Goals for a 'True' 3D Mandelbulb

[Dranorter]

What properties are people hoping for in a 3D Mandelbulb? Intuitively, the idea is simple- it should 'look right', and you know what I mean by 'right'. But what, really, makes it right?

One important property is not looking stretched out- having 'full 3D detail'. I don't know a good way of really defining this one. My best guess is that its fractal dimension should be higher than 2 everywhere, or maybe, higher than 1 for every slice of the thing. But no, that's probably not right.

Anyway another property which comes to mind is that it actually be composed of many near-spheres the way the M-Set is composed of many near-circles (plus many cardioids). Along with this I'd like to add that it should be actually a pinched sphere - and I would argue the pinches should be pinched to points, not lines, so that the shapes can actually be near-spheres. So this would make the interior consist of infinitely many near-spherical openings, though who knows what the outside would look like.

Another property is that it should not touch itself, though I am not so certain about this one. What I mean is, the sphere can be pinched infinitely many times, and therefore touch itself this way, but no point on the sphere should otherwise touch a point it was not already adjacent to- so it should not bend around and touch itself. If I am not mistaken the M-set itself probably has this property but it has not yet been proven or disproven that it does.

The reason I'm unsure about that property is because the M-Set being embedded in 2D space, it couln't touch itself without creating a 'bubble' cut off from the outside. I definitely think the 3D set should not contain bubbles, but maybe bridges would be fine (though this would make it not topologically spherical).

A property the 3D set should have for sure is that its surface area should be infinite, ie it should take up volume.

It should also actually summarize a set of Julias, because summarizing Julias seems to be the source of the infinite variety of the Mandelbrot set. So it should summarize a set of Julias which contains great variety, including some Julias which are dusts (of positive volume?), some which are dendritic, some which are sheets, and some which are topological spheres. (One of the julias should be a perfect sphere.)

So, which properties does the "3D Mandelbulb" have, and which does it not have? (oh no hard maths)

[twinbee]

Like these criteria! It makes an interesting intellectual exercise to guess the properties of the.... Mandelisk

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 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?

[David Makin]

The "perfect" z^2+c 3D Mandelbrot would be made up of a single cardioid (a simple rotation of the 2D one around the real axis) then many near-spherical objects (bulbs) analagous to the near circles in the 2D version. For the circles in the 2D version that are centred on the real axis then the analogous bulbs would again simply be a rotation of the near-circle around the real axis. For the other near-circles that make up the rest of the inside i.e. those that exist symmetrically above and below the real axis then there would be analagous bulbs in the appropriate positions not only with respect to the i axis but also with respect to the j axis (or in the appropriate rotated positions around parent bulbs in both i and j orientations).
The attractors of the orbits in the bulbs would have the same period as the attractors of the orbits in the analogous circles - this gives us a potential method for "artificially" creating the perfect 3D Mandy or maybe even a method of deriving the "correct" 3D algebra for z^2+c
Also of course the X-Y and X-Z cross-sections should both be identical to the 2D version but I think the above rules would give that anyway.

[bib]

It's funny no one has ever tried to do it manually instead of programmatically. At least for the main "apple" and the first level of spheres.

[Dranorter]

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!

[LesPaul]

Hi all, new to this forum.

I found the "Mandelbulb" in an article in New Scientist.  Amazing.  I've been an explorer of the Mandelbrot set itself for nearly 20 years now, and this is the most interesting development yet.

Anyway, to answer the original poster's question, in my opinion:

1) The 2D Mandelbrot set should somehow be a "shadow" of the 3D Mandelbrot set.  That doesn't necessarily mean that the 3D Mandelbrot set should be equivalent to the 2D set for Z=0, although that would be nice, I suppose.  But some very simple transformation should reduce the 3D set to the 2D set.
2) Simplicity.  The beauty of the 2D Mandelbrot is its simplicity.  All it takes is z^2 + c.  I was slightly disappointed to read that the gorgeous pictures in New Scientist were the result of 8th power calculations, and that they involved fairly complex trigonometry.  The 2D set can be calculated without complex numbers at all, in fact, with only multiplication and addition.
3) Self similarity.  The fact that the 2D Mandelbrot set contains near-perfect copies of itself is unfathomably incredible.  And it's even more incredible, perhaps, that these copies aren't identical to the original.

