Talk:List of fractals by Hausdorff dimension/Archive 1

penrose tilings
Could you explain shortly how to determine the Hausdorff dimension of the Penrose tiling? In what sense it can be seen as a true fractal? At some scale its geometry is no longer "fine". The citation has no details. Beaumont 21:02, 2 October 2006 (UTC)


 * The best is to have a look at the external link (bottom of page).Prokofiev2 3 October 2006 (UTC)
 * Thanks! I've changed it into a reference. Beaumont 15:56, 4 October 2006 (UTC)

Coastline of Norway
Felt a fleeting temptation to link it to Slartibartfast... AnonMoos 19:20, 9 December 2006 (UTC)

z-order curve
I believe that this is discontinuous and thus not a "curve", yet I agree it belongs in this context as it shares other important properties with these curves. See &lt;http://cap-lore.com/MathPhys/Zorder/> for discontinuity argument and also non-differentiability. NormHardy 19:18, 26 December 2006 (UTC)

Definition
Several of the listed examples (like the Smith-Volterra-Cantor set, the Mandelbrot set, and the space-filling curves) seem to have the same Hausdorff dimension and topological dimension, and therefore they don't meet the definition of fractal at the beginning of the article. Maybe the definition should be changed to something more like the one on the fractal page. —Preceding unsigned comment added by 160.39.28.117 (talk) 07:18, 30 November 2007 (UTC)

Ok You are right for the Mandelbrot set (corrected, the interresting part is the boundary). For the space-filling curves, the topological and the fractal dimension are actuallay different. Prokofiev. 7 dec 2007. —Preceding unsigned comment added by Prokofiev2 (talk • contribs) 17:19, 7 December 2007 (UTC) Reformatted —DIV (128.250.80.15 (talk) 02:55, 28 April 2008 (UTC))

Deterministic
The whole point of so-called 'natural, random' aggregates etc. is that they are products of deterministic chaos. The classification used here is misleading. —DIV (128.250.80.15 (talk) 02:57, 28 April 2008 (UTC))

Why does the coastline of Norway have more dimesions than the coastline of Britain?
It seems like they would have the same value, as they are both coastlines.128.232.241.115 (talk) 00:38, 10 May 2008 (UTC)

It has been mesured (see reference on the article). Coastlines are more or less irregular and Norway's westcoast is particularily irregular.Prokofiev2 (talk) 08:32, 13 May 2008 (UTC)

Casey Lange fractal
The new article titled Casey Lange fractal seems to fail the google test. Does anybody know anything? Michael Hardy (talk) 04:26, 20 June 2008 (UTC)

Fibonacci fractal
New entry on 60o Fibonacci fractal gives a measured dimension of 1.2075. Seems to me that we can calculate an exact Hausdorff dimension. The kth generation Fk consists of 4 copies of Fk-3 and 1 copy of Fk-6, so the area scaling factor is the positive root of


 * $$x^2 = 4x + 1$$

which is


 * $$A=2 + \sqrt{5} = \phi(1+\phi)$$

- same as the 90o Fibonacci fractal (as expected). Linearly, two of the Fk-3 copies are at an angle of 60o, so linear scaling factor is the positive root of


 * $$x^2 = 3x + 1$$

which is


 * $$B=\frac{3 + \sqrt{13}}{2}$$

So Hausdorff dimension should be


 * $$\frac{\ln(A)}{\ln(B)} \approx 1.2083$$

which is close to the give measured value. I don't have a source for this, however, so it might be discounted as OR. Gandalf61 (talk) 14:19, 29 June 2008 (UTC)

=> You're right !
I was looking for that result all the afternoon. I confirm your calculations are right, Gandalf. Well done ! Prokofiev2 (talk) 19:59, 29 June 2008 (UTC)

Units?
The measurements of all the natural things should have some kind of units. Otherwise they are just meaningless numbers. Asmeurer ( talk   ♬  contribs ) 00:03, 16 October 2008 (UTC)


