Talk:Fibonacci number/Archive 1

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Archive This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page.

Contents

Early discussion

The main page says:

A logarithmic spiral can be approximated as follows: start at the origin of the cartesian coordinate system, move F(1) units to the right, move F(2) units up, move F(3) units to the left, move F(4) units down, move F(5) units to the right etc.

Is it not the case that a logarithmic spiral can be approximated just as well by moving an units at step n for any fixed a whatsoever? For example, I might move 1 unit right, 2 units up, 4 units left, 8 units down, 16 units right, etc., and get an approximation to a logarithmic spiral also? In which case, this is not a property of the Fibonacci numbers except by virtue of the fact that they happen to approximate a geometric sequence.

The main page continues:

This is similar to the construction mentioned in the golden mean article. Fibonacci number s often occur in nature when logarithmic spirals are built from discrete units, such as in sunflowers or in pine cones.

The pine cone thing (phyllotaxis) is related to the Fibonacci numbers, or, more precisely, to the golden ratio. The arrangements of pine cone bracts and other plant parts into Fibonacci spirals can be explained by a number of arguments; see for example pp. 408-412 of A New Kind of Science.

I am going to remove the logarithmic spiral remark from the main page unless someone corrects me soon.

Dominus 09:03, 6 Nov 2003 (UTC)

Well, if you really want to get pernickety about it, it does say a logarithmic spiral, not the or all...I can't rememer exactly, but these logarithmic spirals involve the golden ratio, and are connected to the Fib numbers.
Just some thoughts :) Dysprosia

I removed the logarithmic spiral discussion. I was sad to remove the illustration of the approximation to the logarithmic spiral, because it was very attractive. I hope it can find a more apt home somewhere else.

I will add back discussion of the appearance of Fibonacci numbers in sunflowers, pine cones, pineapples, daisies, and so forth. I have something about that written up already that I think could be adapted for inclusion in this article without much trouble. -- Dominus 14:24, 11 Mar 2004 (UTC)

Isn't the first fibonacci number 1 not 0? //Wellparp

Calculate F_2 by that definition in the article, and with F_0 = 1 in the way you are thinking of and you will see that the two sequences are basically the same. Yet, I will make clear of this in the article. Dysprosia 13:30, 10 Nov 2003 (UTC)
It's all a matter of convention. Having F(0)=0 makes several properties easier to write and more beautiful. For instance, if n divides m then F(n) divides F(m). --FvdP

I removed this paragraph and link (sorry Alex):

Recursive computation of very long Fibonacci numbers in C++ with using STL features can be seen at http://groups.google.com/groups?selm=bnni5p%2412i47o%241%40ID-79865.news.uni-berlin.de. The algorithm was used to calculate Fibonacci [5,000,000] (see http://alexvn.freeservers.com/s1/fibonacci.html)
  • the algorithm is uninteresting (it computes the numbers in sequence F1, F2, ...)
  • the Bignum implementation is strictly limited to the needs of the above algorithm, i.e. not much;

Perhaps it may be interesting to someone who wants to compute Fibonacci numbers, who knows, but then a 5-line Python_programming_language program would do the same computations (at much the same speed)...

a = 0 # or 0L on earlier Python versions
b = 1 # or 1L on earlier Python versions
while 1:
    print a
    a,b = b,a+b

--FvdP 20:28, 19 Nov 2003 (UTC)

Before we'll know it there will be a Fibonacci number program in 137 programming languages in this article. But this article is about math folks, not about you favorite programming language! So I've created a new Fibonacci number program article especially for you, and moved the two sample program from this article there. Lots of work to do guys, at least 135 programs to go!
—Herbee 02:41, 2004 Mar 11 (UTC)


Phyllotaxis

I put a discussion of the appearance of Fibonacci numbers in Phyllotaxis (the distribution of leaves on a plant stem) at Talk:Fibonacci number/Phyllotaxis. The style isn't quite right for an encyclopedia article, but I think large chunks of it could be extracted and added to the Fibonacci article, or maybe even into a separate discussion of the Fibonacci number in phyllotaxis. -- Dominus 15:33, 11 Mar 2004 (UTC)

Looks good. Abigail 15:44, Mar 11, 2004 (UTC)

INCORRECT INFORMATION ON THE PAGE

Please review the page, the Fibonacci sequence begins as: 1, 1, 2, 3, 5, 8, ... NOT 0, 1, 1, 2, ...!!! This needs to be corrected everywhere in the page...

Hmmm ... Fibonacci probably didn't have zero. But then, that means he would start with F(1) = F(2) = 1. So, if we start with F(0) = 0, wouldn't that be OK?

Actually Fibonacci introduced the concept of zero to European Mathematics. Before, except in India, this it was conceptually impossible to have something like cows or dollars and have none, zero of them. see Fibonacci —Charles Matthews 17:47, 30 Apr 2004 (UTC)
He did indeed have zero. The Fibonacci sequence is so-called because it was introduced by Fibonacci in his 1202 book Liber Abaci, which is about how to calculate with Hindu-Arabic numerals. In fact, it was this book that popularized the Hindu-Arabic numerals in Europe and introduced the zero to wide usage in Europe.
He started the sequence with F(1) = 1 and F(2)=2, contrary to your suggestion. Here's the relevant section of the book, taken from L. E. Sigler's recent translation:
A certain man had one pair of rabbits together in a certain enclosed place, and one wishes to know how many are created from the pair in one year when it is the nature of them in a single month to bear another pair, and in the second month those born to bear also.
Because the abovewritten pair in the first month bore, you will double it; there will be two pairs in one month. One of these, namely the first, bears in the second month, and thus there are in the second month 3 pairs; of these in on month two are pregnant, and in the third month 2 pairs of rabbits are born, and thus there are 5 pairs in the month...
You can indeed see in the margin how we operated, namely that we added the first number to the second, namely the 1 to the 2, and the second to the third, and the third to the fourth, and the fourth to the fifth, and thus one after another until we added the tenth to the eleventh, namely the 144 to the 233, and we had the abovewritten sum of rabbits, namely 377, and thus you can in order find it for an unending number of months.
There is also a tabulation in the book, but I don't remember whether it begins with "1, 1, 2, 3" or with "1, 2, 3"---I suspect the latter. Leonardo was almost certainly unaware of such identities as GCD(F(a), F(b)) = F(GCD(a, b)) for which the zero-indexing is important.
I hope this is helpful, or at least interesting. -- Dominus 15:49, 18 October 2005 (UTC)

F(0) = 0 Yes. [[1]]

The Fibonacci sequence can be continued towards negative infinity: F(−1) = 1, F(−2) = −1, F(−3) = 2, … So there is no natural "begin" to the sequence. If one wants the sequence to begin somewhere, the starting index is a matter of definition— it cannot be "correct" or "incorrect".
—Herbee 00:06, 2004 May 10 (UTC)

This discussion results from the anon's incorrect sentiment that Fibonacci's sequence has remained unexplored and should remain so, forever being presented as Fibonacci presented it 800 years ago. Not only is this bad math, science and history, it's just a plain bad idea. Tomertalk 22:03, 27 March 2006 (UTC)

Miles to km conversion paragraph removed

I removed the paragraph about miles to kilometers conversion, I thought it of dubious value and potentially misleading. A5 02:05, 1 Sep 2004 (UTC)

Proven irrational??

Has the sum 1 + 1 + 1/2 + 1/3 + 1/5 + 1/8 + 1/13 + 1/21 + 1/34... (which is equal to 3.35988566624...) known to be irrational as of 2004?? 66.245.105.195 02:11, 9 Oct 2004 (UTC)

Yes, according to http://mathworld.wolfram.com/ReciprocalFibonacciConstant.html, this was proved irrational in 1989. Jwwalker 07:02, 13 February 2006 (UTC)

H'm. "Fibonacci numbers can be obtained from the first term alone, by rounding to the nearest integer" yet "Computing Fibonacci numbers by computing powers of the golden mean is not very practical...since rounding errors will accrue and floating point numbers usually don't have enough precision." Needs elaboration, like, "if you need exact values beyond your machine's float precision, use the matrix method with bignums..." 'cause rounding errors aren't much of a problem if you don't even have to do the subtraction. Kwantus 04:27, 2004 Oct 10 (UTC)

Benford's law

The article has a section describing in details the conformance to Benford's law. But this conformance isn't surprising, (or, if it is surprising, the article doesn't say why) and since many collections of numbers conform to Benford's law, it's not clear what this is doing here. The detailed statistics about the distribution of initial digits in Fibonacci numbers does not seem to me to have any intrinsic interest, and smacks of original research. Am I missing something? -- Dominus 14:54, 10 Jan 2005 (UTC)

I was wondering that myself. I personally don't find Benford's law particularly relevant. And if it's relevant, couldn't the application of Benford's law to the Fibonacci sequence (or any other prominent integer sequence) be discussed in the article on Benford's law instead? --MarkSweep 15:55, 10 Jan 2005 (UTC)
My thoughts exactly. If there is no dissenting opinion in a day or two, I will remove the section. -- Dominus 16:09, 10 Jan 2005 (UTC)
I'd say if you want to remove the section and merge it with Benford's law, there's no loss of information and no need to wait. --MarkSweep 17:24, 10 Jan 2005 (UTC)
You can do what you want, but I'm going to wait. -- Dominus 18:44, 10 Jan 2005 (UTC)

F(n) or Fn?

