Talk:Mandelbrot set/Archive 1

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Opening Paragraph

The opening paragraph is too wordy and too technical. Most Wikipedia articles are:

Title
Brief summary
Table of contents
Detail

I would have fixed it myself, but I can't think of any good way to summarize it briefly -- which I guess is why it's like this in the first place. Does anyone have any ideas? Ravenswood 16:18, August 3, 2005 (UTC)

Agreed - I came here looking for a brief summary to copy and edit into the glossary of a paper I'm working on. Instead I'm going to have to mangle my own one. Hope someone picks up on this - I can't see a way to create a clear intro. 62.253.219.238 14:10, 7 December 2005 (UTC)

I hope the new opening paragraph is more useful --- it is a more mathematical definition of the Mandelbrot set. Suggestions are welcome. --Lasserempe 17:58, 16 March 2006 (UTC)

The new opening paragraph maybe more concise now in terms of a mathematical definition, but it is still way too technical for an opening paragraph. I believe the current opening paragraph should be moved to appear after a new opening paragraph that is more suitable for wider audiences to understand what the Mandelbrot set is.

This encyclopedia exists to spread knowledge as widely as possible and the introductory paragraphs should be understandable by a lay person where possible, especially with something as interesting and beautiful as fractals and the Mandelbrot set.

Also the black and white that was the first image of the article is perhaps the most boring picture of the Madelbrot set I have ever seen in my life. The initial picture ought to be showing the Mandelbrot in all its glory. That black and white image can come later. (I have now done this).

I am not trying to distract from the mathematical aspect of the article at all, I just think this subject will reach a wider audience with the correctly pitched 'front end'. I would gladly write a suitable introductory paragraph myself if I was in a position to do so, but I'm afraid I'm pretty ignorant on the subject. --CharlesC 22:18, 16 June 2006 (UTC)

Having re-arranged the paragraphs in the introduction so the maths comes after some words in plain English, I think it's much better. --CharlesC 23:49, 17 June 2006 (UTC)

Yeah, that's fine. It's better to explain what it is in english before diving into the math. Expecially with such an innocuous looking function that turns out to be this complex. No-one knew what it really looked like until about 1980 because it's not really decipherable by simply reading the formula. - Rainwarrior 05:07, 18 June 2006 (UTC)

I made the black-and-white rendering of the Mandelbrot set because I looked all over the Web and couldn't find a rendering of the set which included the axes. I have a degree in math, so it bugs me when the mathematical nature of the subject is obscured to the point that people don't realize that the Mandelbrot set is a plain old subset of C, that points are either in or not in the set (laypersons confuse the infinite sequence with the set proper), that the set isn't "special" and in itself does not feature "pretty colors," and -- as one generally does with sets of points in C -- it is not unreasonable to render this set with axes. I think it's misleading to have a colored picture labeled "Mandelbrot set" at the top of this article, because the Mandelbrot set is not colored -- it's a set, it has a membership operator that maps to the booleans, it's black and white. However, I'll go along with the colored Mandelbrot image if people like it. Moreover, the article is currently confusing, as it mashes together the formal mathematics of the Mandelbrot set with the hobbyist's perspective of the Mandelbrot set. I mean, read the introduction: "connectedness locus...family of complex quadratic polynomials...Julia set...connected...critical point...n-fold composition." While off to the right is a picture from the hobbyist's perspective (labeled "Mandelbrot set," which is formally incorrect, as it's an image made from the "Mandelbrot sequence" and not the set). I guess I don't really like the edit to the introduction section by User:Lasserempe [1] and I much prefer a succinct introduction which defines the set M from a sequence, such as [2]. The formal math stuff can be moved into a formal math section. The advantage of the "sequence definition" is that it allows the person with only a high school math background to appreciate the beauty and simplicity of the Mandelbrot set's mathematics. I can add an aside to explain to the reader with little math background that the "Mandelbrot set" itself is "black and white," and that the color is generated by looking at the beginnings of the infinite sequence which determines the "true" Mandelbrot set M. What do you guys think? - Connelly 03:42, 3 August 2006 (UTC)

While I would agree that the black and white image with axes is the best image for demonstrating the mathematical definition of the mandelbrot set, I think the current image is more suitable for this page. The reason I say this is that I think the page should proceed from "layperson" information at the start to "expert" information later on, so that a person who does not yet have the skills to interperet the mathematics of it does not have to sift through technical descriptions to get to the information they want to know. I think most people who come to this page probably want to know something about this pretty thing they've heard of or seen called the mandelbrot, and the best information to leave at the top is maybe a bit of its history, and a general description. (We should have the formula there too, but try to explain it in less technical terms.) After the top couple of paragraphs, we should be free to get mathematical, as it is a very mathematical topic, but we seriously need to tone down the jargon. e.g. We should mention that is is a connectedness locus somewhere, but definitely not in the lead. Put the simplest explanations first, and then introduce technical terms after. - Rainwarrior 04:26, 3 August 2006 (UTC)

I agree with Rainwarrior. Also just to say I didn't mean to be rude or dimissive about the black and white image and am aware it is simple and correct and of course it belongs in the article. With the coloured image I was intending to 'draw' people into the subject, then when their interest is piqued they are more likely to get into the maths. -- CharlesC 14:49, 3 August 2006 (UTC)

Bail-out value

I always assumed when I used to play with Fractint that it was "bail-out" value -- as in "after so many iterations, we bail out". The article currently makes it sound as it it's named after something: "Bailout value". Could someone clarify? -- Tarquin 09:22 Aug 24, 2002 (PDT)

It should be "bail out" BEE.

It should be "bail-out", with a hyphen. I've changed it. Michael Hardy 01:30 Feb 16, 2003 (UTC)

Now, who donates a nice picture? AxelBoldt 01:51 Feb 16, 2003 (UTC)

There was already one in the system. It was used in earlier versions of the page, but somehow the link got removed at some point. I've put it back in. You can move the picture if you want. Or possibly replace it with another one - personally, I prefer it with the real axis horizontal and the imaginary axis vertical... -- Oliver P. 02:08 Feb 16, 2003 (UTC)

Where's the real axis right now? Also, the self-similarity isn't very well visible in this one. AxelBoldt 03:25 Feb 16, 2003 (UTC)

Why, it goes right through the middle from left to right, of course! I rotated the image, you see. I'm not sure the self-similarity can be shown in a single small image, though, can it? Maybe we should have a sequence of images, gradually zooming in on some point on the boundary... -- Oliver P. 14:12 Feb 16, 2003 (UTC)

Sometimes, if the picture is a bit larger and shows enough detail, one can actually see that parts of it look similar to the whole thing. But a sequence of zoomings would be really nice too. AxelBoldt 21:34 Feb 16, 2003 (UTC)

Check out Mandelbrot1.jpg, Mandelbrot2.jpg ... Mandelbrot6.jpg. I generated them with a program I created using Cycling 74's Max/MSP/Jitter. They are a nice sequence of zoomings. I wonder if anyone might be able to find information about ways to speed up the generation of the mandelbrot fractal, i.e. good, fast algorithms. Snotwong 15:23 1 Jun 2003 (UTC)

"Whilst it is of no mathematical importance, most fractal rendering programs display points outside of the Mandelbrot set in different colours depending on the number of iterations before it bailed out,"

Why is the number of iterations of 'no mathematical importance'? The original definition of the set, ignored the iteration numbers, but this is it, was there any study about the patterns for different interation number classes? Could someone explain what is ment by 'no mathematical importance'?

