Talk:Mandelbrot set/Archive 1

This is an archive of past discussions about Mandelbrot set. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page.

Archive 1

Archive 2

Archive 3

Archive 4

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. Finkelstein 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)