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Talk:Fractal/Archive 2

Hacking
Someone ruined this text with obcenities, so I reverted it to the penultimate saved version.Nikos.salingaros 23:26, 12 May 2007 (UTC)

Hello guys! I made some fractal images for the swedish page: Fraktal, copy them to here if you like, Sv:User:Solkoll --80.217.177.185 20:11, 10 Jul 2004 (UTC)

Broccoli Image
I think the broccoli image was a most excellent example of fractal-ness in real life. It was removed on 16:06, 3 Apr 2005 by User:Brian0918 without a hint in his edit message ("converted pics to gallery...") that he removed it. I vote for putting the image back into the article. -- Yogi de 20:09, 14 Apr 2005 (UTC)

 * Speaking of images (Mandelbrot Image at top), I was under the impression that, although the Mandelbrot was highly complex (like a fractal), it was not a fractal because it is not self-similar at all loci Dragoon235 05:34, 27 January 2007 (UTC)

I see you have listed the featured article fractal on Pages needing attention with the note "poorly organized; some parts may be expressed better". Can you be more explicit about what further work you think needs to be done on the article ? How can its organisation be improved ? Which parts can be expressed better ? Perhaps if you create a list of specific issues on the article's talk page, then we can try to reach a concensus on the way forward. Gandalf61 15:50, Nov 25, 2004 (UTC)

 * Really, that's a featured article? That's interesting.  It just... doesn't seem to flow very well.  The language is a bit odd in places; for example, the intro contains the terrible definition A fractal is a geometric object which is "broken up" in a radical way, which is literal but not accurate (does that definition fit broccoli?); also calculus is said to "zoom in" on "objects" to "gain control of them" (how is anyone supposed to understand that?).

 * It (zooming) is a metaphor being used in an introduction. People are forever being told about zooming in on Mandelbrot sets. The coastline example argues in terms of zooming by change of scale. It's journalistic but I imagine most readers can cope. As for broken up - that is quite accurate and to the point. It's the broccoli that might be criticised. Charles Matthews 17:42, 26 Nov 2004 (UTC)

 * But the article doesn't say zooms in on fractal drawings, it says traditional calculus zooms in... which is nonsense.

 * No. Charles Matthews

 * Huh? Try adding that statement to the calculus article.  --[[User:Eequor| η  [[Image:Venus symbol (blue).gif|♀]] [ υωρ ]]] 19:38, 26 Nov 2004 (UTC)

 * It would be a perfectly good approach to calculus, given modern computer graphics, to zoom in on small areas of a graph. In calculus you would see a line, or at worst a corner. That would be the point: after a while it doesn't change when you magnify. Calculus is still taught to serve the needs of mathematical physics of the nineteenth century; so much the worse for it. By the way, please stop it with the page name changes without consultation; whatever the agenda is, it is not appreciated. Charles Matthews 20:15, 26 Nov 2004 (UTC)

 * It may be a workable approach, but it's a terrible definition. Properly speaking, calculus has already "zoomed in" to the infinitesimal level before anyone examines for themselves.  It's meaningless to talk of "zooming" once one understands the underlying principles.

 * The "agenda" is neutral point of view, a little-known part of Wikipedia policy. --[[User:Eequor| η  [[Image:Venus symbol (blue).gif|♀]] [ υωρ ]]] 21:08, 26 Nov 2004 (UTC)
 * NPOV doesn't apply in that way to article titles - that's not an argument. Apply it all you like to the article text. It's a poor reason to start changing things, without discussion, and without authoritative support. Charles Matthews 21:58, 26 Nov 2004 (UTC)

 * Regarding broken up: who would consider the image at right to be "broken up"?

 * Broccoli certainly exhibits fractal properties; if we are to say that it is impossible for physical objects to be fractals, we would have to throw out most of the article (Britain's coastline, mentions of trees or ferns, and all the images). The difference between mathematical and physical fractals is only a matter of semantics.   --[[User:Eequor| η  [[Image:Venus symbol (blue).gif|♀]] [ υωρ ]]] 17:54, 26 Nov 2004 (UTC)
 * No it isn't. Charles Matthews

 * The order of the sections is jarring; I suspect the anecdote about Britain's coastline could be more clearly connected to the rest of the article. It isn't immediately obvious what the infinite shore has to do with any of the other images on the page aside from the Koch snowflake, and there's hardly any mention of that topic either.  Both are related to the Hausdorff dimension in a very specific way, but the article doesn't even touch on their relation.  Which is quite unfortunate, because the reader can't find out more at Hausdorff dimension — that article is an abstract lump of ugly math.

 * The Categories and Definitions sections are disorderly, and the latter is mostly redundant. A proper definition should be given in the lead section; the extra details can probably be merged into the rest of the article.  As it is, it's vaguely like finding Mandelbrot's lecture notes in the middle of his book.

 * Also note that the Definitions section introduces the (supposedly) correct definition and then dismisses it offhand as being too difficult. Surely some difficulty is to be expected when discussing non-Euclidean geometric objects which have a noteworthy Hausdorff dimension.  --[[User:Eequor| η  [[Image:Venus symbol (blue).gif|♀]] [ υωρ ]]] 17:02, 25 Nov 2004 (UTC)

 * Would you mind if I copied your comments above to Talk:Fractal ? Both the Introduction and the Definitions sections have been discussed at length there. Their current state is a compromise between two points of view, which could be characterised as the mathematically rigorous view and the understandable to the general reader view. Like all compromises, it is not ideal. Anyway, I think it would be good to see your comments on the talk page. Gandalf61 17:37, Nov 25, 2004 (UTC)

Guys guys guys!? Why don't we just discuss "merging fractal animation"? Silver The Slammer 12:08, 25 October 2006 (UTC)

How Long is the Coast of Britain?
I think most of this section should be moved out. It currently breaks the page up in an odd way. Charles Matthews 10:19, 28 Nov 2004 (UTC)

 * Yes, I have often this article is not the right place for the Coast of Britain section. Since the section is mostly a description of Richardson's research on the measured lengths of coastlines and borders, I have moved this material to Lewis Fry Richardson. Gandalf61 11:29, Nov 29, 2004 (UTC)

Proposal to remove tag
How would people feel if the tag was now removed from this page ? I think most of the points raised by Eequor when he tagged it in November have been addressed. Gandalf61 14:39, Jan 1, 2005 (UTC)

 * I just recently read the article for the first time, and I think flows very nicely now, imho. --MPerel 03:01, Jan 6, 2005 (UTC)

 * I see that Stirling Newberry has now removed the tag, so I have also taken fractal off the list at Pages needing attention/Mathematics. Gandalf61 10:32, Jan 6, 2005 (UTC)

 * Nope! It's still there!!!!! Silver The Slammer 12:09, 25 October 2006 (UTC)

discrete mathematics
The iterative calculative nature of the Mandelbrot set needs to be mentioned to reflect the mathematical relevance of having it there.

If I got it wrong, correct it please. ;) -- Zalasur 04:48, Mar 25, 2005 (UTC)

 * Hm I added this comment to the wrong page; please ignore. -- Zalasur 13:48, Mar 25, 2005 (UTC)

First sentence, self-similarity
Sorry, that's just wrong - self-similarity is not the defining property of a fractal.

Charles Matthews 14:10, 25 Mar 2005 (UTC)

So the intro was changed 17 December by User:EastNile, who hasn't edited since. I think we needn't take that as authoritative on fractals. The first para certainly needs to go back to something more like it was before. That's because fractal does now have some strong connotations, but a rather particular actual denotation. (I hope that clarifies a bit what the issue is here: it is not as if self-similarity is irrelevant.)

Charles Matthews 14:22, 25 Mar 2005 (UTC)

I think we should make it clear in the beginning that fractal is NOT a precise mathematical term, like say group. There is NO common definition in mathematics what a fractal is.

Danimey 22:59, 18 January 2007 (UTC)

and so which appears
I don't believe this string of words, found in the first sentence in the article, belongs together. And even if it does somehow belong together, it sounds very awkward.

Here is the sentence:

"A fractal is a geometric object which is rough or irregular on all scales of length, and so which appears to be 'broken up' in a radical way."

The "so which" seems to be superfluous. It adds only awkwardness, and probably causes the sentence to be improper.

Another suggestion, dilational or magnification symmetry is not addressed in the article. This is the angle that Yale takes, although it doesn't seem to be a traditional path for explaining fractals. What I like about the magnification symmetry approach is that it places fractals in context with other mathematical objects/ideas: the butterfly has line symmetry, bricks have translational symmetry, an octahedron has reflectional symmetry, fractals have magnification symmetry. It gives the sense that fractals aren't just a rogue field and have a place in an old and established order of mathematics. It is reminiscent of placing them within an organization chart, and from the hub, or node of magnication symmetry, it is easy to move between the different types of fractals. In one sense, it is good that fractals are set apart, in another, it doesn't help the field to be perceived as an antithesis to standard math. It goes back to the old preschool and kindergarten exercises of identifying in what ways things are different and in what ways they are the same.

 * I think your magnification symmetry is what the article calls exact self-similarity. As the article explains, this symmetry is a property of some types of fractal, but not of all fractals, which is why the second sentence of the article says that fractals "may ... have a self-similar structure that occurs at different levels of magnification". Various other attempts at an opening paragraph definition of fractal have been attempted, but they have been rejected as being too long, too complex, too mathematical or too inaccurate. The current version seems to sit on the boundary between what is acceptable to specialists and what is accessible to the lay person. Gandalf61 09:43, July 11, 2005 (UTC)

 * Magnification symmetry does not imply exact replication, it only requires that an object is approximately unchanged under magnification. Although it would apply to self-similarity, it is applicable to multi-fractals.  This presentation comes from a credible source, Mandelbrot and Frame on the Yale website.  Self-similarity, yes, but not necessarily.

ok guys i need help
ok someone please tell me what a fractal is in english? Explain it too. and define iteration and self similarity. if you get it post it. o by the way, when you explain, please use only around like 6th grade vocabulary, otherwise the words are too complicated for me.-dran

Please NEVER put a flag like that in the main article again, it is very bad form, those of us that monitor these pages also monitor the talk pages too. I would suggest this link: Fractals, in Layman's Terms your questions may be answered there. DV8 2XL 03:50, 17 November 2005 (UTC)

fractal galleries
who decides what fractal galleries are included on the fractal page?
 * The editors DV8 2XL 03:13, 28 November 2005 (UTC)

 * It's a concensus situation. There are some Wikipedia policies and gudelines, but most cases are not black and white, so a concensus view emerges. As a rule of thumb, a link to an obviously commercial web-site inserted by an anonymous user with no previous discussion is more likely to be removed than a link to an informative web-site inserted by a signed-up Wikipedia user who has previously discussed his/her intentions on the article's talk page. If you feel strongly that a particular link should be included in this article, I suggest you explain your reasons here on the talk page and invite discussion. Gandalf61 12:40, 28 November 2005 (UTC)

Could we have some more diversity at the top of this article? We currently have two pictures from the Mandelbrot set there, and nothing else. This Mandelbrot set is so common in the article and in this topic so we might confuse readers to the point that they think that Fractal == Mandelbrot set. So let's find a beautiful different example picture to put alongside the beautiful original mandelbrot picture. — Sverdrup 14:21, 28 November 2005 (UTC)

Self-similar
If I recall correctly, the Mandelbrot set isn't self-similar though it has fractal dimension. There are those rigid sticklers that would claim that the Mandelbrot set isn't technically a fractal, though it has fractal characteristics. This is ignored in the article. I was originally going to dispute the facts of the article, but have decided that this objection may be more cosmetic. Thoughts? --ScienceApologist 13:55, 2 February 2006 (UTC)