I would prefer if this page used the latter format. Subscript is most common for integer sequences within Wikipedia and elsewhere, and certainly the most common for denoting the Fibonacci sequence. Fredrik | talk 29 June 2005 18:27 (UTC)

Both notations should be mentioned as both are common. Convenience and legibillity of formulae should dictate further use. --MarSch 11:12, 30 August 2005 (UTC)
I agree with MarSch. I find it strange that so many people seem to wish to enforce a nonexistent convention. -- Dominus 13:10, 31 August 2005 (UTC)

Tiling picture sloppy

The tiling picture of the Fibonacci series is rather sloppy. To do it right, each next box, should be adjacent to the previous two, to clearly show the construction. As it is now, the construction is not consistent per step. Box for nr 5 adjoins 1, 1 and 3. It should be on the upper side, along boxes 2 and 3. Then 8 can be two the right of 3 and 5 etc.

Or, alternatively, it should be consistently drawn on the opposite side of the previous two. That way the centers of the boxes follow show the Fibonacci spiral. −Woodstone 16:46, 24 September 2005 (UTC)

Relative Prime

The section below was moved from the main page, because it is incomprehensible as it stands. Can be re-inserted after thorough rework. −Woodstone 14:51, 18 October 2005 (UTC)

"All consecutive Fibonnaci numbers are relatively prime, this is to say that no two consecutive Fibonnaci numbers have a greatest common divisor greater than one. A proof is as follows:
Consider the following facts, if a and b have a common factor then it must also be a factor of a+b. If a and b have a common factor then it is also a factor of b-a if a and b have no common factor then neither do b and a+b. So let us say that the fibonnaci sequence starts with f(1)=a f(2)=b if a and b are relatively prime so are all consecutive numbers in the series. f(1)=1 and f(2)=1 have no common factor other than one therefore no neighbouring pairs in the Fibonnaci series have a common factor."

I see what the author is getting at, although I agree with you that it's a mess. I'll try to write something briefer and clearer and put it into the article. -- Dominus 15:31, 18 October 2005 (UTC)

Done. I hope you like this version better than what was there before. Thanks for pointing this out. -- Dominus 15:41, 18 October 2005 (UTC)

the sequence in music

1 Black 1 then 2 white are 3 all I see 5 in my infancy. 8 red and yellow then came to be, 5 reaching out to me. 3 lets me see. 13 As below, so above and beyond, I imagine 8 drawn beyond the lines of reason. 5 Push the envelope. 3 Watch it bend.

-Lateralus -TOOL

he sings the sylabols.

http://toolshed.down.net/lyrics/latmaster.html cite

Christa rim 06:30, 22 October 2005 (UTC)

Nice. It should also be noted that BT used the Fibonacci sequence in his song--wait for it-- "Fibonacci Sequence". Note that this was not referenced on the main page at all, which is immensly stupid. //user:DJRaveN4x 12:11 EST, 28 December 2005

Bee generations

The statement about the bee generations is incorrect. It assumes that no parentage can be shared. However it is quite likely that two grandmothers of a bee are sisters, so no new parents are added going backwards. −Woodstone 12:59, 24 October 2005 (UTC)

I edited that so that it doesn't state anything as 'fact'. −David Thomas 27 Aug 2006.

I'm skeptical of this:

Fibonacci is also stated as having described the sequence "encoded in the ancestry of a male bee."

Was it known in 1900, let alone 1250, that an unfertilized bee egg is male? —Tamfang 07:09, 11 October 2006 (UTC)

He didn't. Thanks for pointing this out. -- Dominus 08:38, 11 October 2006 (UTC)

Roulette

The roulette page mentions that the fibonacci numbers can be used as a method of gambling. No chance of a mention on this page? (And an extrapolation - including why it is flawed?) --JohnO You found the secret writing! 08:24, 21 February 2006 (UTC)

spirals

Why is the conch picture here? Is the ratio in size between successive turns known to be τ? If not, let's avoid encouraging the popular misconception that all logarithmic spirals have something to do with Fibonacci numbers. —Tamfang 19:30, 21 February 2006 (UTC)

Pseudocode Computer Programs

The current examples of pseudocode are sloppy.

They both will give invalid results for negative input.

That's right, but a check for negative input is not common in CS textbooks. Just assume the input type is natural/unsigned. Qwertyus 14:12, 18 March 2006 (UTC)

The second example shows a line

a,b := a+b,a

I would say that should be

a,b := b,a+b

And it is not elegant either because it calculates one number too many (b is the next one). −Woodstone 21:51, 17 March 2006 (UTC)

It now follows SICP. I'll add a reference. Qwertyus 14:12, 18 March 2006 (UTC)
I see you've changed it; but you've put the inelegance that you complained about back in :) Qwertyus 13:05, 10 April 2006 (UTC)
I did not put it back, because it was still there. There is no difference in cycles though the loop. In the previous form "a" was the excessively computed value, better hidden, but still present. But I'm glad you seem to agree it's not elegant. How about:
var a,b = 0,1
if n=0 return a
while n>1 
  a,b = a,a+b
  n=n-1
return b
Does that strike you as more elegant? −Woodstone 19:00, 10 April 2006 (UTC)
No, because it's not as short as it could be. Elegance lies not in saving a small constant number of processor cycles. Qwertyus 19:07, 10 April 2006 (UTC)
I fully agree. Elegance is much more than that. That's why I didn't put it in. However I still maintain that the show program is not elegant because it computes non-required data. My improvement was just making it slightly more logical, the values a,b being successive results in the right sequence. −Woodstone 22:03, 10 April 2006 (UTC)

I've found a more elegant version at Data Structures and Algorithms with Object-Oriented Design Patterns in C++ and inserted a pseudocode version. I think this is as elegant as it gets. It uses the idiosyncratic loop construct repeat n times because otherwise we'd have to use a counting var, as the referenced book/site does. Using while n≥0, as I did at first, makes the counter stop at -1, thus requiring it to be in signed representation. Qwertyus 23:16, 10 April 2006 (UTC)

The stated version was incorrect because it shifts the result by 1 index number. Now corrected, but it brings back the inelegance of more work then needed (now shifted from the end to the beginning). But surely the "repeat" idea makes it more elegant than the while and decrement. However the trickery for start-up, makes it less obviously correct, so I'm not convinced it's more elegant altogether. I'm inclined to go back to
function fib(n)
var a,b = 0,1
repeat n times
  a,b = a,a+b
return a

−Woodstone 10:22, 11 April 2006 (UTC)

Another question: what is the complexity of algorithms derived from the closed form? When using exponentiation by squaring, I'd say I would be O(logn), and thus not asymptotically better than the algorithm derived from the matrix form. Qwertyus 20:33, 12 April 2006 (UTC)

It is possible to device an algorithm for the fibonacci numbers running in linear time if using dynamic programming. As for simple elegant functions, this is a piece of real sml code:

fun fib(0) = 0

  |fib(1) = 1
  |fib(n) = fib(n - 2) + fib(n - 1);

That's as beautiful as real code can be me thinks :)

Can you explain the math?

I don't understand the last step of the derivation in the relaytion to goldn ratio section. Can someone explain it to me?

How does

1+\frac{1}{\displaystyle\lim_{n\to\infty}\frac{F(n)}{F(n-1)}}

equal

1+\frac{1}{x}

when

x={\lim_{n\to\infty}\frac{F(n+1)}{F(n)}}

?

