Talk:Julia set/Archive 1

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Archive 1

Waist

What's a "waist", as defined in the article? — Preceding unsigned comment added by 209.107.95.230 (talk) 21:32, 29 July 2003 (UTC)

A place where the Mandelbrot set constricts to a point, then expands again. Precisely, it is a point such that two disjoint open neighborhoods whose boundaries are circles tangent at the waist lie inside the Mandelbrot set, while a line segment or arc of a circle whose midpoint is the waist, and which is tangent to said circles at the waist, lies outside the set except for the waist. -phma 21:11, 9 March 2004 (UTC)

Original?

I read in Fractal Geometry: Mathematical Foundations and Applications by Kenneth Falconer, that the set of all points c for which the Julia set of f(z) = z^2 + c is connected was the definition of the Mandelbrot set, and the definition in the article was later proven (in the book, at least). I can't submit any more, for fear of copywright infringement. — Preceding unsigned comment added by Scythe33 (talk • contribs) 23:56, 2 June 2005 (UTC)

Yes, when Benoît Mandelbrot first plotted the Mandelbrot set, he was investigating the values of c for which the Julia set of z^2+c is connected (in fact, he was originally investigating a related question concerning the values of c for which stable limit cycles exist). But Mandelbrot already knew that the Julia set is connected if and only if the critical point at z=0 does not lie in the domain of attraction of the super-attracting fixed point at infinity i.e. if and only if the forward orbit of 0 does not tend to infinity - this is a special case of more general results about rational maps proved by Pierre Fatou and Gaston Julia around 100 years ago. This equivalent definition of the Mandelbrot set is more suited to numerical computation, as well as being more easily understood (it does not involve the concepts of "connected" or "Julia set"), so it is the one that is usually given today. For Mandelbrot's own account of his early work on the Mandelbrot set see Fractals and the Rebirth of Iteration Theory in The Beauty of Fractals; Peitgen and Richter, 1986. Gandalf61 11:32, 4 June 2005 (UTC)

pictures and phi

In the articel in picture descriptions there is often mentioned "phi". For example "Filled Julia set for fc, c=φ−2" or "Julia set for fc, c=(φ−2)+(φ−1)i". But the "phi" isn't explained anywhere in the article. So where does it come from? --EnJx 21:17, 2 April 2007 (UTC)

phi is the golden ratio, either 0.6180339.. or 1.6180339

({\sqrt {5}}\pm 1)/2

, i believe.

One use that number in many ways in fractal geometry because it has some interesting properties.

Thanks for answer, good idea! It seems that in this article the phi (as a golden ratio) is cca 1.6, then c=phi-2=-0.4 and c=(phi-2)+(phi-1)i=-0.4+0.6i. --EnJx 20:40, 3 April 2007 (UTC)

Quaternion Julias

There is no mentioning of any Quaternion Julia Set in this article. Can someone kindly contribute more information on Quaternions? Doomed Rasher 23:05, 13 September 2006 (UTC)

Quaternions

Paxinum 08:59, 5 April 2007 (UTC)

Spoken Word Version

I've stuck a spoken word version of this article up (I was bored and thought it would be fun!) so comments and changes can go here, or preferably my talk page :) Please note I have changed wording very slightly in a couple of places to make it more clear in spoken word. JebJoya 22:53, 18 June 2007 (UTC)

Properties of the Julia set and Fatou set

May I add that $J(f)=J(f^{-1})$ ? dima (talk) 00:59, 30 September 2008 (UTC)

As long as you have a source for this assertion. I think you will have to explain just what you mean by

f^{-1}

. It is usual to define

f^{-1}(z)

as the set of pre-images of z, but there is no function

f^{-1}

because

f

is typically many-to-one. Gandalf61 (talk) 09:29, 30 September 2008 (UTC)

Gandalf61, Thank you; now I see, I was not clever. I mean, any of functions

f^{-1}

such that

f(f^{-1}(z))=z~\forall z\in \mathbb {C}

except perhaps one point. However, in general case, we cannot claim that

f^{-1}(f(z))=z~\forall z\in \mathbb {C}

; after to evaluate

F=f(z)

it is difficult to guess, which of solutions

\phi

of equation

f(\phi )=F

should refer

f^{-1}(F)

. With such explanation, the relation I suggest seems to be correct, although I have no appropriate source on hand. dima (talk) 03:32, 1 October 2008 (UTC)