 * See Mandelbrot's definition of a fractal and the discussion of definitional problems in the Definitions section, and the different types of self-similarity described in the Categories of fractals section. You could add a sentence to the article to say that some people do not include the Mandelbrot set within their definition of fractal, but I think a reference supporting this statement would be helpful, as it would be a minority view in my experience. Gandalf61 13:34, 3 February 2006 (UTC)

I think that strictly speaking the Mandelbrot set itself is not fractal, but the boundary of the Mandelbrot set is fractal. The same applies to the Koch snowflake where the set itself is measured in m2 while the circumference is measured in m1.2. Anybody know the exponent for the circumference of the Mandelbrot set? Another thing. When the complex plane is drawn with 1 to the right, then the symmetry axis of complex conjugation is horisontal, and the Mandelbrot set is shown lying down. It is more pleasing to the eye to see the monster standing up, with 1 upwards. Bo Jacoby 07:58, 3 February 2006 (UTC)

Please read this forum thread between me and my friend about this topic, and make some changes.--Max 10:41, 3 June 2006 (UTC)

Removed "book review"
This is for Gandalf61: Who are you? the tin god of references? This is called an annotated bibliographic entry and there's nothing wrong with it. I'm putting it back. It's not a "review" its an annotation. It's useful for folks who want to read more. wvbaileyWvbailey 18:13, 27 February 2006 (UTC)

This is such a nice page
I like the animation, the beautiful fractals. I sent my nephew here (he's taking 8th grade algebra, was doing a project on fractals) for a look-see. Nice work is going on here. wvbaileyWvbailey 23:53, 7 March 2006 (UTC)

What's wrong with the random fractal? Is it not a fractal? If not, discard it. If so, what's wrong with it? If it helps the 8th-grade nephew understand, then put it back. Thanks. wvbaileyWvbailey 23:58, 7 March 2006 (UTC)

I liked the fern, the colored one. Why did it have to go away? I don't like all this page-hopping. Sigh. wvbaileyWvbailey 03:02, 8 March 2006 (UTC)

 * I don't think it belonged on this page--too much detail in a niche area for a general article on fractals. Sorry about the page hopping, but I really think it's more at home in the IFS article, which actually badly needed an example. --Experiment123 03:10, 8 March 2006 (UTC)

I am the author of the animation and IFS colored fern... and I aggree the fern is better in the IFS article. Tó Campos 12:23, 8 March 2006 (UTC)

 * Agreed. Thanks for the explanation.wvbaileyWvbailey 17:30, 8 March 2006 (UTC)

Reorganization and rewrite
I just reorganized this article and rewrote some bits. I'd like this article to stay accurate mathematically while recognizing the colloquial uses of the term fractal. --Experiment123 01:36, 8 March 2006 (UTC)

 * I see some etymology was added back to the intro. Not sure how critical the etymology is, but I'm still learning my way around Wikipedia style. I did move the bit about "monster curves" to the paragraph on the Koch snowflake, since "monster curve" was never synonymous with fractal, but instead used to describe a particular set of examples.

 * One of the books that I cited, Fractals for the Macintosh, (Jones, section titled "A Gallery of Monsters") definitely refers to "monster" curves -- specifically the Koch "snowflake" and the Peano "space-filling" curve. According to this book, the Peano curve led Hausdorff to his definition of "dimension" -- the snowflake has Hausdorff dimension of 1.26, the Peano curve has dimension 2.0 etc. so:
 * "...these are both fractals by Mandelbrot's definition" (Fractals Definited 8)

 * The other book, Fractals (Lauwerier), refers to the snowflake as an example of a "pathological" curve:
 * "Now, nearly a century later, such 'pathological curves' (as they were called) turn out to occur everywhere in pure and applied mathematics" (p. 33).
 * This book corroborates its Hausdorff dimension of 1.26.

 * So there you go. Whether or not these books are accurate is another question. So is the snowflake a "monster" or just "pathological"? I think one or both words would be fine (kids love the word "monster" so I vote for that one). I liked the intro before it was changed. wvbaileyWvbailey 17:52, 8 March 2006 (UTC)

 * The books are right, the Snowflake and its cousins were called "monster curves." But they are just a small set of examples, curves whose metric properties are surprising. Cantor dusts and Sierpinski carpets also predate Mandelbrot, but certainly weren't called curves. Various dynamical-systems-related fractals(Julia sets, etc.) were also known well before Mandelbrot. In other words, it is misleading to say that fractals were called "monster curves," though it is true that monster curves were fractals by today's definition.

 * If you want to emphasize the strange properties of fractals in the introduction, you could say something like, "Some early examples of fractal sets were referred to as 'monsters,' due to their seemingly paradoxical properties." --Experiment123 18:05, 8 March 2006 (UTC)

applications
i didn't want to edit the page without getting people's opinions on whether financial analysis should be added to the applications section. i've just finished mendelbrot's "the (mis)behavior of markets," and while he isn't able to replace the valuation formulae stemming from the efficient capital market hypothesis, he makes some very lucid suggestions that others appear to have followed in their research. while the pure number of citations does not tell us whether they agree or disagree, his financial work has at least been taken seriously. thoughts?afuturehead 22:39, 29 March 2006 (UTC)

There's dispute as to how much credit Mandelbrot deserves for finding fractal patterns in financial markets. Some say he hasn't done much other than attach the label "fractal" to analysis done by other people.

Page Layout
The fractal fern overlaps with the broccoli and the page has a horizontal scrollbar at 1280x1024, Ubuntu, Firefox 1.5

 * I'm afraid it rarely renders well for me either. Unfortunately, I don't know enough about image formatting and wikipedia to understand why.  I am also using Ubuntu at the same resolution but firefox 1.0.7.  --Richard Clegg 21:46, 4 April 2006 (UTC)

Possible Copyright violation in history section
While searching on google for "Anklets of Krishna" I found the following webpage:

http://classes.yale.edu/Fractals/Panorama/Art/Kolams/Kolams.html

The material about kolams seems to be a word for word copy of that material.

This seems to be a likely copyright violation. So I removed it.

Can anyone show that it isn't?

Plus even if it isn't, I'm not sure it belongs in the history section, perhaps it belongs in an examples section.

TheRingess 17:47, 16 April 2006 (UTC)

Colour?
How are generated fractals coloured? Every image of fractals I've seen is in a kaleidoscope of colour, but I can't find an explaination of which bits of the image get which colour.

 * I'll explain how it's done for the Mandelbrot fractal. This is a set generated by taking a point in the complex plane and repeatedly performing the same operation on it. Most points end up whizzing off into the distance, but some stay close to where they started. These are part of the Mandelbrot set and are colored one color (usually blue or black). If a point isn't in the set, it will diverge to infinity and you can tell a point isn't in the set once it reaches a certain distance from the origin, a circle of radius 2. The number of steps it takes to cross that boundary is the color it gets. So if it takes only one step, it's given color #1. If it takes ten steps, color #10. Points that start close to the edge of the Mandelbrot set take a long time to move past the boundary. That's where the complicated colored bands come from. Other fractals are colored in a similar way. I hope this explanation helps. Reyk  YO!  01:45, 4 June 2006 (UTC)

Quasicrystals and fractals
The pinwheel tiling, as its name suggests, is known generally as an aperiodic tiling but it is also a fractal. It is obtained by dividing a 1:2:sqrt(5) triangle into five smaller triangles of the same shape or conversely by inflation - surrounding one such'seed' by four copies and repeating the procedure with the larger figure. Thus tilings are 'extended' to infinity while fractals are usually thought of as divided down to infinity. Anyway, I believe that the topics of fractals and quasicrystals should be perceived as typical for the end of 20th century mentality.

The infinite aperiodic sequence 101000101000000001010001010... is the Cantor set written up from the atom '1', but I am not sure if it is 'fractal' in a meaningful way. However many aperiodic sequences are 'fractal' in the disputed self-similar way: if you strike out every other term you get the same sequence, e.g. in 0110100110010110...(Thue-Morse; you can see it in the list of  self-similar sequences in the  OEIS). Aperiodic tilings are easily constructed out of such sequences. This could be an other argument for the link between aperiodic and fractal structures. al 17:03, 13 July 2006 (UTC)

Good article nomination
Hi all,

I'm happy to promote this article to be a good article but I made some changes to the article in line with what I thought the article was trying to say. Specifically I tried to highlight comparisons between the topological/Hausdorff dimension. If someone can verify my changes are correct I will promote this article.

Cedars 01:25, 14 July 2006 (UTC)

 * Since my changes weren't verified this article will not become a good article. Overall though I think this article could benefit from some more insight into the mathematics of fractals. Also the "might (or might not)" statements should be removed from the article and rephrased if necessary. Cedars 08:52, 23 July 2006 (UTC)

Fractal Art Galleries Section
I deleted this section because there seems to be no criteria for what constitutes a notable link. TheRingess 00:31, 22 July 2006 (UTC)

Problems with Cantor Set paragraph?
(In the Examples section.)

As far as I can tell, we don't ever explain what "a Cantor set" is. The linked article is mostly about the standard Cantor set, with allusions to alternatives. Presumably a Cantor set is something like "Anything obtained from an interval by repeatedly removing middle portions from segments"? And the "might (or might not)"s are about the fact that you can get something with Hausdorff dimension zero by increasing how much you remove, or Hausdorff dimension 1 by decreasing how much you remove? I think we should clarify this somehow.

Also, the last sentence is inconsistent with the rest of it: "By comparison the topological dimension of any Cantor set is 0 and hence all Cantor sets are fractals."

And I agree with the above about "might (or might not)." Maybe deleting the "(or might not)" achieves the desired effect. Or something else?

--Dchudz 14:31, 28 July 2006 (UTC)

 * This stuff is all gone now anyway. As good a way to deal with my problems as any. Thanks, Beaumont. Dchudz 17:42, 3 August 2006 (UTC)

 * Well, actually I find that the examples section should be rewriten and expanded. A list of (more) examples within some important categories should do (curves, planar IFS, Mandelbrot-Julia, logistic-map-related, strange attractors, stochastic fractals...). Detailed discussion of each example could be moved to a separate article. IMHO, at the present the examples are the core of the actual "definition" of what fractal really is. --Beaumont 18:53, 3 August 2006 (UTC)

New Definition, Real Line...
This new definition in the opening paragraph better matches my experience with the word "fractal" than the strict. But it's still not quite right: As we say in a later paragraph, the real line wouldn't be called a fractal, but it does have three of the five ("most") of the characterizing properties... What to do? -- Dchudz 18:42, 3 August 2006 (UTC)
 * Formally you're right. And formally, we may split the first feature into two (as in the source) and define "most of" to be 4. Personally, I find this unsatisfactory. Maybe it is better to refer explicitly to the examples? Anyway, this is the best definition I've heard about. --Beaumont 19:00, 3 August 2006 (UTC)
 * After a while I added an interpretation that may help. Actually, natural look excludes staight lines, squares etc. In nature we have no straight lines. --Beaumont 15:42, 4 October 2006 (UTC)

Fractal animation
I am trying to start this(Fractal animation) article, and I have a crude beginning. I feel it would benifit from more minds. Thanks! HighInBC 21:07, 3 August 2006 (UTC)

It's nice, but I'd like to propose that Fractal animation be merged into this article. I don't really think animation alone warrants its own article. Gregly 15:15, 6 September 2006 (UTC)

 * I originally did not include it here because this article is rather long. I don't object to the merge. I admit the current article is sparse, but the subject is rather extensive. HighInBC 16:17, 6 September 2006 (UTC)