I also wonder about the lack of mathematical interest. Here's an example of potential mathematical interest: consider the "coastline" example often given as a fractal in nature. If we measure the coastline of an island using gross cartographic techniques we get a certain distance. If we use finer cartographic techniques we get a larger distance. The "closer" we look at the coast, for example, down to the grains of sand on the shore or beyond, the longer the coastline will measure. Now consider a set such as Mandelbrot. If you look at a high iteration number you get an approximation of the circumference of the set, if you take the next lower iteration number you get a larger circumference that sits closer to the edge of the set. This continues as you get closer to the infinitely fractal border of the set. Can we find a mathematical relationship between the difference in circumference and the iteration number? If so there may be applications in Geographic Information Systems (GIS). [mark.dixon@uwa.edu.au]

The contour of the Mandelbrot set is not like a coastline because it is loaded to the brim (and then some) with pinch points (cusps). Coastlines do not have pinch points. --AugPi 01:58, 13 Jun 2004 (UTC)

"Whilst it is of no mathematical importance, most fractal rendering programs display points outside of the Mandelbrot set in different colours depending on the number of iterations before it bailed out," - "Why is the number of iterations of 'no mathematical importance'? ... Could someone explain what is ment by 'no mathematical importance'?"

I can't speak to the mathematical importance or unimporance of the coloring, but the visual importance (besides looking pretty!) is that much of the structure of the Mandelbrot set consists of infinitely-thin filaments. Since they are (believed to be) infinitely thin, they would not show up on any picture, no matter how detailed. However, the bail-out values closely follow the filaments, but give them a bit of thickness, so they can be seen. Ravenswood 18:47, August 4, 2005 (UTC)

Circles, cardioid

AugPi, I have an issue with your June 12 edit. The following statement is false, as can be shown by the infinite number of "mini" Manelbrot sets attached to and surrounding the main set: The Mandelbrot set can be divided into an infinite set of black figures: the largest figure in the center is a cardioid. The rest of the figures are all circles which branch out from this central cardioid. Mackerm 05:49, 26 Aug 2004 (UTC)

I recently found that the circles attached to the central cardiod can be asigned different rational numbers between 0 and 1 in numberical order. What is the mapping from the boundry of the cardiod(excluding the cusp) to the interval (0,1)?--SurrealWarrior 18:19, 1 Jun 2005 (UTC)

I have included a section on the main cardioid, and the hyperbolic components bifurcating diretly from it. --Lasserempe 17:54, 16 March 2006 (UTC)

Mandel and the bifurcation.

Anyone to add a note about this in the article, (it is pretty large and a bit messy so I want try to do that. If I do, somebody (maybe you) WILL change it, I'm sure (because I'm Swedish and my writing in English is not perfect, somebody (maybe you) always change the addings I have done to the fractal articles here at en: ), better you write it from scratch =) // Solkoll 20:58, 8 Jun 2005 (UTC)

Every quadratic polynomial is conformally equivalent to exactly one map of the form z^2+c; in particular this is true for polynomials of the form \lambda z(1-z). I may add a comment on this if I have time. --Lasserempe 17:57, 16 March 2006 (UTC)

Are the filiments of the Mandelbrot set infinitly thin or do they have a finite thickness?--SurrealWarrior 29 June 2005 20:16 (UTC)

In many of them you can find tiny "copy" mandelbrots, which have a definite thickness, but it is possible to find points that appear to be infinitely thin on the mandelbrot; for instance, the "neck" between the large cardioid and the circle to its left I believe is infinitely thin. So, I'm not sure that your question has an answer one way or the other; there are both infinitely thin points and finitely thin points. - Rainwarrior 23:59, 18 June 2006 (UTC)

Questions

Moved following paragraph from article page, as it consists mainly of questions. Gandalf61 09:30, August 12, 2005 (UTC)

It is said that the Mandelbrot set is a cardioid, but it is not clear if this is just looks or if theer's some mathmatical definition. It is also unclear why Mandelbrot set is shaped the way it is. Also, it is not widespread a knowledge, perhaps due to its esteric nature and the flood of computing enthusiasts, about the significance of Mandelbrot set in mathematics.

Why are captial Z and minor z in the formulas mixed?

For no good reason. I've changed it now.

What, exactly, is meant by "infinite set"?

Infinite set is a set which is not finite. -- EJ 10:09, 15 August 2005 (UTC)

Scope and limitations of Mandelbrot art

One can wonder how long it would take a room full of monkeys with computers to reproduce the works of van Gogh. He painted with a finite number of molecules of paint which have a finite number of permutations, while the set has an infinite range of coordinates, so they all may be in there somewhere. Or are there mathematical limits on what a Mandelbrot picture can look like?David R. Ingham

My brother wrote a program to write poetry that had one accepted for publication. When he admitted how he wrote the poem, the periodical changed its mind and rejected it.David R. Ingham 04:07, 1 December 2005 (UTC)

Simply connected / path connected?

The article contains the sentence: "...the Mandelbrot set is connected, and even simply connected. It is conjectured but unproven to be path connected."

According to the simply connected space, a set is simply connected if it is path connected and all loops can be continuously shrunk to points. Therefore, it seems absurd to say that the Mandelbrot set is simply connected but may or may not be path connected. Is this a typo? Or is there another definition of simple connectedness that does not include path connectedness? Reedbeta 01:09, 12 December 2005 (UTC)

See [3], [4]. The Mandelbrot set is connected, and any loop can be deformed to a point. This is indeed a weaker definition of "simply connected" than what is given in the simply connected article. -- EJ 15:59, 12 December 2005 (UTC)

It appears, from looking at it, that the connections do not always have finite width. That is, spirals can be zoomed in on by almost a factor of 10**20 and still remain spirals with apparently point vertices. Does anyone know mathematically if that is true? David R. Ingham 19:18, 10 January 2006 (UTC)

Ideal computer

Except for "accelerator" short cuts, it is a point by point calculation, so it would be best done with parallel arithmetic units. The the requirements for memory and connectivity are minimal. David R. Ingham 04:43, 17 December 2005 (UTC)

Better yet would be a field programmable gate array. David R. Ingham 19:08, 6 January 2006 (UTC)

Change in formula

(I also posted this question on the Talk:Benoît Mandelbrot page) The formula as originally presented by BBM was z -> z² - c but almost every single current reference uses z -> z² + c. Anybody know when and why this was changed? Khim1 14:23, 17 January 2006 (UTC)

Mandelbrot's contribution to The Beauty of Fractals (Peitgen and Richter; 1986) refers to "the quadratic map z -> z² - c". In 1988 Michael Barnsley uses z -> z² - λ in Fractals Everywhere. OTOH Peitgen and Richter's 1986 survey article Frontiers of Chaos, also published in The Beauty of Fractals, uses the map x -> x² + c and includes illustrations of the Mandelbrot set in its "modern" orientation. I don't know why the second form became more popular. Gandalf61 11:55, 23 January 2006 (UTC)

I am not sure about the historic development, but certainly Douady and Hubbard, in the early 80's, defined the Mandelbrot set in its current form. In this parametrization, the parameter c agrees with the singular value of the map

f_{c}\,

, which is useful from a conceptual point of view. --LR 22:56, 17 March 2006 (UTC)

Pseudo code vs QBASIC

An anon user User:86.129.85.127 replaced the pseudo code with this (as being more compact):

QBasic code for plotting a Mandelbrot set.