 * Our dimensionless quantity article has a list of over 80 physical constants and properties that are dimensionless. They can't all be "meaningless". Gandalf61 (talk) 10:21, 16 October 2008 (UTC)

I agree. But in this case, the units are of length. While the fractals on the top table might be dimensionless, the bottom ones require them. Otherwise, how do I know your coast of Britain refers to the real Britain or the map, and is it in feet or miles or kilometers. Also, I do believe that most dimensionless quantities are ratios of dimensioned quantities that have the same units, so they cancel out. That clearly isn't the case here. Asmeurer ( talk   ♬  contribs ) 15:13, 16 October 2008 (UTC)


 * An accurate map of Britain's coastline has the same fractal dimension as the real thing, so it is dimensionless. —Tamfang (talk) 06:54, 17 October 2008 (UTC)

Beryl fractal
Gandalf, this fractal has been studied under the control of mathematics teachers, as mentionned in his site : [] "étudiée dans le cadre de mon TIPE" means "Studied in the frameword of my TIPE". TIPE (see []) is a research work for French scientific students. Ok, it's not the "Annals of mathematics", but, for me, it is serious enough and does not go against Wikipedia policies regarding reliable sources (if not, then we should delete half of the mathematics contributions in all wikipedias...). Prokofiev2 (talk) 10:00, 20 December 2008 (UTC)


 * Sorry, but "under the control of mathematics teachers" does not meet the criteria for being a reliable source. There is only one Google hit for "Beryl fractal", which is a page on SourceForge.net. Without better references (and preferably in English - see WP:NONENG) this looks like original research and also a neologism. But you don't have to just take my word for it - I am going to consult wider opinion at Wikipedia talk:WikiProject Mathematics so that we can determine consensus on this. Gandalf61 (talk) 12:08, 20 December 2008 (UTC)


 * This is WP:OR. A single pdf by the creator of this term is not notable. Arid Zkwelty (talk) 12:55, 20 December 2008 (UTC)

Let me list some arguments why I decided to propose this entry: Now for the weak points: Prokofiev2 (talk) 13:26, 20 December 2008 (UTC)
 * This fractal (close to Mandelbrot one) seems to me interesting enough to be cited here.
 * The details in the reference shows the author knows what he is talking about(if you can read french and have the necessary knowledge). And his result is easily verifiable.
 * The fact that it is a work done in a TIPE gives some garanties about its reliability. That kind of work is read, controled and verified, just like for publication papers.
 * Neologism : Most figure names were given by their author. Is that a problem ? And I far as I have seen this fractal has never been studied and named before.
 * Single reference by the creator: Ok, but is that enough to discard it ?
 * Reference in french : If there were references in english I whould have cited them, but there is none, so far.


 * I don't see any analogue of this in the English language which would be accepted; we don't have the concept of notes for a National exam, as far as I know. If it were published in a peer-reviewed journal, I would (reluctantly, as it still could be a neologism) agree it should be noted here.  But I cannot verify that it's peer-reviewed, and P's comments at WT:MATH seem to indicate it's not analogous to any similar English-language concept.  Is it peer-reviewed?  I don't know.  — Arthur Rubin  (talk) 17:28, 20 December 2008 (UTC)

why cf.?
I think perhaps someone has the wrong idea about the meaning of cf. (Latin confer): it means 'compare' (not 'see'), so (cf. Stephen Wolfram&lt;ref name="stephenwolfram.com"/&gt;) invites me to compare Doc W to the fractal dimension of a Sierpinski triangle, an exercise of dubious value. Am I missing some point? —Tamfang (talk) 02:55, 25 December 2008 (UTC)


 * Cf. means "consult" (or "compare") as anybody can check. —Preceding unsigned comment added by Prokofiev2 (talk • contribs) 10:13, 26 December 2008 (UTC)


 * Exactly. It's wrong in this context.  In general, the reference should be attached to the formula or the description (depending on what, exactly is in the reference), and the author's name should be removed from the body of the table.  — Arthur Rubin  (talk) 20:36, 26 December 2008 (UTC)

delta
The greek letter delta is used both for the Haussdorff dimension and the Feigenbaum constant. Use different symbol for the Haussdorff dimension, e.g. capital D? —Preceding unsigned comment added by 193.174.246.168 (talk) 09:01, 6 January 2009 (UTC)