Thank you. Loom91 12:26, 28 March 2006 (UTC)


It amounts to a change of variable: \lim_{n\to\infty}\frac{F(n)}{F(n-1)} is the same as \lim_{m\to\infty}\frac{F(m+1)}{F(m)} where m = n − 1.

If that explanation is insufficient you can expand the limit statement in terms of the δ and ε definition of a limit. Roger Hui 15:36, 28 March 2006 (UTC)

so you are saying (F(n))2 = F(n − 1)F(n + 1)? Doesn't this require an explanation? Loom91 10:33, 29 March 2006 (UTC)

- No, that is not what I am saying. The two limits are both on the ratio of consecutive elements of a sequence, whether it's the Fibonacci sequence or some other sequence. There is no difference between \lim_{n\to\infty}a_{n} and \lim_{n\to\infty}a_{n+1} and \lim_{m\to\infty}a_m . Roger Hui 16:19, 29 March 2006 (UTC)

Thanks for the explanation, now I get it. But I think someone should make this clearer in the article. Loom91 09:28, 30 March 2006 (UTC)

indexing

68.155.181.96 says "a set cannot start with an index of zero". Must be a Fortran programmer! —Tamfang 06:58, 31 March 2006 (UTC)

Fibonacci primes

I have removed the following contrbution from an anonymous editor from the main article, as I think it is very unclear, and the final sentence sounds like original research:

However, since Fib(p) can share no common divisors exceeding one, with F(n<p).
Using the identity:
http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibmaths.html#fibprimecarm
GCD(Fn, Fm) = FGCD(m,n)
Proof:
Since if we are saying that m is a prime p, and n is less than p, then the
GCD(Fp, Fn ) = F(GCD(p,n)) = F(1) = 1
Which means that all F(p), will be a product of all new prime factor/s, also called characteristic factor/s (Charmichael's theorem), or primitive factor/s. It is reasonable so assume that, F(p) is more likely to be a single prime factor (itself) because the field of potential factors at random has been reduced.

If anyone can sort this tangle out, please feel free to copyedit it and re-insert it. Gandalf61 09:47, 4 April 2006 (UTC)

The section was re-inserted without change, so I have removed it again. My specific objections to this section, as written, are:
  1. It is ungrammatical. For example, the first sentence has no main clause.
  2. It uses inconsistent notation - Fib(n), F(n) and Fn all seem to be used for the Fibonacci sequence.
  3. It does not give adequate references - what is "Charmichael's theorem", for example ?
  4. Its final sentence sounds as if it is an unverified claim, or at best original research.
If all these errors can be fixed, then of course the section could be re-inserted. Gandalf61 13:18, 5 April 2006 (UTC)



What errors? It's rude to have removed this just because you don't understand. This is well known Gandalf61!


A. All prime Fibonacci numbers have a prime index, "p". The converse is not true, because not all Fibonacci with a prime index are prime.

B. Fibonacci numbers with a prime index "p" do not share any common divisors exceeding one, with preceeding Fibonacci numbers, due to the well known identity:

GCD(F(n), F(m)) = F(GCD((m),(n)))

The greatest common divisor between any two Fibonacci numbers, is equal to the greatest common divisor between their index values.

Simple Proof: If m is a prime p from the identity above, and n is less than p, then it is clear that F(p), cannot share any common divisors with preceeding Fibonacci numbers.

GCD(F(p), F(n)) = F(GCD((p),(n))) = F(1) = 1


C. Charmichaels' theorem states, that for every Fibonacci number there will always be at least one unique prime factor, that has not been a factor of a preceeding Fibonacci number.

By combining B&C, a Fibonacci number with a prime index, will be a product of all new prime factor/s called characteristic factor/s(Charmichael's theorem). There is nothing to rule out one single prime factor, as stated by C.


  1. The sentences now have a main clause with a subject, and a verb.
  2. Notation is fixed, and could be by anyone, rather than deleted.
  3. Charmichael's theorem is stated, and there had been a link, but I guess you could not follow it as a reference.
  4. The final sentence just words it correctly, if you call that original.
Well, I don't think the original contributor means this Carmichael's theorem, which agrees with the definition of Carmichael's Theorem in this mathworld entry. Maybe he means a different Carmichael's Theorem ? Maybe this is where he gets the statement GCD(F(n), F(m)) = F(GCD((m),(n))) from ?? This is exactly why this tangle need more clarity. Gandalf61 10:01, 10 April 2006 (UTC)
Fixed this. Carmichael's theorem now describes the theorem that the original contributor was refering to (it was previously a redierct to Carmichael function). Gandalf61 10:44, 11 April 2006 (UTC)

There is a spelling error. That identity is not a made up statement, but is well known. You can find a link and post it, but dont delete this again. http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibmaths.html#primefactor Oh you and your toil and tangle, you british....errrgh As you can tell I don't like you personally now. A am Shane Findley, and this was published in the Fibonacci quarterly. You can also do a google search and find Shane On the Mizar Encoding of the Fibonacci numbers.

I have now included an external link, to the greatest common divisor identity.

Divineprime, I suggest we continue this discussion on your talk page. Gandalf61 08:39, 11 April 2006 (UTC)


Common factors

Any two consecutive Fibonacci numbers are relatively prime. Suppose that Fn and Fn+1 have a common factor g. Then Fn+1 − Fn = Fn−1 similarly. Thus by induction, all preceding Fibonacci number must be a multiple of g. But F1 = 1, so g = 1.

Hello what's this? What does the induction mean? None of this is clear at all, I think I'll delete it then. This needs to be fixed or reworded. —Preceding unsigned comment added by 4.233.134.240 (talk • contribs) 12:29, 11 April 2006

What is unclear about it? It seems to me a perfectly correct proof, by descending induction, albeit exceedingly simple. −Woodstone 21:07, 11 April 2006 (UTC)
Perhaps noname doesn't know what mathematical induction is. —Tamfang 23:13, 12 April 2006 (UTC)
Perhaps Gandolf61 does not know what mathematical induction is.
Noname —Preceding unsigned comment added by 4.233.149.161 (talk • contribs) 13 apr 2006 01:40
Please people, this is no way of arguing. Here's my attempt at clarifying the passage by making it more explicit:
Any two consecutive Fibonacci numbers are relatively prime. Let g be a common factor of Fn and Fn + 1, i.e. there exist positive integers a and b such that Fn = ag and Fn + 1 = bg. Then
Fn − 1 = Fn + 1 − Fn = bg − ag = (b − a)g,
i.e. Fn − 1 is a multiple of g. By descending induction all lower Fibonacci numbers must be multiples of g. But F0 = 0 and F1 = 1, and 0 and 1 are relatively prime, so g = 1.
But maybe this is too long and explicit? Qwertyus 12:22, 13 April 2006 (UTC)
Why bring zero into it? g must be a factor of 1, that's sufficient. Speaking of the (relative) primeness of zero makes me nervous. —Tamfang 17:04, 14 April 2006 (UTC)
"g must be a factor of 1" (equiv. g=1) is what I'm trying to prove there.
What makes you nervous about it? gcd(0,1) = 1 because 1 is the greatest n such that 1|0 and 1|1. Also, try calculating it using Euclid's algorithm. Qwertyus 18:31, 14 April 2006 (UTC)
I also think it is perfectly clear as written, and that the proposed replacement is considerably less so. Making it "more explicit", and so longer and more convoluted and turgid, does not make it clearer, in my opinion. -- Dominus 02:30, 15 April 2006 (UTC)
Personally, I don't think it's "convoluted and turgid", it clearly and precisely states the steps of the proof. Qwertyus 02:34, 15 April 2006 (UTC)
I think the version above is not very precise. The statement "But F0 = 0 and F1 = 1, and 0 and 1 are relatively prime, so g = 1." is not the right way to close the proof. As Tamfang stated above, the relative primeness of 0 and 1 is not directly relevant. The proof should continue: so g is a factor of 1 and consequently g=1 (ignoring the sign). So the only common factor between consecutive Fibonacci numbers is 1 and they are relative prime. Also, talking about "lower" Fibonacci numbers is unnecessarily assuming they are ascending (true, but why use it?). Earlier versions used the more accurate "preceding" (in the sequence). −Woodstone 10:32, 15 April 2006 (UTC)

Broccoli?

