Talk:Mandelbrot set/Archive 2

Mandelbrot Fractal Set Video by Arthur C. Clarke
I would like to request that a page be created describing in detail the topics discussed in the video titled The Colours of Infinity. I feel it was rather important and do not see anything about it here on wikipedia. If this request is "in the wrong place", "or in violation of a rule" I am sorry and was not intending to do "harm to this project" I just wanted to see this video give it's own page like it deserves. —Preceding unsigned comment added by 75.132.0.255 (talk) 04:01, 27 October 2008 (UTC)
 * Anyone can create a new Wikipedia page. A good way to do that is first to get a Wikipedia account, so that you have a name we can recognise you by. Second, put together a draft of your page on your own sandbox. Here would be a good place to invite us to look over the topics that you think are notable, when you have something written and ready for our (constructive!) comments. Cuddlyable3 (talk) 13:48, 6 November 2008 (UTC)

Animation
Hi. I’m not an expert in Mandelbrot sets, but I’m not bad at making rather compact animations and had a Mandelbrot movie-making application sitting around on my hard drive. I added an animation here in the article. As you can see, it is a rather clumsy physical placement as it disrupts the page layout somewhat. Perhaps someone can find a better location. This 196-frame, 15.33-second animation has a variable frame rate. The main portion of the active zoom runs at 16.6 frames per second. The view zooms in 1.540 fold, or 11,057,332:1. I managed to squeeze the thing into a 1.2 MB file and retain rather high quality color. Also, to help prevent that persistent-motion visual effect and motion sickness, I added some some fixed-frame bookends, 750 ms at the start and 2000 ms at the end, and also added a 1000 ms black loop leader. Greg L (talk) 16:59, 4 November 2008 (UTC)


 * I supposed I’ve got the page layout done well enough now. Greg L (talk) 04:12, 5 November 2008 (UTC)


 * Greg that is a nice compact animation you have made. It shows qualitatively much the same features as the preceding section Image gallery of a zoom sequence shows more quantatively i.e. giving some explanation of key features. I think it would be better not to have so much overlapping information. The example given of what "11 million fold" means in terms of life size (?) and a carbon atom is an unnecessary attempt to be impressive; the zoom-ability of Mandelbrot set details is actually infinite, as far as anyone knows. Cuddlyable3 (talk) 13:35, 6 November 2008 (UTC)


 * To your last point first “…Mandelbrot set details is actually infinite”: Yes, that’s why the associated text begins with “Regardless of the extent to which one zooms in on a Mandelbrot set, there is always additional detail to see.” As for “unnecessary attempt to be impressive”, that the zoom actually is an eleven million factor truly is an impressive bit of information and it isn’t easy to intuit from watching the video that the zoom is actually that extensive. The analogy of drilling down to the realm of atoms helps drive home this point. I can hardly see that having the animation detracts whatsoever from the article with “overlapping information”; it’s not at all like adding another set of fixed image gallery images. The image gallery images above the animation are quite impressive in their own right because of their many more colors and the smooth transitions between them. The animation gives readers an entirely different “ah Haa” as to how the Mandelbrot set works and is organized. There is clearly room for both in the article. Greg L (talk) 03:52, 7 November 2008 (UTC)


 * The subject of the page is the M-set entity and not arbitrary specifics of a zoom. Greg please see WP:POINT as it relates to your wish to "drive home" information that is, as you rightly say, already stated. Cuddlyable3 (talk) 13:09, 8 November 2008 (UTC)


 * Cuddlyable3: So you provided a link to WP:POINT (Do not disrupt Wikipedia to illustrate a point) in a balled faced “If it’s blue, it must be true”-fashion. Thus, you characterized my contribution of an animation and its accompanying text as disruptive!?! If you’re going to post fallacious link-based accusations of improper conduct on my part, I strongly suggest you check out WP:OWN, WP:CIVILITY, and Please do not bite the newcomers and consider what all three mean. As to your very first sentence, yes, the subject of the page is the entire M-set. More to the point, the subject of the section in question is clearly the animation. You are certainly welcome to revise the accompanying text to make it more appropriate and better suited for the greater context of the article if you feel the current text is somehow misleading or incorrect. This is, after all, a collaborative&thinsp; writing environment. However, your accusation that mentioning the fact that it is an eleven-million-fold zoom is somehow an “unnecessary attempt to be impressive” makes me question your objectivity here. In the mean time, addressing your first post where you objected to the very existence of the animation itself (citing pure nonsense of “overlapping information”), please accept the obvious reality that the animation clearly improves the article. I ask that you not be so quick to act as the “Mayor of the Mandelbrot set”, where you undertake the role of censor who decides what contributions you will and will not permit here. Greg L (talk) 23:18, 8 November 2008 (UTC)


 * Greg L, please chill. Your overheated responses would be poor etiquette even if the accusations that you imagine had been made. They have not. Nobody has characterised your contribution of an animation as disruptive, that is something you have imagined. Nobody has said that its zoom range may not be stated, and that also is an "accusation" that you have imagined. Nobody has "objected to the very existence of the animation", and you misquoted me in that claim. Notwithstanding, there are WP policies that we must respect, or we get nowhere. Using ad hominem terms such as balled[sic] faced, Mayor of the M-set and censor is inappropriate. Seeing that you have made thousands of contributions to Wikipedia since you were welcomed you are far from a newcomer and could benefit from a well earned WP:WIKIBREAK. Cuddlyable3 (talk) 03:47, 9 November 2008 (UTC)


 * I do not need to “chill” as you put it, and your suggestion that I should do so does not magically establish you as a calm, wise voice of reason here; particularly when it is you who is quite well deserving of my my calm rebuttals of your fallacious assertions. As to your first claim: “Nobody has characterised your contribution of an animation as disruptive”: nice try, but it was you who invited me to


 * (13:09, 8 November 2008 post above). Is that or is that not your signature at the end of the above post wherein you more-than-suggest that my defense of the text accompanying the animation is somehow disruptive to Wikipedia? I won’t be baited with such tactics as stating that “WP policies that must be respected”; that was precisely my point with regard to your conduct. So please just pardon me all over the place for asking you to drop the baiting game and the associated patronizing act here; it won’t work (with me anyway) and Wikipedia’s history provides irrefutable proof of precisely what you wrote. Now, since your arguments here are as elusive as the headless horseman, why don’t you just clearly and precisely explain exactly what you want, or just hold your peace please. Greg L (talk) 06:28, 9 November 2008 (UTC)


 * Cuddlyable, the Mandelbrot set et al. is an enduring fascination for me, as a non-expert. I've read what you've written above, and find your arguments against Greg's animation most uncompelling. Greg, the animation is excellent, and adds significantly to the explanatory power of the article. Please retain it. Tony   (talk)  09:34, 9 November 2008 (UTC)


 * Tony I agree with you that the latest animation should be retained. Indeed my arguments against Greg's animation are more than uncompelling, because no such argument exists. Let us keep this nice compact animation (my words emphasized) and turn to responsible editing of text around it.


 * Clearly there is repetition, i.e. overlapping information, while we have the two sections "Image gallery of a zoom sequence" and "Zoom animation". May we seek to integrate these?


 * I hope it will become clear to Greg that we have a difference of opinion about whether [|this edit] enriches the page or is unnecessary. Eleven million has no essential relevance to the M-set. It is obviously a big number. It is not Wikipedia's mission to "drive home" that fact, just because it is the arbitrary range of the animation.