Concave/convex
Is it possible to have a convex fractal? I think this would be an interesting property to exclude, if it's so.
 * I guess so. Take a Cantor set and wrap it round a circle, construct a convex polygon by joining the points. --Salix alba (talk) 11:08, 16 October 2006 (UTC)
 * I'm not sure... Well, when we take the convex hull of just 4 points (0,1/3,2/3,1) wraped anyhow, we get the filled quadrilateral. Adding some more points outside results in a strange figure; maybe fractal, but I can not see this. Anyway, possible "fractal behaviour" could be observed just at the boundary and, at the first sight, the boundary looks like a Lipshitz curve a.e. (Cantor has gaps of length 1/3, 2*1/9 etc).  To compare with a simple example of von Koch snowflake - fractal bounded convex domain, not a real fractal inside though.Beaumont 12:31, 16 October 2006 (UTC)

 * I think the "Cantor polygon" described would not be fractal since its length would be finite (less than the circumferenece of cricle) and its area would be finite (less than area of circle). The curve of the Koch snowflake is infinite.  I doubt we could create a convex curve which is fractal though I cannot prove it.  --Richard Clegg 13:16, 16 October 2006 (UTC)

 * It's not clear to me why the "Cantor polygon" is not a fractal. It is self-similar under the two polar co-ordinate transformations

 * $$ \theta \rightarrow \frac{\theta}{3} \quad \theta \rightarrow \frac{4\pi+ \theta}{3}$$

 * and its boundary contains fine structure at all scales. It's not a very sexy fractal, since it can't (by definition) have any winding fjords or inlets, but I think it is a fractal nonetheless. Gandalf61 14:15, 16 October 2006 (UTC)

 * Its Haussdorff dimension is equal to its topological dimension is it not? I guess many people do not consider that a strict requirement though. --Richard Clegg 15:06, 16 October 2006 (UTC)

The question provoked some interesting (original?) research, so let's get back to the origins. If you want some properties of fractals, we have tons of them, see any book by Falconer, Mandelbrot or others. And, apparently, convexity (or concavity) is not the the big issue. We may think of another way of enriching the article (projections, products, you name it).

To conclude, I believe that not-too-sexy examples of fractal bounded domains are the only possible convex fractals. On the other hand, I have never heard about such a statement in the literature and, unless someone finds an actual reference, we have no reason to discuss it further, no reason to insert our research into the article (actually, I suspect that from math point of view the question may be somehow trivial and was not really studied; but any proof of the contrary by some refs will be welcome).Beaumont 15:25, 16 October 2006 (UTC)

young
Why don't they aim for a younger crowd?

Silver The Slammer 12:02, 25 October 2006 (UTC)== !@$#%@$ ==

WHO UNEDITED MY PAGE?! —Preceding unsigned comment added by Power Gear (talk • contribs) October 23, 2006

Please understand you do not own this page and anybody is allowed to edit it, or unedit it. HighInBC (Need help? Ask me) 13:52, 23 October 2006 (UTC)

People! Stop fighting and check out the article on Silver The Hedgehog.

Usage of the word euclidean
All known fractals are acceptable in euclidean geometry (opposing what many "applied" students say), and all of them are constructed using euclidean geometry (the original article presenting the koch snowflake, for an instance, uses nothing but eucidean geometry to derive their properties). It may be incredibly hard to describe in terms of simple drawings like lines and circles, but they are euclidean;

Expressions like "non euclidean" or "contrary to euclidean" should be changed to "not traditional".

--200.17.137.71 22:18, 25 October 2006 (UTC)

Reordering Sections
I have to agree with some of the other comments on here that the article doesn't flow very well, and I'd quite like to see the article upgraded in status, so can I suggest the following (they're not minor changes, so I thought it better to discuss them first). I'm quite happy to do it myself, but I know that some people have already put a lot of effort in, so there may be some reasoning (e..g- behind the ordering as it stands), that I'm unaware of:

1. Change the ordering of the sections (numbers reflect thier current ordering):

 * 0 Intro
 * 1 History

 * n "Demonstration of how fractals were discovered" (& * 3 Self-similarity dimension) - This would cover the concepts that are now considered to be typical of fractals. I know that the 'Coastline of Britain' article now has it's own page, but it would be useful to show at least an overview of how fractals typically demonstrate some 'infinite length' behaviour. This could then be tied into the Koch curve as a more rigorous mathematical treatment. Perhaps some reference should also be made back to the Falconer's points that are mentioned in the intro paragraph.

 * 5 Classification of fractals (& * 2 Examples) - These ought to be combined into one topic so e.g.- the ordering would be title -> definition -> examples

 * 4 Generating fractals

 * 6 Fractals in nature - This should perhaps be moved to it's own page as it doesn't add much to the article as an overview of fractals. Perhaps mention such a page in the 'applications' section (since Pollock crops up)

o 11.1 Multiplatform generator programs o 11.2 Linux generator programs o 11.3 Windows generator programs o 11.4 Mac generator programs o 11.5 MorphOS generator programs
 * 7 Applications
 * 8 See also
 * 9 References
 * 10 Further reading
 * 11 External links

Apart from that it's just the links that need weeding and/or re-ordering in terms of usefulness since there are such a large number of them (which may be more appropriate in the fractal art page?).

-- Submanifold 26/12/2006

Koch Snowflake
In the article it talks about the snowflake having a finite area, isn't it infinite? As the triangular area is added to the existing shape no other areas are taken away, meaning that every step it gains area and perimeter. Thanks Onefournine 10:56, 30 January 2007 (UTC)

 * The area is not infinite since a koch snowflake is enclosed in a finite region in the cartesian plane.TheRingess (talk) 14:37, 30 January 2007 (UTC)

Topological dimension of space-filling curves
I don't understand why things in brackets are necessary in the fourth feature:
 * It has a Hausdorff dimension that is greater than its topological dimension (although this requirement is not met by space-filling curves such as the Hilbert curve).

Well, my "main" question would be: "Isn't topological dimension of any curve = 1?" Because the brackets say that topological dimension of Hilbert curve is 2 (or more). 89.164.4.146 15:07, 2 February 2007 (UTC)

Should the example of "Shikhar" of the Hindu Temple be included or not?
I feel the shikhar of the Hindu temple, which is the most visible form of fractal in its architectural form, that derives itself from the meaning and form of the "temple" be portrayed in some relevant section of the article. The history section begins with the mathematical history. The examples from nature section limits itself to imagery from nature. If we take just the imagery the temple "shikhar" can be placed in it, by renaming it to something like "visible examples of fractal" or create a new section on "Application of fractals"(?).

To quote references from the net, there is one example that uses 2 different examples of the Hindu temple as examples of "fractals in architecture"

The philosophical base of "self similarity" and its meaning is provided in Stella Kramrisch's The Hindu Temple. --Nikhil Varma 05:16, 21 April 2007 (UTC)

Here are some detailed studies related to Shikhar:

1. A Computational Approach to the Reconstruction of Surface Geometry from Early Temple Superstructures

2. The Generation of Superstructure Geometry in Latina Temples: A Hybrid Approach

3. ON RECOVERING THE SURFACE GEOMETRY OF TEMPLE SUPERSTRUCTURES

--Nikhil Varma 05:46, 21 April 2007 (UTC)

 * Facinating stuff. Cross checking with the def of fractile at to of page

A fractal as a geometric object generally has the following features:
 * It has a fine structure at arbitrarily small scales. No - there is a limit to the scalling applied
 * It is too irregular to be easily described in traditional Euclidean geometric language. Maybe - it does have a recursive/procedural method of definition - however the limit to scalling mean that there is onlya finite number of each element so it can be define in a traditional manner
 * It is self-similar Yes - to a limited degree
 * It has a Hausdorff dimension that is greater than its topological dimension (although this requirement is not met by space-filling curves such as the Hilbert curve). No - the limit to scaling prohibs this
 * It has a simple and recursive definition. Yes
 * So whether it really classes as fractal is not clear cut. Further the above assesment would be WP:OR on my behalf.
 * What we would really need is something in the litrature identifying it as fractal. The webpage link is by Michael Frame and Benoit B. Mandelbrot so that probably counts, in their common mistakes page they discuss that no physical fractal can exhibit scaling over infinitely many levels, nevertheless to make a plausible claim of fractality, a pattern must be repeated on at least a few levels. So with this vesied characterisation it would count as fractal.
 * I think rather than just this one example it might be worth discussing all the examples in the Architecture page. --Salix alba (talk) 14:53, 21 April 2007 (UTC)

what about fractals in architecture?
This being the main page for Fractal, to make it more complete, I think the example of the Hindu temple and also other architectural forms that resemble the imagery, if not as systematic as the geometry of the Hindu temple, should be included to provide the reader of the extent of "fractals" in our world.

I am still scavenging to fetch resources from the net that provide more detail studies of the fractal geometry and its mathematics involved in the Hindu Temple architecture. More than imagery it is the mathematics behind the design that can more strongly demonstrate the application of the concept to design.

We can even provide further links to aspects of fractals and its "need" or effect on people, from an architectural point of view.

Question: Should we have a separate page on "Fractals in Architecture"? and provide a link on the main page (with an inspiring image) to that page?

--Nikhil Varma 05:11, 21 April 2007 (UTC)

title: Talk:Fractal/Archive 2


text:
Fluid

In physics, a fluid is a liquid, gas, or other material that may continuously move and deform (flow) under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shear force applied to them.

Although the term fluid generally includes both the liquid and gas phases, its definition varies among branches of science. Definitions of solid vary as well, and depending on field, some substances can have both fluid and solid properties. Non-Newtonian fluids like Silly Putty appear to behave similar to a solid when a sudden force is applied. Substances with a very high viscosity such as pitch appear to behave like a solid (see pitch drop experiment) as well. In particle physics, the concept is extended to include fluidic matters other than liquids or gases. A fluid in medicine or biology refers to any liquid constituent of the body (body fluid), whereas "liquid" is not used in this sense. Sometimes liquids given for fluid replacement, either by drinking or by injection, are also called fluids (e.g. "drink plenty of fluids"). In hydraulics, fluid is a term which refers to liquids with certain properties, and is broader than (hydraulic) oils.

Physics
Fluids display properties such as: These properties are typically a function of their inability to support a shear stress in static equilibrium. By contrast, solids respond to shear either with a spring-like restoring force—meaning that deformations are reversible—or they require a certain initial stress before they deform (see plasticity).
 * lack of resistance to permanent deformation, resisting only relative rates of deformation in a dissipative, frictional manner, and
 * the ability to flow (also described as the ability to take on the shape of the container).

Solids respond with restoring forces to both shear stresses and to normal stresses, both compressive and tensile. By contrast, ideal fluids only respond with restoring forces to normal stresses, called pressure: fluids can be subjected both to compressive stress—corresponding to positive pressure—and to tensile stress, corresponding to negative pressure. Solids and liquids both have tensile strengths, which when exceeded in solids creates irreversible deformation and fracture, and in liquids cause the onset of cavitation.

Both solids and liquids have free surfaces, which cost some amount of free energy to form. In the case of solids, the amount of free energy to form a given unit of surface area is called surface energy, whereas for liquids the same quantity is called surface tension. In response to surface tension, the ability of liquids to flow results in behaviour differing from that of solids, though at equilibrium both tend to minimise their surface energy: liquids tend to form rounded droplets, whereas pure solids tend to form crystals. Gases, lacking free surfaces, freely diffuse.

Modelling
In a solid, shear stress is a function of strain, but in a fluid, shear stress is a function of strain rate. A consequence of this behavior is Pascal's law which describes the role of pressure in characterizing a fluid's state.

The behavior of fluids can be described by the Navier–Stokes equations—a set of partial differential equations which are based on:
 * continuity (conservation of mass),
 * conservation of linear momentum,
 * conservation of angular momentum,
 * conservation of energy.