Simplicity of Article

I remember the previous versions of the mandelbrot set and julia set articles, that were far more accessible to those not schooled in advanced mathematics. The mandelbrot and julia sets are of great interest to the less mathematically trained, and the article should reflect this. --86.141.94.84 16:34, 1 August 2006 (UTC)

Maybe one should post a similar article with a more "practical" approach under Recreational mathematics

or something. Just a thought.... Paxinum 19:25, 3 April 2007 (UTC)

The initial outline is quite formal but the language used is no more advanced than that encountered on a undergraduate course in complex analysis —Preceding unsigned comment added by 212.248.196.12 (talk) 09:59, 11 October 2007 (UTC)

I completely agree: what is the point of presenting the most abstract definition possible? It takes too much effort to read it for those who are non-specialists or who want to program the julia set, which is 99% of people who access this page. I think the article needs a very major rewrite. There is no need for all the formalism. Also, the section about plotting is not how it's done in practice! In practice, it's basically done like newton fractal, i.e. for every pixel in the window, iterate until converges; then assign colour based on where it converges to.

Asympt (talk) 17:27, 13 April 2008 (UTC)

I agree. Jonathan coulton was able to explain a julia set (in his wrongly named song "mandelbrot set") as: Take a point called Z in the complex plane. Let Z1 be Z squared plus C. And Z2 is Z1 squared plus C. And Z3 is Z2 squared plus C and so on.... If the series of Z’s should always stay close to Z and never trend away, that point is in the Mandelbrot Set

Wikipedia describes it as In complex dynamics, the Julia set J(f)\,[1] of a holomorphic function f informally consists of those points whose long-time behavior under repeated iteration of f can change drastically under arbitrarily small perturbations (bifurcation locus). The Fatou set F(f)\, of f is the complement of the Julia set: that is, the set of points which exhibit 'stable' behavior. Thus on F(f)\,, the behavior of f\, is 'regular', while on J(f)\,, it is 'chaotic'. These sets are named after the French mathematicians Gaston Julia,[2] and Pierre Fatou[3] who initiated the theory of complex dynamics in the early 20th century.

for a subject that attracts a lot of general public, not well skilled in advanced math, I strongly believe that the inital paragraph should be totally rewritten. The first paragraph is supposed to define the concept the simplest way possible (but not any simpler, to paraphrase), and it is making a disservice by deflecting away any curious that came here to know more about fractals. The complete explanation should come in the articleh but not in the head. --Alexandre Van de Sande (talk) 02:42, 20 January 2009 (UTC)

Alexandre, you are wrong about the description in the song. It does describe generating the Mandelbrot set, not a Julia set as you think. Cuddlyable3 (talk) 13:34, 2 May 2010 (UTC)

Agreed with the need for simplicity. I added the "jargon" tag (just before looking at the talk page); do with it what you will, but I think it's useful to keep up as a signal until this article is a little more accessible to an encyclopedic audience. Thanks. Noble-savage (talk) 05:02, 2 March 2009 (UTC)

Just want to add my vote to the need for a re-write. Would it be unreasonable to reinstate an earlier version of the first paragraph (say from 2006)? Without meaning to sound rude, the article seems to have been hijacked by undergraduates with an overt love of jargon that casual readers have zero chance of comprehending. One of the key attractions of fractals is that they are ubiquitous in nature and have a universal aesthetic appeal. The article in its present form seems somehow to have missed the point. —Preceding unsigned comment added by 82.18.141.162 (talk) 20:35, 18 March 2009 (UTC)

I have re-written the section 'Formal definition' (Gertbuschmann (talk) 20:08, 25 April 2010 (UTC))

Target audience unidentified ⋅ encyclopedia or useless journal?

The purpose of wikipedia.org is lost to those that can breath at the highest altitudes. I agree with the observation that not only this article's jargon and most articles at wikipedia.org that could be technical are stratospheric, drifting into realms not of general comprehension... to a point that those capable of comprehending are so few that the editing audience is lost. Without a meta-perspective guideline that perhaps provides an overview with a drill-down approach to the more complex, I find I can only make sense of the technical wikipedia articles by extensive google research to interpret the wikipedia article — not the definition of an encyclopedia. This is a social phenomena, and we see an indication of where wikipedia will eventually go... into the clouds... if continuing without education guidelines. DonEMitchell (talk) 06:12, 17 April 2010 (UTC)

The problem is not so much the technical altitude of this article as the fact that a general reader should start at the article Fractal and drill down from there. Eventually we will have a tutorial Wikibook but that needs more work. Cuddlyable3 (talk) 13:47, 2 May 2010 (UTC)

Putting faces in field lines

Pictures in the field lines for an iteration of the form $z^{2}+c$

A section "Field lines" was added[1] that shows this remarkable picture. The text explains the faces thus:

A field line is divided up by the iteration bands, and such a part can be put into a ono-to-one correspondence with the unit square: the one coordinate is the calculated from the distance to the centre of the field line, the other is the non-integral part of the real iteration number. Therefore we can put pictures into the field lines (third picture).