I don't mean to diminish the Mandelbulb in any way, because the images really are incredible.  But I get the feeling that it's really more of a first step to uncovering the real deal.

[David Makin]

Hi all, new to this forum.

I found the "Mandelbulb" in an article in New Scientist.  Amazing.  I've been an explorer of the Mandelbrot set itself for nearly 20 years now, and this is the most interesting development yet.

Anyway, to answer the original poster's question, in my opinion:

1) The 2D Mandelbrot set should somehow be a "shadow" of the 3D Mandelbrot set.  That doesn't necessarily mean that the 3D Mandelbrot set should be equivalent to the 2D set for Z=0, although that would be nice, I suppose.  But some very simple transformation should reduce the 3D set to the 2D set.
2) Simplicity.  The beauty of the 2D Mandelbrot is its simplicity.  All it takes is z^2 + c.  I was slightly disappointed to read that the gorgeous pictures in New Scientist were the result of 8th power calculations, and that they involved fairly complex trigonometry.  The 2D set can be calculated without complex numbers at all, in fact, with only multiplication and addition.
3) Self similarity.  The fact that the 2D Mandelbrot set contains near-perfect copies of itself is unfathomably incredible.  And it's even more incredible, perhaps, that these copies aren't identical to the original.

I don't mean to diminish the Mandelbulb in any way, because the images really are incredible.  But I get the feeling that it's really more of a first step to uncovering the real deal.

Actually the formula turns out to be quite simple in terms of real and complex - maybe you're forgetting that complex z^power can be written in trig terms ?
As for self-similarity I don't think there's been enough time to find copies of the whole object yet - finding such is more difficult in 3D than 2D. Of course there is definitely some self-similarity since the buds are made of more buds and so on.

[xenodreambuie]

I don't think the true 3D Mandelbulb is possible, because there's no way to make a version of complex numbers with the desired rotational properties in 3D. Spherical coordinates make sense and give the best visual results so far, but have some problems due to the compression of theta at the poles, and the double covering of phi. Any alternative can only fix these problems by introducing some other kind.

The polar compression causes features to be squashed in some places and I suspect is also why features appear stretched in other places. However it is a success in most other respects. Also, no matter how nice the high degree forms look, the second and other low degrees are where you have to look for comparing with 2D. It's also much easier to judge how well it satisfies criteria such as similarity to corresponding Julias, which have much more variety at low powers.

Looking at the Julias, the right choice of c gives a result with connected spheres in some powers. It does for power 2, 3, 4 but not 5, and some higher powers, but the value of c varies with power. Here's an inverse iteration render of power 4.

[Tglad]

I'm tempted to agree that a strict 3d analogy of the 2d mandelbrot doesn't exist. I think its related to the 'hairy ball problem' that states that there are no coordinates or vector field on a sphere that is singularity free. Also, the 2d mandelbrot is like a pealed open disk which gives the cardoid main shape, you can't peel open a sphere and turn it back on itself like that without stretching and compressing the surface.

However, I think a lot of people think that there does exist a 3d shape that shares all the important properties of the mandelbrot, even though there may exist no cross section that is the mandelbrot. Properties such as:
- infinite complexity, similar but different at different zoom levels
- a boundary surface with a fractal dimension of 3 everywhere; no drop
- no stretching
- the simplest possible object in this family
I would call such an object the true 3d mandelbulb

[Dranorter]

Tglad, those criteria sounds pretty good.

The hairy ball problem is a good analogy, too. There are properties we can't get in 3D. But I'm not sure what you mean with the peeled sphere. Many have proposed a dimple as the analogous warping of the main sphere. But the reason for the cardioid shape in the main Mandelbrot set is basically that when the imaginary part of a number is small and the real part is positive, it is not part of the Mandelbrot set, though the curve of the period 1 attractor does reach into the reals- so the real axis pokes a dimple into it. This is then mathemagically reflected in all the little self-similar minibrots.

