An abstract grail.

[Schlega]

I'm terrible at explainning things, but I'll try to be understandable.

    I've been thinking about Lavaurs' algorithm, and trying to come up with a 3D extension of it. The standard formulation is based on rational points on the unit circle, but I found an alternate form that allows a natural extension to arbitrary dimensions.
    I don't think the abstract mandelbrot set is generally well known, so I'll put a quick summary here:

    To build the set, measure the angles on a circle in radians/2pi (so the angle is in the range [0,1)). Connect the points at 1/3 and 2/3. Increase the denominator until it is possible to pair off the points 1/n, 2/n, ... (n-1)/n in such a way that none of the connection cross the chords connecting other points. In addition, it is also required that the chord from a to b must stay in the semicircle cut by the diameter from a/2 to (a+1)/2. the point with angle 0 is never included in the connections.
    For each pair of connected points, collapse the chord connecting them to a single point. Assuming the mandelbrot set is locally connected, this will be topologically equivalent to the mandelbrot set.
    It turns out that the kth denominator can be defined recursively by . In addition, the component cut off by chords with the kth denominator correspond to period k components of the M-set. You can determine whether a component is a disk or the cardioid of a minibrot by looking at the period of the component and the period of the component it was cut off from. If the smaller component divides the larger, it will be a disk. Otherwise it will be a minibrot.

    My first attempt to generalize was based on the fact that the rational points with denominator n give a solution to the Thomson problem on the circle. If the solutions on a sphere were known, they could be the points used. Instead of connecting points in pairs, they would be connected in triples, since it takes 3 points to specify a unique disk. Due to lack of symmetry, it would be unlikely that this approach would result in a unique fractal.

    Another option I tried was to start with four points arranged tetrahedrally, with one point on the x-axis and a disk connecting the other 3. I labeled those points as 1/4, 2/4, 3/4, then changed to denominator 13 (3*4 +1).  This resulted in four more disks, so I just put them at the center of each face of the tetrahedron. I tried continuing the process, spreading new disks as uniformly as possible, but hit a problem when I reached denominator 364 and needed a way to add 115 new disks.

   Eventually, I realized that the topology is encoded in the denominators. For each , label the fractions {} on the interval (0,1). Connect them in triples in such a way that none of the lines cross each other. If a triple is connected in between points that were connected in previous steps, then the new disk will be contained in the old one.

   In this image, the black disk represents the surface of a punctured sphere (the removed point corresponds to (1,0,0)). In the 3D fractal, this would be the main cardioid. The smaller disks are colored according to the "period" of the component it represents, where "period k" just means that the disk corresponds to the kth denominator:


I'm trying to write a program to automate the process and build a proper 3D model of the result, but so far progress has been slow.

[Sockratease]

I'm trying to write a program to automate the process and build a proper 3D model of the result, but so far progress has been slow.

I have been known to turn images into 3D models with varying degrees of success.

I'd be glad to give that one a shot - it looks simple enough.

I just "vectorize" the image, then import the vectors to a 3D Program and extrude.

I can post a couple examples of this if you want to see some finished product.

But I'd need some idea of what the colors actually represent (a height map of sorts? Holes?  some combination of both?)

I do all this manually and with no math, so just a general explanation is needed.

Interesting design.

[hobold]

I don't think I fully understood the construction, but I find the idea very promising to start from a very different algorithm.

In the image you posted, there seem to be pairs of disks contained in larger disks. But in the text you write that three points are needed to specify a disk. It is not clear to me what the coloured disks represent, and how they are related to the rational points on the interval (0, 1).

[Schlega]

Here's the connection between the disks and the points in (0,1):


Each path joining 3 points represents one of the disks. The disk (1,2,3)/4 (dark red in the previous image) contains {(4,5,6), (7,8,9)}/13 (bright red), {(11,12,19), (13,14,15), (16,17,18), (21,28,29), (22,23,24),(25,26,27)}/40 (orange), and 20 of the /121 (light green) disks.

Similarly, the (11,12,19)/40 disk contains two /121 disks that are not contained in any sub-disk. One of these is empty, while the other contains (4,5,6)/13 and everything below it.

Sockratease: To make the 3D version, the boundary of each disk are collapsed to a single point. The black "period 1" disk becomes the main cardioid. The dark red "period 2" disk should be a sphere attached to it. I'm not sure about the condition for determining whether a component is spherical or a cardioid, but for now, I'd be happy with just making them all spheres to get a general idea of the shape.