 * Regrettably I find Greg's confrontational tone on this page disruptive to our editing work. I stand by the letter of what I have posted. I offer my apology for any unclarity found on my part, such as who has signed what. It is alright to say what we feel is impressive in discussion but that must be restrained by WP:NPOV in constructing the article. Cuddlyable3 (talk) 15:48, 9 November 2008 (UTC)


 * Regrettably, I find your accusing me of being disruptive immediately after my 03:52, 7 November 2008 post as being beyond fallacious and a breathtaking display of gall and incivility. You can’t possibly expect any editor to assume good faith in another after making several shots across the bow as you chose to do here. Your motives for behaving as you’ve done here are baffling and what you are objecting to makes no sense. And please stop quoting Wikipedia policies while pretending to wisely counsel others about proper etiquette here and spout about how something I’ve done is “disruptive to [your] editing work”. Such posturing is truly laughable. You have much to learn about collaborative writing and dealing with others. I am absolutely done here wasting any further time with you as I find your conduct here exceedingly annoying and entirely unproductive. No one should have to put up with so much crap to just contribute to an article. If you have edits to make, go make them. I would suggest that you first sit back and watch how others on this pale blue dot react to the animation and accompanying text. Greg L (talk) 17:06, 9 November 2008 (UTC)


 * Oh… and thanks for weighing in here Tony. Greg L (talk) 03:52, 10 November 2008 (UTC)


 * I found the zoom animation a valuable addition to the article. I must admit the maths goes over my head, but the zoom was one of the parts that made things clearer to a non maths person like me. The static pics were usefull in that each shape was described, but I didnt really get what mandelbrots were about until I saw the animated zoom. I'd say keep it. Metafis (talk) 03:34, 25 November 2008 (UTC)

Here's an Flash application which lets you zoom in for your self: http://veclock.deviantart.com/art/Flash-Mandelbrot-set-106531519 —Preceding unsigned comment added by 85.224.88.254 (talk) 18:05, 2 January 2009 (UTC)

probable error in diagram
There appears to be something wrong in the second diagram, containing the Mandelbrot set relative to the axes in the complex plane. The imaginary axis has units 1 and -1, which are located outside the set, whereas i and -i should lie inside the set. Jorn74 (talk) 12:04, 11 January 2009 (UTC)


 * The points i and -i are right on the boundary of the Mandelbrot set - their Julia sets have an empty interior. They are joined to the main body of the Mandelbrot set by very thin filaments. These filaments show up in some of the images in the article where points are coloured according to escape time. In the first image, for example, you can see two filaments branching off from the top bulb, and i is at the end of the right-hand filament. However, these filaments are too thin to show up in the pure black-on-white image shown in the second diagram. Gandalf61 (talk) 13:27, 11 January 2009 (UTC)

Continuous coloring
The "equations" in the subsection Mandelbrot Set are actually not equations since there are no equals signs! Thus, it is not clear which quantity should be equal to the expressions shown there.--SiriusB (talk) 08:24, 26 August 2008 (UTC)


 * You are right - they are formulae, not equations. I have fixed the text. The formulae associate a real number with each point z in the complex plane outside of the Mandelbrot set. This real number can then be linked to a colour gradient in order to colour the pixels in an image. The algorithm does not seem to be prescriptive about exactly how a colour is derived from a real number value. Presumably the algorithm just ensures that points close to one another have similar real number values, so that colour varies "smoothly", avoiding artificial "contours" where one colour changes abruptly to another. Gandalf61 (talk) 10:30, 26 August 2008 (UTC)

I've taken a closer look on these formulas and I am more convinced than before that they are incomplete. What's missing is an detailed and unambiguous description which values have to be inserted. The most important issue is that the formulas are obviously discontinuous since they invoke the non-smooth n values. The fraction without the n is related to the absolute value, |z|, it is radial symmetric, i.e. yields equal values for equal distances from (0,0). The addition of the iteration number adds a discontinuous component that will far from being smooth, e.g. there may be z1 and z2 with |z1|=|z2| but n1!=n2. Unless one manages to get non-integer n these formulas appear to be useless in the posted forms. However, if one already has non-integer ("smooth") values for n there would be no need for an additional formula (but for another clever algoritm that gives us smooth iteration numbers).--SiriusB (talk) 13:43, 4 November 2008 (UTC)


 * After taking an even closer look on them and their source, I have found (and now fixed) the error. z should be replaced with zn, i.e. not the starting point of z but its final value. Therefore both parts of each formula are no longer independent. However, it remains to show that the resulting real-valued function of z is really continuous.--SiriusB (talk) 14:13, 4 November 2008 (UTC)


 * Finally, I've added a symbol $$\nu$$ for the smoothed value to turn the naked formulas into proper equations. Someone might feel that this edit might be original "research", but at least it is what the author of the cited source seems to expect from the reader.--SiriusB (talk) 14:21, 4 November 2008 (UTC)


 * In addition: I have tested the algorithm in a self-written Mandelbrot program. As expected, the smoothing function is not continuous. However, the jumps that occur at the classical color borders become small if, as suggested in the article, several extra iterations are done, so that the result looks smooth. However, I do not know whether the article may benefit from this since this would clearly qualify as original research, I think.--SiriusB (talk) 08:29, 8 November 2008 (UTC)


 * SiriusB please explain what you meant by maganes which is not an english word. Cuddlyable3 (talk) 12:54, 8 November 2008 (UTC)


 * It seems likely that maganes is a simple typo for manages - Unless one maganes manages to get non-integer... 216.241.205.81 (talk) 19:23, 20 December 2008 (UTC)


 * Oops, sorry, I missed that. Yes, its just a typo. Although it might disrupt the continuity of this thread, I've fixed it now ;-)--SiriusB (talk) 10:00, 23 December 2008 (UTC)

I've just made a fairly clumsy tweak to the simplified formula. I adjusted the base of the inner log rather than the constant before the minus because it would have resulted in an irrational number and in all likelihood a base-two log could be used anyway. Somebody may want to check it; I observed the problem not by thinking hard, but by plotting the functions in gnuplot and then tweaking the values until they made more sense -- specifically, that they run from 1 to 0 between bailout and bailout-squared.

What's not made clear in that section is what the function actually does. Here's my mathematical-halfwit attempt:

If we ignore the $$+ c$$ part of the iteration formula, then the we know that the final length of $$z_n$$ at bail-out will be between $$B$$ and $$B^2$$. We can take our fractional component of the iteration count as an interpolation along this range -- the nearer to $$B$$ we landed, the closer we came to needing another iteration to escape and the higher the fractional part should be. Evidently one log gives us a linear distance between iterations, and another log gives us a curve that more closely resembles the curve we see in the integer part of the escape time.

Bringing $$c$$ back in, $$z_n$$ can be anywhere between $$B$$ and $$B^2+|c|$$, and our fractional part could spill outside of the [0,1] interval. The idea of running on a few extra iterations or using a larger bailout is simply to make the over-spill introduced by $$c$$ proportionally smaller such that it can be ignored. I think this explains why "If iterations cease as soon as z escapes, there is the possibility that the smoothing algorithm will not work." which is a fairly opaque statement as it stands.