SCREEN 12 
FOR sy% = 0 TO 479 
   FOR sx% = 0 TO 639 
      x = (sx% - 320) / 160: ox = x 
      y = (sy% - 240) / 160: oy = y 
      FOR c% = 15 TO 1 STEP -1 
         xx = x * x: yy = y * y 
         IF xx + yy >= 4 THEN EXIT FOR 
         y = x * y * 2 + oy 
         x = xx - yy + ox 
      NEXT 
      PSET (sx%, sy%), c% 
   NEXT 
NEXT

I've rolled it back, (and notified the anon) please discuss here. Rich Farmbrough. 23:46, 21 February 2006 (UTC)

I agree with the revert. I know a lot of programming languages and assembly languages, and one hardware design language, but the QBasic is not clear to me. I once would have assumed that everyone using computers would know Fortran. David R. Ingham 06:13, 22 February 2006 (UTC)

I agree that QBASIC is unreadable, but it would be a nice way if the article could have a pointer to where to find it, since it does serve the purpose of being code that can be run by the reader as a demonstration. However, I wouldn't really select QBASIC as my first choice for this purpose, either. Senatorpjt 12:25, 28 February 2007 (UTC)

illustration of sequences

That makes the mathematics much more intuitive. I only wonder whether it is 5 or 6 cases. David R. Ingham 05:18, 4 March 2006 (UTC)

Visibilty of Image:Mandelset hires.png

I can't see it, when viewing the Mandelbrot set page, on Safari or Netscape, on my Mac. (I read that FireFox is more "politicly correct" but have not tried it yet.) David R. Ingham 05:35, 4 March 2006 (UTC)

I can't see it with Safari or Firefox, nor with IE on the PC. I suspect that this is because of the fact that the image is just an empty, transparent 325x235 frame. The file is only 477 bytes in size. Khim1 21:00, 4 March 2006 (UTC)

I too first thought it was some temporary error or my browser, so I thought it would be better once my or wikipedias cache was reset. Ah well, I fixed it by changing the size of the thumbnail slightly so Wikipedia had to rerender it. --David Göthberg 23:39, 4 March 2006 (UTC)

Thanks for finding a workaround. It is a nice image. David R. Ingham 00:51, 5 March 2006 (UTC)

Generalizations

I have removed the reference to the "Burning ship" fractal, of which I have never heard before (which may not say much), and which seems to have been included by one of its originators. However, a general mention of non-analytic dynamics is well-merited, and I have included a paragraph to this effect. This includes a reference to Milnor's 'tricorn', which I believe is the most popular of these objects. --Lasserempe 15 March 2006 (UTC)

I have also moved this section forward, to keep the mathematical material bundled together. --Lasserempe 18:00, 16 March 2006 (UTC)

Cycles/Pictures

I am going to remove the new animation of a cycle, for the following reason. The Mandelbrot set is *not* the phase space of a dynamical system: it lives in the *parameter space* of quadratic polynomials. For every parameter in a hyperbolic component, the corresponding cycle points live in the *dynamical plane*. Plotting these cycles over the Mandelbrot set confuses more than it clarifies, as it gives the impression that there is some dynamical system in parameter space. The conceptually correct picture would have the dynamical plane of each parameter as a complex *fiber* over the corresponding point of parameter space. However, I suspect that it would require a lot of thought to produce a picture of this type that would really clarify the situation to a layperson.

At some point, when I have more than a few minutes, I might try to write a paragraph on how the position of a parameter in the Mandelbrot set determines the combinatorics of the corresponding Julia set, but to write this in a concise and understandable way is not easy. It might be worthwhile to add a new article "Combinatorics of quadratic polynomials and the Mandelbrot set", but this would require some seriously good expository skills to be useful. (The mathematics behind all this is well-known since Douady and Hubbard's work, although some people have worked to clear it up a bit since then.) --LR 22:44, 20 March 2006 (UTC)

I put the picture in because to an engineer like me it is very interesting to see how these cycles evolve. It is interesting from the point of view of a «practical» engineer. But I can understand your point about the parameter/dynamical ambiguity. But anyway are not both the Mandelbrot set and the dynamic systems both represented in the complex plane? I would think that it could be interesting to let the picture stay. Commented by someone like you. But I am not a mathematician so I think it is not for me to decide and that is why I asked if you thought it was useful. Engineers like it, but maybe that is not relevant. Tó campos 11:57, 21 March 2006 (UTC)

I agree that information about the dynamics of different parameters in the Mandelbrot set is interesting, and not just to engineers. In fact, the way that periodic cycles evolve in the parameter plane organize the combinatorial structure of the Mandelbrot set. However, it is not clear what plotting them over the Mandelbrot set will accomplish --- plotting them over the corresponding (filled) Julia set, and indicating where this Julia set arises in the Mandelbrot set, on the other hand, would be useful. --LR 20:39, 21 March 2006 (UTC)

You are right, of course. I will try to find some time to make plots over Julia sets arising in different p/q bulbs and put them in the page for you to check, ok? Tó campos 19:23, 23 March 2006 (UTC)

I have tried to add an explanation in the article; I hope it is readable. It might be useful to find some way to emphasize the order in which the cycle is being permuted. For example, instead of connecting the orbit points by lines, one could label them 0,1,

\dots

,q-1, starting with the critical point. --LR 19:39, 26 March 2006 (UTC)

Art and the Mandelbrot set

It does not seem clear to me that the section on 'Art and the Mandelbrot set' is really adequate for this article. To me, it seems to be presenting one person's pictures of the Mandelbrot set, and just to be a special case of Fractal Art. I feel that this part of the article should either be removed completely, or moved to the Fractal Art article. However, I will wait for other users' comments on this suggestion. --LR 19:39, 26 March 2006 (UTC)

I think that a small part of the initial text could be kept, and maybe the first line of images, and then refer the Fractal Art article and move the rest there.Tó campos 15:35, 27 March 2006 (UTC)

I posted pictures here without knowing about Fractal Art. So I do not, at this point, have an opinion about the organization. David R. Ingham 05:45, 3 April 2006 (UTC)

I exported the Art and the Mandelbrot set section to Fractal Art as well as the gallery with details of the Mandelbrot set. I think it looks better like this.Tó campos 13:50, 3 April 2006 (UTC)

Smooth iteration counts

The article could probably use a discussion of the ideas described at Renormalizing the Mandelbrot Escape. Basically, the well-known discrete integer iteration count can be elegantly generalized to simple continuous function, which not only yields smooth, accurate color plots of the set, but has useful connections to other areas of mathematics associated with the set. When i have some free time, i'll try and see where/how it can be worked in. --Piet Delport 13:45, 5 April 2006 (UTC)

The natural way to assign a 'continuous' color-coding of the outside of the Mandelbrot set is given by the Green's function of the Mandelbrot set, which can be 'computed' (or at least estimated) by a simple formula. This is the basis for the 'distance estimate' method of plotting the Mandelbrot set. --LR 23:16, 5 April 2006 (UTC)

I have started a new section to cover this (I'm only familiar with the "renormalized" fractional iteration count method). Slowspace 15:28, 4 December 2006 (UTC)

Thanks! --Piet Delport 22:26, 5 December 2006 (UTC)

Commercial venture of interest

I'm not at all affiliated to Bill Boll or anything like that, but his new DVD "The Amazing Mandelbrot Set" (http://www.mandelbrotset.net/) seems worth mentioning here or in Fractal Art. The DVD contains lots of very deep and nicely coloured deep zooms as well as a tutorial. Would it be ok to add a link to it, and if so, where? Opinions, anyone? --Khim1 06:50, 17 May 2006 (UTC)

Wikify tag added

Most wikification needed in the lower half of the article, i.e. fixing heading levels, un-bolding words, fixing references format, all to conform with WP:MoS. J. Finkelst e in 17:09, 16 June 2006 (UTC)

Fixed heading levels; changed bold emphasis to italic emphasis; changed references to standard in-line citations format; removed wikify tag. Gandalf61 10:16, 19 June 2006 (UTC)

Animated zoom

The animated zoom picture is nice, but please don't put a 4.5MB image onto the page. It would be better to link to it instead. - Rainwarrior 03:20, 17 June 2006 (UTC)

Sorry, didn't realise it was so large! --CharlesC 09:37, 17 June 2006 (UTC)

Udo of Aachen

A quick scan of the article seemed to show no mention of this monk, and his [possible] discovery of the mandelbrot set 700 years prior to the eponymous mathematician. Surely this is notable! See: [5] and [6] etc. 84.43.126.11 21:29, 26 October 2006 (UTC)

Udo of Aachen "is a fictional monk, a creation of British technical writer Ray Girvan, who introduced him in an April Fool's hoax article in 1999."