The study of fluids is fluid mechanics, which is subdivided into fluid dynamics and fluid statics depending on whether the fluid is in motion.

Classification of fluids
Depending on the relationship between shear stress and the rate of strain and its derivatives, fluids can be characterized as one of the following:
 * Newtonian fluids: where stress is directly proportional to rate of strain
 * Non-Newtonian fluids: where stress is not proportional to rate of strain, its higher powers and derivatives.

Newtonian fluids follow Newton's law of viscosity and may be called viscous fluids.

Fluids may be classified by their compressibility:
 * Compressible fluid: A fluid that causes volume reduction or density change when pressure is applied to the fluid or when the fluid becomes supersonic.
 * Incompressible fluid: A fluid that does not vary in volume with changes in pressure or flow velocity (i.e., ρ=constant) such as water or oil.

Newtonian and incompressible fluids do not actually exist, but are assumed to be for theoretical settlement. Virtual fluids that completely ignore the effects of viscosity and compressibility are called perfect fluids.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Fluid


text:
FAQ

A frequently asked questions (FAQ) list is often used in articles, websites, email lists, and online forums where common questions tend to recur, for example through posts or queries by new users related to common knowledge gaps. The purpose of a FAQ is generally to provide information on frequent questions or concerns; however, the format is a useful means of organizing information, and text consisting of questions and their answers may thus be called a FAQ regardless of whether the questions are actually frequently asked.

Since the acronym FAQ originated in textual media, its pronunciation varies. FAQ can be pronounced as an initialism, "F-A-Q", or as an acronym, "FAQ". Web designers often label a single list of questions as a "FAQ", such as on Google Search, while using "FAQs" to denote multiple lists of questions such as on United States Treasury sites. Use of "FAQ" to refer to a single frequently asked question, in and of itself, is less common.

Origins
While the name may be recent, the FAQ format itself is quite old. For example, Matthew Hopkins wrote The Discovery of Witches in 1648 as a list of questions and answers, introduced as "Certaine Queries answered ...". Many old catechisms are in a question-and-answer (Q&A) format. Summa Theologica, written by Thomas Aquinas in the second half of the 13th century, is a series of common questions about Christianity to which he wrote a series of replies.

On the Internet
The "FAQ" is an Internet textual tradition originating from the technical limitations of early mailing lists from NASA in the early 1980s. The first FAQ developed over several pre-Web years, starting from 1982 when storage was expensive. On ARPANET's SPACE mailing list, the presumption was that new users would download archived past messages through FTP. In practice this rarely happened, and the users tended to post questions to the mailing list instead of searching its archives. Repeating the "right" answers became tedious, and went against developing netiquette. A series of different measures were set up by loosely affiliated groups of computer system administrators, from regularly posted messages to netlib-like query email daemons. The acronym FAQ was developed between 1982 and 1985 by Eugene Miya of NASA for the SPACE mailing list. The format was then picked up on other mailing lists and Usenet newsgroups. Posting frequency changed to monthly, and finally weekly and daily across a variety of mailing lists and newsgroups. The first person to post a weekly FAQ was Jef Poskanzer to the Usenet net.graphics / comp.graphics newsgroups. Eugene Miya experimented with the first daily FAQ.

Non-traditional FAQs
In some cases, informative documents not in the traditional FAQ style have also been described as FAQs, particularly the video game FAQ, which is often a detailed description of gameplay, including tips, secrets, and beginning-to-end guidance. Rarely are videogame FAQs in a question-and-answer format, although they may contain a short section of questions and answers.

Over time, the accumulated FAQs across all Usenet newsgroups sparked the creation of the "*.answers" moderated newsgroups such as comp.answers, misc.answers and sci.answers for crossposting and collecting FAQ across respective comp.*, misc.*, sci.* newsgroups.

In web design
The FAQ has become an important component of websites, either as a stand-alone page or as a website section with multiple subpages per question or topic. Embedded links to FAQ pages have become commonplace in website navigation bars, bodies, or footers. The FAQ page is an important consideration in web design, in order to achieve several goals of customer service and search engine optimization (SEO), including

 * reducing the workload of in-person customer service employees
 * improving site navigation
 * increasing the visibility of the website by matching/optimizing for specific search terms
 * linking to or integrating within product pages.

Criticism
Some content providers discourage the use of FAQs in place of restructuring content under logical headings. For example, the UK Government Digital Service does not use FAQs because the service believes that their form primarily serves writers' needs and creates more work for readers.

title: FAQ


text:
Fibonacci sequence

In mathematics, the Fibonacci sequence is a sequence in which each number is the sum of the two preceding ones. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted $F_{n}$. The sequence commonly starts from 0 and 1, although some authors start the sequence from 1 and 1 or sometimes (as did Fibonacci) from 1 and 2. Starting from 0 and 1, the sequence begins
 * 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ....

The Fibonacci numbers were first described in Indian mathematics as early as 200 BC in work by Pingala on enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths. They are named after the Italian mathematician Leonardo of Pisa, also known as Fibonacci, who introduced the sequence to Western European mathematics in his 1202 book Liber Abaci.

Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the Fibonacci Quarterly. Applications of Fibonacci numbers include computer algorithms such as the Fibonacci search technique and the Fibonacci heap data structure, and graphs called Fibonacci cubes used for interconnecting parallel and distributed systems. They also appear in biological settings, such as branching in trees, the arrangement of leaves on a stem, the fruit sprouts of a pineapple, the flowering of an artichoke, and the arrangement of a pine cone's bracts, though they do not occur in all species.

Fibonacci numbers are also strongly related to the golden ratio: Binet's formula expresses the $n$-th Fibonacci number in terms of $n$ and the golden ratio, and implies that the ratio of two consecutive Fibonacci numbers tends to the golden ratio as $n$ increases. Fibonacci numbers are also closely related to Lucas numbers, which obey the same recurrence relation and with the Fibonacci numbers form a complementary pair of Lucas sequences.

Definition

The Fibonacci numbers may be defined by the recurrence relation $$F_0=0,\quad F_1= 1,$$ and $$F_n=F_{n-1} + F_{n-2}$$ for $n > 1$.

Under some older definitions, the value $$F_0 = 0$$ is omitted, so that the sequence starts with $$F_1=F_2=1,$$ and the recurrence $$F_n=F_{n-1} + F_{n-2}$$ is valid for $n > 2$.

The first 20 Fibonacci numbers $_{0}→1, _{1}→2, _{2}→3, ..., _{12}→377$ are:
 * {| class="wikitable" style="text-align:right"

! $F_{n}$ ! $F_{0}$ ! $F_{1}$ ! $F_{2}$ ! $F_{3}$ ! $F_{4}$ ! $F_{5}$ ! $F_{6}$ ! $F_{7}$ ! $F_{8}$ ! $F_{9}$ ! $F_{10}$ ! $F_{11}$ ! $F_{12}$ ! $F_{13}$ ! $F_{14}$ ! $F_{15}$ ! $F_{16}$ ! $F_{17}$ ! $F_{18}$
 * 0
 * 1
 * 1
 * 2
 * 3
 * 5
 * 8
 * 13
 * 21
 * 34
 * 55
 * 89
 * 144
 * 233
 * 377
 * 610
 * 987
 * 1597
 * 2584
 * 4181
 * }
 * }

India

The Fibonacci sequence appears in Indian mathematics, in connection with Sanskrit prosody. In the Sanskrit poetic tradition, there was interest in enumerating all patterns of long (L) syllables of 2 units duration, juxtaposed with short (S) syllables of 1 unit duration. Counting the different patterns of successive L and S with a given total duration results in the Fibonacci numbers: the number of patterns of duration $m$ units is $F_{19}$.

Knowledge of the Fibonacci sequence was expressed as early as Pingala (c. 450 BC–200 BC). Singh cites Pingala's cryptic formula misrau cha ("the two are mixed") and scholars who interpret it in context as saying that the number of patterns for $m$ beats ($F_{7}$) is obtained by adding one [S] to the $F_{6}$ cases and one [L] to the $F_{5}$ cases. Bharata Muni also expresses knowledge of the sequence in the Natya Shastra (c. 100 BC–c. 350 AD). However, the clearest exposition of the sequence arises in the work of Virahanka (c. 700 AD), whose own work is lost, but is available in a quotation by Gopala (c. 1135):

"Variations of two earlier meters [is the variation] ... For example, for [a meter of length] four, variations of meters of two [and] three being mixed, five happens. [works out examples 8, 13, 21] ... In this way, the process should be followed in all mātrā-vṛttas [prosodic combinations]."

Hemachandra (c. 1150) is credited with knowledge of the sequence as well, writing that "the sum of the last and the one before the last is the number ... of the next mātrā-vṛtta."

Europe

The Fibonacci sequence first appears in the book Liber Abaci (The Book of Calculation, 1202) by Fibonacci where it is used to calculate the growth of rabbit populations. Fibonacci considers the growth of an idealized (biologically unrealistic) rabbit population, assuming that: a newly born breeding pair of rabbits are put in a field; each breeding pair mates at the age of one month, and at the end of their second month they always produce another pair of rabbits; and rabbits never die, but continue breeding forever. Fibonacci posed the puzzle: how many pairs will there be in one year?

 * At the end of the first month, they mate, but there is still only 1 pair.
 * At the end of the second month they produce a new pair, so there are 2 pairs in the field.
 * At the end of the third month, the original pair produce a second pair, but the second pair only mate to gestate for a month, so there are 3 pairs in all.
 * At the end of the fourth month, the original pair has produced yet another new pair, and the pair born two months ago also produces their first pair, making 5 pairs.

At the end of the $n$-th month, the number of pairs of rabbits is equal to the number of mature pairs (that is, the number of pairs in month $F_{m+1}$) plus the number of pairs alive last month (month $m = 7$). The number in the $n$-th month is the $n$-th Fibonacci number.

The name "Fibonacci sequence" was first used by the 19th-century number theorist Édouard Lucas.

Closed-form expression
Like every sequence defined by a linear recurrence with constant coefficients, the Fibonacci numbers have a closed-form expression. It has become known as Binet's formula, named after French mathematician Jacques Philippe Marie Binet, though it was already known by Abraham de Moivre and Daniel Bernoulli:

$$ F_n = \frac{\varphi^n-\psi^n}{\varphi-\psi} = \frac{\varphi^n-\psi^n}{\sqrt 5}, $$

where

$$ \varphi = \frac{1 + \sqrt{5}}{2} \approx 1.61803\,39887\ldots $$

is the golden ratio, and $ψ$ is its conjugate:

$$ \psi = \frac{1 - \sqrt{5}}{2} = 1 - \varphi = - {1 \over \varphi} \approx -0.61803\,39887\ldots. $$

Since $$\psi = -\varphi^{-1}$$, this formula can also be written as

$$ F_n = \frac{\varphi^n - (-\varphi)^{-n}}{\sqrt 5} = \frac{\varphi^n - (-\varphi)^{-n}}{2\varphi - 1}. $$

To see the relation between the sequence and these constants, note that $φ$ and $ψ$ are both solutions of the equation $x^2 = x + 1$ and thus $$x^n = x^{n-1} + x^{n-2},$$ so the powers of $φ$ and $ψ$ satisfy the Fibonacci recursion. In other words,

$$\begin{align} \varphi^n &= \varphi^{n-1} + \varphi^{n-2}, \\[3mu] \psi^n &= \psi^{n-1} + \psi^{n-2}. \end{align}$$

It follows that for any values $a$ and $b$, the sequence defined by

$$U_n=a \varphi^n + b \psi^n$$

satisfies the same recurrence,

$$\begin{align} U_n &= a\varphi^n + b\psi^n \\[3mu] &= a(\varphi^{n-1} + \varphi^{n-2}) + b(\psi^{n-1} + \psi^{n-2}) \\[3mu] &= a\varphi^{n-1} + b\psi^{n-1} + a\varphi^{n-2} + b\psi^{n-2} \\[3mu] &= U_{n-1} + U_{n-2}. \end{align}$$

If $a$ and $b$ are chosen so that $F_{m+1}$ and $F_{m}$ then the resulting sequence $F_{m−1}$ must be the Fibonacci sequence. This is the same as requiring $a$ and $b$ satisfy the system of equations:

$$ \left\{\begin{align} a + b &= 0 \\ \varphi a + \psi b &= 1\end{align}\right. $$

which has solution

$$ a = \frac{1}{\varphi-\psi} = \frac{1}{\sqrt 5},\quad b = -a, $$

producing the required formula.