WP:FL worthy?
Anyone considered nominating this fine list at Featured lists? -- &oelig; &trade; 07:57, 5 June 2009 (UTC)

Rational numbers
Does any source qualify rational numbers as a fractal? The example with the box dimension is quite interesting, but it belongs in a diffferent place (Box dimension, say). I find the rational numbers on this list a bit surprising, so please give some references as to their fractal nature. ptrf (talk) 15:10, 5 June 2009 (UTC)
 * Good remark. The point of this item is precisely to show that the set of rational numbers is NOT fractal, in spite of what some authors can suggest(for examples http://alphard.ethz.ch/Hafner/Diverses/r-fract.htm or http://linas.org/math/chap-rat/chap-rat.html) or what is sometimes said about the Farey sequence. Its Hausdorff dimension value closes the debate and it is worth mentioning this somewhere and I think this is a good place. Now, you are right, it must be stated more clearly, I'll add a clear comment.Prokofiev2 (talk) 08:41, 7 June 2009 (UTC)
 * Well, I'm not sure whether the Hausdorff dimension=0 closes the debate. First, there are different dimensions in use. Second, one can construct countable self-similar sets. Are they fractals? The answer is not crystal clear (although they seem not rich enough to be called fractals). Third, despite the links you provided, I don't think there is a serious debate about fractal nature of rational numbers. They are not fractals and we agree on this point. Therefore, it's not so important to mention the problem in the article. Fourth, I'm not convinced that opening the list of fractals by a non-fractal set is a good idea.  Sometimes less is better and the list would be just better without it. What about moving the example somewhere else? ptrf (talk) 09:30, 9 June 2009 (UTC)
 * Actually, the problem is that there is no real concensus about the definition of "Fractal" (Falconer writes it is like defining "life", there are always counter-examples). The definition chosen on top of the article is that its "Hausdorff dimension is greater than its topological dimension" => In that sense rationals are not fractals, ... but neither are the Blancmange curve nor the Smith-Volterra Cantor set (also on the list), which are self-similar and defined as "fractal" by many authors... Should we remove them as well ? I don't think so. In the absence of real consensus I think it is worth mentionning all those cases with their dimension. Now the case of the set of rationals described as "fractal" is, I understand, questionnable. (and starting the list with a non-fractal is strange) but I don't see for the moment where to transfer it.Prokofiev2 (talk) 12:36, 9 June 2009 (UTC)
 * You're right about the lack of a formal definition of fractals. My point is that, contrary to what you're suggesting above, in most cases there is an obvious consensus what is and what isn't a fractal. Almost universally, blancmange and Smith-Volterra Cantor are called fractals, while the rational number are not.
 * Moreover, if there were doubts, we should just report that there are diverging opinions or specific situations--and not take sides stating that "rationals can never be fractal", see WP:NPOV. If there are no doubts, just suppress the entry. Otherwise, we're rather creating a confusion than explaining anything. In any case, the inclusion of rational numbers in this list needs to be referenced by a source; at least one not self-published "serious" publication, see WP:RS.
 * PS. I'd propose to merge this with the box dimension or Hausdorff dimension —Preceding unsigned comment added by Ptrf (talk • contribs) 13:19, 9 June 2009 (UTC)
 * All right, there is a risk of creating confusion, I think Box Dimension will be a good place. I remove the entry.Prokofiev2 (talk) 19:45, 9 June 2009 (UTC)
 * Thanks for your consideration. BTW, I too think this list is really not bad. I do not know exactly what are requirements for Feature Lists, but you could eventually consider a nomination. ptrf (talk) 20:09, 9 June 2009 (UTC)
 * Well I don't really see the point about those feature lists. This is supposed to be an encyclopedia. A medal doesn't add value to good information (even on bad information). And I notice this list contains very few articles on mathematics.Prokofiev2 (talk) 11:19, 10 June 2009 (UTC)

