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.

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.

Here are some more slices of the Hopfbrot. The slice at zc = 0 looks like it might be worth further investigation. I think the slice at yc = 0 looks like a Christmas tree. And the slice at xc = 0 is just, weird.

Hi Paul,

Although I made a movie some time ago about the Hopf fibration, I don't understand what you are doing here.

Here are some things I know about it, and I don't really see how it relates to you formula.

Let z1 and z2 be complex numbers, related as z2=a.z1 (a is a complex number also). The intersection of the plane z2=a.z1 with the three-sphere gives a circle in R^3 under stereographic projection from S^3 to R^3. As there are an infinite number of complex numbers a this gives an infinity of circles (and every one of those circles is linked with all the other ones)

See http://www.dimensions-math.org/Dim_reg_E.htm (see the films Dimensions_7 and Dimensions_8)

Can you clarify?

It's "Sheep may safely graze" by J.S.Bach

What does the zc=0 slice look like for power 8?

great videos

That way, both the quaternionean (or offspring of that) Mandelbrot as well as the full Mandelbrot set, where also the c-plane changes, could be downprojected during rotations...

I'm not totally sure but I *think* I really never saw a stereographic projection of the full 2+2iD Mandelbrot object... - only inversed Mandelbrots, I think...

I think you're right in eliminating the Abs.  It is just an artefact of using ArcCos (which has a range of only PI, not 2PI).  Just to be sure, if you feed Mathematica something based on arctan(y/x) it should produce formulas without Abs.

Have you taken a look at my quaternion thing? Can you try feeding it to Mathematica and see what it simplifies to?...