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Hi,

I'd like to be able to get some really big magnifications of Julia sets and Mandelbrot sets. Do any of these programs have "arbitrary precision" calculating, as Mathematica does? That is, not limited by "machine-size" numbers or that kind of thing. I've tried using Mathematica on my old iMac to graph fractals, but it's very frustrating. I can't figure out how to save intermediate results, and it goes on for hours with no indication of how long it will take. I've also got a MacBook Pro (with 2.33 GHz Intel Core 2 Duo, 2 GB 667 MHz DDR2 SDRAM, running Mac OS X) and so I think I've got a lot more computing power than with the iMac. However, I can't afford to upgrade Mathematica so I can use it on this machine. (I got a student license for Mathematica which was tied to one computer.)

Any info on how much detail I can expect with the various programs at Gandreas, such as Quadrium | flame and Quadrium | prime?

I'm interested first and foremost in trying to understand where all the points are "going" as we increase iterations. Any way to color points by their "escape velocity" and that sort of thing? And are there ways to add more information to the plots of the points in the Julia and Mandelbrot sets, such as plotting their orbits as we increase the number of iterations, or showing their velocity?

I may be a bit confused about the nature of the points in the Julia and Mandelbrot sets, but just from the graphics point of view it would be nice to have something other than a grim-loooking black blog in the middle. I think Quadrium | Prime had some of the most interesting plots of Julia sets and Mandelbrot sets I've seen, and I'm leaning towards this.

Any info appreciated.

Thanks,

Chris Young
c56y@comcast.net

Hi Chris,

There are a number of programs that do arbitrary precision (though I'm not sure exactly which apart from Ultra Fractal), but I don't know of any native to the Mac OS.

However there are ways of running Windows software on your Mac Pro, to quote from the Ultra Fractal faq:

"Ultra Fractal is only available for Windows. There are currently no plans to support other platforms. However, you can run Ultra Fractal on a Mac with VMware Fusion, Parallels, Boot Camp or Virtual PC (PowerPC only)."

I'm pretty sure that other Windows based fractal software would work in a similar way - the only other programs I'm fairly sure do arbitrary precision are Fractint and ChaosPro. Even then I'm not sure - I tend to avoid arbitrary precision because of the render times involved (I very rarely zoom in beyond *1e10).

As to colouring methods based on things like "escape velocity" then using either Fractint or Ultra Fractal if you can think of a method of colouring a fractal (either "inside" or "outside") then you'll probably find that there's a colouring formula using that method already written and publicly available. Also ChaosPro can use Ultra Fractal formulas up to version 3/4. Ultra Fractal can use most Fractint formulas and I think ChaosPro probably can too but I'm not sure.

Hope that helps
Dave

Consider that the programs zoom in to areas the size of hydrogen atoms.

How much to trust the pictures of the programs is a question, even with the use of arbitrary precision libraries, and many iterations.

It is discussed in this article; (it's in PDF format)

http://mathdl.maa.org/mathDL/22/?pa=content&sa=viewDocument&nodeId=2690 (http://mathdl.maa.org/mathDL/22/?pa=content&sa=viewDocument&nodeId=2690)

Just to point out that assuming no actual bugs in the software then the greater the precision used the more accurate the results and hence the deeper you can go into the Mandelbrot without noticeable errors.

I think the article just states the obvious i.e. that there's no way of finding some infinite limits to 100% accuracy.

The only point I think should be taken from the article is that no matter how accurately it's calculated, it's impossible to represent the "exact" Mandelbrot Set so all such fractals are approximations - but then again so is just about anything represented by computer graphics.

Hi folks  O0

Very interesting paper. Some thoughts :

Why Mandelbrot images are nice is mostly due to the coloring formula, not really the shape of the M-Set itself. I'm a bit provocative here  :evil1:

Although I'm not an expert neither in computer science nor in mathematics (although I have solid knowledge in both) I'm convinced that the deep zoom pictures produced by computer programs CAN be trusted.

Indeed, the M-set has an infinitely complex boundary, while being connected and compact. Imagine you visualize a perfect picture (I mean calculated with an infinite number of iterations) of an "infinitely" small part of the boundary. Color in black the points that stay bounded, the points that really stay bounded after an infinite number of iterations, which is very theoretic I know, and in white the points that escape. I think you should only see very simple shapes : cardioids (or kind of) and circles (bulbs), no filaments at all, because filaments have no thickness, although it's not demonstrated (is it?).

So for example you should not see spirals at all, because spirals are only made of filaments, each part of any filament having of course an infinite number of minibrots on it, but these are too small and sparse to distinguish the global spiral shape. Maybe you could see it if human eye precision was higher and if we could build screens with pixel smaller than human eye resolution and computers that could calculate an (almost?) infinite number of iterations.

I agree you never know if some points will escape at some very very high number of iterations, but it's quite predictable graphically: when you have navigated the M-set a few tens of hours, let's say you can "anticipate" what the next image will look like when you zoom further and further, thanks to self-similarity and inheritance of previous level graphical properties and shapes. OK the M-set describes some chaotic behaviour, but it's determinist.

So is the question of the area now solved ? That question is really interesting, to me it's almost philosophical  :angel1:

I'm probably too lazy to browse google (I like better browsing fractals :evil1:), and I really appreciate this forum, where this question of the area has probably already been asked, so could anyone please provide some links for my education ?

Another connected question : How do we explain that the M-set shape can be found in other types of fractals when you choose good parameters and zoom a little bit? (e.g. in the Herman ring family)

Thanks  ;)

bib

First of all, Greetings, and Welcome to this particular Forum !!!     :D

I understand what you are looking for, so it can run on a MAC.  I also understand your predicament concerning the spending of money, since you are a student.

But... have you thought about acquiring an old PC that some company, organization, or person is getting rid of??  I get them all the time for FREE, just because they do not have the time to deal with trying to dispose of or sell something that is not worth the trouble.

You could then run a FREE application such as FractInt (http://www.nahee.com/spanky/www/fractint/fractint.html) or ChaosPro (http://www.chaospro.de/) without any problems.

FREE computer + FREE software = FREE large fractals using "arbitrary precision" !!     ;)    :D
 
And if you can get your hands on enough of these older machines, then you could start producing animations with them.  Just let them run until they finally cannot run any longer.