He was already linked in the "see also" section, but i've added a quick description to put it in context. --Piet Delport 05:07, 27 October 2006 (UTC)

Links to sites requiring login

What does everyone think about providing links to pages/sites that require a login to access the information? For an example, we currently have "PHP Source Code PHP Mandelbrot Class" listed as an external link. However, in order to actually download this PHP class, one must first register and login. I am against this. Such information should be freely downloadable(?). I don't know. At least, it would be nice to provide a free/open source example as well. Thoughts? --Thorwald 23:10, 28 October 2006 (UTC)

Remove the link. - Rainwarrior 03:06, 29 October 2006 (UTC)

Image gallery of a zoom sequence

Nice article, but it consists of many mathematical facts and equations which makes it difficult to understand for non professional readers. On the other hand the Mandelbrot set is known in the general public by an extend which is unusual for a mathematical object like this one. Therefore we should add more of the fascinating facts which can be described simply by words. So I have inserted this commented zoom sequence. Perhaps I have some time next weekend for en extended general introduction ;-). --Wolfgangbeyer 23:52, 5 December 2006 (UTC)

?

Elatanatari 02:29, 25 January 2007 (UTC)Is there a version of this article for dumb ppl?

I assume you mean "people who do not know a lot about mathematics", right? I made an attempt at a layman's description of how to produce a Mandelbrot image, but I believe I failed since it turned out way too verbose. Still, here it is, if someone wants to use it as a starting point or something. Khim1 05:50, 25 January 2007 (UTC)

Failed GA

Quick summary:

GA review (see here for criteria)

It is reasonably well written.
a (prose): b (MoS):
It is factually accurate and verifiable.
a (references): b (citations to reliable sources): c (OR):
It is broad in its coverage.
a (major aspects): b (focused):
It follows the neutral point of view policy.
a (fair representation): b (all significant views):
It is stable.
It contains images, where possible, to illustrate the topic.
a (tagged and captioned): b lack of images (does not in itself exclude GA): c (non-free images have fair use rationales):
Overall:
a Pass/Fail:

In a nutshell, insufficient referencing. MER-C 07:57, 9 March 2007 (UTC)

External Links Section

I reduced the External Links Section extensively. There seems to be a habit of people contributing massive numbers of links to JAVA applets, fractal generator programs and galleries. I believe that there should be, at the very most, two of these. After all, Wikipedia is an encyclopedia, and not a collection of links. Before adding links, please consult Wikipedia:Links and consider whether they add to the value of the article. If you feel that another link of these types should be added, or that one of the present links should be replaced by a more appropriate one (preferable), please discuss on this page first. --LR 15:27, 1 May 2006 (UTC)

I think that these links ( below) contain further research that is accurate and on-topic and should be not removed from external link section to dmoz where are so many links that it is hard to find liks of such of extraordinary value.

The Mandelbrot and Julia sets Anatomy by Evgeny Demidov
papers of G. Pastor and M. Romera
FAQ on the Mandelbrot set

(Adam majewski 16:25, 19 April 2007 (UTC)).

I added a link to my own applet (Refract) before reading this (sorry), but I think it's a better applet than some of the ones already linked and it also has freely available source code which the other ones don't. Rowanseymour 14:11, 2 October 2006 (UTC) I have written an article that could be made the target of an external link here. I am very nervous about just adding the link myself, especially as there is so much good stuff out there already.

The article in question is here:

    http://www.tooby.demon.co.uk/Log/DidYouKnow.html#TheAmazingMandelbrot

The intention of the article is give someone with a non-mathematical background an insight into what is going on, not by ignoring or skimming the maths, but by trying to explain what the maths is really doing. Whether the article is successful in this, only other people can judge! I am new to Wikipedia, so if someone would like to give me guidance as to whether this article is "up to standard" for an external link, I would appreciate it. AirToob 18:12, 26 March 2007 (UTC)

I don't have time currently to look through your article, but does it add anything that's not already in this wikipedia article? If so, why not consider contributing to this article instead? If you think something is overly technical and could be said more simply, go ahead and change it. Fact is there are thousands of Mandelbrot websites out there, and we can't very well have links to all of them :) You could request to have yours added to DMOZ, and then indirectly it would be linked from here. Ciotog 18:57, 26 March 2007 (UTC)

I added a link to my java implementation (there is currently now only one java link in there), which has source code, special effects, and ability to permalink specific regions. - Eric oldneuro

Basic properties - lemniscates

If Abs(z)=1 the one gets circle of radius 1 centered at the origin . Mandelbrot set has a point c=-2 so it is out of that circle. IMHO proper definition should be Abs(z)=EscapeRadius where EscapeRadius>=2. (Adam majewski 15:58, 19 April 2007 (UTC))

I think we need a specific source for this, to avoid OR. It appears to me that any constant threshold (i.e. |p_n(z)|=k for any k>0) may work to define these curves so that the limit of their sequence is the Mandelbrot set, but I'm no fractal expert. My guts (via computational experiment) tell me that 1 works: the set grows as n increases, to include points outside of the unit circle. From a pedagogical point of view, the expression "EscapeRadius" has not been introduced in the article yet at this point. Doctormatt 20:10, 19 April 2007 (UTC)

Someone should check this...

[this edit] was made by an ip address. Maths isn't my field, so someone should check that. —The preceding unsigned comment was added by Fluck (talk • contribs) 16:46, 6 May 2007 (UTC).

Logistic map

I believe the logistic map should send z to lambda*z*(1 - z), not lambda*z*(z - 1) as it was in the article, so I changed it. Sorry if I was mistaken on that. Kier07 16:49, 6 May 2007 (UTC)

Formatting of the zoom gallery

Hi Fropuff. The formatting you tried is very space consuming. From esthetical point of view I think it is not a very satisfactory solution, because of the large difference of size for the figure captions. Furthermore it makes it impossible to see all thumbs on one screen. I think its nice to see the initial image and a 60,000,000,000 fold magnification together with all intermediate zoom steps to get a visual idea of this enormous zoom factor. Ok, it makes it somewhat easier to correlate the text with the corresponding thumb, but those who want to understand the text in detail have to open the larger images anyway in order to recognise some of the structures described in the figure captions. In this case they see each images together with a copy of the figure caption from the article and can navigate easily from one image to the next. The only reason why the figure captions should stay also in the article at all is that the figure captions below the large images are not part of the article itself. After all on usual monitors any thumb and the corresponding text fits to one screen without scrolling. Finally you have eliminated the zoom step numbers, but the figure captions refer to them. --Wolfgangbeyer 23:01, 28 December 2006 (UTC)

Hi Jeandré, did you notice my comments above, before formatting the image gallery on 4 April 2007? --Wolfgangbeyer 22:42, 24 May 2007 (UTC)

Position of the image gallery

Hi Shiftchange, the chapter "Computer drawings of the Mandelbrot set" deals with algorithms and program code. The main point of the "Image gallery of a zoom sequence" is not to demonstrate algorithms or to show some nice "Computer drawings" but to present different typical geometrical structures and the underlying rules. Therefore I have moved it to "Geometry of the Mandelbrot set". In principle I would prefer a place earlier in front of the chapters with hard mathematics which I guess only few readers will follow ;-). --Wolfgangbeyer 22:47, 24 May 2007 (UTC)

Color Mapping

We need a section on how the colors are defined. Wiseoldowl 06:31, 4 June 2007 (UTC)

Do you mean more than what is in the article already:

A picture of the Mandelbrot set can be made by coloring all the points c which belong to M black, and all other points white. The more colorful pictures usually seen are generated by coloring points not in the set according to how quickly or slowly the sequence diverges to infinity

What do you have in mind? I think it is important that people realize that the coloring is to a great extent arbitrary: points that diverge at the same rate are usually the same color, and that's about it. Maybe something like that would help? Doctormatt 06:38, 4 June 2007 (UTC)

What do you think about including a text like the following in the end of the For Programmers section to give an suggestion of what kind of coloring schemes can be used?

Instead of coloring each pixel according to the number of iterations corresponding to an orbit that has modulus larger than 2, other possible coloring schemes can be used as, for instance, coloring each pixel according to an exponential expression as 0.1 n^v, where n is the number of iterations and v is an integer which can be chosen according to the zooming factor. For this purpose, n can be defined as modulo some number (e.g. n mod 200) [7].

