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This is an old revision of this page, as edited by Lowercase sigmabot III (talk | contribs) at 00:18, 7 April 2020 (Archiving 2 discussion(s) from Talk:Mandelbrot set) (bot). The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision.
Archive 1Archive 2Archive 3Archive 4

Smooth coloring formula wrong

I don't know the right formula (it seems no one does) but the one given here is clearly wrong. Just take P-2, N=2, c=1.8 for example. In that case, one iteration (or two, depending how you count) gives z=5.04, which is over the bailout value of 2. Now, the log formula gives 1.2224, which is not in the range [0, 1) as specified. I wish someone who knows the correct algorithm (if any such person really exists) would fix this. Wikipedia readers are spreading this misinformation all over the web. 70.116.13.152 (talk) 03:31, 28 April 2013 (UTC)

Update: the August 2009 version seems to contain better information (it's certainly a different formula from this one, even after simplification, by at least a factor of 2 inside the outermost log, and it also explains that unusual bailouts are required for the smooth coloring algorithm, a crucial piece of information left out in the very poor article that exists today). I will nonetheless wait for a knowledgeable person to do something about it. 70.116.13.152 (talk) 03:49, 28 April 2013 (UTC)

Limit of. Iteration

Your last sentence of your. second paragraph should say the pixels are colored according to the number the sequence approaches. — Preceding unsigned comment added by Claustro123 (talk • contribs) 04:17, 8 March 2013 (UTC)

That is hardly possible, since the usual coloration is applied to points where the iteration diverges to infinity.--LutzL (talk) 13:55, 11 March 2013 (UTC)

Ugly picture

Look at http://en.wikipedia.org/wiki/File:Blue_Mandelbrot_Zoom.jpg. If you look closer, you can notice, that the rectangle in one picture does not strictly correspond to the following picture. The last picture is completely out of the blue, it has no telation whatsoever to the previous one and also has to few iterations. I find it disturbing. The picture has another version: http://commons.wikimedia.org/wiki/File:Mandelzoom.jpg which is slightly better, but suffers similar issues. Janek37 (talk) 21:21, 19 June 2013 (UTC)

It goes to a "personal" site. Is it ok? Tony (talk) 02:21, 9 July 2013 (UTC)

I know this is a hot topic and I know it has million examples on the web. Yet I believe a live demo that the readers can explore without any knowledge of programming languages or even the need to download anything, has a real value.

Please review this page: Interactive live demo (HTML + Javascript) It is very simple, interactive and totally open source.

Leeron-s (talk) 07:14, 8 September 2013 (UTC)

Relation to Julia Sets

Mandelbrot as classification of Julia sets
My (LL) earlier version of it

Having just finally learned, via sources outside of wikipedia, exactly what the difference between the Mandelbrot Set and Julia sets are, equation-wise, I think that this information should go into the article. I'm just not sure how to word it, or where to stick it in.

  • zn+1 = zn2 + c where z0 = 0 (or c) and you map out the variable c, is the mandelbrot set.
  • zn+1 = zn2 + c where c is a fixed constant complex number and you map out the variable of all z0 points, is the julia set of that given c.

The fact that the equation is the same, but which term is the variable, is of great importance to understanding what on earth the equation IS in the first place, and was the chief hindrance in my prior understanding. Any ideas how to put this into the article? Fieari (talk) 14:19, 21 March 2013 (UTC)

You are right. The basic information that the Mandelbrot set is a classification of Julia sets in that it consists of those parameters c such that the Julia set for c is connected resp. contains interior points resp. has z=0 as interior point is completely missing from this article. I found Alan F. Beardon, Iteration of Rational Functions, (Springer 1991) to be a helpful resource.--LutzL (talk) 11:09, 7 April 2013 (UTC)

This article says the Mandelbrot set is actually part of the Julia Set. Anyone else care to read? http://www.relativitybook.com/CoolStuff/julia_set_4d.html Shroobtimetraveller (talk) 06:08, 21 September 2013 (UTC)

Of course you can do that. But the big set is neither a Julia nor a Mandelbrot set. A certain set of parallel cross section gives the Julia sets and another cross section orthogonal to them is the Mandelbrot set. However, the presentation at the website ignores that the point z=0 is a special point for every Julia set in that it is the root of the derivative of the iteration. And in consequence, the iteration starting at z=0 alone already decides if the Julia set is a connected set. The other cross sections have the same importance as other Mandelbrot related pictures, like the Buddhabrot, log-escape maps etc. They look nice, but hold no further fundamental mathematical insight.--LutzL (talk) 07:51, 21 September 2013 (UTC)

POV?