Is any part of that worthy of explanation on the main page? --ToobMug (talk) 16:17, 21 February 2009 (UTC)


 * Also, my guess is that these extra iterations can be done in comparatively limited precision, because the truncation errors will be small in comparison to values known to be greater than B -- and it's just an approximation for display anyway. I haven't confirmed this yet, and I'm not sure when I'll get around to testing it (yes, I could just think hard -- but I won't).  Is it worthy of mention if it's correct? --ToobMug (talk) 16:24, 21 February 2009 (UTC)

flash mandelbrot set (external)
i stumbled upon this page and i thought it as a good example. It has fantastic zoom but it requires a fast broadband connection.
 * 

This is a very interesting subject, and if anyone has any more user friendly fractal animations, post them! —Preceding unsigned comment added by 68.209.202.24 (talk) 03:51, 14 February 2009 (UTC)

Pseudocode error
There seems to be a problem with the pseudocode.

while ( x*x + y*y <= (2*2) AND  iteration < max_iteration )

Only the first couple pixels will succeed this comparison. At pixel (1,2) for example, the equation will be: (1)*(1) + (2)*(2) <= (2*2), which of course fails the comparison.

151.112.23.68 (talk) 19:21, 29 January 2009 (UTC)


 * x and y are not pixel counts, they are the horizontal and vertical coordinates of a pixel. The whole M. set is within the range -2<x<1 and -1<y<1 and only your chosen pixel resolution limits the number of pixels within the set. Cuddlyable3 (talk) 21:13, 29 January 2009 (UTC)

I agree, this is very confusing. Maybe the range should be clarified? --xAXISx (talk) 23:34, 7 May 2009 (UTC)
 * iX for integer coordinate ( screen pixels) and rX for real coordinat ( world). HTH --Adam majewski (talk) 08:59, 17 May 2009 (UTC)

Honestly the whole pseudo-code implementation is poor quality. I'd suggest writing the algorithm out using complex primitives which would make the whole thing a lot clearer. —Preceding unsigned comment added by 208.103.225.213 (talk) 05:12, 17 May 2009 (UTC)

another external link
I've written another Mandelbrot viewer in Flash: If anyone thinks it warrants inclusion, please add it to the main page. Jamo777 (talk) 17:16, 14 February 2009 (UTC)

Boundaries
What are the extreme values of the real and imaginary components of the points contained in the set? Lucas Brown (talk) 02:25, 22 March 2009 (UTC)

Do you mean root point and apex point ? Informations about boundaries of hyperbolic components :

--Adam majewski (talk) 06:59, 22 March 2009 (UTC)

What do you mean by "root point" and "apex point?" Lucas Brown (talk) 04:19, 27 March 2009 (UTC)

Root and apex point are points where 2 hyperbolic components meet. (It is not precise definition) WHen you analyse components on real axis then root and apex point are extreme real values for this component.--Adam majewski (talk) 09:15, 28 March 2009 (UTC)

In that case, yes. Lucas Brown (talk) 20:36, 28 March 2009 (UTC)

Much too technical language used in the article
In my opinion, this article is written in a much too technical language to be readliy understood by people with a general education. The opening pargraph of the article reads: "the set of complex values of c for which the orbit of 0 under iteration of the complex quadratic polynomial zn+1 = zn2 + c remains bounded". I have more education in mathematics than most people, and still that does not tell me very much about what the Mandelbrot fractal really is. The first paragraph of any article on wikipedia must be understandable by the average generally-educated internet-literate user, or it will not be interesting to anyone but a select few. For example, phrasings like: $$M = \left\{c\in \mathbb C : \sup_{n\in \mathbb N}|P_c^{\circ n}(0)| < \infin\right\}.$$ are not likely to be readily understood by anyone who does not have a degree, not only in mathematics, but specifically in set theory. My experience is that people who hold university degrees in certain subjects rarely cite wikipedia in their academic work.

For anyone who is interested in learning about this subject in a more pedagogic way: 83.209.10.168 (talk) 22:04, 27 May 2009 (UTC)


 * The Fractal article is better for a non-specialist reader. This wikibook that needs work should approach the subject in a more tutorial way. Wikipedia being a tertiary source is not well suited to citing in academic articles but in my experience is one of the best sources of secondary sources. Cuddlyable3 (talk) 09:54, 28 May 2009 (UTC)
 * About math eqations. I think that below such equation should be an expanation in simple language. Such 2 ways of presentation of ideas can be good. --Adam majewski (talk) 12:09, 29 May 2009 (UTC)


 * An anonymous user has added C-code to the article. I think the pseudocode alone is adequate to explain the iteration and that actual software is better placed in this | wikibook. Wikipedia is not a software repository. Does anyone think the C-code should be kept in the article? Cuddlyable3 (talk) 14:06, 31 May 2009 (UTC)

Fractal or not?
I am far from beeing a mathematician but I may have found a discrepancy between the english and the german article. The english claims that the boundary of a Mandelbrot set forms a fractal while the german one claims that it is not.
 * Since the term fractal was coined by Benoit Mandelbrot the germans have some explaining to do. Cuddlyable3 (talk) 20:18, 23 June 2009 (UTC)

Internal Distance estimation
Hi. How to expand math notation of internal distance estimation to computer program or pseudocode ? Regards --Adam majewski (talk) 19:40, 25 July 2009 (UTC)

Boundary
Can the boundary be generated by reverse iteration? If so, how? --72.197.202.36 (talk) 00:22, 20 September 2009 (UTC)
 * Here is Inverse Mandelbrot Iteration by R Munafo. I do not know example images and do not know what is the result of this method. HTH. --Adam majewski (talk) 22:01, 27 September 2009 (UTC)

Loading time
You should create other articles on the parts of this article, as it's too slow to load.114.76.185.200 (talk) 12:03, 26 September 2009 (UTC)

3D Mandelbrot/Mandelbulb?

 * ''Moved new section to bottom of page. 69.228.171.150 (talk) 17:25, 16 November 2009 (UTC)

Informal description
I've added such a section near the beginning of the article, because (speaking as a high-school student) in reading the article as it was, I was unable to obtain even a vague idea what the Mandelbrot set was. Five minutes reading an off-wikipedia description found of Google, and I was crystal clear on the subject.

If my definition contains any gross errors (or if it's against Wikipedia policy), then by all means remove or edit it, preferably the latter. But I think the purpose of any non-technical introduction is to provide an intuitive understanding of the subject, and a certain amount of rigor and accuracy can and should therefore be sacrificed in favor of clarity. Gaiacarra (talk) 19:14, 27 October 2009 (UTC)

can someone explain what the color in the pictures means?
I'm guessing it's a result of downscaling an image of the function with some kind of averaging function and then assigning a series of colors to the values in the downscaled image. Am I right and if so shouldn't the article be clear on this point? Plugwash (talk) 00:38, 16 November 2009 (UTC)


 * There is an explanation of various colouring methods in the Computer drawings section of the article. Colours typically represent an encoding of escape time or some other measure of "how far" a point is from the boundary of the Mandelbrot set. Gandalf61 (talk) 10:59, 16 November 2009 (UTC)

missing practical uses ?!?
Is the article missing practical uses ? --89.152.177.195 (talk) 16:05, 16 November 2009 (UTC)

3D Mandelbrot/Mandelbulb?

 * ''Moved from top of page by 69.228.171.150 (talk).