Tó campos 17:45, 4 June 2007 (UTC)

I don't think the wording here is right: the scheme you suggest is "coloring each pixel according to the number of iterations" (by the way, your scheme ought to generate horribly large color values, which subsequently would need to be converted, and rather slow down the calculations: what is the benefit?) But, yes, including something along these lines could be helpful. It might be a good idea to find a decent reference in order to address why one would choose one of the various schemes that anyone might invent. It's mostly just aesthetics, isn't it? Doctormatt 18:13, 4 June 2007 (UTC)

You are right, what I said above was really not very clear at all. Sorry. Let me correct that:

To use different color schemes without slowing down the calculations, the number of executed iterations, corresponding to the first orbit that has modulus larger than 2, can be used as an entry to a look-up color palette table initialized at startup. If the color table has, for instance, 500 entries, then you can use n mod 500, where n is the number of iterations, to select the color to use. You can initialize the color palette matrix in various different ways (e.g. using an exponential expression as 0.1 n^k, where k is an integer), depending on what special feature of the escape behavior you want to emphasize graphically[8]. Tó campos 10:15, 5 June 2007 (UTC)

Mathematics is not arbitrary, so the coloring methodology needs to be defined. This article starts off with standard mathematical notation and then the color maps just suddenly appear with no explanation of how they are obtained. Each color is a defined by some criteria, which needs to be stated. There can be several ways to color the maps, but I think that they need to be explained. The attractiveness of fractals and the Mandelbrot set images to the lay person lies not in the mathematics but in the color images. So the article must include the methodology used to define the colors. This will bridge the abstract world of mathematics to the real world of imagery and color and beauty. It's an important point. There are many programmers that are very skillful, but are not really mathematicians.Wiseoldowl 03:58, 5 June 2007 (UTC)

Regardless of whether or not mathematics is arbitrary, there is no fixed definition of the "coloring methodology". The coloring schemes are, to a large extent, arbitrary, as I said. The basic idea is that there is an integer (let's call it m) associated to each pixel in your picture; this integer is a measure of how rapidly the point corresponding to that pixel diverges to infinity. The color you use to color that pixel is a function of m: that function is then the "coloring methodology" for the picture you are making. What function you use to do this is arbitrary. And that is a heck of a lot of arbitrary. That's (one reason) why you see so many different pictures of the Mandelbrot set. Please read what it says about color in the sections Formal Definition, Escape Time Algorithm, For Programmers, and Continuous (smooth) colouring. Can you explain further what more you think the article needs to say on this? Would it help to add a bit about how coloring is arbitrary and up to the programmer, and not a function of the set itself (per se, or something), just to make that point explicit? Doctormatt 07:49, 5 June 2007 (UTC)

Initial value

Ok, so this article says that the set is the orbit of 0, but Wolfrom Mathworld on [9] page defines it as $z_{0}=c$ . I mean, this is really nitpicky, because both definitions generate the same set, because for $z_{n+1}=z_{n}+c$ , if it's the orbit of 0, then $z_{1}=c$ , but if it's the orbit of c, then $z_{0}=c$ , so it's just a difference of the subscripts, but it would be nice to firm that up... I'm not an expert on the subject, I'm just a high-school math nerd, so forgive me if I'm being stupid. Sbrools ^{(talk . contribs)} 19:10, 4 June 2007 (UTC)

The article does not quite say that the Mandelbrot set is the orbit of 0 - it says that it is the set of values of c for which the orbit of 0 is bounded. This is equivalent to the MathWorld definition, because c lies in the orbit of 0, so the fate of the orbit of 0 and the fate of the orbit of c are the same. Mathematical objects can have different but equivalent definitions - the article just happens to use one particular definition (with a source) in its opening paragraph. Gandalf61 04:42, 5 June 2007 (UTC)

Yeah, that's what I said. I was just wondering. Sbrools ^{(talk . contribs)} 11:38, 5 June 2007 (UTC)

Distance estimates

The image : http://commons.wikimedia.org/wiki/Image:Demm_2000.jpg shows the distance estimation method applied to the complement of Mandelbrot set. I don't know why it is strange .--Adam majewski 16:08, 3 July 2007 (UTC)

The trouble is that linking directly to a header doesn't give the reader an explanation of what the image represents. Perhaps you could explain how this image differs from other Mandelbrot images, so it is more clear what the "distance estimation method" does. Cheers, Doctormatt 17:15, 3 July 2007 (UTC)

Right; sorry about the terse of the undo message. External image links are unusual, and doubly so inside section headings. (Both are discouraged.)

It's hard to see a place for the image to be added, but to do so you would use the normal image syntax, as described here. --Piet Delport 04:15, 4 July 2007 (UTC)

Infinite Zooming Feature

I noticed that the infinite zooming feature of the Mandelbrot Set is hinted at but it's never shown to its best possible degree. A 6*10¹⁰× magnification is no problem for any modern computer to calculate and antialias. I've got an image, albeit not antialiased, on my desktop at a magnification of 2¹⁰⁵⁰×≈1.206×10³¹⁶×. At this level of magnification, I've noticed all the self-similar features that have been zoomed into begin to "stack up" on each other with every new center zoomed into having several "rings" around of smaller julia sets from lower zoom levels. Hopefully, this makes sense I'm not an expert in fractal theory, but I do enjoy exploring them, so I'm sorry if I've misused any terms. I'm not exactly a layperson in math and programming either though, but I am a newbie wikipedian, so I wouldn't dare upload the image without some expert advice though. I do feel, though, that this would be a nice, probably small addition to the article. Any comments especially expert comments would be greatly appreciated.--Plynch22 21:09, 12 July 2007 (UTC) You can find a midget magnified by a factor of | $10^{359}$ . What program have you used to create this image ?--Adam majewski 06:24, 15 July 2007 (UTC)

Checked the link. Unfortunately, he's got me beat on that one. The coordinates I'm looking at actually continuously degenerate into smaller julias with new centers and never into a midget (At least not yet). The program I use is Fractal eXtreme by Cygnus Software who copyrights their software but not the images created by it, however I think I should ask them to directly state this on the website, because it is only alluded to in a few spots. For example, read bullet 5 here.--Plynch22 16:23, 15 July 2007 (UTC)

Optimizations

I wonder if anyone might be able to find information about ways to speed up the generation of the mandelbrot fractal, i.e. good, fast algorithms. Snotwong 15:23 1 Jun 2003 (UTC)

Yeah, I noticed the pretty bad section on optimisations in the article. Back in 1988 I implemented and tested more or less all the then known methods of speedups for making Mandelbrots. Both speedups to make the calculation of a single pixel faster and speedups that avoid calculating most pixels by "guessing" contiguous colour areas. (I even independently invented and managed to implement "contour following" / "boundary tracing" for Mandelbrots.) Snotwong, when I am in the mood some day I might update the optimisations section with a description of the 10 best optimisation methods or so that I know of. (I know of at least 20 but I think we should limit ourselves to the more interesting ones.) --David Göthberg 11:20, 20 November 2005 (UTC)

I think this article could do with some pseudocode so it is more readable for non-mathematicians. I'll think about that too. --David Göthberg 11:29, 20 November 2005 (UTC)

I just removed the redundant assignments to x2 and y2 in the while block. I would also like to check on the 2*2 calculation in the while expression, though. While a good optimizing compiler will replace this with 4 automatically, I don't think we want to rely on any target language a reader may want to translate this to having such a good compiler. Any objections to adding the line "r2 = 2*2" with the remaining x2 and y2 assignments and using r2 for 2*2 in the while expression? Intchanter 07:01, 15 April 2007 (UTC)

First: that is not redundant. x and y change each iteration, and likewise so will x2 and y2. Second: this is pseudocode, intended to be understandable. Optimizing this will not aid comprehension. Third: if you don't have a compiler that optimizes 2*2, you likely don't care about optimizing anyway. Fourth: more significant optimizations (such as the ones described above by David Göthberg) of the algorithm are done outside of this basic pixel loop. Optimizing this loop itself won't give a great deal of speed gain by comparison. If we really wanted to optimize it properly, though, we'd probably want to get down to the assembly level, and maybe even use SIMD, but this is a kind of programming we really shouldn't have in the article. - Rainwarrior 07:58, 15 April 2007 (UTC)