Comments
MISTAKE —Preceding unsigned comment added by 85.221.173.203 (talk) 20:17, 9 September 2009 (UTC)

Comments
Hausdorff dimension of Menger sponge is log(20)/log(3)!!! —Preceding unsigned comment added by 85.221.173.203 (talk) 20:19, 9 September 2009 (UTC)

Comments
There are some fractal vegetables more wonderful than a simple white cauliflower, as a Search on Google will show. I was past 50 before I first saw a Romanesco brocolli or cauliflower - what a wasted life! P.r.newman (talk) 10:15, 16 April 2010 (UTC)
 * Amazing vegetable. I'd like to find a measure of its Fractal dimension. Surprizing to see that, apparently, nobody tried that calculation.Prokofiev2 (talk) 21:51, 17 April 2010 (UTC)

Pascal triangle modulo 3
Is it me or does it scale really badly? Maybe we need a custom thumbnail? --80.175.250.220 (talk) 11:07, 24 January 2011 (UTC)

Cauliflower Hausdorff dimension
http://icpr.snu.ac.kr/resource/wop.pdf/J01/2005/046/R02/J012005046R020474.pdf stated the dimension to be ~2.8, can someone proof that? —Preceding unsigned comment added by 88.65.198.120 (talk) 19:28, 19 May 2011 (UTC)

dimension Cantor set wrong
Shouldn't the interval removed from the Cantor set be γ((1-γ)/2)m-1 instead, in contrast to the Smith-Volterra-Cantor set?

--User:Dom220.255.1.136 (talk) 13:27, 25 May 2011 (UTC)

The 3d Greek Cross Fractal IS the Octahedron fractal
The list shows two fractals next to each other (3d Greek Cross and Octahedral), these are two different ways of creating the same limit set. So, I don't think they should be listed as two different fractals. — Preceding unsigned comment added by 98.225.20.68 (talk) 17:21, 12 November 2012 (UTC)


 * hm, looks like you're right —Tamfang (talk) 04:44, 13 November 2012 (UTC)

Analytic formulas for boundary of 'tame twindragon' and others
Hi all, the same method used to calculate analytically dimension of the boundary of twindragon is also used for infinite family of such simple IFS - for example I've just seen that one of them is called 'tame twindragon' in the table: just use

Solve[Det[fin - IdentityMatrix[ie]*x] == 0, x]

and take 2Log[x]/Log[n] of the only reasonable solution in the demonstration getting another nasty formula for 'tame twindragon':

$$\textstyle{\frac{2\log\left(\frac{1+\sqrt[3]{27-3\sqrt{78}}+\sqrt[3]{27+3\sqrt{78}}}{3}\right)} {\log(2)}}\cong 1.21076053328852339502586750643 $$

These formulas can be written in plenty of ways - maybe there is some nicer(?)

Cheers, Jarek —Preceding unsigned comment added by Jarek Duda (talk • contribs) 07:04, 18 July 2010 (UTC)
 * Hi. If we have no nicer formula in the form Log[x]/Log[n], we could add it to the table.Prokofiev2 (talk) 07:55, 23 July 2010 (UTC)


 * Jarek, there is a typo in your formula. It should be,

$$\textstyle{\frac{2\log\left(\frac{\sqrt[3]{27-3\sqrt{78}}+\sqrt[3]{27+3\sqrt{78}}}{3}\right)} {\log(2)}}\cong 1.21076053328852339502586750643 $$

Titus III (talk) 05:05, 15 May 2013 (UTC)

Universe Model Fractal
Are there estimated fractal dimensions of galaxies, galactic superclusters, or the current model of the known universe? If so, those would be excellent additions to the natural fractals section.

