Some questions

[Cobbles]

Hello everybody,

first time poster here, and probably my questions have been answered before (but I can't find where). So, if you know the answer (or a link to the answer), I would be very grateful.

Let me start with my first question:

The simple, quadratic Julia sets can be found with the iteration z_(i+1)=z_i^2+c, where c is a (complex) parameter, and we want to answer the question whether a given z_0 (the starting point of the iteration) is a member of the Julia set or not.

Now, we know that the Julia set is connected when z=0 is a member of the Julia set. So, we can change the question a bit and ask for which parameters c do we get connected Julia sets. The answer to this question is obviously the Mandelbrot set, which we can get through the iteration z_(i+1)=z_i+c with the starting point z_0=0.

Now, in practice, we could ask the question whether a specific combination of (z_0,c) is member of the respective Julia set or not. My question is: would all the members (z_0,c) form a four-dimensional fractal? And if so, would some three-dimensional sections of this four-dimensional fractal produce a three-dimensional fractal? Obviously some two-dimensional sections of this object produce fractals. So I could imagine that some three dimensional sections could do the same.

Sorry if the answer is obvious.

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Since this is my first post, let me also tell a bit about myself. I work in physics (I have an advanced degree) and although I've seen my share of higher math, I'm an amateur when it comes to fractals. Recently I have seen an article about the 3D mandelbulb with reference to this forum, and I'm fascinated by the beauty of these objects. While I was a student (late 1980s to early 1990s) I remember producing a fair number of mandelbrot images for fun. It's nice to see a bit of a resurgence of the topic.

[bib]

Hi and Welcome to the forums!

The fractal you describe is a "Julibrot". It is famous for its whipped cream aspect, which can be nice, but at the same time disappointing!

Here is a video by Dave Makin:
<a href="http://www.youtube.com/v/gr-ul7sZDwc&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/gr-ul7sZDwc&rel=1&fs=1&hd=1</a>

[Cobbles]

Hello bib, thank you for the fast reply and the video. It's still a nice object but I see what you mean by disappointing.

So I was thinking about something else which I'm unclear on. Generally, Julia sets can be obtained for f(z)=p(z)/q(z) where p and q are polynomials (of finite order, hence no Taylor or Laurent series allowed, I assume). Now looking through the forum, I've seen a few fractals where f(z) wasn't a ratio of two polynomials. In particular, there were some graphs based on f(z)=z^p where p was not an integer. Now, strictly speaking, these things are not Julia sets and I assume one could also produce fractals with a bunch of other functions such as the usual transcendental and other higher functions.

My question is, how much of what we know of Julia sets and their connection to their respective Mandelbrot sets can be generalized to these kind of sets? Or, maybe it is better to ask the other way around: what cannot be generalized?

Actually, I'm also a bit unsure whether the quadratic function z^2 is in any way special among the rational functions to produce Julia/Mandelbrot sets. I know it's by far the most popular one and maybe I'm generalizing too much from that. Are there any pitfalls, i.e., things we know from z^2 but which are not true for proper Julia/Mandelbrot sets produced by other rational functions?

[Tglad]

Yes, the Mandelbrot set has a very important property, more important than any other, which makes it different to all other mandelbrot type sets and julia sets; it is universal.
http://www.math.harvard.edu/~ctm/papers/home/text/papers/muniv/muniv.pdf
Universal means that each iteration you can translate, scale or rotate the point by any amount and you still end up with a mandelbrot set (which might be translated or rotated or scaled). You can't say the same about any other fractal.
It is why you find mandelbrot sets turning up everywhere, like inside newton fractals and inside other polynomial mappings.

<Actually, all the multibrots (z^3 + c etc) are also universal, but the mandelbrot is the simplest non-trivial fractal in that family>
 

[Cobbles]

Hello Tglad, thanks for the link. Now, I can't say that I understand everything in the paper. It's been some time since I could remember the precise definition of all the mathematical terms and symbols. Some of them I might never have learned. So, in very, very simple and mathematically not precise terms, let me try to paraphrase (and maybe you can tell me whether I'm really off with this):

If you take some general polynomial of order 'n', locally, it might look like z^n, or z^(n-1) or any lower order such as z^2 and hence it will produce the respective multibrots. Now, some special polynomials such as z^3 probably are too 'symmetric' (that one for instance has one triple root) so probably it has no local region where it might look like z^2. So I guess if you want a nice variety of multibrots, you should use a function with several distinct roots which are not in any kind of symmetric configuration? For instance, using the function Product_i (z-z_i) with i=1..N and z_i=some random numbers, should produce all multibrots up to order N somewhere?

