Like these criteria! It makes an interesting intellectual exercise to guess the properties of the.... Mandelisk
Thanks! I am just trying to approach it from a more mathematical point of view, because it may well provide inspiration for interesting shapes. It could also provide quick ways to 'disqualify' certain ways of looking for solutions, though I would never want to discourage someone from rendering a particular proposal!
I'd just add that the outside would look like spheres surrounding other spheres.
Inside there'd be long cavernous pathways, and maze-like tunnels, and it'd look so cool. Maybe we can expect a main large 3D cardioid shape for the spheres to surround as well.
I disagree here! The original Mandelbrot set consists of purely near-circles and near-cardioids. I would not be surprised if there were a third basic shape introduced in three dimensions, but I would hope it wouldn't be maze-like except when the iterations are turned low.
I definitely think the 3D set should not contain bubbles, but maybe bridges would be fine (though this would make it not topologically spherical).
Interesting! I like that. Like you, I expect no islands/bubbles, but bridges, now that's a whole different matter! I don't really know, but I'm *guessing* not... what would the 2D equivalent be?
Ah yes, I forgot to list the requirement that the shape be connected. Locally connected in fact, or at least it should look "locally connected" (see Wikipedia) (since the Mandelbrot set itself looks locally connected)!
What I'm uncertain about is extending the "simply connected" property into three dimensions. Wikipedia says, though, that a torus is not simply connected, so unless there is a really good proposal for generalizing things differently, I suppose there should be no bridges.
Dave Makin, I don't think the perfect version would necessarily have to follow all those rules- ie the x-y and x-z cross-sections might not be the same, and who knows, maybe the biggest bud (circle) off every cardioid is something other than a sphere in 3D (unlikely, but I would have said cardioids were unlikely so hey, whatever).
However I absolutely agree with your statement about the attractors! One interesting note regarding them is that the Mandelbrot set z^2+c can be thought of as representing the point 0+0i from each Julia set, or the point c. In any circle, the orbit of c in the Julia set is of the same period as the order in that circle.
So I know using points from the Julia set has been tried (ie cross sections from the full 4D ... thing...), but what if we choose our cross-sections carefully, looking for the properties described above?
So ok, I need some terminology to ask my next question. I've seen the near-circular and near-cardioids called the 'hyperbolic components'. (I think the locally connected conjecture is equivalent to conjecturing that the interior is only composed of hyperbolic components?) There is something called a root of a hyperbolic component -- hyperbolic components are either near-cardioids, in which case their root is the point where they ... inflect ... you know, the sharp point; or they are near-circles, in which case their root is the point at which they attach to a larger circle or a cardioid. (All the hyperbolic components are in a hierarchy; they form a tree, with the main cardioid of the whole set as its root. It is one crazy, infinite tree. There are roots arbitrarily close to any point on the border, but I guess, not all the points on the border are roots.)
It makes sense to talk about hyperbolic points and roots on the Julias, though their hyperbolic components aren't circular.
So if we want to use cross-sections of Julias to fill out the 3D set, we know (do we? I think so) we want the roots in the Mandelbrot set to correspond to places in the Julia sets where there is exterior on either side. (I hope you know what I mean! I can draw some pictures later maybe.) At the very least this means the point must be on the border. I'm pretty sure these places would necessarily be roots.
If we try 0 or c as the 'point of attachment', the point is surrounded by interior- so it can't work. However, c is is the middle of a hyperbolic component, and it would be nice to find the root of that component and try using that.
So my question is, anyone know where, for a Julia set given by complex number c, the root of the hyperbolic component surrounding the point c is? Is there an equation for predicting it? I think I know of a method of finding it but it is really slow and only an approximation.
Note: if this question were answered I think it would actually provide a totally alternative formulation of the Mandelbrot set but still in terms of the Julias. This might be evidence that I am totally wrong.
If any other points of math are wrong please correct me!