I just scanned the sci.fractals newsgroup for the past few weeks and found this post from Brian Quincy Hutchings <QncyMI@netscape.net>, dated 19 september 2003 23:27: <snip> but folks should know that the "universality of the M-set," that is the recurrence of "mini-bugs" or cardioids, at every level of "magnification," is just an artifact of the floating-point ops (IEEE-755, -855, I think). this was (really/partially) confirmed by monsieur M, when he glroriously begged my (only) technical question at a talk for a "general audience." <snip> Is this just some "besserwisser"-talk or might it imply that images containing minibrots/midgets (and there certainly are a *lot* of those on the Net and in the litterature) are just artifacts from some weird bug in the floating-point algorithms? The fact that the main cardoid itself stems from a very simple formula make it seem, if not probable so at least possible, that some oddities in the FPU could produce unexpected results. I find this a bit disturbing. Could anyone please care to comment? Regards Kenneth Lifvenborg _________________________________________________________________ The new MSN 8: advanced junk mail protection and 2 months FREE* http://join.msn.com/?page=features/junkmail
Is this just some "besserwisser"-talk or might it imply that images containing minibrots/midgets (and there certainly are a *lot* of those on the Net and in the litterature) are just artifacts from some weird bug in the floating-point algorithms?
The fact that the main cardoid itself stems from a very simple formula make it seem, if not probable so at least possible, that some oddities in the FPU could produce unexpected results.
I find this a bit disturbing. Could anyone please care to comment?
The Mandelbrot set comes from a very simple calculation. If you really want to, it's quite possible to sit down with pencil and paper and draw the M-set yourself, the oLd-SkHOoL way... It has nothing to do with bugs in floating-point algorithms. (If it did, I imagine computer's wouldn't work terribly well! Besides, FractInt for example has a fixed-point algorithm that produces the exact same results.) Plus there are mathematical explainations for the frequent occurrance of these figures - explanations that have nothing to do with floating-point bugs or anything else... Thanks. Andrew. PS. If there a Guinniss Book for Records entry to the highest-resolution M-set ever plotted by hand? If there isn't, there should be!!
Brian (quoted by Kenneth) wrote:
<snip> but folks should know that the "universality of the M-set," that is the recurrence of "mini-bugs" or cardioids, at every level of "magnification," is just an artifact of the floating-point ops (IEEE-755, -855, I think). this was (really/partially) confirmed by monsieur M, when he glroriously begged my (only) technical question at a talk for a "general audience." <snip>
This is patently ridiculous and easily refuted. Fractint has a number of different numerical methods of computation: IEEE floating point, integer (fixed point) math, and arbitrary precision, and one can see the same "mini-bugs" with all of them (within their range of magnification). If these "mini-bugs" were artifacts of IEEE floating point, if one changed the math to non-IEEE (e.g. integer math or arbitrary) the "mini-bugs would go away. But they don't. All three math implementations show very similar images. SImilarly, with fractint one can force a higher precision using aribrary precision. Typically when one does this, one gets the same (or very nearly the same) image. When increasing precision does not change an image, that is pretty good evidence that the image is not an artifact of the math implementation. Just goes to show you that anyone can write anything on the internet, whether or not it makes sense. Tim
Just goes to show you that anyone can write anything on the internet, whether or not it makes sense.
Precisely. (Just like when we were at school, and everyone I knew seemed to use the logic "it's in a book, so it MUST be true"...) ((I went to school with some amazingly stupid people!))
["Minibrots" of Mandelbrot set a floating point artifact?] Well, *most* real number cannot be represented exactly by floating point math because of the finite width of the mantissa (think of 1/3 in decimal or 3/5 in binary). Therefore nearly any iteration calculation will accumulate errors, which causes the computed values to "drift away". But: As others already have pointed out, if the "fine structure" of the Mandelbrot set were an artifact, then a change in number format (64 bit, 80 bit, arbitrary precision) should at least shift those minibrots around, if not deform them beyond recognition. Strangely enough, whatever hard-, whatever software is used, they always reside in the same spot, easily found on machine B using coordinates calculated on machine A. In "Chaos and Fractals: New Frontiers of Science" by Peitgen/Jürgens/Saupe (1992) there's a whole chapter dedicated to this question (p. 575 ff. under "Numerics of Chaos: Worth the Trouble or Not?"): As it turns out, although the small rounding errors while iterating cause the calculated orbit to wander off the real one, it is provable that, *because* of the limited size of each error at every iteration, there *always* will be *another* real orbit close to the calculated one - and its starting point will be near the starting point of the calculated orbit, too! In short, we might not get the behavior of the point at the *exact* pixel coordinates, but of one that is so close, it in all likelihood will lie in the area of the same pixel anyway. The true reason why there are tiny Mandelbrots in the Mandelbrot set (and in other complex iteration maps, too) is the fact that polynomials of high degree (and many other functions) often have small areas in their domain where they look just like the quadratic map. BTW, an interesting site regarding the math of the MSet: http://www.people.nnov.ru/fractal/MSet/Contents.htm Regards, Gerald
participants (4)
-
Andrew Coppin -
Gerald K. Dobiasovsky -
Kenneth Lifvenborg -
Tim Wegner