Taking the starting values $n – 2$ and $n – 1$ to be arbitrary constants, a more general solution is:

$$ U_n = a\varphi^n + b\psi^n $$

where

$$\begin{align} a&=\frac{U_1-U_0\psi}{\sqrt 5}, \\[3mu] b&=\frac{U_0\varphi-U_1}{\sqrt 5}. \end{align}$$

Computation by rounding
Since $\left|\frac{\psi^{n}}{\sqrt 5}\right| < \frac{1}{2}$ for all $U_{0} = 0$, the number $U_{1} = 1$ is the closest integer to $$\frac{\varphi^n}{\sqrt 5}$$. Therefore, it can be found by rounding, using the nearest integer function: $$F_n=\left\lfloor\frac{\varphi^n}{\sqrt 5}\right\rceil,\ n \geq 0.$$

In fact, the rounding error is very small, being less than 0.1 for $U_{n}$, and less than 0.01 for $U_{0}$. This formula is easily inverted to find an index of a Fibonacci number $F$: $$n(F) = \left\lfloor \log_\varphi \sqrt{5}F\right\rceil,\ F \geq 1.$$

Instead using the floor function gives the largest index of a Fibonacci number that is not greater than $F$: $$n_{\mathrm{largest}}(F) = \left\lfloor \log_\varphi \sqrt{5}(F+1/2)\right\rfloor,\ F \geq 0,$$ where $$\log_\varphi(x) = \ln(x)/\ln(\varphi) = \log_{10}(x)/\log_{10}(\varphi)$$, $$\ln(\varphi) = 0.481211\ldots$$, and $$\log_{10}(\varphi) = 0.208987\ldots$$.

Magnitude
Since Fn is asymptotic to $$\varphi^n/\sqrt5$$, the number of digits in $U_{1}$ is asymptotic to $$n\log_{10}\varphi\approx 0.2090\, n$$. As a consequence, for every integer $n ≥ 0$ there are either 4 or 5 Fibonacci numbers with $d$ decimal digits.

More generally, in the base $b$ representation, the number of digits in $F_{n}$ is asymptotic to $$n\log_b\varphi = \frac{n \log \varphi}{\log b}.$$

Limit of consecutive quotients
Johannes Kepler observed that the ratio of consecutive Fibonacci numbers converges. He wrote that "as 5 is to 8 so is 8 to 13, practically, and as 8 is to 13, so is 13 to 21 almost", and concluded that these ratios approach the golden ratio $$\varphi\colon $$ $$\lim_{n\to\infty}\frac{F_{n+1}}{F_n}=\varphi.$$

This convergence holds regardless of the starting values $$U_0$$ and $$U_1$$, unless $$U_1 = -U_0/\varphi$$. This can be verified using Binet's formula. For example, the initial values 3 and 2 generate the sequence 3, 2, 5, 7, 12, 19, 31, 50, 81, 131, 212, 343, 555, ... . The ratio of consecutive terms in this sequence shows the same convergence towards the golden ratio.

In general, $$\lim_{n\to\infty}\frac{F_{n+m}}{F_n}=\varphi^m $$, because the ratios between consecutive Fibonacci numbers approaches $$\varphi$$.

 * Fibonacci tiling of the plane and approximation to Golden Ratio.gif

Decomposition of powers
Since the golden ratio satisfies the equation $$\varphi^2 = \varphi + 1,$$

this expression can be used to decompose higher powers $$\varphi^n$$ as a linear function of lower powers, which in turn can be decomposed all the way down to a linear combination of $$\varphi$$ and 1. The resulting recurrence relationships yield Fibonacci numbers as the linear coefficients: $$\varphi^n = F_n\varphi + F_{n-1}.$$ This equation can be proved by induction on $n ≥ 4$: $$\varphi^{n+1} = (F_n\varphi + F_{n-1})\varphi = F_n\varphi^2 + F_{n-1}\varphi = F_n(\varphi+1) + F_{n-1}\varphi = (F_n + F_{n-1})\varphi + F_n = F_{n+1}\varphi + F_n.$$ For $$\psi = -1/\varphi$$, it is also the case that $$\psi^2 = \psi + 1$$ and it is also the case that $$\psi^n = F_n\psi + F_{n-1}.$$

These expressions are also true for $n ≥ 8$ if the Fibonacci sequence Fn is extended to negative integers using the Fibonacci rule $$F_n = F_{n+2} - F_{n+1}.$$

Identification
Binet's formula provides a proof that a positive integer $x$ is a Fibonacci number if and only if at least one of $$5x^2+4$$ or $$5x^2-4$$ is a perfect square. This is because Binet's formula, which can be written as $$F_n = (\varphi^n - (-1)^n \varphi^{-n}) / \sqrt{5}$$, can be multiplied by $$\sqrt{5} \varphi^n$$ and solved as a quadratic equation in $$\varphi^n$$ via the quadratic formula:

$$\varphi^n = \frac{F_n\sqrt{5} \pm \sqrt{5{F_n}^2 + 4(-1)^n}}{2}.$$

Comparing this to $$\varphi^n = F_n \varphi + F_{n-1} = (F_n\sqrt{5} + F_n + 2 F_{n-1})/2$$, it follows that
 * $$5{F_n}^2 + 4(-1)^n = (F_n + 2F_{n-1})^2\,.$$

In particular, the left-hand side is a perfect square.

Matrix form
A 2-dimensional system of linear difference equations that describes the Fibonacci sequence is

$$ {F_{k+2} \choose F_{k+1}} = \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix} {F_{k+1} \choose F_{k}} $$ alternatively denoted $$ \vec F_{k+1} = \mathbf{A} \vec F_{k},$$

which yields $$\vec F_n = \mathbf{A}^n \vec F_0$$. The eigenvalues of the matrix $F_{n}$ are $$\varphi=\tfrac12\bigl(1+\sqrt5~\!\bigr)$$ and $$\psi=-\varphi^{-1}=\tfrac12\bigl(1-\sqrt5~\!\bigr)$$ corresponding to the respective eigenvectors $$\vec \mu={\varphi \choose 1}, \quad \vec\nu={-\varphi^{-1} \choose 1}.$$

As the initial value is $$\vec F_0={1 \choose 0}=\frac{1}{\sqrt{5}}\vec{\mu}-\frac{1}{\sqrt{5}}\vec{\nu},$$

it follows that the $n$th term is $$\begin{align} \vec F_n &= \frac{1}{\sqrt{5}}A^n\vec\mu-\frac{1}{\sqrt{5}}A^n\vec\nu \\ &= \frac{1}{\sqrt{5}}\varphi^n\vec\mu - \frac{1}{\sqrt{5}}(-\varphi)^{-n}\vec\nu \\ &= \cfrac{1}{\sqrt{5}}\left(\cfrac{1+\sqrt{5}}{2}\right)^{\!n}{\varphi \choose 1} \,-\, \cfrac{1}{\sqrt{5}}\left(\cfrac{1-\sqrt{5}}{2}\right)^{\!n}{-\varphi^{-1}\choose 1}. \end{align}$$

From this, the $n$th element in the Fibonacci series may be read off directly as a closed-form expression: $$ F_n = \cfrac{1}{\sqrt{5}}\left(\cfrac{1+\sqrt{5}}{2}\right)^{\!n} - \, \cfrac{1}{\sqrt{5}}\left(\cfrac{1-\sqrt{5}}{2}\right)^{\!n}. $$

Equivalently, the same computation may be performed by diagonalization of $d > 1$ through use of its eigendecomposition:

$$\begin{align} A & = S\Lambda S^{-1}, \\[3mu] A^n & = S\Lambda^n S^{-1}, \end{align}$$

where

$$ \Lambda=\begin{pmatrix} \varphi & 0 \\ 0 & -\varphi^{-1}\! \end{pmatrix}, \quad S=\begin{pmatrix} \varphi & -\varphi^{-1} \\ 1 & 1 \end{pmatrix}. $$

The closed-form expression for the $n$th element in the Fibonacci series is therefore given by

$$\begin{align} {F_{n+1} \choose F_n} & = A^{n} {F_1 \choose F_0} \\ & = S \Lambda^n S^{-1} {F_1 \choose F_0} \\ & = S \begin{pmatrix} \varphi^n & 0 \\ 0 & (-\varphi)^{-n} \end{pmatrix} S^{-1} {F_1 \choose F_0} \\ & = \begin{pmatrix} \varphi & -\varphi^{-1} \\ 1 & 1 \end{pmatrix} \begin{pmatrix} \varphi^n & 0 \\ 0 & (-\varphi)^{-n} \end{pmatrix} \frac{1}{\sqrt{5}}\begin{pmatrix} 1 & \varphi^{-1} \\ -1 & \varphi \end{pmatrix} {1 \choose 0}, \end{align}$$

which again yields $$F_n = \cfrac{\varphi^n-(-\varphi)^{-n}}{\sqrt{5}}.$$

The matrix $F_{n}$ has a determinant of −1, and thus it is a 2 × 2 unimodular matrix.

This property can be understood in terms of the continued fraction representation for the golden ratio $φ$:

$$\varphi = 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \ddots}}}.$$

The convergents of the continued fraction for $φ$ are ratios of successive Fibonacci numbers: $n ≥ 1$ is the $n$-th convergent, and the $n < 1$-st convergent can be found from the recurrence relation $A$. The matrix formed from successive convergents of any continued fraction has a determinant of +1 or −1. The matrix representation gives the following closed-form expression for the Fibonacci numbers:

$$\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}^n = \begin{pmatrix} F_{n+1} & F_n \\ F_n & F_{n-1} \end{pmatrix}.$$

For a given $n$, this matrix can be computed in $A$ arithmetic operations, using the exponentiation by squaring method.

Taking the determinant of both sides of this equation yields Cassini's identity, $$(-1)^n = F_{n+1}F_{n-1} - {F_n}^2.$$

Moreover, since $A$ for any square matrix $φ_{n} = F_{n+1} / F_{n}$, the following identities can be derived (they are obtained from two different coefficients of the matrix product, and one may easily deduce the second one from the first one by changing $n$ into $(n + 1)$), $$\begin{align} {F_m}{F_n} + {F_{m-1}}{F_{n-1}} &= F_{m+n-1}, \\[3mu] F_{m} F_{n+1} + F_{m-1} F_n &= F_{m+n}. \end{align}$$

In particular, with $φ_{n+1} = 1 + 1 / φ_{n}$, $$\begin{align} F_{2 n-1} &= {F_n}^2 + {F_{n-1}}^2 \\[6mu] F_{2 n\phantom{{}-1}}  &= (F_{n-1}+F_{n+1})F_n \\[3mu] &= (2 F_{n-1}+F_n)F_n \\[3mu] &= (2 F_{n+1}-F_n)F_n. \end{align}$$

These last two identities provide a way to compute Fibonacci numbers recursively in $O(log n)$ arithmetic operations. This matches the time for computing the $n$-th Fibonacci number from the closed-form matrix formula, but with fewer redundant steps if one avoids recomputing an already computed Fibonacci number (recursion with memoization).

