3D Mandelbrot Formula based on the Hopf Map

[bugman]

Here are some 8th order renderings of the 3D slices using this Hopfbrot formula.

[TedWalther]

Coolness! Even though it has whipped cream / taffy, I like this idea. Thanks for the lesson.

In general, I think the taffy problem happens because the power function isn't conformal. The 3D White-Nylander power functions are least conformal near the poles. In experimenting with 2D escape time fractals over the years, I think I've noticed that fractals that look all stretched out come from formulas that aren't conformal, and fractals that look "nice" invariably come from formulas that are. I think this makes sense because that's what a non-conformal map does--it "stretches out" space.

As pointed out by Tglad on another thread, Liouville's theorem says that the only smooth conformal maps possible in > 2 dimensions are Moebius transformations, which unfortunately don't become complicated when iterated since they form a closed group. I suspect that if one could find a 3D map that was somehow "close" to being conformal, it still might not be good enough because any deviations from conformality might get magnified when iterated lots of times. I think completely getting rid of the taffy is likely a hopeless task, but there may be other nice ways to compromise. In my opinion, the high-degree Mandelbulb fractals are nice ones because they "spread the taffy around" so that there are lots of areas that aren't dominated by it. It seems kinda hard to find much variety in them though from what I've seen so far.

Just out of curiosity, and because you KNOW I'm going to have to go now and find a 3d conformal function, can you explain in what sense the 2d mandelbrot function is conformal?  I read on wikipedia that conformal functions preserve angles; if I may ask, which angle is being preserved by the 2d mandelbrot function?

[Tglad]

"if I may ask, which angle is being preserved by the 2d mandelbrot function?"
Hi Ted, I read up on this recently so I'll try and forge an answer. By conformal they mean that internal angles of any set of points that are close together stay the same. So if you choose three points in say a right angled triangle near to 0+i, then the map Z^2 will map those 3 points to a new right angled triangle near to -1+0i. In fact Z^2+c will convert any small shape in the complex plane into the same shape, just rotated and scaled, but importantly, not stretched.
Another way to visualise it is, if you apply the function to a regular grid, then the resulting distorted grid lines will still cross at right angles.

According to Liouville's theorem in 3d there are only 5 mappings that are conformal: translation, rotation by a fixed angle, uniform scaling, reflection and inverse (scaling vector by 1/length^2)... and combinations of these. Though combinations basically cancel down back to this simple set.

I looked a little at 'quasi-conformal' maps, which nearly preserve angles, so have an upper bound on their stretch. In particular, folding in a tetrahedral pattern seems quasi-conformal. None of the mandelbulb formulas are quasi conformal as their stretch goes to infinity at the poles.

[bugman]

Very nice! The quest goes on...

As I posted somewhere else, I think everybody "imagines" the ideal 3D Mandelbrot, it would look like this famous picture: http://www.renderosity.com/mod/gallery/index.php?image_id=1308487&member

So why not try a completely different approach, and of course completely wrong and purely artistic, for example by using a 3D software to draw a kind of apple (a 3D cardioid) and then plug spheres all around, etc... ??

Here is a link to a similar artistic approach:
http://www.fractalforums.com/theory/mandelbrot-pearls/

[faxingberlin]

In search of the ideal 3D Mandelbrot set, a couple of people on this forum have mentioned the hairy ball theorem. Basically, the problem with the Mandelbulb formula has something to do with the poles of the spherical coordinate system. Everything gets distorted near the poles. An ideal formula would have no poles, but this is only possible in 4 dimensions (because of the hairy ball theorem). This got me thinking, maybe we could create the ideal 4D Mandelbrot using the Hopf map (a mapping for the surface of the 4D sphere with no poles), I'll call it the "Hopfbrot". So I came up with the following formula and I tried rendering it. Unfortunately, it doesn't look as I had hoped, although it does contain the Mandelbulb. But perhaps this will provide some more food for thought.

Thanks for the information!
I made Hopfbrot Fractal using Houdini.