"Romanesco Broccoli clearly illustrating the fractal repeating spiral pattern found within nature"
"Romanesco Broccoli clearly illustrating the fractal repeating spiral pattern found within nature"

I've removed the image to the right from the article, as it makes no claim to have anything to do with Fibonacci numbers. In fact, the whole "Fibonacci numbers in nature" section needs citations. Melchoir 22:31, 23 April 2006 (UTC)

Ratio proof

The current proof that F(n+1)/F(n) \to \varphi is not rigorous, because it assumes that this ratio x exists, and under that assumption proves that x must equal \varphi. Therefore, I changed the proof to one that is rigorous (see my edit "Put rigurous proof of ratio claim"). But Woodstone reverted it, asking "Why does d(n)/F(n) to 0?". Well, it tends to zero because dn tends to zero and F(n) tends to infinity. Pretty simple. Gabn1 06:49, 18 May 2006 (UTC)

I agree that the earlier proof is not rigorous, but neither is the newer one. You would have to prove that F(n) tends to infinity. We all know that, but the proof is not shown. Note that the convergence to φ is independent of the starting values (even negative ones). It would be nice to keep that in. −Woodstone 08:23, 18 May 2006 (UTC)
The fact that F(n)\to\infty follows easily from the definition of the Fibonacci sequence. It's normal in math writing to omit easy steps that the reader can fill in for himself. So that's not a problem with the newer proof. On the other hand, the current proof has a real hole: It starts by assuming that the limit x exists. Therefore, I recommend switching to the newer proof. Gabn1 10:14, 18 May 2006 (UTC)
Nevertheless I would like to see your proof (for any starting values) that the series tends to infinity. At least you might want to remark that it can also converge to minus infinity. −Woodstone 14:25, 18 May 2006 (UTC)
I agree that the new proof is better. I think Woodstone's objection is silly. The old proof assumes that the ratio of succeessive Fibonacci numbers actually converges to a limit. This is not only a tricky matter, but it is very close to what is being proved. In contrast, the fact that Fn increases without bound is both trivial and obvious. -- Dominus 14:32, 18 May 2006 (UTC)
Thanks, Dominus. Woodstone asks for a proof that F(n) tends to infinity. Well, first prove by induction that F(n) is positive for all n>=1, and then use that to prove that F(n)>F(n-1) for all n>=3. Regarding alternative starting values: If you change the starting values then it's not the Fibonacci sequence anymore; it's a different sequence (Lucas or whatever else). Gabn1 15:18, 18 May 2006 (UTC)
I considered giving a proof, but I think that really misses the point of why the objection is silly. The objction would not be so silly if the fact for which the proof was demanded weren't so very trivial. -- Dominus 16:52, 18 May 2006 (UTC)
I have clarified and polished the proof a little. Indeed it is almost rigorous. However I still don't like that it is not made clear that the limit is independent of the starting values. It is easy to define starting values that lead to a zero value at an arbitray position in the series. Of course from that point on it will never be zero again (unless all zeroes). So the proof still holds, but it is not easy to formulate that concisely in the article. −Woodstone 20:50, 19 May 2006 (UTC)
I really think the proof with the limit of consecutive Fibonacci numbers set = to x was fine, and much more clear, and certaintly more concise. So long as you prove that phi is equal to x and substitute that value in, you're not assuming the ratio exists. I think reverting, or possibly having both would make this article more accessible to the general public. snwbrdr642
No, that "proof" was not fine, because in it you first assume that the limit ratio x exists. You show that if x exists, then it must equal phi. But who says that the limit ratio exists in the first place? Limits don't always exist. That "proof" might have been more simple, but it is more a hand-waving argument than a real proof. Gabn1 11:28, 6 June 2006 (UTC)

A much simpler proof can be given by using the explicit expression shown (and proven) just below the proof. First remark that \left |{\frac{1-\varphi}{\varphi}}\right | < 1. Then set

\frac{F(n+1)}{F(n)}
=\frac{\varphi^{n+1}-(1-\varphi)^{n+1}}{\varphi^n-(1-\varphi)^n}
=\frac{\varphi-(1-\varphi)(\frac{1-\varphi}{\varphi})^n}{1-(\frac{1-\varphi}{\varphi})^n} \to \varphi as n tends to infinity

−Woodstone 21:51, 12 June 2006 (UTC)

Elements of "Fibonacci numbers in nature" needing verification

As of now, this is the progress in verifying the claims made in "Fibonacci numbers in nature" that various things are connected to the Fibonacci numbers, with smoking guns:

  • General notability in biology done
    • Douady p.255: "The predominance of Fibonacci order in botany..."
  • Flowers done
    • Douady p.268: "...the spectacular ordering of sunflower heads... Remarkably high Fibonacci numbers can be reached..."
  • Trees done
    • Douady p.272: "...even a normal vertical pine tree branch, with a Fibonacci structure..."
  • Bushes
  • Pine cones done
  • Raspberry seeds
  • Horns
  • Shells
  • Lindenmayer grammars done
    • Prusinkiewicz p.??: "???"
  • Fractal Fuchsian groups
  • Kleinian groups
  • Belousov-Zhabotinsky reaction
  • Gene regulatory networks
  • Sphere packing

Rather than hit every of those with a {{citeneeded}} tag, it's easier to apply {{not verified}} to the section and list the items here. Please update the above as necessary, crossing off items as they are either verified or removed, or adding items as the article grows. Feel free to remove the tag when every item is crossed out. Melchoir 11:12, 22 May 2006 (UTC)

...Is there any work being done on this, or shall I delete the unverified bits? Melchoir 00:29, 20 August 2006 (UTC)

I'll be surprised if someone can show Fibonacci relevance in sphere packing. —Tamfang 02:29, 21 August 2006 (UTC)
Eh, I'll just delete the problems. For the record, the unverified claims can still be found here. Melchoir 02:54, 21 August 2006 (UTC)
I removed "segments of snail shells"; I'm guessing someone was confused by the Nautilus, and by the annoyingly common misconception that the golden ratio appears in every exponential spiral. —Tamfang 07:16, 23 October 2006 (UTC)
Hmm, are you saying that some exponential spirals have the golden ratio, and that some don't? Or that None of them do? McKay 19:36, 23 October 2006 (UTC)
The ratio between successive elements of an exponential progression (of which a spiral shell is an example) can be τ, φ, π, e, 2, √2, 3, √3, or any other number >1. —Tamfang 07:18, 24 October 2006 (UTC)

Original research

Sorry, I thought the {{OR}} tag was self-explanatory in this case, but I'll go through it in detail. It refers to the following paragraph, specifically:

  • "practical use of these algorithms is limited": Floating-point operations are costly. Why wouldn't programmers prefer integer algorithms for small n also?
  • Which practical uses need exact values for large n? How large, exactly, do they usually need?
  • "must be represented by a floating point number"? Isn't it straightforward to do exact computations in \mathbb{Q}[\sqrt5]? Aren't many computer algebra systems capable of doing it out of the box?
  • "n = 75": Did we figure that out, or is there a source that actually calculated it?
  • "32 bit integers will run into an overflow situation before that." How much before? Is there an application where you'd need such large values but you wouldn't already be using arbitrary-precision arithmetic?
  • Okay, if they abandon the closed form, what algorithm is commonly used by programmers who need large values? Is it the one listed above?

Melchoir 17:47, 23 August 2006 (UTC)

I've heard that operations on floating-point coprocessors can be cheaper than integer arithmetic. Non c'è più religione!
My own bit of OR: F(46) = 0x11e8d0a40 > 2^32. —Tamfang 05:14, 25 August 2006 (UTC)
That would sure explain a couple of things! I don't suppose you can divine where you might have seen it? Melchoir 05:29, 25 August 2006 (UTC)
What, the bit about floating point? It was a conversation with Dad, I think, several years ago. And it occurs to me that the advantage is likely to flip back and forth as new chips appear. —Tamfang 06:53, 25 August 2006 (UTC)
Ah. Oh well. Melchoir 07:17, 25 August 2006 (UTC)

Meh. Melchoir 19:01, 1 September 2006 (UTC)

After the discussion on computation has been deleted, the programs might go as well. They are absolutely trivial. Anyone with any programming knowledge can write them down in a few minutes. They do not add any real information to the article. −Woodstone 21:17, 1 September 2006 (UTC)

Really? I deleted the discussion that was critical of the algorithms' practical use, but the discussion of their use as educational tools was still there. Given all the space the article already spends on equation slinging, I thought the computation section was a nice break.
Maybe there's a compromise to be had: we could cut the first two pseducode sections, and keep the prose and the third algorithm? Melchoir 22:01, 1 September 2006 (UTC)
Perhaps we should consider transferring the computation material to a separate entry. While they certainly make this already-long main page even more tedious, they may also add some value as educational tools, esp. as examples for those without much programming knowledge. JXM 23:54, 1 September 2006 (UTC)