There is a 3d picture, but no mention of these pretty objects. Perhaps it should be an article of its own, but even then there ought to be links. http://www.skytopia.com/project/fractal/mandelbulb.html —Preceding unsigned comment added by 130.238.15.194 (talk) 10:56, 16 November 2009 (UTC)


 * I don't think there's grounds for a separate article. I saw that link on slashdot and was thinking of putting it into this article, but I'm a bit hesitant because it offers prints for sale and could be considered a retail link.  The sales part is fairly low-key though, and the images on the page are nice. 69.228.171.150 (talk) 17:25, 16 November 2009 (UTC)
 * Someone else made a stub article with just that link, so I added a section to this article and redirected the stub here. 69.228.171.150 (talk) 10:51, 18 November 2009 (UTC)

making the Mandelbrot (iterative construction images?)
I've understood the general principle of fractals and how you create them for years, but not one article I've read has ever explained this visually for the Mandelbrot set - until now.

http://www.skytopia.com/project/fractal/2mandelbulb.html#iter has a set of eight images running from iteration 2 (which is elliptical - at least to my eye - while iteration 1 is a circle) to iteration 5000.

Wikipedia has such iteration illustrations already for things like the Koch snowflake, Sierpinski triangle and Peano curve. Could we illustrate some iterations of the Mandelbrot too - pretty please? It really made the whole subject (of the Mandelbrot in particular) click for me. As I'm not a mathematician or programmer, fractal formulae say very little to me, while their iteration diagrams speak oceans. 87.194.30.190 (talk) 23:20, 18 November 2009 (UTC)
 * The Mandelbrot set is not a geometric recursion like the Koch snowflake. It's the set of points c for which iterating a certain function paramatrized by c never escapes to infinity.  In practice the way you tell whether the function actually escapes for a given c is by actually computing up to (say) a few hundred iterations and seeing whether the value exceeds some threshold at any stage of the iteration (see the "Escape time algorithm" section of the article).  If it does, c is definitely in the set, so you can stop iterating; otherwise, c is probably not in the set, so render it in your graphic as being not in the set.  What you're seeing in that diagram is not different stages of the construction of the Mandelbrot set.  It's just the stages of the escape time algorithm, which itself is just a computation heuristic.  69.228.171.150 (talk) 06:09, 20 November 2009 (UTC)


 * Is this animation what you are looking for? Cuddlyable3 (talk) 17:24, 20 November 2009 (UTC)

Senses resonating with fractals?
Merry Christmas to you all. Has anyone tried to make a 3D Mandelbrot that is similar to a human form? That is to say, if we live in a fractal world and the frequency of the five senses is resonating with this fractal formation, then it should be possible to make a rought reverse engineering plotted on a computer screen. —Preceding unsigned comment added by 94.254.60.60 (talk) 13:27, 22 December 2009 (UTC)
 * Greetings but please sign your posts. Our fractal world encompasses many other kinds of fractals than the Mandelbrot set so something similar to a human form might well exist at some level of magnification. We can speculate about that but your hypothesis about our senses resonating with fractal formation would have to be reported in a peer-reviewed source before it can be put in Wikipedia. See WP:RS and WP:OR. I added the title above this thread. Cuddlyable3 (talk) 20:22, 27 February 2010 (UTC)

Animation problems
The two deep zoom in section Mandelbrot_set are no longer correctly subtitled (problem with new.gif). Can someone put htings back in order (without losing the new new.gif, of course)--Dfeldmann (talk) 15:22, 23 November 2009 (UTC)

Other 'versions' of the Mandelbrot Set
I followed the separate links to the other articles, i.e. 'versions' of the Mandelbrot Set (e.g. "Mandelbar" set, Newton fractal) and they all seem to be either start or stub class (i.e. articles with relatively little or no necessary or important content.) Should we just remove all the links and combine the articles (move them) into one single article named something like Other "versions" of the Mandelbrot Set or something like that? &#124; TelCoNaSpVe :&#124; (talk) 02:15, 5 March 2010 (UTC)


 * I am not in favour of this. Stubs can be expanded; Other "versions" of the Mandelbrot Set is not a term that anyone is goling to search on; and generic articles tend to become too long and too unstructured. Gandalf61 (talk) 09:12, 19 March 2010 (UTC)


 * I agree with Gandalf61. This article should retain links to articles or stubs about fractals that are (historically) derived from the Mandelbrot set. An example of a derived fractal article that has been expanded is Multibrot set. Some articles to develop in the future (i.e. redlinks) may be Mandelbulb set, Mandelset in the quarternions or even Mandelset in the octonions. One cannot call any fractal a "version of the Mandelbrot set" without citing a WP:RS for the connection to Benoît Mandelbrot. Cuddlyable3 (talk) 16:20, 31 March 2010 (UTC)

Update to the "History" section?
According to The Mandelbrot Monk, Udo of Aachen examined the Mandelbrot system as well. This is important as he died in 1200AD. If the references in the article could be checked, that would certainly deserve a paragraph on his achievements. —Preceding unsigned comment added by 62.190.51.12 (talk) 10:20, 24 March 2010 (UTC)
 * Have a look at Udo of Aachen it seems that this is a fictional character. It did seem suspicious to me as complex numbers are necessary to calculate the set and these came considerably later. --Salix (talk): 11:21, 24 March 2010 (UTC)
 * User 62.190.51.12 did you look at the date at the bottom of your link? April 1st 1999. It's already a 10+ year old hoax. Cuddlyable3 (talk) 16:24, 31 March 2010 (UTC)

Mandelbrot Set and applications
Are there any instances of a 3-4 dimensional representation of the Mandelbrot Set actually appearing in reality, and if so, what are the implications of such a representation as regards to its 2-dimensional counterpart, described here? P.S. Should this question be included on the Mandelbulb or what-ever page as well? 71.108.11.25 (talk) 03:40, 26 February 2010 (UTC)


 * Actually, this question shouldn't even be here in the article's talk page (see Talk page guidelines). In short, an article's talk page serves as a forum for discussions about the content or format of the article, not for discussions about the subject of the article. Questions like these belong on one of our reference desks. This one in particular should of course go to the mathematics instance. Good luck! - DVdm (talk) 08:24, 26 February 2010 (UTC)
 * Oh, thanks! 71.108.11.25 (talk) 08:45, 26 February 2010 (UTC)
 * That reminds me... There is an unusually high amount of images on this particular article, and while they are very intriguing and beautiful 'iterations', they are not completely necessary in understanding the depth of the content, which is why I suggest removing them and instead adding a link next to the headlines instead to get the general idea. I recently commented about how Wikipedia should not include this in its articles. 71.108.11.25 (talk) 09:01, 26 February 2010 (UTC)
 * I agree that this particular page is a bit of a mess. So you could propose a list of images for deletion and see if you find wp:consensus for this kind of cleanup. Be prepared to face some opposition. Good luck. - DVdm (talk) 09:09, 26 February 2010 (UTC)