Rememberlands (talk) 18:23, 15 November 2011 (UTC)

Subsequent measurements have revealed that on larger scales, the power spectrum of galaxies fails to follow the (R/R0) ~ R^1.8 curve. Therefore, the distribution is not a fractal. — Preceding unsigned comment added by SCS137 (talk • contribs) 14:31, 14 February 2015 (UTC)

Include
Could the Infinity mirror and Kaleidoscope by added to this list? Also Harmonograph and Spirograph. - Shiftchange (talk) 01:08, 7 May 2017 (UTC)

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Space filling curve a fractal ?
I haven't quite understood why a space-filling curve is considered a fractal. Ok, I have an iterative procedure yielding a self-similar geometric object. But in the limit, I end up just with the unit square, which is defnitely not a fractal since topological and Haussdorff dimension both equal 2.


 * Do you mean the Peano curve? Hyacinth (talk) 05:37, 3 July 2018 (UTC)

'swiss cross fractal'
I'm a little puzzled both by the inclusion of this shape and it's linked picture. The limit of the fractal described is just a square. It seems a little strange to consider this a fractal. I know some 'fractal curves' cover the unit square in the limit, but these have other interesting properties in general (continuity, parametrization of the square..) I'm not sure the 'greek cross fractal' needs to be mentioned so much given that it doesn't seem to be a particularly interesting shape in any obvious way. Furthermore, the picture of the 2d greek cross fractal shows an early stage approximation to it. As mentioned the limit would just be a square. Obviously it's not possible to represent the limit strictly, but I find the picture is a little misleading given the above points. A mathematically unsophisticated reader might think the 'fractal' has some fine structure not present in the square. I'll review the page on this fractal and consider changing or removing it from here.
 * Do you mean the "Quadric cross"? Hyacinth (talk) 05:38, 3 July 2018 (UTC)

Why was half the page removed
A month ago an unknown user removed half the page with no reason and no discussion I can find. Why? It seems like such a large change to the page should have some discussion first. 173.229.9.179 (talk) 18:06, 1 February 2019 (UTC)


 * I reverted it. Hopefully if they had a good reason for the removal they will notice and discuss it here. Qbt937 (talk) 04:44, 9 February 2019 (UTC)

Proposal
I propose expanding List of fractals to include more information so that it looks like this and then moving the page to Table of fractals. Thoughts? &#x27A7; datumizer   &#9742;  19:07, 5 July 2021 (UTC)


 * I don't know much about fractals, but: (A) Your table looks better, upon first impression. (B) Both your version and the standing version are 50% too wide for my screen (as I like to have it). Cheers. Mgnbar (talk) 21:34, 5 July 2021 (UTC)

Why use nonstandard parameters in the Lorenz attractor?
The standard parameters for the Lorenz attractor are σ = 10, β = 8/3, ρ = 28.

So: What is the point of listing the Lorenz attractor in this article with the non-standard parameters σ = 16, β = 4, ρ = 40 ???

(To make matters even more confusing, the parameters are listed in nonstandard order: first ρ, then σ, then β.)

That makes no sense at all.

Possible Resolutions
I understand the sentiment. However, note that the meaning that humans attribute to unordered sequences of symbols are purely a matter of culture and that, for example, in cultures that read in different directions than English, such as from right to left, the "natural" order would be reversed. In short, there is no notion of absolute correctness.

It is nice when an offspring source maintains the symbological standard of its citation. I encourage you to be bold and make these changes whenever you encounter them and simply check the box that says "minor edit" and make a note in the change field summary that you have re-ordered the parameters or renamed them to fit a certain style or source. If it is based on a source, please cite that source both in the change and in the article. The change form can just have the URL/DOI; the article obviously needs to have the citation inserted more carefully, and in the format that is currently used for that article.

Thank you for reading, and thank you for caring. As a fellow human, your insights and observations are equally important to us all. I also encourage you to sign your edits and talk-entries. The signing contains the time stamp that allows others to know the rough timeline and history of the development of an article. In short, it is a key piece of the historical record of changes and suggestions, which is very important. (Incidently, the information is still there; it is just not on the page, but rather in the metadata. Putting it on the page helps only humans.) MMmpds (talk) 17:02, 4 May 2023 (UTC)