"Erroneously"

Bakasuprman, please stop trying to describe the naming of the Fibonacci numbers as "erroneous". It is enough to note that Fibonacci did not first discover them and to describe the first known treatments. Any judgement that this information alters what the numbers should be called is POV. Melchoir 02:57, 11 September 2006 (UTC)

User:Melchoir, Pls be civil. The word "erroneous" is really necessary in the opening sentence itself.Bharatveer 05:28, 11 September 2006 (UTC)Bharatveer 05:30, 11 September 2006 (UTC)

Why? Melchoir 05:31, 11 September 2006 (UTC)
It is so because till very recently it was thought that he was the mathematician who originally contributed to the discovery of this . As it is known now ; the so called fibonacci numbers was studied long before fibonacci. So the information regarding this ,should be mentioned in this article.Bharatveer 05:39, 11 September 2006 (UTC)
I don't believe that it is true that anyone ever thought that Fibonacci was the mathematician who originally contributed to the discovery of the Fibonacci numbers. Fibonacci's contribution is limited to a single paragraph in Liber Abaci, and he does not discuss any of the special properties of the Fibonaaci sequence. The sequence is not named for him because he studied it, or because he proved any theorems about it, but simply because it appears in his book, and attracted attention thereafter. So the attribution to Fibonacci, while perhaps undeserved, is not "erroneous". -- Dominus 13:36, 11 September 2006 (UTC)
Yes, any objective, verified facts should be covered in full detail. But judgements and opinions must be left out. Melchoir 05:45, 11 September 2006 (UTC)
The word "Erroneous" is neither an opinion nor a judgement .Bharatveer 06:54, 11 September 2006 (UTC)
That's your opinion. —Tamfang 07:31, 12 September 2006 (UTC)

Names are arbitrary. Many things and concepts are named for someone who did not discover them; yet the skies stubbornly refuse to fall. When we learn that something has been misnamed – even when the name is a bad description, rather than a mere misattribution – usually we go on using the erroneous name, with a footnote.
Leonardo here, even if he stole the idea from some superior Indian intellect, deserves some credit for bringing it to the world's attention. It appears that even India might not be aware of it if not for him, given that apparently the previous discoverer was only recently rediscovered! Or have Indian mathematicians known the sequence all along, and called it something else? —Tamfang 07:31, 12 September 2006 (UTC)

I really would like an answer to that. —Tamfang 06:00, 19 September 2006 (UTC)
Pls read the article once more and also the references.-Bharatveer 07:35, 19 September 2006 (UTC)
A) if you make a request for someone to read an article you should probably reference which article you want to read. B) The reference used to support these claims (Math Ed.) is a journal, and journals are primary sources, and WP:CITE discourages it's use. There isn't really any reason to read the questionable sources. McKay 16:15, 19 September 2006 (UTC)
User:McKay,I guess User Tamfang is fully capable of asking me those question if he had any.This also means he is not foolish to ask these type of silly questions.
FYI , Pls read What is a Primary Source.Regarding allegations of sources being "questionable" i have already explained with additional references in the subsection given below.-Bharatveer 16:42, 19 September 2006 (UTC)
I retract my previous statement. I think that what I was reading on WP:CITE and WP:RS was a little ambiguous initially, but on a second more-thourough read. My assertion is incorrect. There still remains the "reputability" of the publications in question. Below, you did make a request for someone to verify the reputability of these sources, but such a verfication has not been made. His question still remains, and I think needs to be answered. Have Indian mathematicians known the sequence all along, or is it something new to them?—The preceding unsigned comment was added by Mckaysalisbury (talk • contribs) .
Read this -" The "Fibonacci" numbers first appear, under the name maatraameru (mountain of cadence), in the work of the Sanskrit grammarian Pingala (Chhandah-shāstra, the Art of Prosody, 450 or 200 BC). Prosody was important in ancient Indian ritual because of an emphasis on the purity of utterance. The Indian mathematician Virahanka (6th century AD) showed how the Fibonacci sequence arose in the analysis of metres with long and short syllables. Subsequently, the Jain philosopher Hemachandra (c.1150) composed a well known text on these. A commentary on Virahanka by Gopala in the 12th c. also revisits the problem in some detail. "
Another reference for Indian Mathematician's work on so called Fibonacci series can be seen in this paper .
Subhash Kak -Bharatveer 04:04, 20 September 2006 (UTC)
I don't think there's even an issue here as to whether or not Indian mathematicians considered the Fibonacci numbers before Leonard of Pisa. Evidence on that question is irrelevant. Do you have a reference that specifically comes out and says that the naming of the Fibonacci numbers is in error? Melchoir 05:03, 20 September 2006 (UTC)
As per Users Tam,Mckay etc, They have issues regarding that question also.All the references given by me clearly shows that so called fibonacci was in fact a misattribution (due to Eurocentrism, and the series should have been called Gopala-HemaChandra Series.-Bharatveer 06:10, 20 September 2006 (UTC)
The Parmanand Singh paper does not appear to me to say anything about "Fibonacci" being a "misattribution", and does not seem to assert that it "should have been called Gopala-HemaChandra Series". Where do you think it says this? -- Dominus 21:55, 22 September 2006 (UTC)
No, reading this paper, the phrase "Gopala-Hemachandra numbers" does appear. This phrase is not meant to be the fibonacci numbers, but any mathematical sequence of the form F(n) = F(n-1) + F(n-2) (though his statement of this isn't nearly as mathematically clean. Of which the fibonacci sequence is one. So the Fibonacci numbers as a set, are a member of the set of Gopala-Hemachanrda numbers. So are the Lucas Numbers. But he doesn't make the proposal, he claims that others have made this proposal. Also, this paper is probably considered a primary source arxiv is just a library of papers. Also, the paper isn't clear as to where he's getting his information for which parts of his paper. So to verify individual facts, the entire bibliography would have to be retrieved (which goes against all stylebooks that I've seen). McKay 14:01, 20 September 2006 (UTC)
User: McKay , Pls read that paper once more so that you dont have to retract your statements again.-Bharatveer 14:24, 20 September 2006 (UTC)
Things I don't like this time around:
  • "Adolf Zeising (1854) appears to have been the first to propose the romantic idea that in the golden ratio “is contained the fundamental principle of all formation striving to beauty and totality in the realm of nature and the field of the pictorial arts.”" The Golden ratio has been proposed as a fundemental idea of aesthetics, for 2400 years in the western world.
  • "Mount Meru" vs "Pascal's Triangle" this is the same discussion we're having on fibonacci, but this one is actually claimed in the paper. He gives reason for not calling it Pascal's triangle, claiming that Mount Meru is in English use, but Google("pascal's triangle") generates 149,000 hits. Google("Mount Meru") generates 183,000 hits, but none on the first page are of the math concept. There is a Mount Mero in tanzania Google("mount meru" -tanzania) removes 1/2 the hits Google("mount meru" math) returns a piddly 572 results.
  • see what I mention again about Gopala-Hemachandra numbers below He isn't very good at pattern problems.
What the paper does state:
  • Hemachandra discovered fibonacci numbers before Leonardo of Pisa.
  • "It has been suggested that the name Gopala-Hemachandra numbers be used for the general sequence: a, b, a+b, a+2b, 2a+3b, 3a+5b, ...". This is the same thing that I said above. "mathematical sequence of the form F(n) = F(n-1) + F(n-2)" I criticised his form for not being very clearly this recursive definition. The pattern in his symbology is not clear, and is in fact self referencing:
    the a terms are 1, 0, 1, 1, 2, 3, 5, 8, ..., F(n+2)
    the b terms are 0, 1, 1, 2, 3, 5, 8, ..., F(n+1)
    so finding the pattern in his "a, b, a+b, a+2b, 2a+3b, 3a+5b, ..." is non-trivial, because it requires knowledge of the pattern.
    a=b=1 is fibonacci, but the lucas numbers are here too. QED: the set of Fibonacci Numbers is an element of the Gopala-Hemachandra number set.
What I do retract
  • that all of his statements don't tell us where he got the material from. Only most of his statements are likewise unreferenced.
Any questions? McKay 16:09, 20 September 2006 (UTC)
I am very much amused to see your arguments.Especially the bit about " Subhash Kak's recursive definition not being clear ".Yes it is more amusing than the previous "primary sources" bit.So no more questions this time , but still,I would like to see more such amusements.-Bharatveer 04:06, 21 September 2006 (UTC)
Here's a series for you, you tell me what's next in the pattern.
  • a, a + b, 2a + 2b, 3a + 4b, 5a + 7b, 8b + 12b
A recursive definition is much more clear.
Also, you should aim to be more civil. Especially after calling someone else uncivil. McKay 06:10, 21 September 2006 (UTC)
Pls do more homework before asking me to solve problems for you.Further this is going off-topic.-Bharatveer 06:40, 21 September 2006 (UTC)
No, I think this is still very much on topic. I'm talking about the validity of the paper. Particularly it's notability. Your "original" claim was that the fibonacci sequence was named erroneously, "the series should have been called Gopala-HemaChandra Series" with this paper as a reference. I then made two kinds of comments:
  1. that the paper didn't actually claim that
  2. that the paper wasn't written all that clearly, and that it wasn't written in an established academic fashion. This isn't directly related to the topic, but it still relates to the notability of the work.
You replied thinking that I couldn't be correct in calling you in your error, so you told me to reread. Making an ad hominem reference to support your claim.
So, I did reread it, and responded this time with quotes from the paper. I criticized the paper, but also firmly established the fact that I was, in my original statement, correct.
This caused you to laugh at me, making an argumentum ad verecundiam, to someone that I had never even heard of.
To back up my statement, I presented a series in a similar form to the way the venerable Subhash Kak presented. I also commented on your apparent incivility.
To this, you replied that I should "do [my] homework", and were apparently displeased with the problem that I constructed for you. And claimed that we were going off topic.
So, this is my reply. I think that it is still on topic, because I am still talking about the paper, and criticising it's notability, though you're right, I've somewhat been making an ad hominem attack on Subhash Kak, rather than attacking his arguments, but it was you who laughed at my comment, so I feel somewhat justified in this regard. Also, I don't think that it really is an ad hominem attack. Subhash Kuk is noted for his out of India theories (on his Wikipedia article at least), and I think that his reason for publishing a paper saying these things may be stretches because he has an alterior motive in claiming that India is the source of all things good. He calls it Mount Maru, and wants to name the group of fibonacci/lucas numbers indian as well, supporting his out of india theories.
So, back to topic. There still remains an unanswered question -- You claim that the paper says that the fiboancci sequence should be called Gopala-Hemachandra sequence, I questioned that fact, you told me to reread, so I replied quoting the paper, and you haven't replied to that. McKay 14:16, 21 September 2006 (UTC)
Oh, one more thing. The paper, on several occasions, refers to the Fibonacci numbers, as a series (mathematics), when in fact, it is a sequence (mathematics). I understand, it's a mistake I make (I've made it twice here on this page, but mostly because I was coping him, and I was referencing his work), but I would expect someone who claims his line of work as mathematics, to not make this kind of mistake in a scholarly paper. McKay 04:35, 25 September 2006 (UTC)
Mckay, You have been making a lot of retractions and mistakes here.It is always beneficial to correct one's self before going for critical evaluation of scholars-Bharatveer 05:22, 25 September 2006 (UTC)
I have made exactly 3 retractions.
  1. A comment on primary vs secondary sources. Not relevant to the discussion of the content of the article.
  2. A clarification on a hyperbole I made earlier. The substantitive portion of my original comment remains.
  3. After coping text from Subhash Kak's paper. I used his terminology. After reflection, I commented his terminology was wrong, and I mistakenly thought he was an authority on the subject.
Also, any scientist should be perfectly content with a critical evaluation of his work. It's a part of the scientific process. I shouldn't have to be perfect before criticising an academic paper on any subject. You on the other hand haven't replied to any real criticisms I've (and others have) made of your work. Questions have been asked that remain unanswered. You laugh at something you don't really know anything about. You make an appeal to authority when the authority is the one in question, and isn't deserving of such an appeal. You, by your own admission shouldn't be criticizing me. I'm saying that criticizm is important to the review process. McKay 14:17, 25 September 2006 (UTC)
Mckay,I have seen many lame excuses and questions in this discussion.Firstly it was about the 'reputation' of the sources.When the references from three different journals were given; the next question was to give the page and line of that paper.
I dont really want to argue anymore regarding this; not atleast with some one who gets confused with primary and secondary sources.Subhash kak does not need your certificate , and the statement that you mistakenly thought he was an authority on the subject is just another example of intellectual dishonesty.-Bharatveer 06:51, 26 September 2006 (UTC)Bharatveer 06:54, 26 September 2006 (UTC)
I wasn't questioning the "reputation" of the source. I only pointed out that if we wanted to provide a reference for the well-known and widely-acknowledged fact that Indian mathematicians studied the Fibonacci sequence before Fibonacci, it should be possible to do so with a reference that is better known than the (extremely obscure) Siwan one.
I think a much more serious issue with this discussion is that the papers you've cited don't appear to say what you claimed that they did. -- Dominus 09:40, 26 September 2006 (UTC)