Comments? Cuddlyable3 (talk) 20:38, 30 April 2010 (UTC)

I have written this and it is not very clear, I have re-written it a little, but we cannot use more words on this construction in a Wikipedia article. (Gertbuschmann (talk) 10:20, 4 May 2010 (UTC))

The image is a striking demonstration that the field lines are orthogonal to the iteration contours, by applying them to conformally map images of human faces. Cuddlyable3 (talk) 11:59, 4 May 2010 (UTC)

In many cases chaotic layout of information

I was trying to find just how the image is plotted, and the information is missing. The article is chaotic when compared to e.g. article about mandelbrot set. —Preceding unsigned comment added by 84.10.174.127 (talk) 22:43, 29 March 2011 (UTC)

If you want find infomations about : "how the image is plotted" then you should look at wikibooks : Fractals, Fractals//Iterations_in_the_complex_plane/Julia_set or Pictures of Julia and Mandelbrot Sets. Regards --Adam majewski (talk) 08:18, 30 March 2011 (UTC)

Bug in the DEM/J sample

There is a bug in the source code, which may explain why the center of the image appears noticeably blurred.

   nz = 2*(x*xp - y*yp) + 1; // bug!
   yp = 2*(x*yp + y*xp); 
   xp = nz;

For Julia sets, z is the variable and c is a constant. Therefore df[n+1](z)/dz = 2*f[n]*f'[n] -- you don't add 1.

For the Mandelbrot set on the parameter plane, you start at z=0 and c becomes the variable. df[n+1](c)/dc = 2*f[n]*f'[n] + 1. That is what's being (incorrectly)used in the source code. Incidentally it's called "DEM/M", with M standing for the Mandelbrot set.

With the correct DEM/J algorithm, the image should be uniformly sharp.

— Preceding unsigned comment added by Hskim000 (talk • contribs) 01:11, 27 May 2011 (UTC)

You are right, byu could you expand explanations :

df[n+1](c)/dc = 2*f[n]*f'[n] + 1

Why not :

df[n+1](z)/dc = 2*f[n]*f'[n] + 1.

starting point is:

z=0

It is constant , right ?

Could you describe a few steps of computing derivetive, like :

$z_{0}=0$

$z_{1}=f_{c}(z_{0})=c$

$z_{2}=f_{c}(z_{1})=c^{2}+c$

and

$z'_{0}={\frac {d}{dc}}f_{c}^{0}(z_{0})=?$

$z'_{1}={\frac {d}{dc}}f_{c}^{1}(z_{0})=?$

$z'_{2}={\frac {d}{dc}}f_{c}^{2}(z_{0})=?$

???--Adam majewski (talk) 20:01, 27 May 2011 (UTC)

Let's start with $f_{c}(z)=z^{2}+c$ .

For a Julia set

$z_{0}(z)=z$

$z_{1}(z)=f_{c}(z)=z^{2}+c$

...

$z_{n}(z)=f_{c}^{n}(z)=z_{n-1}(z)^{2}+c$

${\frac {d}{dz}}z_{n}(z)=2z_{n-1}{\frac {d}{dz}}z_{n-1}$

For the Mandelbrot set, there is no variable z.

$z_{-1}(c)=0$

$z_{0}(c)=c$

$z_{1}(c)=f_{c}(c)=c^{2}+c$

$z_{n}(c)=f_{c}^{n}(c)=z_{n-1}(c)^{2}+c$

...