"The Mandelbrot set has become popular outside mathematics both for its aesthetic appeal and as an example of a complex structure arising from the application of simple rules, and is one of the best-known examples of mathematical visualization." -- A little searching would support the popularity of the set (and then documenting it), but is a alleged esthetic appeal really a factor? I find the math great, but the visual image is extremely repugnant to me. It alone almost turned me away from studying fractals. Even today, years later, I get a negative visceral reaction when I see the Mandelbrot set (as opposed to, say, the Julia set).211.225.33.104 (talk) 04:46, 4 October 2013 (UTC)

The whole set is indeed rather boring, the esthetic interest is more on magnifications of the escape map. For an extreme example, I have seen step-colored megnifications as print on a dress. You may argue that those magnifications are close to magnifications of the corresponding Julia set.--LutzL (talk) 07:01, 4 October 2013 (UTC)
Indeed, as soon as you zoom in you'll see variety equivalent to the Julia set, the zoomed-out view is just a starting point. So you can't really say you like the Julia set but not the Mandelbrot, it would be like saying you like trees but don't like forests. Egrange (talk) 10:04, 10 January 2014 (UTC)

I've ran a project last year to pre-compute the Mandelbrot Set as a four Terapixel image:

Terapixel Mandelbrot Set Image

I'm suggesting this as an external link. The difference with the myriad of "classic" exploration programs being the pre-computation, meaning the exploration is fast and interactive even on tables and low-power devices. The raw pre-computed Data is also available through an API and downloads. Egrange (talk) 10:14, 10 January 2014 (UTC)

Possible omission?

Stephen Wolfram uses fractals in a unique way to describe how nature can create complex patterns by repeating a simple algorithm, like a cellular automaton. A sample of images from A New Kind Of Science demonstrates the underlying algorithms similar to that of the Mandelbrot set and fractals in general. Would it be appropriate to mention that herein, or is there a more general Wiki entry on fractals that might benefit from this? http://www.wolframscience.com/downloads/colorimages.html http://www.wolframscience.com/index.html


Hpfeil (talk) 19:22, 28 January 2014 (UTC)

Is the Mandelbrot set a fractal?

Here's what Mandelbrot himself had to say about this: http://www.webofstories.com/play/benoit.mandelbrot/86 — Preceding unsigned comment added by Lbertolotti (talk • contribs) 00:31, 16 February 2014 (UTC)

Dynamically generated DeepZoom image of Mandlbrot (set) fractal

Self similarity type?

The Mandelbrot set is said to possess quasi self-siimilarity, however I am not sure if that applies to the whole fractal or certain regions. Whoever knows the actual answer should make this clear. — Preceding unsigned comment added by 121.72.115.77 (talk) 07:24, 29 April 2014 (UTC)

See the link I provided above:

"The Julia sets again are self-similar; each part is like any other part, but one of the observations made about the Mandelbrot set was the following: that a small corner of the Mandelbrot set looks in many ways like the Julia sets corresponding to it; therefore the Mandelbrot set included as parts of it reduced scale images of an immense variety of Julia sets; therefore it is not self-similar in that respect. It is self-similar in the sense that the islands are like the continent but slightly deformed; but not in a sense that the way in which the islands are arranged, the kind of strings that link them together are the same; in fact, they're different in every point. Therefore the complication of a set goes beyond fractals. Julia sets are self-similar, they are fractals by every definition and by intuition, except when they are just straight intervals which happens for some cases, in which case they are too simple to be fractal. The Mandelbrot set is a complication which includes a huge number of different fractals in its structure and is therefore beyond any fractal. It is a paradox of sorts that this has become the icon of fractality, whereas it does not fit the definition of the concept at all. " Lbertolotti (talk) 14:18, 24 May 2014 (UTC)

All Julia sets?