Images Nominated for Deletion
Here are a few images that I have proposed for deletion, self-explanatory. Anyway, please feel free to add or remove suggestions to the list until the deletion has been enacted. (The first one clearly does not belong on this article.)
 * Buddhabrot Method. Underneath the Computer Drawings headline.
 * Buddhabrot is a fairly important way of representing the same information so there is some justification for keeping it. (talk): 09:07, 5 March 2010 (UTC)
 * I think there is already an article entitled: Buddhabrot set, as shown directly in the caption for the picture. We should move the entire picture there instead. &#124; TelCoNaSpVe :&#124; (talk)
 * No objection to taking the Buddhabrot image. Objection to disappearance of the link to Buddhabrot from the article. Cuddlyable3 (talk) 15:25, 31 March 2010 (UTC)
 * Done. I already added Buddhabrot to the bottom of the article under See Also, whilst taking special care in deleting the image. &#124; TelCoNaSpVe :&#124; (talk) 22:54, 12 April 2010 (UTC)
 * Attracting cycle in 2/5 bulb animation. Underneath the Other properties: The main cardioid and period bulbs headline. There are already enough pictures to its right.
 * 1) I think this one has some merit as it shows a repeating cycle, which is not illustrated well by other images. The Julia sets which are closely related could do with some explanation.--Salix (talk): 09:07, 5 March 2010 (UTC)
 * Two pictures: Misiurewicz point and quasi-self-similarity. Underneath the Other properties: Self-similarity headline. The animation to the right seems informative enough.
 * I think these illustrate that there are different types of self similarity.--Salix (talk): 09:07, 5 March 2010 (UTC)
 * Then should we delete the animation to the right? Or keep all of them together? &#124; TelCoNaSpVe :&#124; (talk)
 * Agree with Salix. The animation shows a self similarity which is different from that of the Misiurewicz points. So keep it. Cuddlyable3 (talk) 08:18, 12 April 2010 (UTC)
 * Picture: An "embedded Julia set". Underneath the Other properties: Relationship with Julia sets headline. The pictures below it are enough to show them in the Mandelbrot set.
 * Tend to agree, not quite enough explanation to be able to understand the image.--Salix (talk): 09:07, 5 March 2010 (UTC)
 * One of the zoom animations. Under same-name headline. Two is simply one too many.
 * Agree--Salix (talk): 09:07, 5 March 2010 (UTC)
 * Agree with deleting one of the zoom animations. The deletion was done in a slipshod way leaving text that related to the zoom depth and placement of the deleted image. I have fixed that.Cuddlyable3 (talk) 12:22, 13 April 2010 (UTC)
 * One of the 'ray-traced' 3D Mandelbulbs. At bottom of page under same-name headline. See comment above.
 * Agree--Salix (talk): 09:07, 5 March 2010 (UTC)
 * The three pictures of Escape Time Algorithm and Normalized Iteration Count Algorithm. Underneath the Computer Drawings headline. I do not understand the 'information' they provide. At all. —Preceding unsigned comment added by 71.118.39.219 (talk) 06:00, 27 February 2010 (UTC)
 * Methods of rendering the Mandelbrot Set are an important subject in their own right, and I think these illustrate the difference between a nieve algorithm and a more sospticated one well.--Salix (talk): 09:07, 5 March 2010 (UTC)
 * Okay then, it seems like there are still many pictures in the article, despite my deletion of them. Do we have consensus on deleting at least one of the Normalized Iteration Count Algorithm pictures (there is more than one of them) :| TelCo  NaSp  Ve :|  23:05, 18 May 2010 (UTC)
 * I agree with removal of the "Another example" picture of Normalized Iteration Count Algorithm which seems to have been included only for its beauty. No big fonts please. Cuddlyable3 (talk) 13:16, 19 May 2010 (UTC)
 * Picture: Mandelbrot set: cardioid and period-2. Underneath the Computer Drawings: Optimizations headline. Such a picure is already in the section: Other properties: The main cardiod and period bulbs. 71.118.39.219 (talk) 06:26, 27 February 2010 (UTC)
 * Agree.--Salix (talk): 09:07, 5 March 2010 (UTC)
 * Map of Julia sets for points on the complex plane, in Relationship with Julia sets section. Not quite sure that this is displaying.--Salix (talk): 09:07, 5 March 2010 (UTC)
 * It is displaying, but somehow it is also halfway inside the Geometry section as well. &#124; TelCoNaSpVe :&#124; (talk) 18:50, 7 March 2010 (UTC)
 * The tricorn fractal and Burning Ship fractal. Underneath the Other non-analytic mappings headline. They should be moved to their respective articles. &#124; TelCoNaSpVe :&#124; (talk) 19:08, 7 March 2010 (UTC)
 * When you say "nominated for deletion", do you just mean that you are proposing to remove these pictures from the article (as opposed to permanent deletion of the image files themselves) ? Gandalf61 (talk) 09:36, 27 February 2010 (UTC)
 * Oh, yes, of course. &#124; TelCoNaSpVe :&#124; (talk) 08:34, 28 February 2010 (UTC)
 * User(s) 71.108.11.25 or 71.118.39.219 you (both?) suggest removing many images. Please if possible get an account with a name and sign your posts because that helps us follow who is saying what.
 * In general I don't think the proposed images should be deleted without a coresponding restructure of text. Images and text are supposed to complement each other. I advise cautiously reviewing case-by-case each image with all the associated text before deleting either. The test is whether a change is a nett improvement for a reader. The article title Mandelbrot set (not Mandelbrot fractal) uses terminology "set" that belongs to mathematics, which should be the tone of the article.
 * I agree that one of the zoom animations can go. Keep the deeper one. Cuddlyable3 (talk) 17:31, 2 March 2010 (UTC)
 * I find that while the pictures do correlate somewhat with the text, that the entire article does not have to be inundated with a repetition of two or more images. Furthermore, I find that other articles which do not have the same amount of images but still contain useful and informative content have the same impact and necessary information a reader needs to know about. So I believe this article needs not rely on an amazing array of images to convey its message to the reader; in fact, if you want to keep the pictures, you could probably provide some sort of link down at the bottom of the page, in some other section (e.g. the 12+ images in the zoom gallery seem entirely useless in light of the zoom animations down below). Wikipedia is not an artistic essay used in conjunction with images to provide some sort of artistic neatness; it is an encyclopedia, and must be treated as such. As stated above, feel free to add or remove suggestions. You could even delete the images right now, read the article again to see if it does not work, and then revert the edit.
 * And yes I have signed up for an account. This is my current username, right here, without the changing IP addresses: &#124; TelCoNaSpVe :&#124; (talk) 01:39, 5 March 2010 (UTC)
 * Say, why don't we set up some sort of deadline for this kind of thing? Say, how about, April 5-9, since that is a convenient time to review this article again. &#124; TelCoNaSpVe :&#124; (talk) 01:43, 5 March 2010 (UTC)

I began deletion of the proposed images on this day 00:09, 19 March 2010 (UTC)

If anyone has any objections to the deletions, they are free to re-submit the pictures. &#124; TelCoNaSpVe :&#124; (talk)


 * TeleComNasSprVen thank you for signing up for an account and welcome to Wikipedia. TeleComNasSprVen you must stop what you are doing. It is clear that you 1) think images should be removed from the article, 2) you have started to delete images and 3) you are deleting images with no corresponding attention to text and with no more care for consensus than saying in effect "Now I deleted it so fix it yourselves if you don't like that." Reactions to the above are 1) That's a fair opinion that can be discussed case-by-case. You must allow that discussion. It is arrogant and unilateral of you to postulate a review deadline (4 weeks) for major deletions from an article that has been 9 years in the making by many editors. ( 1) and 2) are your own statements. ) Your action 3) is most worrying and is how edit warring gets started. You declare the zoom gallery to be "entirely useless" and neglect the facts that there are 15 images each of which is tied to a supporting text about a prominent feature. It is impossible to attach that information to a zoom sequence. Your distaste for the artistic "neatness" of the mathematical set is not the fault of our article about its many noteworthy aspects. If you wish to rail against Wikipedia content about Fractal art see that mentioned article. Cuddlyable3 (talk) 16:02, 31 March 2010 (UTC)
 * Okay. "Now I deleted it so fix it yourselves if you don't like that." I'm allowing that to happen because if I delete an image someone posted I may not know how to retrieve it back again if they wished it included. You can explain to me what is wrong with that policy. I believe I had allowed enough time for consensus and discussion, but I could be wrong. I haven't deleted anything Salix challenged, now, have I? I'll concede that I was wrong in posting a deadline; not very Wikipedian of me. But I don't think I am that arrogant; that's an issue for another time. I do not particularly support the 15 images and their content, and if I wanted to I could probably delete the entire section with the accomodating texts: (Step 1..., Step 2..., etc.) But as they are featured images on a featured Wikipedian article, I would leave it alone for article's sake. Nine years in the making by many contributors: I admit I do appreciate the effort being put into the article, and I try to make more perfect whatever endeavor someone has in an article in my own way; but I'm not a deletionist, yet. And please do not mention that in the discussions; I believe that it is a very good article by itself already. I did not know that edit warring can get started in that way, and I am glad that if my actions were to lead into that, that it hasn't happened yet and if so, that it was stopped before it did. And my comment was about the artistic quality of the article itself, not necessarily about the mathematical set. &#124;  TelCoNaSpVe :&#124; (talk) 04:37, 14 April 2010 (UTC)