${\frac {d}{dc}}z_{n}(c)=2z_{n-1}{\frac {d}{dc}}z_{n-1}+1$

Make sense? — Preceding unsigned comment added by Hskim000 (talk • contribs) 21:33, 30 May 2011 (UTC)

yes. Thx. I have made some chages. Please look here. If you will find more bugs or know how to improve it let me know. Regards. --Adam majewski (talk) 18:02, 12 June 2011 (UTC)

Trouble understanding statement

I'm having trouble interpreting the following: "The termini of the sequences of iterations generated by the points of F_i ... are finite cycles of finite or annular shaped sets that are lying concentrically." Is each point cycling between annuli, or confined to one annulus which is concentric to any annulus that terminates another point's iteration? ᛭ LokiClock (talk) 19:37, 21 July 2011 (UTC)

Problem with formal definition of Fatou domains

Right now it is asserted that "there are a finite number of open sets such that..." in the definition of F_i. Does it mean to say "There exists a finite collection of open sets such that..."? In other words, it definitely isn't possible to play game with dividing up the Fatou domains just slightly differently so as make an infinite amount of candidate F_i's? —Preceding unsigned comment added by 69.123.96.13 (talk) 04:58, 17 October 2010 (UTC)

Later on it reads, "The last statement means that the termini of the sequences of iterations generated by the points of F_i are either precisely the same set, which is then a finite cycle, or they are finite¹ cycles of finite² or annular shaped sets that are lying concentrically." I feel as though ¹ should be "infinite" and ² should be "circular". — Preceding unsigned comment added by Sorin2120 (talk • contribs) 06:17, 1 May 2012 (UTC)

Missing explanation of term DEM/J

The term "DEM/J" is used in the article without any explanation. Please give some explanation, otherwise the whole section, currently only a gallery of some b/w images, is completely meaningless.--SiriusB (talk) 20:57, 5 February 2013 (UTC)

uadratic map as a pseudorandom number generator

I removed the following statement about the quadratic map $f(x)=4(x-1/2)^{2}$ (which is similar to $f(x)=x^{2}-2$ under an affine mapping of $[-2,+2]$ to $[0,1]$ ):

This can be used as a method for generating pseudorandom numbers.

There is no clear connection in the linked article. Presumably, the intended method is: start with an arbitrary "seed" $x$ and iterate $x\gets f(x)$ some number of times to get the "random number" $y$ . However, $f$ is not one-to-one, and in fact will lose some information about the seed $x$ at every iteration. Worse, it stretches part of the interval and compresses other parts; so, if the iteration is performed with fixed precision, additional information will be lost in the compressed parts at every iteration. Therefore, one should not recommend this method for practical use. --Jorge Stolfi (talk) 02:15, 19 August 2013 (UTC)

Presentation of 400 Julia sets

I want to introduce the Julius Ruis set, being a smart presentation of 400 Julia sets, showing that the Mandelbrot set is the parameter basin of all closed Julia Sets. — Preceding unsigned comment added by Julesruis~enwiki (talk • contribs) 21:16, 11 April 2005 (UTC)

Sorry, but this is not the place to do it: our no original research policy forbids this sort of content. - jredmond 21:44, 11 April 2005 (UTC)

I do not understand your remark. See File:Julius-Set.jpg. This is an example of the Julius Set for z'=z^2+c. — Preceding unsigned comment added by Julesruis~enwiki (talk • contribs) 22:18, 11 April 2005 (UTC)

From WP:NOR:

If you have a great idea that you think should become part of the corpus of knowledge that is Wikipedia, the best approach is to publish your results in a peer-reviewed journal or reputable news outlet, and then document your work in an appropriately non-partisan manner.

Basically, it says that we're sorry but this is not the appropriate place to publicize your new approach to Julia sets. - jredmond 22:24, 11 April 2005 (UTC)

Why then published the image with 121 Julia sets?

I am very sorry. I thought I was giving a service to Wikipedia. — Preceding unsigned comment added by Julesruis~enwiki (talk • contribs) 22:29, 11 April 2005 (UTC)

I disagree, and think that the picture makes sense, it is something that is very natural to do to a trained mathematician, and the picture is a great illustration that definitely SHOULD be on wikipedia. Paxinum (talk) 20:05, 19 August 2015 (UTC)

All the little Julia set pictures are inappropriate, unless:

All the little Julia set pictures are inappropriate, unless they are labeled so that the reader know where the Julia set is — and what the colors mean.

Currently there is no indication of either one, so I hope someone removes them — or labels them — soon.Daqu (talk) 13:04, 18 August 2015 (UTC)

I agree, it requires a trained eye to extract some information from these pictures. I suggest to replace with cleaner, more straigtforward, and less "artistic" images. Paxinum (talk) 20:11, 19 August 2015 (UTC)