In the NOVA documentary Fractals, Hunting The Hidden Dimension (around 21:45) the narrator claims the Mandlebrot set represents all the Julia sets with a single formula. If this is true, shouldn't the distinction be made in the lead rather than just saying it is closely related? - Shiftchange (talk) 04:47, 15 June 2014 (UTC)

National varieties of English

In reading this article, I noted multiple varieties of English were used, especially with respect to the spelling "color/colour". The WP Manual of Style guideline recommends consistency. In cases where inconsistent usage is found, it calls for discussion to establish a consensus. When discussion cannot resolve the issue, "the variety used in the first non-stub revision is considered the default."

The first revision using a variety of English was [1] on February 15, 2003. It used the "colour" spelling, but the article was a stub (though not tagged as such) at that time. Subsequent edits over the next year predominantly used the "color" spelling. It's debatable whether or at what point during this time the article ceased to be a stub.

At present, the "color" spelling outnumbers the "colour" spelling 4 to 1 (47 vs. 12). I would consider that a deciding factor in establishing the American spelling as the de facto standard, but am opening the question for discussion. Unconventional (talk) 16:19, 25 July 2014 (UTC)

Linking to FractalJS

Solendil (talk) 10:06, 8 September 2015 (UTC)  Hi, FractalJS is an open-source interactive fractal explorer in javascript, working right in the browser. It's certainly not the first of its kind, but AFAIK, it's the fastest and most user-friendly. I created FractalJS to promote fractal exploration by everyone : you don't need to install anything, just click, scroll and explore; you can also share links of your favorite places.

I believe FractalJS would be a nice companion to fractal related web pages. I just modified the "Image gallery of a zoom sequence" to include FractalJS links to the same picture. I was wondering if we could add FractalJS in the External Links category of the Mandelbrot article? And how about providing FractalJS links in other relevant articles, like Burning Ship fractal, etc?

This article and Wikipedia "book-creator"

I have a rather obscure problem with this article. As you may know, Wikipedia includes a facility for compiling PDF "books" out of a set of related articles. They do say that the system is "crippled" but I've used it with some very successful results. However, my latest venture was to be on fractals, the first chapter being Wikipedia's "Fractal" article and with subsequent articles on various types of fractals stc. This has all worked fine - except that including the "Mandelbrot set" article causes the book-creation process to fail. (Other articles on fractals are fine.) Does anyone have any idea why this should be? GeoffHope (talk) 10:55, 13 February 2016 (UTC)

I would guess its either the math formula or the total size of the article causing the break. Considering that creating a book is not a widely used feature its probably better to raise it elsewhere. - Shiftchange (talk) 12:33, 13 February 2016 (UTC)

Thanks for such a quick and helpful response. Your thoughts coincide with mine, that it's either the length of the article or the mathematical symbols. I'll follow your suggestion. GeoffHope (talk) 12:52, 13 February 2016 (UTC)

It was caused by a broken HTML fragment. I fixed it and it complies now. — Cheers, Steelpillow (Talk) 15:38, 13 February 2016 (UTC)

Many thanks! I had just logged on to say "It's working now!" GeoffHope (talk) 09:41, 14 February 2016 (UTC)

a question

what program was used to make the images from the "image gallery of a zoom sequence"? — Preceding unsigned comment added by 178.187.248.198 (talk) 11:47, 28 August 2014 (UTC)

Periodicity checking

Dieser Abschnitt ist in mehreren Punkten falsch. Sollten zwei aufeinander folgenden Iterationen die selben Koordinaten haben, dann beträfe das nur das Hauptkartiodit der MBM. Alle anderen Teile innerhalb der MBM haben andere Periodizitäten. Es müßte jede 2. 3. 4. usw. mit der ersten verglichen werden. Innerhalb der MBM werden nie mögliche aufeinanderfolgende Iterationen die exakt selben Koordinaten haben. Die Konvergenz ist unendlich lang. Das Programmbeispiel ist zudem sehr abhängig vom verwendeten Computer-Zahlenformat, was jeweils ein anderes Verhalten zur Folge hat, was aber im erzeugten Bild nicht erkennbar ist. — Preceding unsigned comment added by 178.26.205.207 (talk) 17:39, 18 September 2014 (UTC)

Google-translated: This section is wrong on several points. If two successive iterations have the same coordinates, then would concern only the Hauptkartiodit [main cardioid?] [of] the MBM. All other parts within the MBM have different periodicities. It would have to be compared every second third fourth etc. with the first. Within the MBM never possible successive iterations will have the exact same coordinates. The convergence is infinitely long. The sample program is also very dependent on the used computer numerical format, which each has a different user experience, which is not visible but in the generated image.
AndrewWTaylor (talk) 20:32, 30 September 2014 (UTC)

Query

In technical terms this article goes way beyond my knowledge. But I can tell that it's a really first-rate article.