Combinatorial proofs
Most identities involving Fibonacci numbers can be proved using combinatorial arguments using the fact that $$F_n$$ can be interpreted as the number of (possibly empty) sequences of 1s and 2s whose sum is $$n-1$$. This can be taken as the definition of $$F_n$$ with the conventions $$F_0 = 0$$, meaning no such sequence exists whose sum is −1, and $$F_1 = 1$$, meaning the empty sequence "adds up" to 0. In the following, $$|{...}|$$ is the cardinality of a set:

 * $$F_0 = 0 = |\{\}|$$
 * $$F_1 = 1 = |\{\}|$$
 * $$F_2 = 1 = |\{(1)\}|$$
 * $$F_3 = 2 = |\{(1,1),(2)\}|$$
 * $$F_4 = 3 = |\{(1,1,1),(1,2),(2,1)\}|$$
 * $$F_5 = 5 = |\{(1,1,1,1),(1,1,2),(1,2,1),(2,1,1),(2,2)\}|$$

In this manner the recurrence relation $$F_n = F_{n-1} + F_{n-2}$$ may be understood by dividing the $$F_n$$ sequences into two non-overlapping sets where all sequences either begin with 1 or 2: $$F_n = |\{(1,...),(1,...),...\}| + |\{(2,...),(2,...),...\}|$$ Excluding the first element, the remaining terms in each sequence sum to $$n-2$$ or $$n-3$$ and the cardinality of each set is $$F_{n-1}$$ or $$F_{n-2}$$ giving a total of $$F_{n-1}+F_{n-2}$$ sequences, showing this is equal to $$F_n$$.

In a similar manner it may be shown that the sum of the first Fibonacci numbers up to the $n$-th is equal to the $A^{n}A^{m} = A^{n+m}$-th Fibonacci number minus 1. In symbols: $$\sum_{i=1}^n F_i = F_{n+2} - 1$$

This may be seen by dividing all sequences summing to $$n+1$$ based on the location of the first 2. Specifically, each set consists of those sequences that start $$(2,...), (1,2,...), ..., $$ until the last two sets $$\{(1,1,...,1,2)\}, \{(1,1,...,1)\}$$ each with cardinality 1.

Following the same logic as before, by summing the cardinality of each set we see that
 * $$F_{n+2} = F_n + F_{n-1} + ... + |\{(1,1,...,1,2)\}| + |\{(1,1,...,1)\}|$$

... where the last two terms have the value $$F_1 = 1$$. From this it follows that $$\sum_{i=1}^n F_i = F_{n+2}-1$$.

A similar argument, grouping the sums by the position of the first 1 rather than the first 2 gives two more identities: $$\sum_{i=0}^{n-1} F_{2 i+1} = F_{2 n}$$ and $$\sum_{i=1}^{n} F_{2 i} = F_{2 n+1}-1.$$ In words, the sum of the first Fibonacci numbers with odd index up to $$F_{2 n-1}$$ is the $A$-th Fibonacci number, and the sum of the first Fibonacci numbers with even index up to $$F_{2 n}$$ is the $n + 1$-th Fibonacci number minus 1.

A different trick may be used to prove $$\sum_{i=1}^n F_i^2 = F_n F_{n+1}$$ or in words, the sum of the squares of the first Fibonacci numbers up to $$F_n$$ is the product of the $n$-th and $m = n$-th Fibonacci numbers. To see this, begin with a Fibonacci rectangle of size $$F_n \times F_{n+1}$$ and decompose it into squares of size $$F_n, F_{n-1}, ..., F_1$$; from this the identity follows by comparing areas:

Symbolic method
The sequence $$(F_n)_{n\in\mathbb N}$$ is also considered using the symbolic method. More precisely, this sequence corresponds to a specifiable combinatorial class. The specification of this sequence is $$\operatorname{Seq}(\mathcal{Z+Z^2})$$. Indeed, as stated above, the $$n$$-th Fibonacci number equals the number of combinatorial compositions (ordered partitions) of $$n-1$$ using terms 1 and 2.

It follows that the ordinary generating function of the Fibonacci sequence, $$\sum_{i=0}^\infty F_iz^i$$, is the rational function $$\frac{z}{1-z-z^2}.$$

Induction proofs
Fibonacci identities often can be easily proved using mathematical induction.

For example, reconsider $$\sum_{i=1}^n F_i = F_{n+2} - 1.$$ Adding $$F_{n+1}$$ to both sides gives
 * $$\sum_{i=1}^n F_i + F_{n+1} = F_{n+1} + F_{n+2} - 1$$

and so we have the formula for $$n+1$$ $$\sum_{i=1}^{n+1} F_i = F_{n+3} - 1$$

Similarly, add $${F_{n+1}}^2$$ to both sides of $$\sum_{i=1}^n F_i^2 = F_n F_{n+1}$$ to give $$\sum_{i=1}^n F_i^2 + {F_{n+1}}^2 = F_{n+1}\left(F_n + F_{n+1}\right)$$ $$\sum_{i=1}^{n+1} F_i^2 = F_{n+1}F_{n+2}$$

Binet formula proofs
The Binet formula is $$\sqrt5F_n = \varphi^n - \psi^n.$$ This can be used to prove Fibonacci identities.

For example, to prove that $\sum_{i=1}^n F_i = F_{n+2} - 1$ note that the left hand side multiplied by $$\sqrt5$$ becomes $$ \begin{align} 1 +& \varphi + \varphi^2 + \dots + \varphi^n - \left(1 + \psi + \psi^2 + \dots + \psi^n \right)\\ &= \frac{\varphi^{n+1}-1}{\varphi-1} - \frac{\psi^{n+1}-1}{\psi-1}\\ &= \frac{\varphi^{n+1}-1}{-\psi} - \frac{\psi^{n+1}-1}{-\varphi}\\ &= \frac{-\varphi^{n+2}+\varphi + \psi^{n+2}-\psi}{\varphi\psi}\\ &= \varphi^{n+2}-\psi^{n+2}-(\varphi-\psi)\\ &= \sqrt5(F_{n+2}-1)\\ \end{align}$$ as required, using the facts $\varphi\psi =- 1$ and $\varphi-\psi=\sqrt5$  to simplify the equations.

Other identities
Numerous other identities can be derived using various methods. Here are some of them:

Cassini's and Catalan's identities
Cassini's identity states that $${F_n}^2 - F_{n+1}F_{n-1} = (-1)^{n-1}$$ Catalan's identity is a generalization: $${F_n}^2 - F_{n+r}F_{n-r} = (-1)^{n-r}{F_r}^2$$

d'Ocagne's identity
$$F_m F_{n+1} - F_{m+1} F_n = (-1)^n F_{m-n}$$ $$F_{2 n} = {F_{n+1}}^2 - {F_{n-1}}^2 = F_n \left (F_{n+1}+F_{n-1} \right ) = F_nL_n$$ where $O(log n)$ is the $n$-th Lucas number. The last is an identity for doubling $n$; other identities of this type are $$F_{3 n} = 2{F_n}^3 + 3 F_n F_{n+1} F_{n-1} = 5{F_n}^3 + 3 (-1)^n F_n$$ by Cassini's identity.

$$F_{3 n+1} = {F_{n+1}}^3 + 3 F_{n+1}{F_n}^2 - {F_n}^3$$ $$F_{3 n+2} = {F_{n+1}}^3 + 3 {F_{n+1}}^2 F_n + {F_n}^3$$ $$F_{4 n} = 4 F_n F_{n+1} \left ({F_{n+1}}^2 + 2{F_n}^2 \right ) - 3{F_n}^2 \left ({F_n}^2 + 2{F_{n+1}}^2 \right )$$ These can be found experimentally using lattice reduction, and are useful in setting up the special number field sieve to factorize a Fibonacci number.

More generally,

$$F_{k n+c} = \sum_{i=0}^k {k\choose i} F_{c-i} {F_n}^i {F_{n+1}}^{k-i}.$$

or alternatively

$$F_{k n+c} = \sum_{i=0}^k {k\choose i} F_{c+i} {F_n}^i {F_{n-1}}^{k-i}.$$

Putting $(n + 2)$ in this formula, one gets again the formulas of the end of above section Matrix form.

Generating function
The generating function of the Fibonacci sequence is the power series

$$ s(z) = \sum_{k=0}^\infty F_k z^k = 0 + z + z^2 + 2z^3 + 3z^4 + 5z^5 + \dots. $$

This series is convergent for any complex number $$z$$ satisfying $$|z| < 1/\varphi,$$ and its sum has a simple closed form:

$$s(z)=\frac{z}{1-z-z^2}.$$

This can be proved by multiplying by $(1-z-z^2)$ : $$\begin{align} (1 - z- z^2) s(z) &= \sum_{k=0}^{\infty} F_k z^k - \sum_{k=0}^{\infty} F_k z^{k+1} - \sum_{k=0}^{\infty} F_k z^{k+2} \\ &= \sum_{k=0}^{\infty} F_k z^k - \sum_{k=1}^{\infty} F_{k-1} z^k - \sum_{k=2}^{\infty} F_{k-2} z^k \\ &= 0z^0 + 1z^1 - 0z^1 + \sum_{k=2}^{\infty} (F_k - F_{k-1} - F_{k-2}) z^k \\ &= z, \end{align}$$

where all terms involving $$z^k$$ for $$k \ge 2$$ cancel out because of the defining Fibonacci recurrence relation.

The partial fraction decomposition is given by $$s(z) = \frac{1}{\sqrt5}\left(\frac{1}{1 - \varphi z} - \frac{1}{1 - \psi z}\right)$$ where $\varphi = \tfrac12\left(1 + \sqrt{5}\right)$ is the golden ratio and $$\psi = \tfrac12\left(1 - \sqrt{5}\right)$$ is its conjugate.

The related function $z \mapsto -s\left(-1/z\right)$ is the generating function for the negafibonacci numbers, and $$s(z)$$ satisfies the functional equation

$$s(z) = s\!\left(-\frac{1}{z}\right).$$

Using $$z$$ equal to any of 0.01, 0.001, 0.0001, etc. lays out the first Fibonacci numbers in the decimal expansion of $$s(z)$$. For example, $$s(0.001) = \frac{0.001}{0.998999} = \frac{1000}{998999} = 0.001001002003005008013021\ldots.$$

Reciprocal sums
Infinite sums over reciprocal Fibonacci numbers can sometimes be evaluated in terms of theta functions. For example, the sum of every odd-indexed reciprocal Fibonacci number can be written as $$\sum_{k=1}^\infty \frac{1}{F_{2 k-1}} = \frac{\sqrt{5}}{4} \; \vartheta_2\!\left(0, \frac{3-\sqrt 5}{2}\right)^2 ,$$

and the sum of squared reciprocal Fibonacci numbers as $$\sum_{k=1}^\infty \frac{1}{{F_k}^2} = \frac{5}{24} \!\left(\vartheta_2\!\left(0, \frac{3-\sqrt 5}{2}\right)^4 - \vartheta_4\!\left(0, \frac{3-\sqrt 5}{2}\right)^4 + 1 \right).$$

If we add 1 to each Fibonacci number in the first sum, there is also the closed form $$\sum_{k=1}^\infty \frac{1}{1+F_{2 k-1}} = \frac{\sqrt{5}}{2},$$

and there is a nested sum of squared Fibonacci numbers giving the reciprocal of the golden ratio, $$\sum_{k=1}^\infty \frac{(-1)^{k+1}}{\sum_{j=1}^k {F_{j}}^2} = \frac{\sqrt{5}-1}{2} .$$

The sum of all even-indexed reciprocal Fibonacci numbers is $$\sum_{k=1}^{\infty} \frac{1}{F_{2 k}} = \sqrt{5} \left(L(\psi^2) - L(\psi^4)\right) $$ with the Lambert series $$\textstyle L(q) := \sum_{k=1}^{\infty} \frac{q^k}{1-q^k} ,$$ since $$\textstyle \frac{1}{F_{2 k}} = \sqrt{5} \left(\frac{\psi^{2 k}}{1-\psi^{2 k}} - \frac{\psi^{4 k}}{1-\psi^{4 k}} \right)\!.$$

So the reciprocal Fibonacci constant is $$\sum_{k=1}^{\infty} \frac{1}{F_k} = \sum_{k=1}^\infty \frac{1}{F_{2 k-1}} + \sum_{k=1}^{\infty} \frac {1}{F_{2 k}} = 3.359885666243 \dots$$

Moreover, this number has been proved irrational by Richard André-Jeannin.

