First of all, I have to admit I don't understand what's being described in the options in the pole... Perhaps I'm too visual in my thinking to pick it up without a picture example.

But I do have thoughts on 3d spirals, unfortunately not backed up by any rigorous investigations. But I've spent a long time thinking about this problem, which comes from many days of when I was working in a cafe that was doomed to fail, and therefore having nothing better to do for hours at a time than to wonder about geometry.

I suspect that a spiral that fits the properties of 2d spirals can only exist in even-numbered dimensions. There are shapes in 3d that are similar, but they're almost all based on 2d spirals extruded or manipulated in some way. For example, a spiral that moves along the y axis at the same rate as it's radius increases will create a 2d spiral projected on a cone. (depending on the type of spiral, I guess).

A snail shell would be an example of this, with the 2d spiral being the path of a sphere-sweep that increases in radius as well as Y direction, and intersects.

A rolled up plane is another option, which is simply a spiral that's been extruded infinitely.

A proper spiral equivalent, I think, (at least for logarithmic spirals) would repeat both by scaling and by rotating, and it would behave like a spiral from any angle. It would also resemble a sphere with smooth and constant curvature changes. Any of the sphere-sweep style spirals I can think of have an axis, like the stem down the center of a snail shell.

I think this is possible in 4 dimensions because 3d extrusion of a spiral could be extruded in a spiraly shape, if it didn't self-intersect in the process... Making use of a 4th dimension would allow this, as far as I can tell. Imagine we take a spiral that's int he x/z plane, and extrude it along a curve... and that curve is a spiral in the w/y plane.

I can't picture this clearly enough to predict anything about dimensions 5 and 6... but intuitively even numbered dimensions make sense.

The closest that the third dimension can do, I think, is a helix, but that still feels like my first example manipulation of a spiral with specific settings. (A spiral that has its radius increase by 0 every turn. Aka a circle being drawn forever.) The nice thing about this in the context of a Mandelbrot analog, though, is that it can maintain a volume of 0. I assume that you're talking about the filiments that come off of the circles of the Mandelbrot set, and are infinitely thin, aside from the infinite number of minibrots contained within...

This became long. I apologize.