What I came here to find out is the potential uses of this advance in other areas, whether mathematical or further afield. But alas, there's no section briefly describing that. Tony (talk) 13:32, 28 June 2016 (UTC)

Hi @Tony1: Excellent question. Probably better asked at Wikipedia:Reference desk/Mathematics but I'll give it a shot here.
The Mandelbrot Set is an example of fractal geometric shape: an infinitely complex shape generated from an extremely simple formula. Benoit Mandelbrot, who discovered it, is credited with introducing the general concept of fractal geometry as a means to explain complex shapes in nature, such as clouds, coastlines, tree bark, and so on. The Mandelbrot Set is probably the most famous fractal, but that particular fractal doesn't really have any practical uses that I know of, other than as a pedagogical device. A related formula, the logistic map, has been used as a model of population cycles (and the nodes of the logistic map coincide with the cusps in the Mandelbrot set).
This BBC article] offers a general description of the more practical uses of fractal mathematics. That wouldn't quite be in the scope of this article, however, since this article is about one specific (and famous) fractal object. ~Amatulić (talk) 05:55, 29 June 2016 (UTC)
Update: I found another one that actually shows relationships between real-world phenomena and the Mandelbrot Set here: http://www.sgtnd.narod.ru/science/complex/eng/main.htm -- not easy to understand, but one can get the general idea. ~Amatulić (talk) 06:03, 29 June 2016 (UTC)
Thanks for this. BBC article a little non-specific about the spin-offs (earthquakes, but ... how?). The other link pretty hard to understand, though, for anyone but an expert. But the beauty arising from simplicity is possibly enough to behold! Tony (talk) 07:27, 29 June 2016 (UTC)
My favorite natural fractal that actually looks like a geometric shape, rather than a random thing like a cloud or a coastline, is Romanesco broccoli. I recall reading an article in which someone came up with a 3-dimensional formula to replicate that fractal shape. Unfortunately the Wikipedia article doesn't link to that paper. ~Amatulić (talk) 07:36, 29 June 2016 (UTC)

Please do not remove pseudo-code from this article.

For many people (myself included) it is easier to understand concepts expressed in pseudo-code than mathematical notation.

So far two individuals have reversed removal attempts by 84.249.83.20. Until better reasons can be given, please stop. Thank you. — Preceding unsigned comment added by 173.212.143.233 (talk) 23:02, 12 May 2016 (UTC)

Location in Article

Please feel free to give suggestions as to where the pseudo-code should be located. The section will remain collapsed by default to prevent intrusiveness. — Preceding unsigned comment added by 173.212.143.233 (talk • contribs) 00:21, 14 May 2016‎

Quality

Please feel free to give suggestions for how it could be improved if you feel it is lacking in some way.

These sources were used as guidelines on how to write good pseudocode:

— Preceding unsigned comment added by 173.212.143.233 (talk • contribs) 00:36, 14 May 2016‎

It is definitely misplaced in the lead section. But also, "The section will remain collapsed by default" violates MOS:DONTHIDE. —David Eppstein (talk) 01:55, 14 May 2016 (UTC)
I support the pseudocode removal from the lead section for those reasons, and because it's redundant. There's already sufficient pseudocode in the section Mandelbrot set#Escape time algorithm. ~Amatulić (talk) 21:02, 14 May 2016 (UTC)
I have a problem with "If the above loop never terminates return true". Even in very abstract pseudocode, I don't think you can get away with a test that cannot be implemented in any language. Gandalf61 (talk) 12:21, 15 May 2016 (UTC)
The pseudocode that remains in the article doesn't have this dubious feature. ~Amatulić (talk) 20:52, 15 May 2016 (UTC)
(Pseudocode can be seen in this revision) Jimw338 (talk) 03:43, 5 April 2017 (UTC)