I thank User:DVdm, User:Salix alba, User:Gandalf61, and User:Cuddlyable3 for all their help and support. &#124; TelCoNaSpVe :&#124; (talk) 04:44, 14 April 2010 (UTC)
 * In response to your striking of text I have struck out my sentence about "arrogant".Cuddlyable3 (talk) 16:54, 14 April 2010 (UTC)

Is there a thing like a Mandelbrot set ?
Benoit B. Mandelbrot constructed his sets M in this way: for a family of iterations $$f(z, c)$$ and two critical points $$z_1$$ and $$z_2$$ for $$f(z, c)$$ (see Julia set), M is the set of points c such that the two sequences of iteration starting in the critical points, do not have the same terminus. The very first set he constructed was for the family $$c(1+z^{2})^{2}/(z(z^{2}-1))$$ and two critical points that were opposite real numbers, and the iterations were towards finite cycles. The picture was "blotchy", but "It sufficed to show that the topic was worth persuing, but had better be persued in an easier context". He ended up with the family $$z^{2} - c$$ having the two critical points 0 and ∞, and iterations towards ∞ outside M.

I say this, because I have had a controversy with Cuddlyable3, who has deleted a remark of mine in the article Multibrot set (now to be found on Talk:Multibrot set) on the grounds that "The term Mandelbrot set must be reserved for set(s) actually introduced by Benoit Mandelbrot. It is incorrect to call other iterated sets "mandelbrot sets" without reputable sources". If you are of the same oppinion as Cuddlyable3, you should contribute to corrections in Wikipedia articles. I have seen several places where "Mandelbrot set" is used for "generalized Mandelbrot set", but I would prefer that the latter term is reserved for a construction that differs from Mandelbrots idea. (Gertbuschmann (talk) 12:33, 28 April 2010 (UTC))


 * Only the set that Mandelbrot ended up with: $$z\mapsto z^2 + c$$ has attained such notability that it now bears his name and its own Wikipedia article. Benoit Mandelbrot is also known for coining the word "fractal". Do we have consensus on the following naming?
 * Maps of many iterated polynomials of a complex variable show fractal characteristics. We do not claim that Benoit Mandelbrot was ever unaware of any of them but it is incorrect to call them all Mandelbrot sets. I agree with Gertbuschmann that some corrections to articles are needed to redact unsourced mentions of Mandelbrot. I see a difference between "a generalization of the Mandelbrot set" and "generalized Mandelbrot set".
 * Reappearances of the characteristic cardioid-plus-circles shape in magnified images present a naming problem. In the section "Image gallery of a zoom sequence" they are consistently called satellites. I have seen them called mandelbrots or mini-mandelbrots; a capital M seems inappropriate here. IMHO our encyclopedia should not lead a rush to turn a living person into a noun, or to paraphrase Winston Churchill (unsourced): [Some] have spoken of [him] in terms that a fellow ought not expect to hear until after he is dead". Cuddlyable3 (talk) 11:57, 1 May 2010 (UTC)

High resolution image
Please provide high resolution image of Mandelbrot set, so I can see all the details. 188.123.243.103 (talk) 08:28, 2 May 2010 (UTC)


 * You can never see all the details of the Mandelbrot set, because it is a fractal. The section of the article headed "Image gallery of a zoom sequence" shows a sequence of increasingly magnified images of one part of the Mandelbrot set's border - the final magnification is about 10,000,000,000 to 1. Gandalf61 (talk) 08:52, 2 May 2010 (UTC)


 * You might enjoy making your own program using the pseudocode in the "For programmers" section that will let you enlarge for viewing any of the details of the set. I remember my delight when I first wrote a few lines of BASIC that made the beautiful Mandelbrot fractal appear on my PC screen. Cuddlyable3 (talk) 13:18, 2 May 2010 (UTC)


 * It would be great to be enable visitors to zoom and navigate fractals, for example like on this page: http://www.wikiwebserver.org/page/example/FractalGallery.class —Preceding unsigned comment added by 90.213.250.41 (talk) 18:23, 14 December 2010 (UTC)

3d Mandelbulb
I notice the 3D Mandelbulb section got moved to a separate article. (Note, the section was originally added by me a few months ago, and I changed an article someone had created at that time into a redirect to the section). IMO splitting the section to a separate article was not really necessary, so my inclination would be to move it back. The separate article makes the most sense if quite a bit more external sourcing has appeared since the section was written (which is possible) and expansion is planned based on the new sources. Even with a separate Mandelbulb article, I think it's still worth having a summary-style description in the main article, rather than just a "see also" cross-reference. 69.228.170.24 (talk) 19:56, 27 May 2010 (UTC)

Deep Magnification
Any way we can get http://vimeo.com/12185093 into the article? Particularly noteworthy and definitely interesting. 68.38.100.91 (talk) 06:36, 15 August 2010 (UTC) Here is better animation fractal 2 throught not for an article. Edo 555 (talk) —Preceding undated comment added 13:46, 11 September 2010 (UTC).

Needs re-writing for the non-mathematician
The authors of this page are so clever and brainy and it's nice that they've been able to show off by writing in a language only comprehensible to other mathematicians - but this page is probably now (the day that Benoit Mandelbrot sadly passed away) being visited by non-mathematicians who would like to find out more about the great man's work. They wil learn precisely nothing from this page, since it assumes a colossal amount of mathematical knowledge and mastery of advanced mathematical terminology. What a shame. 90.207.65.50 (talk) 09:20, 17 October 2010 (UTC)

Watch the video 'Hunting the Hidden Dimension', It's a good explanation of Mandlebrot's work in layman's terms: http://video.google.com/videoplay?docid=-6917200224135375895# — Preceding unsigned comment added by 27.32.234.35 (talk) 03:20, 19 June 2011 (UTC)


 * Can you be more precise ? Start with the first section of the article - the "lead". Where exactly does this assume "a colossal amount of mathematical knowledge and mastery of advanced mathematical terminology" ? Granted it assumes that the reader knows what complex numbers and the complex plane are - but without assuming this much knowledge, you can't say much more than "The Mandelbrot set is a pretty shape". Gandalf61 (talk) 09:42, 17 October 2010 (UTC)
 * @90.207.65.50, Wikipedia has fine articles about the great man Benoit Mandelbrot and about the Fractal. I think that an average reader would find both those articles accessible. However the title of this article "Mandelbrot Set" should have tipped you off at the beginning that it is about a mathematical concept. If you want to see this described with a minimum of math jargon, see this and this at Simple Wikipedia. Cuddlyable3 (talk) 13:20, 17 October 2010 (UTC)