I do see that it's not redundant. I may have misunderstood the point of the comparison in the while expression. It makes a big difference whether you want to declare escape on total change or change for one iteration. The code as it presently stands escapes only if the positional change for a single iteration is greater than 2. Do we know which it should be? Intchanter 08:18, 15 April 2007 (UTC)

What do you mean by change? We don't compute any difference between this iteration and the last (or this iteration and the first either). We compute a new absolute position each iteration. If its magnitude is greater than 2, it has been proven that it can never converge to zero (this is mentioned further up in the article under "basic properties"). For any other position, it's still up in the air, so we keep iterating to see if it will escape. I'm not quite sure what you mean by which it "should be", but the code given does what it should do, which is: 1. square the complex number and add the original complet number to the result (z^2+c). 2. check if it's within the boundary with a radius of two (stop if it isn't). 3. repeat. - Rainwarrior 08:55, 15 April 2007 (UTC)

Ouch, the number of times people have misunderstood that pseudocode and tried to "fix" it makes me wonder. Perhaps we should not have any optimisations what so ever in that example and keep it as "simple" as possible? After all, the main purpose of that pseudocode should be to show the algorithm, not to show optimisations. (Note, I was the one that wrote that code and included those simple optimisations already in the first version.) Instead we should perhaps only show such optimisations in a separate article about "Mandelbrot optimisations"? --David Göthberg 20:19, 17 April 2007 (UTC)

I tested the source code on some friends and they too were confused by the optimisation in the code. So I removed the optimisation. Now that pseudocode is about as simple as it can be. But I bet some people will not understand the variable "xtemp" and remove it. (But the xtemp is necessary since we need to use the old x in the calculation of y, not the new x.) --David Göthberg 18:59, 31 May 2007 (UTC)

The optimization section claims:

To prevent having to do huge numbers of iterations for other points in the set, one can do "periodicity checking"—which means check if a point reached in iterating a pixel has been reached before....This is most relevant for fixed-point calculations, where there is a relatively high chance of such periodicity—a full floating-point (or higher accuracy) implementation would rarely go into such a period.

Is the claim that periodicity checking is most relevant for fixed-point calculations true? My (limited) understanding of the math is that periodicities are real, and that they'll show up even if you're doing infinite-precision calculations. —Preceding unsigned comment added by 69.114.54.71 (talk) 2007-08-01 16:50:03

I agree, that statement is wrong. I did periodicity checking back in 1988 and I think I remember the details about it (but note, that was 19 years ago so...): As far as I remember I had periods pretty quickly even when using floating-point calculations. Even when not rounding of the values. When I did round of the values I could discover the periods in lesser rounds, but it didn't pay of since doing the rounding of costed more CPU than doing some extra rounds. But note, as I understand it the values in theory never exactly repeats, just that the floats I used had a limited size and thus kind of automatically did round off. So if you use infinite-precision calculations then you have to do round off when doing the comparisons or you will never discover the periods. So I say that periodicity checking can be a nice thing even when using infinite-precision calculations.

Oh, and by the way: It only pays off to turn periodicity checking on when/if the last few pixels you calculated was black (in the Mandelbrot set). That is, when it is is a high probability that the pixel you are currently calculating is black.

There are lots of more details on how to do the check itself in the most efficient manner. But that is a long story.

--David Göthberg 20:07, 1 August 2007 (UTC)

If using floating point calculations you should probably use an epsilon value when doing the comparison for periodicity (i.e. don't use equals, instead check if the difference is smaller than some small value, epsilon). Eventually even floating point numbers will becomes periodic, regardless of rounding mode, but it will happen MUCH faster with an epsilon. (Note, though, that using an epsilon will reduce precision somewhere, though my guess would be that it is lost somewhere inside the edge of the set where it wouldn't be seen (the periodicity check will probably start failing as you approach the edge anyway). - Rainwarrior 07:11, 22 September 2007 (UTC)

Changes to lead section

User:Shiftchange made the following changes (in bold) to the second paragraph of the lead section:

Unlike traditional shapes of geometry, Mandelbrot set equations are generally impossible to deduce a shape from. The Mandelbrot set, which is usually shown in black, ...

I reverted these changes and User:Shiftchange has asked for an explanation of my reversion. I reverted the changes because:

The term "traditional shapes of geometry" is vague, and not well defined.
It is clearly not "impossible" to deduce a shape from the definition of the Mandelbrot set - there is a straigthforward algorithm for determining whether a given point is a member of the set and there are many illustrations in the article showing parts of the set and its borders.
The Mandelbrot set is sometimes shown in black, but to claim this is "usual" is too much of a generalisation. Indeed, there are many illustrations in this article in which the Mandelbrot set is shown in blue.

Gandalf61 09:33, 18 July 2007 (UTC)

Add math material

the bulk of this article focuses on the computer drawing of Mandelbrot set, which is quite shallow. Could someone add some mathematical material? e.g. is it connected set? why? what is its significance? what are the fundamental theories? how it connects to other branches of math, who are the major contributors and what they contributed...etc? Thanks. Xah Lee July 6, 2005 11:04 (UTC)

I Remember hearing that it was connected and has Hausdorff dimension 2. I don't want to post it though because I don't know any official references for it.--SurrealWarrior 8 July 2005 02:20 (UTC)

it is said that the main shape is cardioid. Is this real or just by the looks? If real, what are math definitions that makes it so? on this, are there references? for example, why is it so? Also, other math questions being why the set is a fractal the way it is? any top-level explanation at all? Xah Lee July 8, 2005 06:48 (UTC)

I have rewritten the article, but without including more mathematical details than I feel suitable for an encyclopedia article. Suggestions are welcome. I have left the part on computer drawings unchanged for the moment, but that should be revised some time as well.

A proof missing

In the beginning of the article it is said that we get:

x_{n+1}={x_{n}}^{2}-{y_{n}}^{2}+a\,

and

y_{n+1}=2{x_{n}}{y_{n}}+b\,

after reformulating some formulas. Can someone place a proof of this in the article. I tried to proof it and I could only proof that if one of this formulas is right then the other is right too.

Here is a proof, although I think it is too elementary for the main article. If we have

z_{n}=x_{n}+y_{n}i

z_{n+1}=x_{n+1}+y_{n+1}i

c=a+bi

then

z_{n+1}=z_{n}^{2}+c

=(x_{n}+y_{n}i)(x_{n}+y_{n}i)+a+bi

=x_{n}^{2}+2x_{n}y_{n}i+(y_{n}i)^{2}+a+bi

=x_{n}^{2}+2x_{n}y_{n}i-y_{n}^{2}+a+bi

=(x_{n}^{2}-y_{n}^{2}+a)+(2x_{n}y_{n}+b)i

Equating the real and imaginary parts of z_n+1 (which is perhaps a step that the original questioner did not realise was legitimate) gives the two equations quoted above. Gandalf61 15:33, July 30, 2005 (UTC)

I got that 2 but the end result (

x_{n+1}+y_{n+1}i=(x_{n}^{2}-y_{n}^{2}+a)+(2x_{n}y_{n}+b)i

) just proofs that if one of the 2 statements is true then the other is also true. But you still need a proof that one of the statements is true.

Since x_n+1, y_n+1, x_n, y_n, a and b are all real, you can equate the real and imaginary parts on each side of the equation, so you turn one equation with complex variables into two equations with real variables. Gandalf61 09:25, July 31, 2005 (UTC)

Mandelbrot art, my recent additions

Some people don't like my colors. I tend to approach it from a scientific data visualization point of view and show as much detail as I can. That may not be the best thing to do artistically. My only artistic training is with pottery, and a bit of jewelry, so I don't know much about what colors look good together. The color-sets are chosen to have wide ranges of bright colors, and then the coordinates are chosen so that these colors are mixed in the down-sampling.

