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This is from a formula using basically the same concept as the original Mandelbulb but in a simpler way.
Here's the z^16+c Mandy:

(http://www.fractalforums.com/gallery/1/141_22_11_09_2_42_55.jpg)

So far it appears to have considerably less "whipped cream"...but I'll have to leave further investigations for now, it's 2.45am and I have a headache :)

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?

Here's part of the degree 8 using the same formula as the degree 16 "Walnut":

(http://www.fractalforums.com/gallery/1/141_22_11_09_2_01_47.jpg)

I realised an obvious change can be made to the method that *should* give even more interesting results, I'm just about to try it....

hey, cool :D
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 a render using the "obvious" modification, but I think it goes a bit too far, things get a little bit chaotic, here's a degree 9:

(http://www.fractalforums.com/gallery/1/141_22_11_09_3_13_51.jpg)

One more thing to try then I'll let you know the formulas :)

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

(http://www.fractalforums.com/gallery/1/141_22_11_09_3_52_22.jpg)

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.

I should add that all the renders in this thread so far were done using my "delta DE" method, I'm just going to see if an analytical DE will work by adapting the method Jos Leys produced for the original Mandelbulbs.

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

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 :)

I should add that the reason these have never been publicly "announced" before even though the idea is so simple is that the lower degree versions are particularly ugly - especially the z^2+c and since just about everybody tries z^2+c first they were probably discarded as uninteresting :)

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).

(http://fc07.deviantart.net/fs51/f/2009/328/e/6/Hairy_Ball_by_MakinMagic.jpg)

If no image above then look here:

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

haha, nice :) it feautures a lot of bridges which someone took as a requirement for the 3D-Version of Mandelbrot, if I recall correctly...

Though, I don't really like the colour-sheme...
just a thought, maybe you can roughly do a gradient so it looks like skin and hair on it? (if not, then hair only^^)

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):

(http://fc06.deviantart.net/fs50/f/2009/330/9/e/A_Martian_Delicacy_by_MakinMagic.jpg)

If no image above then look here:

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

I got the analytical DE working for these.
I'm just going to add all the non-trig formulas bugman gave for the versions of the Mandelbulb to my WIP 3D formula for UF before I update it, it may be a couple of days as each one needs checking and I want to optimise them as much as possible before release.
Note that, unlike the GPU implimentations, on the CPU/FPU using the non-trig versions of the formulas is considerably faster especially using my delta DE method (I still don't know the non-trig calculations for calculating the derivative using the analytical method - if I did that would undoubtedly be fastest).