos(ph))) + cri
              magn = |zri| + sqr(zj)

This produced the original Walnut and the follow up degree 8 image.

Then I wondered what if you do the same for all three axes simultaeneously:

              r = (magn=sqrt(magn))^@mpwr
              om = @mpwr*atan2(zj + flip(real(zri)))
              ph = @mpwr*atan2(imag(zri) + flip(zj))
              th = @mpwr*atan2(zri)
              zj = r*sin(ph)*cos(om) (edit: there should be +cj here)
              zri = r*(cos(th)*sin(om) + flip(sin(th)*cos(ph))) + cri
              magn = |zri| + sqr(zj)

The answer was the Chaos 9 image.

So then I simply tried using the rotations around the z axis and the y axis:

              r = (magn=sqrt(magn))^@mpwr
              om = @mpwr*atan2(zj + flip(real(zri)))
              th = @mpwr*atan2(zri)
              zj = r*cos(om) (edit: there should be +cj here)
              zri = r*(cos(th)*sin(om) + flip(sin(th))) + cri
              magn = |zri| + sqr(zj)

And got the walnut 2

I will be updating the downloadable version of my WIP 3D UF formula on my website with these added.

 @ triple angle:
well, that certainly adds fractal details to it and gets rid of any smoothness

Very nice stuff

Now I wonder, how that'd look like on a torus as some already had the idea for their formula...

Or any other primitive variation

Ok, first of all an apology - you may notice I've added an edit to the formulas with a "+ cj" in the calculation of the next zj value - this was an oversight on my part (fancy forgetting the constant !).
Anyway it (surprisingly) didn't significantly affect the results, at least not qualitatively.

I did a hi-res render of a minor zoom into the z^14+c of the version that rotates around all 3 axes together (with the constant done correctly) and achieved a result that I termed "Hairy Ball" after the "hairy ball" problem mentioned elsewhere in the Mandlbulb threads (I just liked the fact that the name fits the image though I have no idea what the "hairy ball" problem is).



If no image above then look here:

http://makinmagic.deviantart.com/art/Hairy-Ball-144652915

I realised that the issue causing the hairiness was that using the "simultaeneous" rotation around all 3 angles causes the trig part to vary in magnitude. So I tried normalising the trig part before multiplying by the new (power) magnitude, using (1,0,0) for the trig part if the magnitude of the trig part was zero. This removed the hairiness, here's an example render showing the "active" area of a degree 8 (i.e. the earthquake zone when the power is changed):



If no image above then look here:

http://makinmagic.deviantart.com/art/A-Martian-Delicacy-144901891