External links of interest?

Hi, I made this generator [10] some time ago. Now I come to the conclusion, that I can share it:) I'll be glad when people will be used and enjoy it:)

Hi, I was going to add this google video "Fractals - The Colors of infinity" [11] by Arthur C. Clark to the "external links". But I was met with some very unfriendly directions telling me to post here first - so I'll do that and let it be up to you guys if you find it relevant - because clearly you are the ones to be the judge of that. ;)

--80.196.177.35 17:14, 1 August 2007

According to Wikipedia:Spam#Tagging_articles_prone_to_spam these directions are a preemptive measure against spam. I like to suggest to post the link Animations of the zoom sequence corresponding with the image gallery in this article again, which is obviously no spam but closely related to the article content. Unfortunately it can not be uploaded to Wikipedia because with 37MB due to 1024x768 pixel it exceeds the 10MB limit for media size. --Wolfgangbeyer 21:36, 30 August 2007 (UTC)

Obviously no objections ... --Wolfgangbeyer 15:21, 2 September 2007 (UTC)

--89.55.187.209 14:38, 3 November 2007 (UTC)
I want to point to our new fractal generator Fractalizer
It's a fast freeware program to compute fractal pictures and zoom videos of the Mandelbrot- and Julia-sets, with many examples, formula adaption and ample truecolor palettes with the great quality of "Supersampling Antialiasing".

Optimizations: the cardioid

One way to improve calculations is to find out beforehand whether the given point lies within the cardioid or in the period 2 bulb.

Without giving a formula to decide whether x + iy is in the cardioid in terms of x and y, that advice is quite useless. --Army1987 18:55, 23 September 2007 (UTC)

It's not that great an optimization anyway; it only really helps when you're drawing the whole mandelbrot. When you're zoomed in on a little piece of it, this kind of approximation isn't going to help much. - Rainwarrior 23:54, 23 September 2007 (UTC)

The article used to have the formula for the cardioid next to that statement, but that has been edited out by someone since then. But I agree, that is not much of an optimisation anyway since it only pays of when making the first images and those are not a problem for modern computers anyway. Also the other optimisation mentioned in that "Optimizations" section, the "periodicity checking" isn't especially good. Well, it is good if used alone, but other optimisations are much better and make it almost useless. So the article currently only mention the two worst optimisations. Ah well, but who am I to complain? Since it's probably my job to write the Mandelbrot optimisations" article. But I have more urgent Wikipedia matters to tend to.

--David Göthberg 08:33, 24 September 2007 (UTC)

Here is pseudocode that allows one to avoid repeating unproductive iterations for pixels inside the cardioid or some major bulbs.

For each pixel (x,y) on the screen do:
{
 t=arctangent(y,x) REM: use a 4-quadrant function that returns pi to -pi
 if ((x-.25)**2+y**2) < (.5 * ( 1-cos(t) )) then INCARDIOID
 if ((x+1    )**2 + y**2) < .0625  then INBULB1
 if ((x+1.310)**2 + y**2) < .0036  then INBULB2
 if ((x+1.381)**2 + y**2) < .00017 then INBULB3

 REM Do Mandelbrot iteration
}

The self-explanatory cases INCARDIOID, INBULB1, etc. are pixels in the set that are usually left black.Cuddlyable3 17:35, 3 November 2007 (UTC)

Distance estimates

I was so happy I've found the formulas, implemented them in a program, and seen them to really work, that I just had to add them to the main page. Feel free to improve typesetting, or move them around, but I feel it should not be deleted as it's pretty hard to find.

The babbling about integer multiples of p₀ and the error arising is all mine, found through experiments with my program. It should also be mentioned that the interior distance estimate can in fact be negative for points at boundary of the Mandelbrot set, e.g. -2 or 1+i. I haven't found any explanation for this. Milan va 01:22, 1 December 2007 (UTC)

Do you have any sources for these formulas? Wikipedia is not a place for original research, so your estimations found by a program might be hard to keep, unless you write a book or publish them in an article. Paxinum 09:19, 1 December 2007 (UTC)

My official source would be Bounding the Area of the Mandelbrot Set but it's somewhat hard to understand the meanings of the variables in the equations. That paper in turns points to "Fisher, Y. (1988): Exploring the Mandelbrot Set. The Science of Fractal Images, Peitgen, H.-O., Saupe, D., Editors, Springer-Verlag, New York, pp. 287-296." but that's not to be found anywhere. Milan va 15:42, 1 December 2007 (UTC)

How do you open this paper? (I usu GSView without effect)--Adam majewski 07:01, 2 December 2007 (UTC)

I downloaded it and decompressed with Total Commander. WinRAR will work as well, and maybe other decompressors too. —Preceding unsigned comment added by Milan va (talk • contribs) 11:10, 3 December 2007 (UTC)

I have tried to use Total Commander 6.56 : error in archive file . ??--Adam majewski (talk) 20:33, 13 December 2007 (UTC)

Great. Thx for improving article. I have added image made with coloring the complement of Mandelbrot set with DEM. Can you add images with interior case or limitly periodic (i.e., inner) points. Dou you have your own page with your program ? Maybe it should placed here also a pseudocode for programmers ( math notations is grat but psudocode is easier to read). Milan, maybe you can improve also external ray article ?

Sadly, I know nothing about the external rays. My main goal was to speed up the painting of inner pixels, by detecting a period as early as possible. I want to rewrite my program to Java applet and write a page about it, but I think that will take some time. Currently it's written in Delphi 7, if anybody is willing to translate. I've uploaded it to a free webspace if anybody's interested but it needs much work to be usable by common folks. Just left-click and right-click in the image to navigate. Milan va 19:22, 1 December 2007 (UTC)

Maybe it is time to start a new article about it ?--Adam majewski 09:44, 1 December 2007 (UTC)

Notation

The mathematical notation in this section uses $F_{c}^{n}$ but the rest of the article uses $f_{c}^{n}$ . Is there a reason why to use capital F? Paxinum 16:12, 3 December 2007 (UTC)

I don't know why. Many people use diffrent notation. I think that it should be same notation in articles: mandelbrot est, julia set, external ray, complex quadratic polynomial. --Adam majewski 16:25, 3 December 2007 (UTC)

I've copied the big-F notation of the paper describing the estimator equations. I don't know where the small-f comes from. It was introduced on March 15 2006 by Lasserempe but there's no such user to ask. Milan va 12:20, 4 December 2007 (UTC)

I believe it is different from different books; me myself prefer to use the small one in my research, and the difference is minor. But I believe consistency is preferable, since there is no cons or pros for using one before the other. Hmm, a capital F might be preferable, since it would be consistent with a capital R for rational maps (I have seen that in some books), but an r just feels wrong for using as a function. On the other hand, F is used to denote Fatou components( see here) Paxinum 13:42, 4 December 2007 (UTC)

Oops, I've checked it again and they're actually using $P_{c}^{\circ {}n}(z)$ to denote n compositions of P_c(z)=z² + c. I'm going to fix it. Milan va (talk) 00:25, 5 December 2007 (UTC)

New article?

I think a separate article where the different algorithms (especially colouring algorithms etc) can be added. This article should focus more on the mathematical view on mandelbrot, and the hobby/aesthetical views on mandelbrot fractal can be moved to this new article. Paxinum 12:18, 3 December 2007 (UTC)

For myself, I would just drop the whole algorithms section apart from the initial glossary list with internal links. The rest of the section is unsourced and tends towards POV descriptions of "my favourite algorithm". A lot of it appears to be original research. It is interesting, but I am not convinced it is enclyclopedic. Gandalf61 12:48, 3 December 2007 (UTC)

I actually partially agrees, the article as it is now is quite bloated, and those (partially including me), that are interested in documenting good drawing algorithms and colouring parameters could do so in the wikibook here. Paxinum 15:54, 3 December 2007 (UTC)

IMHO in article should be a math background and maybe a pseudocode. Examples may be in wikiboks. --Adam majewski 16:27, 3 December 2007 (UTC)

Sources for "Other Properties"?

