The Christmas Tree 3D Mandelbrot Set

[bugman]

I was thinking it might be worth exploring different kinds of 3D Mandelbrot sets using power formulas that travel the same distance around the sphere no matter what the angle. Other people have already thought of this idea, but I came up with the following power formula and figured I'd give it a try. As it turns out, this formula is identical to the 3D slice at y = 0 of my 4D Hopfbrot formula (the one that looks like a Christmas Tree):
http://www.fractalforums.com/theory/3d-mandelbrot-formula-based-on-the-hopf-map/

As it turns out, this method is very similar to the cosine method. Here is the power formula:

[bugman]

We can also define a multiplication operator which is commutative but not associative (same as the Mandelbulb multiplication operator):

[bugman]

We can generalize the above power formula to define other power formulas that travel the same distance around the sphere no matter what the angle. So in general we can define phi = n*f(atan2(z,y)) where the function f can be almost anything. For example, if I use the sine function for my function f, then I get the following quadratic and 8th order Mandelbrot sets:

[BradC]

I'm having trouble picturing how this is different from the "cosine method" (formula #3) on your summary page (http://www.fractalforums.com/theory/summary-of-3d-mandelbrot-set-formulas/?action=dlattach;attach=527;image), with the x-axis here playing the role of the z-axis there. The formulas do look different I think, but I can't picture what the difference is geometrically. Aren't they both using spherical coordinates, but with the elevation angles measured as angle away from a pole rather than angle away from the equator as in the usual formula? In the picture for formula #3, I can almost imagine a Christmas tree pointing downward, maybe.

Edit: Try starting with formula #3 from the summary page and make the replacements , and I think it becomes the formula given on this page.

[bugman]

Wow, you're right. The formulas become equivilent if in my formula you define phi = pi/2 - n*atan2(y, z). But the renderings look distinctly different, so apparently that makes a difference.