Journal reference

User Bharatveer recently added a reference that he claims supports the assertion that the Fibonacci numbers are "erroneously" named. The reference is:

Parmanand Singh. Acharya Hemachandra and the ( so called ) Fibonacci Numbers. Math . Ed. Siwan , 20(1):28-30,1986.ISSN 0047-6269

It took me a little trouble to find out the full name of this obscure journal, so I thought I would post it here to help out others who might want to follow up the reference. The full title is "The Mathematics Education". "Siwan" is appended to distinguish it from similarly-titled journals, because it is published in Siwan.

It seems to me that there are two issues here. One is whether Fibonacci was the first person known to have considered the Fibonacci numbers. I think that everyone can agree that it is well-documented that he was not, and that there is consensus in the scholarly community that he was not. So we should be able to come up with a reference for this fact that is in a more accessible, reputable, and prominent source.

The other issue is whether the attribution of the Fibonacci numbers to Fibonacci is "erroneous". I believe that to the extent that this evaluation is true, it is an editorial evaluation, which should not appear in the encyclopedia.

If there were a large number of scholars who considered the attribution to be "erroneous", it would be worth while for Wikipedia to report that. But such a report would require more than one reference from such an obscure journal.

-- Dominus 13:53, 11 September 2006 (UTC)

  • Acarya Hemcandra and the (so-called) Fibonacci Numbers, Int. J. of Mathematical Education, Vol. 20 (1986), pp. 20 – 30.
  • Parmanand Singh,"The So-called Fibonacci numbers in ancient and medieval India. Historia Mathematica v12 n3, 229-244,1985.

User: Dominus , Pls verify the "reputation" of these references . --Bharatveer 14:17, 11 September 2006 (UTC)

Everyone agrees that Fibonacci was not the first person to study, or even to write about, Fibonacci numbers. But many concepts in mathematics are not named after their discoverers. Pascal did not discover Pascal's trinagle. Farey did not discover the properties of Farey sequences. The name given to a concept or theorem is just a label. The standard label for these numbers is Fibonacci numbers, and unless and until there is consensus on an alternative label, it cannot be erroneous to call them Fibonacci numbers. Gandalf61 15:29, 11 September 2006 (UTC)
Gandal ,I had given thoses references to support that statement.So i would like to know why these references were removed without any 'consensus'? —The preceding unsigned comment was added by Bharatveer (talk • contribs) 15:49, 11 September 2006 (UTC)
Please see Wikipedia:Neutral point of view. If you have a reference in which the author states that the naming is erroneous, that proves only that someone thinks it's erroneous. It doesn't prove that the naming actually is erroneous, nor even that a substantial population agrees. Melchoir 16:22, 11 September 2006 (UTC)
I am obtaining copies of these papers, to see what they say. It has taken some time, because they are rather old, and not available online, and I will not be able to look at them before October, because I am on vacation in Korea. I will report on the contents of the papers when I return in early October. -- Dominus 23:19, 17 September 2006 (UTC)
I have read the Historia Mathematica paper, and I do not see anywhere that it advances, or even supports, your contention that the name is "erroneous". It consists mostly of an explanation of how to enumerate certain kinds of Sanskrit prosody, and says "The name of Leonardo of Pisa, also called Fibonacci, is attached to the sequence 1, 1, 2, 3, 5, 8, 13, ...". Have you read this yourself? If so, what part of this paper do you believe advances your claim that the name is "erroneous"? -- Dominus 02:34, 20 September 2006 (UTC)
It now appears that your citation for the other journal article is incorrect. My library reports:
Dear Mark Dominus:
International journal of mathematical education in science and technology // Acarya Hemcandra and the (so-called) Fibonacci Numbers // 20 (1986), 20-30
Please double check your citation or provide additional information about this material. Lenders have indicated that the article/chapter you specify does not appear in the volume cited.
Since the citation is incorrect, I will remove it from the Wikipedia article. Please do feel free to put it back when you have the correct citation. Thanks. -- Dominus 09:51, 26 September 2006 (UTC)
I have reverted the article to remove this reference again. t I am not claiming that the article does not exist. I believe you when you say the article exists. But the citation, as given, is incorrect, and so is useless, because nobody can use it to find the article. The citation must be corrected before it can be included. There is no point in putting incorrect citations in articles. -- Dominus 18:22, 26 September 2006 (UTC)