File:Mandelbrot sequence new.gif to appear as POTD soon

Hello! This is a note to let the editors of this article know that File:Mandelbrot sequence new.gif will be appearing as picture of the day on June 15, 2017. You can view and edit the POTD blurb at Template:POTD/2017-06-15. If this article needs any attention or maintenance, it would be preferable if that could be done before its appearance on the Main Page. — Chris Woodrich (talk) 01:33, 3 June 2017 (UTC)

Mandelbrot zoom sequence
A zoom sequence illustrating the set of complex numbers termed the Mandelbrot set. Images of the set, which was defined and named by Adrien Douady in tribute to the mathematician Benoit Mandelbrot, may be created by sampling the complex numbers and determining whether the result of iterating for each sample point goes to infinity. Images exhibit an elaborate and infinitely complicated boundary that reveals progressively ever-finer recursive detail at increasing magnifications. Consequently, the Mandelbrot set has become popular outside mathematics both for its aesthetic appeal and as an example of a complex structure arising from the application of simple rules.Animation: Simpsons contributor
Archive – More featured pictures...
Its lovely, a great example of infinity visualised. Would the addition of one more link be inappropriate? I wanted to suggest changing the text from created to generated and adding the link to Fractal-generating software. - Shiftchange (talk) 06:30, 3 June 2017 (UTC)

Mandelbrot Curves in Desmos

In case anyone finds this useful, I have plotted the first few Mandelbrot curves on the website Desmos. --  Denelson83  04:18, 25 July 2017 (UTC)

uncoloured pictures showing the set .

Is there a corresponding collection somewhere of uncoloured pictures? I would like to be able to see what the set itself looks like! Davidnugget (talk) 20:36, 8 January 2018 (UTC)

Not specifically, but if you follow the "commons" link at the bottom of the article you will find some mixed among the colored ones. —David Eppstein (talk) 20:39, 8 January 2018 (UTC)

Thanks, but I suppose I was being sarcastic that the article effectively has no detailed pictures of the set at all. Maybe it should be retitled. Davidnugget (talk) 22:18, 8 January 2018 (UTC)

The set itself is clearly visible in many (but not all) of the images in the article, including some of finely detailed regions and some that are purely monochrome. Perhaps as well as being sarcastic you are exaggerating for effect? Perhaps it would be more effective to make constructive suggestions for improvement rather than oblique and veiled criticism? —David Eppstein (talk) 22:25, 8 January 2018 (UTC)

Come off it, the set is hardly "clearly" visible in any of the complicated pictures where the set itself is very wispy. The constructive suggestion is that it would be worthwhile to include a few of just the set itself in the outlying regions. Davidnugget (talk) 22:34, 11 January 2018 (UTC)

Ok, so you don't want to *add* uncolored pictures, you want to *remove* the color ones? Or otherwise, how am I to interpret it when I say that it is clearly visible in some of the pictures and your response is "but what about the other ones where it's not?" —David Eppstein (talk) 23:25, 11 January 2018 (UTC)

Reference to the animation in Generalizations section

Single frames of the animation placed in Generalization section has been created using mandelbrot.ovh.org online generator. Mandelbrot.ovh.org is a previous adress of the website http://e-mandelbrot.com: https://waybackmachine.org/web/20140625105918/http://mandelbrot.ovh.org/ (Capture from 25th February, 2014)

There is/was a statement on the pages: "You can use the results from this site in all of ways, under one condition, you have to include the site address and the author name in your work."

According to reserved copyright, the address and the author name have to be mentioned.

If it were created using a particular tool it is for the original uploader to note that, on the image page. But according fo the page they created it themselves, which is quite plausible as such images are easy to create using a variety of tools or even as a programming exercise.
In addition, whatever its source, it is not something that needs noting in the article. Wikipedia contains many images with source information, including images on this page, but the source is almost always put on the image page, whether a web link, what program was used, or the source code that was used to generate it. Again, this is for the uploader to decide on.--JohnBlackburnewordsdeeds 00:27, 23 January 2018 (UTC)
----------------------------------------------------------
You have just broken the copyright! IT IS ILLEGAL!
---------------------------------------------------------- — Preceding unsigned comment added by 157.158.47.16 (talk) 07:12, 23 January 2018 (UTC)