I've noticed that there is a substantial lack of sources for the majority of the "Other Properties" section on this page, and was wondering if there was a reason for it. Would it be reasonable to request that a source (or sources) for the information in that section be provided? AtramentousAmaranth (talk) 12:21, 30 December 2007 (UTC)

formal definition

Why is here critical point not equal to zero ? I t should be zero !!!!--Adam majewski (talk) 10:23, 6 January 2008 (UTC)

This was an incorrect edit by Cyclone231 - I have fixed it. Gandalf61 (talk) 17:26, 6 January 2008 (UTC)

Continuous (smooth) coloring

Hi. Thx for new item.

Can you add the code to the wikibooks ?
As I know continues escape time uses low ierations ? Is it true in your version ?
Can you write more about gradients ?

--Adam majewski (talk) 17:55, 28 December 2007 (UTC)

Hi. There appears to be an error in the first formula for the normalized iteration count algorithm. It should be n + ( ln(ln(B)) - ln(ln(|z|)) ) / ln(P). See the original paper.

Also, the 2nd formula is not equivalent to the first for P and B == 2. The first formula is designed to match up with the colors from discrete iteration count algorithms, while the second is not. —Preceding unsigned comment added by 63.147.134.2 (talk) 20:19, 23 February 2008 (UTC)

smoothing

"Note that this new equation is simpler than the first, but it only works for Mandelbrot set's with a bailout radius of 2 and a power of 2." I dont know about Mathematics too much, but I think the bailout radius of the formula should be e, not 2.--Luzi82 (talk) 16:34, 2 February 2008 (UTC)

for programmers

There are many change in code. Evrey change is intresting and I think that should be described. Maybe in wikibooks.--Adam majewski (talk) 12:25, 17 February 2008 (UTC)

There is a wikibook about fractals, where pseudocode is encouraged! Paxinum (talk) 07:13, 18 February 2008 (UTC)

so please add coomment to this book if you are making changes (:-))--Adam majewski (talk) 09:10, 19 February 2008 (UTC)

Another External Link

I, Theta4, wanted to know if i could link to the wallpapers page on my website. As of April 30th, 2008, all of the wallpapers are from either the Mandelbrot Set or the Burning Ship Fractal. I'm going to come out and say that I'm doing this primarily because I really want some publicity on my website. I won't get very far telling kids at my school as most of the kids there don't appreciate or know what the Mandelbrot Set and family is. Also, a simple search for 'mandelbrot set wallpapers' on google isn't too helpful. There are some nice pictures that are wallpaper size, but they don't look too good as a wallpaper in my opinion. Keep in mind that I am in no way an expert on the subject of Mandelbrot Set imagery. I know my way around the set and understand both its mathematical and programmatic construction, but I can't compare to the expertise of whoever created that zoom sequence. Talk about digression! Theta4 (talk) 03:06, 1 May 2008 (UTC)

Please read WP:NOT and WP:SPAM and WP:EL, those should answer your question.TheRingess (talk) 03:26, 1 May 2008 (UTC)

Figures. Oh well.

External Link

what about linking http://sourceforge.net/projects/quickman ? it's an open source mandlebrot set generator 151.57.77.203 (talk) 22:06, 18 February 2008 (UTC)

Or this: Fractalizer Freeware Fraktal Program for Microsoft Windows Turm (talk) 08:16, 7 April 2008 (UTC)

How about this: Mandelbrot set explorer, a Java applet for Mandelbrot set generator and explorer for educational material, and is an open source program. This applet has been accepted some textbooks and journals. --219.187.59.138 (talk) 16:15, 17 April 2008 (UTC)

I've written a program that generates both the Mandelbrot Set and the associated Julia Sets using the Flash AS3 engine. I also include all the code so people can duplicate it themselves if they choose. My site is non-commercial. Kitoba (talk) 01:44, 26 June 2008 (UTC)

See archive :

http://en.wikipedia.org/wiki/Talk:Mandelbrot_set/Archive_1#External_Links_Section

I think that

The Mandelbrot and Julia sets Anatomy by Evgeny Demidov is better and also open source. --Adam majewski (talk) 07:47, 18 April 2008 (UTC)

"Quasi-"self-similarity?

Ok, we all know the behavior of the primary satellites: Zoom in on a fiber with two other fibers extending out almost perpendicular to the first and you will see 8 fibers, then 16, 32, till they finally degenerate into this smaller version of the original set that is almost broken off from the original except for that first fiber connecting it. Well, in looking around the set, I've noticed that these satellites are almost exactly similar to the original except for the fact that it has all these fibers from the original forming it. They even have their own satellites placed in the same analogous location to the original, but these satellites could have differently shaped fibers from the satellite their "orbiting" along with the fibers from the original set all degenerating into them. As a matter of fact, the Julias seem to be simply an amalgamation of this compound degeneration. The largest two amoebal structures in the Julias seem to be simply a largely exaggerated version of the fibers emanating from the primary bulbs, but these smaller bulbs would have a periodicity of something like a million instead of just 3. Zoom deeper and the amalgamation continues leading to more and more complex structures, because of this, when you zoom in on any detail of any structure, every detail zoomed into after that, must show a sign of that original structure it's inside of. Because of this, if you know how to read them, every picture obtained from the original Mandelbrot set will tell you the story of how it was zoomed in upon.

I guess my point is: Shouldn't we expand this section to show how the mandelbrot set is used to create fractal art. Any ideas? Plynch22 (talk) 00:32, 28 August 2008 (UTC)

This page ostensibly about the mathematical object is already overstretched by extra matter about renderings and generalisations, while Fractal art needs more good material. This Wikibookhas an incomplete chapter 10 Applications which would benefit from a contribution about artistic fractals. See also [12] Cuddlyable3 (talk) 18:39, 28 August 2008 (UTC)

Multibrot sets - new page

The content about Generalisations with other exponents than 2 has been moved to Multibrot sets where there is also new material. Discussion of multibrots has been moved to Talk:Multibrot_sets. Cuddlyable3 (talk) 11:16, 29 September 2008 (UTC)

Requesting: Convex Hull

Sorry to be a bother, but would someone kindly describe the set's convex hull? This would make mentally visualizing the set much easier, at least for me. Thanks! --24.29.87.24 (talk) 23:49, 15 July 2008 (UTC)

This animation may help. [13] Cuddlyable3 (talk) 17:46, 9 October 2008 (UTC)

Universality of M

This image shows a Mandelbrot set occuring in a very different process: the analysis of Newton's method for cubic polynomials by means of the fate of their critical point. Maybe it's interesting for the article because this M does not originate in z→z²+c but comes from a meromorphic function. The M in the closeup has its bulbs and buds at places that are a bit different from these of the original M, but it is complete with all bulbs, buds, antennas, filaments, etc. of the original M. The main cardioid is linked to an attractive cycle of period 2. This is very astonishing because the body of M has period 1 and the only part of M which has period 2 is the "head". -- Georg-Johann (talk) 17:25, 23 October 2008 (UTC)

In Popular Culture

User 66.188.104.173 had reverted by XLinkBot a link to a video on www.youtube.com. There are art works based on the Mandelbrot set at http://www.jonathancoulton.com/songdetails/Mandelbrot%20Set# Tab: Images. Is this a link we can use? Cuddlyable3 (talk) 06:35, 20 May 2008 (UTC)

There is also a band called Mandelbrot Set. http://www.last.fm/music/Mandelbrot+Set 66.65.221.82 (talk) 18:54, 4 July 2008 (UTC)

Mandelbrot sets can be found in some of the video presentations that the grateful dead would play while performing in there concerts. [needs citation] —Preceding unsigned comment added by 75.132.0.255 (talk) 03:53, 27 October 2008 (UTC)

And Jonathan Coulton has recorded a song entitled "Mandlebrot Set" which is all about various fractal/chaos ideas. Darac Marjal (talk) 20:47, 4 October 2008 (UTC)