Repeated 3RR violation

Could some admin please block Bharatveer? This is getting ridiculous. Fredrik Johansson 12:45, 16 September 2006 (UTC)

I think you are being ridiculous here; Can't you see the last edit was my third one? And each edit supported by references?-Bharatveer 12:48, 16 September 2006 (UTC)

From WP:3RR: "This does not imply that reverting three times or fewer is acceptable." You have repeatedly re-instated the same opinion piece, refusing to listen to the consensus established by other editors. Your appeal to references doesn't help; the opinion is an opinion regardless of whether you put a footnote on it. Fredrik Johansson 13:06, 16 September 2006 (UTC)
Your statement about inserting references looks childish.Bharatveer 13:11, 16 September 2006 (UTC)
Who is it that keeps urging others to be "civil"? —Tamfang 07:19, 17 September 2006 (UTC)

Sunflower count

How was the number 55 obtained for the sunflower image? I can easily count to 21 and 34 (outwards clockwise and counter-clockwise, respectively), but I'm having trouble identifying a set of spirals adding up to 55. Fredrik Johansson 12:06, 17 September 2006 (UTC)

I think I checked it out in Photoshop by drawing curves around the outside. For 55, you really have to work around the edge. Melchoir 18:53, 17 September 2006 (UTC)

"Fibonacci" numbers

Who decided to start calling this sequence the "Fibonacci" numbers, and when? The OED has no citation before 1891, and all the early citations concern phyllotaxis. Isn't that interesting? -- Dominus 10:34, 27 September 2006 (UTC)

Although I do not have a primary source, I have seen the name attributed to Edouard Lucas, which would fit the 1891 citation. Finell (Talk) 21:33, 15 October 2006 (UTC)

Links to first numbers in sequence?

In the lead, the first numbers in the Fibonacci sequence are wikilinked (with some links duplicated). In my opinion, this does not contribute to the article and is a distraction. If a reader clicks on any of the links, the reader will not find relevant elucidation. What do others think? Unless there is concesnus to the contrary, I intend to remove the links after allowing time for other editors to respond. Finell (Talk) 21:41, 15 October 2006 (UTC)

  • Support removal of number links. In my view linking numbers is almost never useful. −Woodstone 08:24, 16 October 2006 (UTC)
  • Neutral. I think linking numbers is useful, but "when" is the real question. All of these numbers we're linking to will just link us back to here, as well as other locations. McKay 19:12, 16 October 2006 (UTC)

Is one number a Fibonacci sequence?

If you have just one number, 1, in a sequence, can that be technically called a Fibonacci sequence? --Waterchan 05:25, 29 October 2006 (UTC)

Technically, you could say that {1} is a finite sub-sequence of the Fibonacci sequence which contains only one term, but this is really stretching the definition of sequence. It would be more usual to describe a single term from the Fibonacci sequence as a Fibonacci number. Gandalf61 09:40, 29 October 2006 (UTC)

Proposal: Move long sections to new articles

I think the article is too long. Wikipedia:Article_size#Science_articles recommends no more than 10 printed pages. I suggest moving the two long sections "Generalizations" and "Popular culture" with their many subsections into two new articles called "Generalizations of Fibonacci numbers" and "Fibonacci numbers in popular culture". They would both be longer than many of the other articles in Category:Fibonacci_numbers. PrimeHunter 00:50, 15 November 2006 (UTC)

I personally am almost always in favor of reducing the length of articles by farming out some material to other pages. I would think this is a good idea. The main reason for this in my mind is to make the articles more accessible and more inviting and not so overwhelming and intimidating. --Filll 01:18, 15 November 2006 (UTC)
PrimeHunter, I agree with your idea - and especially so if you are volunteering to do the work ;) Gandalf61 11:52, 15 November 2006 (UTC)
I can do it after a few days if there is still consensus. PrimeHunter 19:09, 16 November 2006 (UTC)
I have now created Generalizations of Fibonacci numbers and Fibonacci numbers in popular culture, and replaced the corresponding article sections with a summary. PrimeHunter 02:55, 19 November 2006 (UTC)

diacritics

User:PrimeHunter tacitly removed the diacritics on Chhandah-shāstra and Lozanić. Why? —Tamfang 23:00, 19 November 2006 (UTC)

Oops, my bad. I copied the source to a text editor while editing. Somewhere the software dropped diacritics (and other details). I will fix it now (assuming my software allows it). Thanks for pointing it out. PrimeHunter 01:20, 20 November 2006 (UTC)
I have fixed this, and a similar problem with Ernő in Fibonacci numbers in popular culture. Somebody already corrected that my software had renamed the film π to p when it was moved to that article! I will be careful now, and check other edits. PrimeHunter 02:20, 20 November 2006 (UTC)

Fibonacci Zeros

On http://mathworld.wolfram.com/FibonacciNumber.html under formula 9, which generalizes the fibonacci function for real numbers, they list that the fibonacci function has zeros at each negative number approaching n + 0.5. Should this be included in the article at some point? CodeLabMAster 15:55, 20 November 2006 (UTC)

Reliable sources

Hi,

I am proposing this link to be added to the article:

http://fibonaccicalc.freehostia.com/fibonacci.cgi

People says that it violates wp policies about Reliable sources.

First of all Reliable sources policy applies to what is written in the article, and not the links.

Second, many links on the Fibonacci number article are not so authoritative:

http://www.goldenmeangauge.co.uk/fibonacci.htm site of a dentist. So, are dentists Fibonacci experts?

http://www.upl.cs.wisc.edu/~bethenco/fibo/ second year Ph.D. student I'm going to get the degree in computer science, do two years make the differece?

And I do not want to wast more time looking for who are the people writing articles linked here.

So, what's the difference, please?

I didn't get any answere ... why people so fast remove my link, isn't as much fast answering my coment?

PAolo —The preceding unsigned comment was added by 213.156.35.243 (talk) 21:04, 26 January 2007 (UTC).

There is a policy that applies to the links, not what is written in the article. It can be found at WP:EL. It is likely that some of the other links in the article violate that policy, so comparing your link to the others is not helpful: instead, you should either justify your link as satisfying the policies in WP:EL, or accept that it is not appropriate. —David Eppstein 21:18, 31 January 2007 (UTC)

Golden Ratio

I do not know if the following results have been reported. Previously I put them under Section 25 "Ratio Proof".

1=\frac{1}{\varphi}+\frac{1}{\varphi^3}+\frac{1}{\varphi^5}+\dots
\frac{1}{2}= \frac{1}{\varphi^2}+\frac{1}{\varphi^5}+\frac{1}{\varphi^8}+\dots

209.167.89.139 20:01, 29 January 2007 (UTC)

I removed the series involving the golden ratio from the article as they seemed to have little to do with Fibonacci numbers per se. But in general, if you don't know whether a result has been reported in the mathematical literature, you shouldn't be including it here; see WP:OR. —David Eppstein 19:39, 29 January 2007 (UTC)
Thanks for the info. I take it that "... shouldn't be including it here ..." means not in the Article but allowed in the Discussion.
Right, unsourced material on a discussion page is fine as long as you're discussing e.g. whether a source can be found to allow the material to be put into the article. Putting stuff on the discussion page as a way of publishing your own research on the subject is not a great idea, though — you should get a blog or other web site of your own for that sort of thing. —David Eppstein 21:55, 29 January 2007 (UTC)


The closed form solution can be obtained directly from:
{F_{n+1} \choose F_{n}} = \begin{pmatrix} (1-\varphi) & \varphi \\ 1 & 1 \end{pmatrix} \begin{pmatrix} (1-\varphi)^n & 0 \\ 0 & \varphi^n \end{pmatrix} \begin{pmatrix} (1-\varphi) & \varphi \\ 1 & 1 \end{pmatrix}^{-1} {1 \choose 0}

209.167.89.139 20:26, 29 January 2007 (UTC)

Recent change to limit of consequtive quotients

A recent change replaced this:

Johannes Kepler pointed out that the ratio of consecutive Fibonacci numbers
\frac{F(n+1)}{F(n)}\,
converges to the golden ratio \varphi as the limit.