 * I think the original commenter has a point. Yes, you can't describe some concepts without certain terminology, but the emphasis should be firmly placed on making it as easy to understand as possible. I'm not convinced many lay readers would understand this at present and that's a problem that we should at least attempt to minimise. —Preceding unsigned comment added by 94.195.174.51 (talk) 22:08, 18 October 2010 (UTC)
 * General comments are all very well, but some specific suggestions would be better. For example, how exactly would you suggest the lead section of the article should be simplified ? Or the first paragraph ? Or the first sentence ? Gandalf61 (talk) 12:31, 19 October 2010 (UTC)
 * I tried to make some changes to the Formal Definition section, which have now been reverted. Some of the set theory notation looks very intimidating to the non-specialist although it actually is almost trivial. For example, the first statement $$P_c:\mathbb C\to\mathbb C$$ says only that $$P_c$$ is a function that transforms one complex number into another, something that most readers would surely find easier to understand if expressed in words. The following definition is more significant but still depends on the reader understanding the special use of the colon and the   $$\mapsto$$ symbol. Neither uses the same formalism as the algebraic notation in the introduction. The polynomial notation $$P_c(0)$$ is different again. All of this is contrary to the basic principle of starting simple and then developing ideas in a consistent way. Friv (talk) 21:34, 17 December 2010 (UTC)
 * I made some simplifications to the second paragraph in the "For Programmers" section because it talked about the midpoint in a pixel as if it were something special and because it didn't use the notation that had been set up in the first paragraph. That first paragraph may need similar treatment due to its use of the term "critical point". I would have turned that phrase into a link but I'm not sure that it is the correct term. Cutelyaware (talk) 02:32, 7 March 2011 (UTC)

Came looking for a good introduction to the Mandelbrot Set for my blog readers. The article is still way over spec at the start, and unusable for me. I did some maths at university and I cannot understand anything after "Mandelbrot Set". The article needs to assume no mathematics in the open paragraphs - this is a general encyclopedia, not a reference for mathematicians. Nothing about set theory is "trivial" to me. Nor would it be to, say, a 12 year old school kid. The authors need to think about who Wikipedia is for. By all means bring in the technical stuff lower down, but as it stands the page may as well be in a foreign language - I can read all the connectors, but none of the nouns or verbs. mahaabaala (talk) 07:23, 2 June 2011 (UTC)


 * You might like to compare these examples both of which are much better.

mahaabaala (talk) 07:23, 2 June 2011 (UTC)
 * http://www.ddewey.net/mandelbrot/
 * http://www.miqel.com/fractals_math_patterns/visual-math-mandelbrot-magic.html

Example of working code?
In my opinion, what makes the Mandelbrot set so particular and fascinating for programmers, is how easy it is to write a program that will exhibit so beautiful color shapes. In light of this, how would it be to include a working example in the article, like the following? The algorithm here is naive and sub-optimized, but it's easily understandable and pretty powerful for easily experimenting at home...

That is easily compiled and run under Linux with "gcc `sdl-config --cflags --libs` -O3 mandelbrot.c && ./a.out" (and probably as easy under Windows, if someone wishes to tell how).

Gc (talk) 21:14, 17 October 2010 (UTC)


 * Adding your own code to the article would be an example of original research, which is strongly discouraged in Wikipedia. Gandalf61 (talk) 21:17, 17 October 2010 (UTC)
 * However this is good tutorial material for the Wikibook on Fractals. Cuddlyable3 (talk) 07:54, 18 October 2010 (UTC)
 * Fine, but I think there's a lot of that already in wikipedia. For example in french one you can see a Logo program on bottom of http://fr.wikipedia.org/wiki/Courbe_de_Gosper so why it is ok on some pages and not ok on some others? Gc (talk) 09:41, 18 October 2010 (UTC)


 * Thx for your code. I have put it here. I think also about adding to wikibooks about fractals page/pages about computer graphic techniques, like drawing on the screen, direct creating image files. drawing in the memory. Your program would be good there. Regards --Adam majewski (talk) 14:34, 18 October 2010 (UTC)


 * added an informal definition, showing each step, while it lacks math tags it really helps i think. 2011/04/10 (YMD) For Mathamatics, Leave Science behind (talk) 21:32, 10 April 2011 (UTC)
 * @Gc: Adding your own code is not per se original research. Implementing a well-known algorithm (which is what I think you did) involves no more original research than writing an article about it. Rather less than that. Btw. I like the point you chose, or rather, the journey. Thanks a lot! --84.177.51.103 (talk) 19:00, 30 May 2011 (UTC)

"the Yoccoz parapuzzle"
The basic properties section says:
 * The dynamical formula for the uniformisation of the complement of the Mandelbrot set, arising from Douady and Hubbard's proof of the connectedness of $$M$$, gives rise to external rays of the Mandelbrot set. These rays can be used to study the Mandelbrot set in combinatorial terms and form the backbone of the Yoccoz parapuzzle.

The link is to the biographical article on Jean-Christophe Yoccoz, which is barely more than a stub and provides no information about "the Yoccoz parapuzzle". I'm adding a reference to The Mandelbrot set, theme and variations (Tan 2000), which deserves a link here anyway and which has more about it than I can understand (but the expression itself may be due to Tan rather than to Yoccoz). -- Thnidu (talk) 19:14, 27 November 2010 (UTC)

incorrect statement about point i
This (in the 3rd paragraph) is apparently incorrect.

"On the other hand, c = i (where i is defined as i2 = −1) gives the sequence 0, i, (−1 + i), −i, (−1 + i), −i, ..., which is bounded and so i belongs to the Mandelbrot set." —Preceding unsigned comment added by 141.210.135.115 (talk) 12:53, 25 March 2011 (UTC)


 * Is it ? Let's see:
 * $$z_0=0$$
 * $$z_1=z_0^2+i=0^2+i=i$$
 * $$z_2=z_1^2+i=i^2+i=-1+i$$
 * $$z_3=z_2^2+i=(-1+i)^2+i=(1-2i+i^2)+i=1-2i-1+i=-i$$
 * $$z_4=z_3^2+i=(-i)^2+i=-1+i$$
 * etc.
 * Looks fine to me. Gandalf61 (talk) 13:07, 25 March 2011 (UTC)

"If you have a complex data type in your programming language, you should use that instead."
That would most likely be less efficient than a purely-real algorithm, even with modern optimising compilers.

