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I am currently rendering new views of a power 2 variant of the Riemandelettuce. That reminded me of a general question about the holy grail:


What is the three dimensional generalization of a spiral?


Of the top of my head, I can think of three different structures that could play this role:

1. A tube that follows the path of a flat spiral. Similar to a snail's shell. Simply a spiral with thickness.

2. A two dimensional sheet rolled up like a scroll of parchment.

3. A length of string rolled up into a chaotic ball, i.e. a compact three dimensional curve, i.e. a knot in three dimensions (but with some distinguished center point somewhere in the middle).

There are some other alternatives that IMHO do not really qualify:

4. A helix. This one has no center, so it lacks an important feature of spirals.

5. Other curves that curl up around more than one single center. This is "too general" for my taste, but I don't have a good theoretical argument against it.


I am interested in any and all answers. From wishes and expectations to what you think the holy grail should look like; from arguments for and against the suitability of the mentioned (or other) structures; to theoretical considerations about qualities and consequences and side effects of those geometrical structures; all the way to examples you already found during your explorations.


I do not have imagery ready to offer right now. But I can say that the power 2 Riemandelettuce heavily tends to case 2 above, the scroll of parchment, in ways that surprised me. I cannot produce clean renderings, despite wasting even more processor cycles. But eventually I should be able to show you a short noisy animation to illustrate the striking difference to the power 2 Mandelbulb.

What if we try to make a spiral along each of the axes and then connect those spirals somehow?
Or a ''dot'' spinning around a center with x for horizontal rotation and y for vertical rotation and z for how close the ''dot'' is to the center(?)

Maybe a spiral is something that inherently takes up one extra dimension per surface dimension. (That'd be a requirement, not a clear definition)
If that's the case, I'd classify the spiral you probably have in mind as 2D as actually being 1D - its "surface" is 1D.
Then a 2D spiral would be something that "spirals" from/to a single point similarly along two different, orthonormal axes. - The object then would actually be embedded in 4D space.
Furthermore you could presumably distort it further by something that works more like a helix-type transform. And finally you could "project" that 2D-in-4D-space-spiral down to a 3D space in some way. But if I had to guess, it would probably be impossible to do such a transformation in a way that the 2D-spiral remains intersection free.

Another approach might be to look at special spirals. An Archimedean Spiral is very different from a Logarithmic Spiral, for instance. But in general, they are defined by some relationship :
Archimedean Spiral:
Logarithmic Spiral:
Hyperbolic Spiral:
etc.

So by that analogy, instead of polar coordinates, you can take an arbitrary parametrization of the sphere and two arbitrary (perhaps monotonous for good measure) functions in the two parameter variables to define some surface that would end up describing a 2D spiral in a 3D space. (Once again, though, intersections are unlikely to be avoidable except for maybe a small subset of functions.)

So to give one example, simply take spherical coordinates (similarly to how it's been done for the original Mandelbulb set) and define a doubly logarithmic 2D spiral to be:

And finally a third option: You don't like helixes as spirals, but how about, instead of having a straight path as the "center" of your "spiral", that "center" would be yet another spiral?

So you take a spiral, say, the logarithmic spiral, and then, along the normal direction of that, you define another, say, logarithmic spiral. At the center point both would meet but the whole thing would occupy 3 dimensions.

A couple of examples from the gallery here:
(pretty interesting also because we posted them within a months time, similar colors, similar disposition, similar shapes... but that is another topic )

Ross Hilberts 2D version:



And my 3D version:

 

...but OK I get it, maybe the 3D spiral should spiral away along all three axes... I dunno, I think the one in my image is a "true" 3D spiral...  

You have more imagination than I had, thank you!

I don't have a final answer to the original question, and I didn't set out to obtain one. I am really looking for new paths to explore in the "quest for the holy grail". The Mandelbrot set probably has more 2D features that don't have an obvious analogue in three dimensions. I am hoping that asking for these things gives us ideas what else to try.

There is no "scroll of parchment" case among the examples presented so far, so I give you an unfinished and noisy animation of the power 2 Riemandelettuce:

http://vectorizer.org/rmdltc/power2polepart001.mp4

This variant has the fixed point at the single pole, located at north pole in this animation. So there are two 2D Mandelbrots embedded on orthogonal planes, with the main cardioid in the upper part of the images (the "Elephant Valleys" are near the north pole). Towards the south pole, where the embedded Mandelbrots have their main antenna, there are very thin sheets, like butterfly wings, that cause a lot of noise in this rendering.

Surprisingly, a lot of things are happening outside the planes of the embedded Mandelbrots: there are branching trees and more scrolls of parchment.

(I am currently undecided to spend another three weeks on finishing this animations. There are one or two more variants that have equal rights of being "the" power 2 version of the Riemandelettuce.)