Kismetjim: this page used to have equations that non-mathematicians could understand. Either there should be a short explanation with a link to what the mathematical notation in this article means, or there should be widely-understood notation. This is true of virtually every interesting mathematical article on Wikipedia. — Preceding unsigned comment added by Kismetjim (talk • contribs) 06:28, 25 March 2018 (UTC)

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False statement removed from an image caption

The following picture under the tile 'Relationship with Julia sets' has had the following label: 'A zoom into the Mandelbrot set illustrating a Julia "island" and the corresponding Julia set of the form , in which c is the center of the Mandelbrot set zoom-in'

This statement is false. Obviously so, in fact, as the Julia island is surrounded by projections which are characteristic of the right side of Seahorse Valley, while the Julia set is generated from the left side of the valley. The creator of the picture, Simpsons contributor, only mentions that the Julia set 'corresponds' to the Julia island in the picture above it, which is centered at -0.743643887037151 + 0.131825904205330i. He makes no claim that it is generated from its center point.

The anon that originally edited it out was right, the Julia set at -0.743643887037151 + 0.131825904205330i truly is connected, and looks like the image below.

Julia Set at -0.743643887037151 + 0.131825904205330i

Please do not revert the page back to the false statement again.

Starprizm (talk) 04:32, 17 December 2017 (UTC)

What makes you sure that this Julia set is connected ? How can you prove it ? --Adam majewski (talk) 14:38, 24 May 2018 (UTC)
check parameter plane : -0.743643887037151 0.131825904205330 2.158333333333333e-16 . The test parameter is in the center image and is outside Mandelbrot set = Julia set is disconected. --Adam majewski (talk) 15:58, 24 May 2018 (UTC)

Physics of the Mandelbrot set

I once built a Mandelbrot set out of flower pots and odds and ends. The cardioidal point was composed of a pumpkin vine growing southward.

I was not expecting a result, but one day, when the snow was a few centimeters thick with patches of snow melt, the snow was melted around my sculpture within an ellipse just as if it were calculated on the computer.

I am unsure what to conclude: fluke? Universal conservation of position? reception of radiation?

Is there any physics research out there like this experiment? I am very curious to know. — Preceding unsigned comment added by 45.72.152.47 (talk) 21:59, 2 January 2019 (UTC)

Question regarding "Histogram coloring"

This section is not written very well in terms of conveying the correct meaning to the reader. For example, the section does not state that an iteration count must be paired with each pixel to be plotted, which may significantly increase the memory footprint and slow down the code. Jdbtwo (talk) 15:41, 11 June 2019 (UTC)

Mandelbrot zoom sequence color palette?

Does anyone know what color palette the Mandelbrot set zoom sequence uses? (starting with this file :

) I'd like to know for the purposes of a section I'm editing. Jdbtwo (talk) 15:05, 22 June 2019 (UTC)

"Histogram coloring" section

I've re-written parts of this section and added a reference. I don't think this section is OR, as the same algorithm is mentioned all over the web, but during my cursory search, I couldn't find an academic source -- just a forum post. Perhaps this can be improved in the future. Jdbtwo (talk) 18:21, 13 June 2019 (UTC)

If someone could provide more reliable sources for the histogram coloring section, I'd be grateful. Jdbtwo (talk) 15:28, 1 July 2019 (UTC)

Connection to non-linear dynamics

The Mandelbrot set is often implicated as providing insight into non-linear dynamics, i.e., into the solution of non-linear PDEs. But I have not encountered a source yet that makes this connection explicit, i.e., that provides a particular system of PDEs for which the Mandelbrot set is precisely the domain of definition, or something like that. Does such a source exist? If so, I think it should be referenced somewhere in the article. Eleuther (talk) 22:44, 10 August 2019 (UTC)

Ellipsoids

I made some functions to skew the set. https://www.youtube.com/watch?v=WrLirCwsxh0 — Preceding unsigned comment added by Tejolson (talk • contribs) 04:43, 14 August 2019 (UTC)

Splitting the article

This article has become way too long. I propose splitting it into at least two articles, the first mainly devoted to the mathematics of the set, and the second, and others if needed, to programming issues related to creating images from the set. However, I obviously can't take on this project by myself. It needs a lot of initial discussion and consensus about how it should be carried out. Please help. Thanks. Eleuther (talk) 02:48, 7 January 2020 (UTC)

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