So what I'm saying is, if a proposed number system has that same property regarding the positive reals, it will show either a dimple or a crease on the sphere; otherwise it will make something else.

Xenodreamblue. I would have said before seeing the Julia set above that it is hard to imagine a sphere coated with fractal spheres. However, I should have expected to see it given that there are infinitely many Mandelbrot julias composed entirely of circles- every near-cardioid of the Mandelbrot set contains a point which generates an all-spherical julia. so for any given Julia set there is a similar-looking Julia which is composed entirely of spheres.

(But there are some non-spheres on that one- squashed spheres. Are there Julia sets composed completely of near-perfect spheres?)

So now I'm curious which properties out of our collective list can be proven to be impossible, given that we have a couple intuitive arguments that a couple are.

Hi LesPaul, I totally agree that the Mandelbrot set shouldn't need to be a cross-section of a good 3D version. In fact in some ways it seems to me it shouldn't be. I would expect a 3D Mandelbrot set to have dendritic tendrils reaching out in all directions, and they would probably cause many cross-sections of the object to show disconnected areas.

Good to hear the thing got in New Scientist!

[LesPaul]

Actually the formula turns out to be quite simple in terms of real and complex - maybe you're forgetting that complex z^power can be written in trig terms ?

Good point.  Maybe the fact that the power-2 version is less "interesting" than the higher powers is what turns me off, personally.  It's a matter of opionion, I guess.  It's just that I never cease to be amazed that just  z' = z² + c produces such remarkable complexity in 2D.

So far, though, my own vote for best potential "true" 3D Mandelbrot has to be this:
http://www.fractalforums.com/3d-fractal-generation/true-3d-mandlebrot-type-fractal/msg7346/#msg7346
http://www.fractalforums.com/gallery/?sa=view;id=700

The animation showing the cross sections is extremely promising.  I think that it would be very worthwhile to see some much higher zooms of this object, with much higher iteration counts.  I also expect that really digging into this object is going to require arbitrary-precision arithmetic.

I fully understand how difficult this is in 3D, believe me.  I've done my fair share of fractal coding and am tempted to implement some software tools to take this on.  I just can't resist seeing a 3D zoom animation into the complex structures.

[bib]

@xenodreambuie : very nice julia render. The "bubbles on bubbles" effect is the most similar to what I would imagine for a 3D mandelbrot set.

Quick question about attractors and self-similarity : in the 2D M-Set, do the attractors in minibrots have the same properties as the main shape and surrounding circles ? (i.e fixed point in the cardioid, period 2 in the main circle, etc...)

[xenodreambuie]

(But there are some non-spheres on that one- squashed spheres. Are there Julia sets composed completely of near-perfect spheres?)

So now I'm curious which properties out of our collective list can be proven to be impossible, given that we have a couple intuitive arguments that a couple are.

The squashed spheres are due to the polar compression, which of course gets replicated elsewhere. There can't be Julia sets with all near-perfect spheres because there's no escape from the polar effects. In fact there are only zero, one or two Julia sets with the spheres for a given power. Moving away from those points gives less perfect spheres, and I didn't get all that close because the inverse render is slower there.

I don't do mathematical proofs; intuitive arguments and observations are my lot and I'm happy to see evidence otherwise.

[David Makin]

Quick question about attractors and self-similarity : in the 2D M-Set, do the attractors in minibrots have the same properties as the main shape and surrounding circles ? (i.e fixed point in the cardioid, period 2 in the main circle, etc...)

No, I believe I'm correct in saying that in the replicas the main mini-cardioid will have some period say p (in general the smaller the minibrot the higher the value of p) and then all the mini-bulbs (and mini-cardioids) on the minibrot have a period of n*p where n is the period of the corresponding full-size bulb - someone please correct me if I'm wromg.

[illi]

Hi guys, I made a new 3d mandelbrot topic if anyone is interested:
http://www.fractalforums.com/3d-fractal-generation/truerer-true-3d-mandelbrot-fractal-(search-for-the-holy-grail-continues)/