[Schlega]

I put together a few spheres to approximate the 3D version. I still need to work on a way to determine the size and placement of each bulb, but this is the best I can do for now:

[Schlega]

The results so far have not been what I expected, but I have found an explicit mapping from the rationals to the surface of the sphere that gives a much more promising result. (There are some period 5 bulbs in this, but I was too lazy to put all of them in.)

[hobold]

Well, there will be a general problem that the alternative algorithm "only" produces the topology, not the actual shape (in the case of the 2d Mandelbrot). So this new approach might not give us the complete 3d analogue. But it would provide us with a significant amount of information. Possibly enough to deduce the "true" shape from it.

[Schlega]

True, it won't be the real grail, but I think it can at least serve as a map to Castle Aaargh 

I'm hoping to find a spherical analogue of the doubling function that gives the right topology. Once that is found, defining z -> z² by r -> r², (theta,phi) -> D(theta,phi) at least has a chance of being the real thing.

[kram1032]

no matter how close that would be, it definitely looks nice

Just wondering, I'm not an expert on that subject in any way but I read in  couple of places, the Mset's border is topologically equivalent to a circle. If now the abstract Mset has the same topology, wouldn't that mean basically the same thing? It's topologically a circle?

[hobold]

We're actually being a bit sloppy with terminology here. The Mandelbrot Set border being equivalent to a circle would be the strict mathematical meaning of "topology". Unfortunately, we've used both the mathematical and the informal meaning, where the informal meaning could be formalized to "what is close to what". In that informal sense, the top and bottom spikes with the seahorse valleys, for instance, are regarded as touching each other at their tips - but in the strict mathematical sense they do not touch. (In the case of the elephant valley, both definitions of topology agree, as there is nothing else around that could touch the tip of that other spike. That means the new algorithm might not enable us to distinguish between disks and cardioids.)

As for the overall mathematical topology of the Holy Grail ... well, we don't know that yet, because we don't know yet what the Holy Grail looks like exactly. But we might be able to generalize an algorithm that can derive the informal topology of the 2D Mandelbrot Set, and lead us to a collection of touching spheres that would be one "arbitrary" example of a uniquely defined topology.

[Schlega]

Sorry, I should have been more specific. I meant the topology of the union of the circle and the chords drawn by Lavaurs' algorithm. I don't think that is topologically equivalent to a circle, but I could be mistaken. Perhaps the topology of its compliment would be more appropriate?
  In the 3D case, it would be the union of the sphere and disks connecting periodic points of the doubling map analog.

hobold - you're right that it even if it works, it would probably not be "the" grail, since there are still other ways of generalizing the M-set. I just got a bit over excited because this has the same general shape as my personal mental image of what the 3D M-set "should" look like.

[fracmonk]

Schlega- I see you've been busy. Cool.  I saw your construction, and it kinda reminds me of when I was in the first grade, doodling a brick wall on a page, brick by brick...it was the hard way, but it passed the time.  Wouldn't it be grand to have the spheres automatically positioned?  I have a notion of an interesting construction that says something about proportion.  Gravity helps here: you have 4 equal size spheres and arrange them so that each touches the other 3, and their centers become vertices of a tetrahedron.  What is the largest 5th sphere that can fit in the space between them, touching them all?

I doubt if that has much to do with this, though, but it's related to 3-d apollonian gaskets directly.

[twinbee]

Schlega, your latest image looks, well, enticing to say the least. Have you tried building a 2D version to see if it starts to match the 2D Mandelbrot? It would be quicker to see the results.

[Schlega]

Schlega- I see you've been busy. Cool.  I saw your construction, and it kinda reminds me of when I was in the first grade, doodling a brick wall on a page, brick by brick...it was the hard way, but it passed the time.  Wouldn't it be grand to have the spheres automatically positioned?  I have a notion of an interesting construction that says something about proportion.  Gravity helps here: you have 4 equal size spheres and arrange them so that each touches the other 3, and their centers become vertices of a tetrahedron.  What is the largest 5th sphere that can fit in the space between them, touching them all?

I doubt if that has much to do with this, though, but it's related to 3-d apollonian gaskets directly.

That's an interesting idea. I wonder if there's a 2-d apollonian gasket that mimics the scalings in the M-set.

Schlega, your latest image looks, well, enticing to say the least. Have you tried building a 2D version to see if it starts to match the 2D Mandelbrot? It would be quicker to see the results.

Here's a figure from a paper I found while I was reserching Lavaurs' algorithm:

[KRAFTWERK]

Must admit I do not get the maths here, but it shure looks interesting.

@Schlega, in the link in your first post there is a link to this:


http://classes.yale.edu/fractals/MandelSet/MandelCombinatorics/LavaursAlgorithm/Periods.html

Kind of the same as the image in your latest post.

...just my 5 cents...