[David Makin]

From the point of view of Fractal Art rather than Fractal Math the definition of Mandelbrots and Julias is a lot looser - specifically a Mandelbrot is simply considered as any render where the orbits used to produce the image all start with a fixed value of z and some constant in the iterated formula (c) is taken from the position, whereas for Julias the start value of z is taken from the position and the constant (c) is fixed for a particular Julia Set.
In other words a Mandelbrot (as in most fractal programs) is the result of iterating a related set of different functions (generally the functions differing by the variation of the single constant c) and a Julia Set (again as in most fractal programs) is the result of iterating a single function just with different starting points.
Whether the functions involving c are rational, irrational, transcendental, use conditionals etc. makes no real difference provided one remembers that some attractors/attractor sets are at infinity, some single fixed points, some periodic (set of points), some non-periodic but finite (that may or may not be connected e.g. strange attractors) and some (that fit into none of the previous categories) simply an infinite set (that also may or may not be connected).
The last one is *very* difficult to distinguish from the others for instance this attractor (a single orbit) looks like a strange attractor even after many, many iterations but does gradually drift possibly even being divergent in the end....

http://makinmagic.deviantart.com/art/The-Infinite-Harp-40419447

As for formulas that reproduce copies of Mandelbrots, my personal favourites are the classic Magnet formulas and one of my own that came about by trying (against all reason) to find the true half-iterate formula for z^4+c which ended up as "The Magic Formula" in mmf3.ufm for Ultra Fractal (in the Ultra Fractal formula database - http://formulas.ultrafractal.com/ ) in which for certain versions of the formula (as set by the user parameters) there are n^i minibrots within each iteration band starting with the second (where i is either iteration or iteration-1 I forget which) - each being at the centre of a disconnected mini-Julia that matches the true Julia shape from around that location on the Mandelbrot.
If interested then see the comments at the start of "The Magic Formula" in the ufm text file.

[David Makin]

Are there any pitfalls, i.e., things we know from z^2 but which are not true for proper Julia/Mandelbrot sets produced by other rational functions?

I should have added a little more to the point I made regarding orbital attractors - I'm pretty sure that for standard z^p+c where p is an integer >=2 then the attractors involved are simply type 1 (divergent to infinity), type 2 (convergent to a point) and type 3 (periodic i.e. convergent to a finite set of points apart from those that are infinitesimally "outside") and it's possible to analytically derive a maximum magnitude such that if at any time the magnitude of z exceeds this value then we can know for certain that the orbit concerned will diverge.
Admittedly points immediately adjacent to the "inside" (i.e. infinitesimally outside the set) would of course require infinite iterations to identify

Unfortunately things get more complicated with other formulas - for instance for sin(z)+c if using the standard bailout test then the magnitude allowed needs to be very high and even then we know that the result is not correct because the fractal itself is infinite along the real axis.
In practice using a very large magnitude does work reasonably well for sin(z)+c *but* fails as the viewed area travels up or down the real axis away from the origin. If course it's certainly possible to use a more appropriate bailout test for sin(z)+c rather than simply using the magnitude.

Knowing the attractors involved and devising a means to identify them essentially allows any I(f(z,c)) to be viewed "correctly" at least to a finite resolution.

[Tglad]

Cobbles- I think that is right (and I find the paper impenetrable too, just get the drift from it).
Any smooth function f(x) can be build from an infinite polynomial, its Taylor expansion being a + b*x + c*x^2 +d*x^3 + ...
So in complex numbers its the same and any complex mapping can be written as a + b*z + c*z^2 + ....

So all such mappings are built from the multibrots, it is like the multibrots are the natural numbers of fractals, and the mandelbrot set is the number 1.
That's my interpretation anyway

[Cobbles]

Hello again.

So I haven't had time to answer yet, because I have hacked together my own little 2-d iteration program for J and M sets to try out a few things. I also discovered the thread http://www.fractalforums.com/new-theories-and-research/is-there-anything-novel-left-to-do-in-m-like-escape-time-fractals-in-2d/ which discusses some of the questions I have. Right now I'm following and recreating some of that stuff to see and learn. Then I'll come back here for more questions.