217.42.250.131 (talk) 22:27, 26 March 2011 (UTC)
 * Yes. I have changed "you should use that instead" to "using it can simplify your program". The subject of speed optimization does not belong in the introduction to the pseudocode. Cuddlyable3 (talk) 07:34, 12 April 2011 (UTC)

Hyperlink Proposal
Dear members of the English Wikipedia, I would like to suggest a new external link to be added to the article. It should direct to http://webcode-blog.org/data/mandelbrot.html. This is an online visualization of the Mandelbrot set which I created. It uses HTML5 Canvas technology. It makes it possible for everyone to explore the Mandelbrot set without installing additional software. Regards, 84.154.116.136 (talk) 19:32, 9 December 2011 (UTC)
 * It's nice, but it isn't intuitive to use, so at this time it wouldn't be an enhancement to this article. When I tried going there, I was unable to zoom in on places I clicked (it would zoom other places) and there was no obvious way to pan or zoom out. It also would not fit on a 1024x768 display. Some additional text explaining what to do, rather than presenting the viewer with a blank screen, would also be helpful. ~Amatulić (talk) 20:08, 9 December 2011 (UTC)
 * Thank you for your feedback. I will fix the problems you mentioned and repeat the proposal when I'm done. 84.154.120.154 (talk) 09:00, 10 December 2011 (UTC)
 * The problems should now be fixed. 84.154.120.154 (talk) 09:37, 10 December 2011 (UTC)
 * In other words: I now repeat my hyperlink proposal as I have reworked the whole user interface of the program in according to your criticism. 80.135.173.47 (talk) 18:03, 12 December 2011 (UTC)
 * See WP:EL.  It's not really necessary to link to every fractal generator out there.TheRingess (talk) 18:42, 12 December 2011 (UTC)
 * Perhaps it would fit better as a demonstration in the article on HTML5 (canvas).--LutzL (talk) 11:49, 13 December 2011 (UTC)
 * Please also consider the new version. Maybe it also didn't become clear why I proposed this to be added to the article as a hyperlink. This online application makes it possible for everyone to create high-quality images of the mandelbrot set and zoom into it for free. Thus I think now, it would be a great enhancement to this article. Best regards --84.154.119.123 (talk) 13:29, 16 February 2012 (UTC)
 * There are really very many free generators of the mandelbrot set out there. --Dylan Thurston (talk) 15:00, 15 July 2012 (UTC)

"Little copy"
The section 'Hyperbolic components' contains the following ambiguous statement (emphasis mine):

"Not every hyperbolic component can be reached by a sequence of direct bifurcations from the main cardioid of the Mandelbrot set. However, such a component can be reached by a sequence of direct bifurcations from the main cardioid of a little Mandelbrot copy (see below)."

By 'little Mandelbrot copy', does this statement apply to any pseudo-self-similar subset, or to some specific one (eg the n=2 bulb)? Also, was 'see below' intended to refer to the section 'Self-similarity'? Could someone with knowledge of the topic please rewrite this paragraph to be clearer?

--Quantum7 18:26, 16 March 2012 (UTC)

Neat Zoomable Version
There is a zoomable version of this fractal at. Would this be useful to link? Inter change  able  15:44, 19 April 2012 (UTC)

"large bailout radius N (e.g. 10^100)"
I think this must be a typo: first, the use of the word "radius" is probably not what was meant, in that in this context, it implies a distance, e.g., from the origin, and no one is going to wait 'til |z| > 10^100 before they move on to the next pixel. Likewise (actually, more so), no one is going to wait 10^100 iterations (which is what is implied by the variable name N) before "bailing out." Methinks what was/is meant was "large bailout iteration count (e.g., 100, or perhaps 1000)." If I'm missing something, i.e., this is correct the way it is, I feel it needs clarification; otherwise, if there's agreement that it's in error, I'd be happy to change it. — Preceding unsigned comment added by 67.168.97.70 (talk) 06:47, 6 July 2012 (UTC)

Section 'Continuous (smooth) coloring' needs clarification
I'm trying to understand the section 'Continuous (smooth) coloring' well enough to implement it. It's difficult - I do not think this section is well worded.

1. I'm guessing that "the number we subtract from n" is a reference to the equation immediately above those words, which contains a pronumeral n. Can someone confirm this? If that's the case, I think it would be much clearer to say either "the second term of the right hand side of the above equation", or perhaps to repeat the second term verbatim.

2. "using the connection of the iteration number with the potential function" Was this (as I suspect) intended to mean "defining a potential function, given by..."? Or is the word "connection" being used in its formal mathematical sense here? This function doesn't seem to me to have anything to do with a mathematical 'connection', but I'm not a mathematician.

In the course of attempting to implement this algorithm, and with some trial and error, I will probably figure out what this section was trying to say. When and if I reach that point, I'll come back here, clarify the wording and add the missing explanation.

And yes, the part about "large bailout radius N (e.g., 10^100)" also implies to me that we are doing an infeasible amount of computation, so is probably wrong. I suspect that either 'bailout radius' doesn't here mean what that usually means (ie we often bail out before 10^100 iterations), or that 10^100 is much larger than the author intended. — Preceding unsigned comment added by 180.200.140.87 (talk) 13:41, 14 February 2013 (UTC)

Z0 = 0 or Z0 = c?
The page says that Z0 = 0, but the book "The Science of Fractal Images" (ISBN: 0-387-96608-0 / ISBN: 3-540-96608-0) says in Appendix D (by Yuval Fisher) that Z0 = c. Additionally, http://warp.povusers.org/Mandelbrot/ says that Z0 = c.

I'm trying to find an answer to this confusion now in "The fractal geometry of nature" (ISBN: 3-7643-1771-X) by Benoît B. Mandelbrot, but I'm afraid he uses Julia-Sets to define M. I will keep on searching for an answer to this though, and would be happy for anyone to provide additional sources.

EDIT: Nevermind the last paragraph, "the fractal geometry of nature" apparently was written before the Mandelbrot-Set was defined. Silly me.

84.74.169.204 (talk) 15:44, 13 November 2012 (UTC)


 * If you start with Z0 = 0, then Z1 = c. Since membership of the Mandelbrot set is determined by the long-term behaviour for each value of c, it doesn't matter whether you start iterating at 0 or at c. Gandalf61 (talk) 16:05, 13 November 2012 (UTC)
 * Maybe better is to say that z0 is a critial point. It will be more general ( can be applied to other functions, not only z^2+c ). --Adam majewski (talk) 11:03, 7 April 2013 (UTC)

Too technical
While providing a lot of detail isn't a bad thing, this article has become so technical that it's no longer understandable for the average person. Unless you're a mathematical expert, this page is simply incomprehensible. It uses far too many technical terms and advanced mathematics. While this may be very interesting and understandable for a small number of people, I imagine it is well beyond the average wikipedia audience.BabyNuke (talk) 04:42, 23 February 2013 (UTC)
 * Readability could be improved, possibly in the lead section, but then again, not by much. When you're writing an article on complex mathematics, there will be technical terms. As far as I can tell, if you were to pare it down much more, you'd be stripping the article of valuable information. That being said, some sections might benefit from revision. Yangosplat222 (talk) 01:41, 10 March 2013 (UTC)
 * The article is well-written and i cant see anything inaccessible more than the language of number and sets, the lead sections are perfect. Wikipedia is an encyclopedia, it is not for general audience for every page. We should include notable properties for mathematicians, as long as it is notable enough, no matter how technical, it should be appeared somewhere, no matter how brief. If you cant understand the bottom section, then don't. --Mylittleanon (talk) 08:12, 7 April 2013 (UTC)

Fractals seem to be one area of mathematics that attracts a lot of New Agers and cranks who like to cherrypick fruits off the western rational edifice that they don't, at bottom, understand at all. But "there is no royal road to geometry"! I spend a lot of time thinking about how to communicate 'higher' mathematics with a more or less lay audience. To a certain point, it's desirable to render things into more 'understandable' language, but if you take the watering-down too far then nothing meaningful is being communicated. If this page is hard to understand, there are links to follow to make sense of the base concepts. If you don't understand or want to know more, follow them. That's exactly what wikipedia is for. I really don't agree that this article is 'too technical'; it's technical because it's a technical subject, but I think it's appropriately so. 06:03, 8 April 2013 (UTC) — Preceding unsigned comment added by Rhswain (talk • contribs)