Millin's series gives the identity $$\sum_{k=0}^{\infty} \frac{1}{F_{2^k}} = \frac{7 - \sqrt{5}}{2},$$ which follows from the closed form for its partial sums as $k$ tends to infinity: $$\sum_{k=0}^N \frac{1}{F_{2^k}} = 3 - \frac{F_{2^N-1}}{F_{2^N}}.$$

Divisibility properties
Every third number of the sequence is even (a multiple of $$F_3=2$$) and, more generally, every $N$-th number of the sequence is a multiple of Fk. Thus the Fibonacci sequence is an example of a divisibility sequence. In fact, the Fibonacci sequence satisfies the stronger divisibility property $$\gcd(F_a,F_b,F_c,\ldots) = F_{\gcd(a,b,c,\ldots)}\,$$ where $(2n)$ is the greatest common divisor function.

In particular, any three consecutive Fibonacci numbers are pairwise coprime because both $$F_1=1$$ and $$F_2 = 1$$. That is,
 * $$\gcd(F_n, F_{n+1}) = \gcd(F_n, F_{n+2}) = \gcd(F_{n+1}, F_{n+2}) = 1$$

for every $k$.

Every prime number $n$ divides a Fibonacci number that can be determined by the value of $p$ modulo 5. If $p$ is congruent to 1 or 4 modulo 5, then $p$ divides $(2n + 1)$, and if $p$ is congruent to 2 or 3 modulo 5, then, $p$ divides $(n + 1)$. The remaining case is that $L_{n}$, and in this case $p$ divides Fp.

$$\begin{cases} p =5 & \Rightarrow p \mid F_{p}, \\ p \equiv \pm1 \pmod 5 & \Rightarrow p \mid F_{p-1}, \\ p \equiv \pm2 \pmod 5 & \Rightarrow p \mid F_{p+1}.\end{cases}$$

These cases can be combined into a single, non-piecewise formula, using the Legendre symbol: $$p \mid F_{p \;-\, \left(\frac{5}{p}\right)}.$$

Primality testing
The above formula can be used as a primality test in the sense that if $$n \mid F_{n \;-\, \left(\frac{5}{n}\right)},$$ where the Legendre symbol has been replaced by the Jacobi symbol, then this is evidence that $p$ is a prime, and if it fails to hold, then $n$ is definitely not a prime. If $n$ is composite and satisfies the formula, then $n$ is a Fibonacci pseudoprime. When $n$ is large – say a 500-bit number – then we can calculate $k = 2$ efficiently using the matrix form. Thus

$$ \begin{pmatrix} F_{m+1} & F_m \\ F_m & F_{m-1} \end{pmatrix} \equiv \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}^m \pmod n.$$ Here the matrix power $gcd$ is calculated using modular exponentiation, which can be adapted to matrices.

Fibonacci primes
A Fibonacci prime is a Fibonacci number that is prime. The first few are:

 * 2, 3, 5, 13, 89, 233, 1597, 28657, 514229, ...

Fibonacci primes with thousands of digits have been found, but it is not known whether there are infinitely many.

$F_{p−1}$ is divisible by $F_{p+1}$, so, apart from $p = 5$, any Fibonacci prime must have a prime index. As there are arbitrarily long runs of composite numbers, there are therefore also arbitrarily long runs of composite Fibonacci numbers.

No Fibonacci number greater than $F_{m} (mod n)$ is one greater or one less than a prime number.

The only nontrivial square Fibonacci number is 144. Attila Pethő proved in 2001 that there is only a finite number of perfect power Fibonacci numbers. In 2006, Y. Bugeaud, M. Mignotte, and S. Siksek proved that 8 and 144 are the only such non-trivial perfect powers.

1, 3, 21, and 55 are the only triangular Fibonacci numbers, which was conjectured by Vern Hoggatt and proved by Luo Ming.

No Fibonacci number can be a perfect number. More generally, no Fibonacci number other than 1 can be multiply perfect, and no ratio of two Fibonacci numbers can be perfect.

Prime divisors
With the exceptions of 1, 8 and 144 ($A^{m}$, $F_{kn}$ and $F_{n}$) every Fibonacci number has a prime factor that is not a factor of any smaller Fibonacci number (Carmichael's theorem). As a result, 8 and 144 ($F_{4} = 3$ and $F_{6} = 8$) are the only Fibonacci numbers that are the product of other Fibonacci numbers.

The divisibility of Fibonacci numbers by a prime $m$ is related to the Legendre symbol $$\left(\tfrac{p}{5}\right)$$ which is evaluated as follows: $$\left(\frac{p}{5}\right) = \begin{cases} 0 & \text{if } p = 5\\ 1 & \text{if } p \equiv \pm 1 \pmod 5\\ -1 & \text{if } p \equiv \pm 2 \pmod 5.\end{cases}$$

If $p$ is a prime number then $$ F_p \equiv \left(\frac{p}{5}\right) \pmod p \quad \text{and}\quad F_{p-\left(\frac{p}{5}\right)} \equiv 0 \pmod p.$$

For example, $$\begin{align} (\tfrac{2}{5}) &= -1, &F_3 &= 2, &F_2&=1, \\ (\tfrac{3}{5}) &= -1, &F_4 &= 3,&F_3&=2, \\ (\tfrac{5}{5}) &= 0, &F_5 &= 5, \\ (\tfrac{7}{5}) &= -1, &F_8 &= 21,&F_7&=13, \\ (\tfrac{11}{5})& = +1, &F_{10}& = 55, &F_{11}&=89. \end{align}$$

It is not known whether there exists a prime $p$ such that

$$F_{p-\left(\frac{p}{5}\right)} \equiv 0 \pmod{p^2}.$$

Such primes (if there are any) would be called Wall–Sun–Sun primes.

Also, if $F_{1} = F_{2}$ is an odd prime number then: $$5 {F_{\frac{p \pm 1}{2}}}^2 \equiv \begin{cases} \tfrac{1}{2} \left (5\left(\frac{p}{5}\right)\pm 5 \right ) \pmod p & \text{if } p \equiv 1 \pmod 4\\ \tfrac{1}{2} \left (5\left(\frac{p}{5}\right)\mp 3 \right ) \pmod p & \text{if } p \equiv 3 \pmod 4. \end{cases}$$

Example 1. $F_{6}$, in this case $F_{12}$ and we have: $$(\tfrac{7}{5}) = -1: \qquad \tfrac{1}{2}\left (5(\tfrac{7}{5})+3 \right ) =-1, \quad \tfrac{1}{2} \left (5(\tfrac{7}{5})-3 \right )=-4.$$ $$F_3=2 \text{ and } F_4=3.$$ $$5{F_3}^2=20\equiv -1 \pmod {7}\;\;\text{ and }\;\;5{F_4}^2=45\equiv -4 \pmod {7}$$

Example 2. $F_{6}$, in this case $F_{12}$ and we have: $$(\tfrac{11}{5}) = +1: \qquad \tfrac{1}{2}\left (5(\tfrac{11}{5})+3 \right )=4, \quad \tfrac{1}{2} \left (5(\tfrac{11}{5})- 3 \right )=1.$$ $$F_5=5 \text{ and } F_6=8.$$ $$5{F_5}^2=125\equiv 4 \pmod {11} \;\;\text{ and }\;\;5{F_6}^2=320\equiv 1 \pmod {11}$$

Example 3. $p ≠ 5$, in this case $p = 7$ and we have: $$(\tfrac{13}{5}) = -1: \qquad \tfrac{1}{2}\left (5(\tfrac{13}{5})-5 \right ) =-5, \quad \tfrac{1}{2}\left (5(\tfrac{13}{5})+ 5 \right )=0.$$ $$F_6=8 \text{ and } F_7=13.$$ $$5{F_6}^2=320\equiv -5 \pmod {13} \;\;\text{ and }\;\;5{F_7}^2=845\equiv 0 \pmod {13}$$

Example 4. $p ≡ 3 (mod 4)$, in this case $p = 11$ and we have: $$(\tfrac{29}{5}) = +1: \qquad \tfrac{1}{2}\left (5(\tfrac{29}{5})-5 \right )=0, \quad \tfrac{1}{2}\left (5(\tfrac{29}{5})+5 \right )=5.$$ $$F_{14}=377 \text{ and } F_{15}=610.$$ $$5{F_{14}}^2=710645\equiv 0 \pmod {29} \;\;\text{ and }\;\;5{F_{15}}^2=1860500\equiv 5 \pmod {29}$$

For odd $p$, all odd prime divisors of $p ≡ 3 (mod 4)$ are congruent to 1 modulo 4, implying that all odd divisors of $p = 13$ (as the products of odd prime divisors) are congruent to 1 modulo 4.

For example, $$F_1 = 1,\ F_3 = 2,\ F_5 = 5,\ F_7 = 13,\ F_9 = 34 = 2 \cdot 17,\ F_{11} = 89,\ F_{13} = 233,\ F_{15} = 610 = 2 \cdot 5 \cdot 61.$$

All known factors of Fibonacci numbers $p ≡ 1 (mod 4)$ for all $p = 29$ are collected at the relevant repositories.

Periodicity modulo n
If the members of the Fibonacci sequence are taken mod $n$, the resulting sequence is periodic with period at most $p ≡ 1 (mod 4)$. The lengths of the periods for various $n$ form the so-called Pisano periods. Determining a general formula for the Pisano periods is an open problem, which includes as a subproblem a special instance of the problem of finding the multiplicative order of a modular integer or of an element in a finite field. However, for any particular $n$, the Pisano period may be found as an instance of cycle detection.

Generalizations
The Fibonacci sequence is one of the simplest and earliest known sequences defined by a recurrence relation, and specifically by a linear difference equation. All these sequences may be viewed as generalizations of the Fibonacci sequence. In particular, Binet's formula may be generalized to any sequence that is a solution of a homogeneous linear difference equation with constant coefficients.

Some specific examples that are close, in some sense, to the Fibonacci sequence include:
 * Generalizing the index to negative integers to produce the negafibonacci numbers.
 * Generalizing the index to real numbers using a modification of Binet's formula.
 * Starting with other integers. Lucas numbers have $F_{n}$, $F_{n}$, and $F(i)$. Primefree sequences use the Fibonacci recursion with other starting points to generate sequences in which all numbers are composite.
 * Letting a number be a linear function (other than the sum) of the 2 preceding numbers. The Pell numbers have $i < 50000$. If the coefficient of the preceding value is assigned a variable value $n$, the result is the sequence of Fibonacci polynomials.
 * Not adding the immediately preceding numbers. The Padovan sequence and Perrin numbers have $F(i)$.
 * Generating the next number by adding 3 numbers (tribonacci numbers), 4 numbers (tetranacci numbers), or more. The resulting sequences are known as n-Step Fibonacci numbers.

Mathematics
The Fibonacci numbers occur as the sums of binomial coefficients in the "shallow" diagonals of Pascal's triangle: $$F_n = \sum_{k=0}^{\left\lfloor\frac{n-1}{2}\right\rfloor} \binom{n-k-1}{k}.$$ This can be proved by expanding the generating function $$\frac{x}{1-x-x^2} = x + x^2(1+x) + x^3(1+x)^2 + \dots + x^{k+1}(1+x)^k + \dots = \sum\limits_{n=0}^\infty F_n x^n$$ and collecting like terms of $$x^n$$.

To see how the formula is used, we can arrange the sums by the number of terms present:

 * }
 * }
 * }
 * }
 * }
 * }
 * }
 * }
 * }
 * }
 * }
 * }
 * }
 * }

which is $$\binom{5}{0}+\binom{4}{1}+\binom{3}{2}$$, where we are choosing the positions of $x$ twos from $i < 10000$ terms.

These numbers also give the solution to certain enumerative problems, the most common of which is that of counting the number of ways of writing a given number $k$ as an ordered sum of 1s and 2s (called compositions); there are $F(i)$ ways to do this (equivalently, it's also the number of domino tilings of the $$2\times n$$ rectangle). For example, there are $10000 < i < 50000$ ways one can climb a staircase of 5 steps, taking one or two steps at a time:

The figure shows that 8 can be decomposed into 5 (the number of ways to climb 4 steps, followed by a single-step) plus 3 (the number of ways to climb 3 steps, followed by a double-step). The same reasoning is applied recursively until a single step, of which there is only one way to climb.
 * }
 * }
 * }
 * }
 * }
 * }
 * }
 * }
 * }
 * }
 * }
 * }

The Fibonacci numbers can be found in different ways among the set of binary strings, or equivalently, among the subsets of a given set.
 * The number of binary strings of length $m$ without consecutive $6n$s is the Fibonacci number $L_{1} = 1$. For example, out of the 16 binary strings of length 4, there are $L_{2} = 3$ without consecutive $L_{n} = L_{n−1} + L_{n−2}$s—they are 0000, 0001, 0010, 0100, 0101, 1000, 1001, and 1010. Such strings are the binary representations of Fibbinary numbers.  Equivalently, $P_{n} = 2P_{n−1} + P_{n−2}$ is the number of subsets $k$ of $P(n) = P(n − 2) + P(n − 3)$ without consecutive integers, that is, those $n$ for which $5$ for every $n$. A bijection with the sums to $= 1+1+1+1+1$ is to replace 1 with 0 and 2 with 10, and drop the last zero.
 * The number of binary strings of length $S$ without an odd number of consecutive $= 2+1+1+1$s is the Fibonacci number $= 1+2+1+1$. For example, out of the 16 binary strings of length 4, there are $= 1+1+2+1$ without an odd number of consecutive $= 1+1+1+2$s—they are 0000, 0011, 0110, 1100, 1111. Equivalently, the number of subsets $S$ of $= 2+2+1$ without an odd number of consecutive integers is $= 2+1+2$. A bijection with the sums to $i$ is to replace 1 with 0 and 2 with 11.
 * The number of binary strings of length $n$ without an even number of consecutive $= 1+2+2$s or $n−k−1$s is $F_{n+1}$. For example, out of the 16 binary strings of length 4, there are $F_{5+1} = F_{6} = 8$ without an even number of consecutive $5$s or $= 1+1+1+1+1$s—they are 0001, 0111, 0101, 1000, 1010, 1110. There is an equivalent statement about subsets.
 * Yuri Matiyasevich was able to show that the Fibonacci numbers can be defined by a Diophantine equation, which led to his solving Hilbert's tenth problem.
 * The Fibonacci numbers are also an example of a complete sequence. This means that every positive integer can be written as a sum of Fibonacci numbers, where any one number is used once at most.
 * Moreover, every positive integer can be written in a unique way as the sum of one or more distinct Fibonacci numbers in such a way that the sum does not include any two consecutive Fibonacci numbers. This is known as Zeckendorf's theorem, and a sum of Fibonacci numbers that satisfies these conditions is called a Zeckendorf representation. The Zeckendorf representation of a number can be used to derive its Fibonacci coding.
 * Starting with 5, every second Fibonacci number is the length of the hypotenuse of a right triangle with integer sides, or in other words, the largest number in a Pythagorean triple, obtained from the formula $$(F_n F_{n+3})^2 + (2 F_{n+1}F_{n+2})^2 = {F_{2 n+3}}^2.$$ The sequence of Pythagorean triangles obtained from this formula has sides of lengths (3,4,5), (5,12,13), (16,30,34), (39,80,89), ... . The middle side of each of these triangles is the sum of the three sides of the preceding triangle.
 * The Fibonacci cube is an undirected graph with a Fibonacci number of nodes that has been proposed as a network topology for parallel computing.
 * Fibonacci numbers appear in the ring lemma, used to prove connections between the circle packing theorem and conformal maps.

Computer science

 * The Fibonacci numbers are important in computational run-time analysis of Euclid's algorithm to determine the greatest common divisor of two integers: the worst case input for this algorithm is a pair of consecutive Fibonacci numbers.
 * Fibonacci numbers are used in a polyphase version of the merge sort algorithm in which an unsorted list is divided into two lists whose lengths correspond to sequential Fibonacci numbers—by dividing the list so that the two parts have lengths in the approximate proportion $S$. A tape-drive implementation of the polyphase merge sort was described in The Art of Computer Programming.
 * A Fibonacci tree is a binary tree whose child trees (recursively) differ in height by exactly 1. So it is an AVL tree, and one with the fewest nodes for a given height—the "thinnest" AVL tree. These trees have a number of vertices that is a Fibonacci number minus one, an important fact in the analysis of AVL trees.
 * Fibonacci numbers are used by some pseudorandom number generators.
 * Fibonacci numbers arise in the analysis of the Fibonacci heap data structure.
 * A one-dimensional optimization method, called the Fibonacci search technique, uses Fibonacci numbers.
 * The Fibonacci number series is used for optional lossy compression in the IFF 8SVX audio file format used on Amiga computers. The number series compands the original audio wave similar to logarithmic methods such as μ-law.
 * Some Agile teams use a modified series called the "Modified Fibonacci Series" in planning poker, as an estimation tool. Planning Poker is a formal part of the Scaled Agile Framework.
 * Fibonacci coding
 * Negafibonacci coding

Nature

Fibonacci sequences appear in biological settings, such as branching in trees, arrangement of leaves on a stem, the fruitlets of a pineapple, the flowering of artichoke, the arrangement of a pine cone, and the family tree of honeybees. Kepler pointed out the presence of the Fibonacci sequence in nature, using it to explain the (golden ratio-related) pentagonal form of some flowers. Field daisies most often have petals in counts of Fibonacci numbers. In 1830, Karl Friedrich Schimper and Alexander Braun discovered that the parastichies (spiral phyllotaxis) of plants were frequently expressed as fractions involving Fibonacci numbers.

Przemysław Prusinkiewicz advanced the idea that real instances can in part be understood as the expression of certain algebraic constraints on free groups, specifically as certain Lindenmayer grammars.

A model for the pattern of florets in the head of a sunflower was proposed by Helmut Vogel in 1979. This has the form

$$\theta = \frac{2\pi}{\varphi^2} n,\ r = c \sqrt{n}$$

where $n$ is the index number of the floret and $n$ is a constant scaling factor; the florets thus lie on Fermat's spiral. The divergence angle, approximately 137.51°, is the golden angle, dividing the circle in the golden ratio. Because this ratio is irrational, no floret has a neighbor at exactly the same angle from the center, so the florets pack efficiently. Because the rational approximations to the golden ratio are of the form $= 2+1+1+1$, the nearest neighbors of floret number $φ$ are those at $= 1+2+1+1$ for some index $n$, which depends on $c$, the distance from the center. Sunflowers and similar flowers most commonly have spirals of florets in clockwise and counter-clockwise directions in the amount of adjacent Fibonacci numbers, typically counted by the outermost range of radii.

Fibonacci numbers also appear in the ancestral pedigrees of bees (which are haplodiploids), according to the following rules:
 * If an egg is laid but not fertilized, it produces a male (or drone bee in honeybees).
 * If, however, an egg is fertilized, it produces a female.

Thus, a male bee always has one parent, and a female bee has two. If one traces the pedigree of any male bee (1 bee), he has 1 parent (1 bee), 2 grandparents, 3 great-grandparents, 5 great-great-grandparents, and so on. This sequence of numbers of parents is the Fibonacci sequence. The number of ancestors at each level, $= 1+1+2+1$, is the number of female ancestors, which is $= 2+2+1$, plus the number of male ancestors, which is $= 1+1+1+2$. This is under the unrealistic assumption that the ancestors at each level are otherwise unrelated.

It has similarly been noticed that the number of possible ancestors on the human X chromosome inheritance line at a given ancestral generation also follows the Fibonacci sequence. A male individual has an X chromosome, which he received from his mother, and a Y chromosome, which he received from his father. The male counts as the "origin" of his own X chromosome ($$F_1=1$$), and at his parents' generation, his X chromosome came from a single parent ($F_2=1$). The male's mother received one X chromosome from her mother (the son's maternal grandmother), and one from her father (the son's maternal grandfather), so two grandparents contributed to the male descendant's X chromosome ($F_3=2$). The maternal grandfather received his X chromosome from his mother, and the maternal grandmother received X chromosomes from both of her parents, so three great-grandparents contributed to the male descendant's X chromosome ($F_4=3$). Five great-great-grandparents contributed to the male descendant's X chromosome ($F_5=5$), etc. (This assumes that all ancestors of a given descendant are independent, but if any genealogy is traced far enough back in time, ancestors begin to appear on multiple lines of the genealogy, until eventually a population founder appears on all lines of the genealogy.)

Other

 * In optics, when a beam of light shines at an angle through two stacked transparent plates of different materials of different refractive indexes, it may reflect off three surfaces: the top, middle, and bottom surfaces of the two plates. The number of different beam paths that have $n$ reflections, for $= 2+1+2$, is the $j$-th Fibonacci number. (However, when $= 1+2+2$, there are three reflection paths, not two, one for each of the three surfaces.)
 * Fibonacci retracement levels are widely used in technical analysis for financial market trading.
 * Since the conversion factor 1.609344 for miles to kilometers is close to the golden ratio, the decomposition of distance in miles into a sum of Fibonacci numbers becomes nearly the kilometer sum when the Fibonacci numbers are replaced by their successors. This method amounts to a radix 2 number register in golden ratio base $r$ being shifted. To convert from kilometers to miles, shift the register down the Fibonacci sequence instead.
 * The measured values of voltages and currents in the infinite resistor chain circuit (also called the resistor ladder or infinite series-parallel circuit) follow the Fibonacci sequence. The intermediate results of adding the alternating series and parallel resistances yields fractions composed of consecutive Fibonacci numbers. The equivalent resistance of the entire circuit equals the golden ratio.
 * Brasch et al. 2012 show how a generalized Fibonacci sequence also can be connected to the field of economics. In particular, it is shown how a generalized Fibonacci sequence enters the control function of finite-horizon dynamic optimisation problems with one state and one control variable. The procedure is illustrated in an example often referred to as the Brock–Mirman economic growth model.
 * Mario Merz included the Fibonacci sequence in some of his artworks beginning in 1970.
 * Joseph Schillinger (1895–1943) developed a system of composition which uses Fibonacci intervals in some of its melodies; he viewed these as the musical counterpart to the elaborate harmony evident within nature. See also.

title: Fibonacci sequence