With this:

Johannes Kepler pointed out that the ratio of consecutive Fibonacci numbers
\lim_{n\to\infty}\frac{F(n+1)}{F(n)}=\varphi,
converges to the golden ratio \varphi as the limit.

The intention in the original text was for the phrase "the ratio of consecutive Fibonacci numbers" to be followed by \frac{F(n+1)}{F(n)}\, the expression for the ratio of consecutive Fibonacci numbers. This expression is the subject of the sentence. The following clause then explains that this expression converges to the golden ratio \varphi.

After the change, however, the sentence is at least ungrammatical, and arguably incorrect. \lim_{n\to\infty}\frac{F(n+1)}{F(n)}=\varphi is not "the ratio of consecutive Fibonacci numbers", as it says; it is an equation. And the equation does not "converge to the golden ratio" as it says, because equations do not converge; they assert convergence.

It would have been correct to have said something like this:

Johannes Kepler pointed out that
\lim_{n\to\infty}\frac{F(n+1)}{F(n)}=\varphi,

because then the formula itself asserts the convergence. But that is not the change that was made. The change that was made was redundant, since it say both in symbols and in words that \lim_{n\to\infty}\frac{F(n+1)}{F(n)}=\varphi,. It was also incorrect, as I explained above.

I reverted, but shortly thereafter another editor reverted it again. I am about to put it back the way it was originally, and point to this explanantion. -- Dominus 15:32, 15 February 2007 (UTC)

-edited July 16-

I've been messing around with the golden ratio part of this article with a little bit of computer programming. When first trying to verify that the ratio of successive quotients does come to the Golden Ratio, I had the ratio flipped - F(n-1)/F(n) instead of the proper F(n)/F(n-1). I made a bit of a discovery when I first tested the code. This particular sequence (the original sequence flipped) appears to converge at 1 - phi (my code did show the correct ratio to converge to phi as well). At a certain point the accuracy of the calculation was limited by the size of the variable, but the closest result I got was 0.6180339887498949 after approximately 48 ratios. This is longer than the estimate given on the Wiki page, but is accurate to as many decimal places, and matches the longer estimate given here: http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/phi.html#phi2000 (their value is 1.61803 39887 49894 84820). Is there a well known proof for this limit? Should it be added to this page? A summary form of this concept would be that Phi = F(n)/F(n-1) = F(n-1)/F(n) - 1 (or, I mean to say the limit as x goes to infinity for the two ratios). I've never edited anything here on Wikipedia before and I don't have a profile, so it would probably be best for someone else to do so. Regardless, I think this is an interesting bit of info that should be included. 74.68.58.71 05:42, 16 July 2007 (UTC)

This is one of the properties of φ - see golden ratio. Because φ2 − φ − 1 = 0, it follows that \frac{1}{\phi}=\phi-1. Gandalf61 08:59, 16 July 2007 (UTC)

Combinatorial proof of the identities

Hi I'm new to all this so forgive me if I'm doing this all wrong

In the combinatorial proof section, for the "Proof of the third identity" the page says

     [F(n + 2) − 1] + [F(n + 1) − 1] + … + [F(2) − 1]
   = [F(n + 2) − 1] + [F(n + 1) − 1] + … + [F(2) − 1] + [F(1) − 1] + F(0)
   = F(n + 2) + [F(n + 1) + … + F(1) + F(0)] − (n + 2)
   = F(n + 2) + F(n + 3) − (n + 2).

Should the last line not be = F(n + 2) + F(n + 3) − (n + 2) − 1 instead?


Also the proof for the fourth identity F(0)2 + F(1)2 + F(2)2 + … + F(n)2 = F(n) F(n + 1) is currently lacking.

Thanks —The preceding unsigned comment was added by Jclistens (talk • contribs) 21:46, 20 February 2007 (UTC).

Sequence from difference of two preceding terms

Just curious, does anyone know if there are results about the sequences G(n+2)=G(n+1)-G(n)? If G(0)=0 and G(1)=1, G is a periodic trapezoidal wave with a period of 6 and an amplitude equal to 1, for both positive and negative n. Expressed in a matrix form, the transition matrix has complex eigenvalues and complex eigenvectors. The sequence H(n+2)=H(n)-H(n+1) is a backward Fibonacci sequence. As such, it is trivial. 209.167.89.139 15:39, 22 February 2007 (UTC)

The G(n) sequence can be viewed as sampled 1.1547[sin(πx/3)] at integers x.Zymogen 16:58, 27 February 2007 (UTC)

Spiral?

What is a Fibonacci spiral? This article features a drawing of such a spiral, but includes no text explaining what it is. I can kind of make the jump from Golden sprial, I think, but the image seems out of place here with no text anchoring it to the article. -- Mikeblas 21:26, 24 February 2007 (UTC)

I've expanded the caption on the image and linked to golden spiral. Gandalf61 10:38, 25 February 2007 (UTC)
Thanks! I think that helps the piece. -- Mikeblas 15:27, 25 February 2007 (UTC)

Programming

Why is there very, very little about calculating the series on a computer in here? It seems like it would be an important area. mrholybrain's talk 03:24, 10 March 2007 (UTC)

I added a reference to this (the Dijkstra one). I won't say there's little to say about this topic, because I often use it as the first lecture in my algorithms classes, but I'm not sure what more is appropriate to add here. —David Eppstein 04:01, 10 March 2007 (UTC)
The history is mostly found in Wikipedia:Articles for deletion/List of hello world programs, Wikipedia:Articles for deletion/Fibonacci number program (2 nomination), b:Transwiki:Fibonacci number program, b:Fibonacci number program, and this old version of the article. Melchoir 04:33, 10 March 2007 (UTC)

Structure of the article

With the recent move of applications and arts downwards, the structure of the article has not improved. In an earlier discussion, we agreed to have the definition and very simple math at the start, then go into the applications and arts, finally come back to more math details. The problem with bringing all math together at the start is that most readers will never reach the softer part. We should restructure again. −Woodstone 17:19, 15 May 2007 (UTC)

Yes I do kind of agree. It did seem a bit odd to go from maths to applications and then maths then popular culture and then a bit more maths. I'm not sure what the ideal structure at the moment is. Plus I think some of the maths could do with a prune, or moving to a sub /Proofs page. --Salix alba (talk) 22:45, 15 May 2007 (UTC)
When I split out Generalizations of Fibonacci numbers and Fibonacci numbers in popular culture, I considered doing something similar for proofs and maybe the more advanced math. It is intimidating to some people and proofs may be of interest to few readers. This is a long article and the central in a whole category. (I don't want to do split work again) PrimeHunter 01:13, 16 May 2007 (UTC)
Which reminds me that the sub article is on AfD at the moment Wikipedia:Articles for deletion/Fibonacci numbers in popular culture. --Salix alba (talk) 07:34, 16 May 2007 (UTC)

Simplification

Can anyone simplify the article, at least slightly? It seems to me that anyone who is capable of reading all of the mathematical equations on the page probably already knows about Fibonacci number and the golden ratio. There are most likely a lot of people who are very interested in the concept yet who are less inclined to understand such mathematical equations. I, for one. Any help for the less mathematically inclined? --Bentonia School 17:30, 23 May 2007 (UTC)

I can try. Let's start with the opening. Any problems there? Yes, it's defined as a recurrence relation, but that definition is explained directly below. —METS501 (talk) 20:50, 6 June 2007 (UTC)
I suggest to create Fibonacci number/Proofs and move the proofs there. See Wikipedia:WikiProject Mathematics/Proofs for general discussion and Category:Article proofs for examples. PrimeHunter 00:18, 7 June 2007 (UTC)
We don't use the virgule to make subtopics in the wikipedia article namespace. The more appropriate way would be something like Fibonacci proofs or Proofs relating to Fibonacci numbers or something like that. McKay 14:37, 7 June 2007 (UTC)
The fibonacci numbers show up originally in problem 79 of the Rhind Papyrus Eves "In Mathematical Circles" 1969, Prindle Weber and Schmidt, pp 10-11 and the Mathematical Leather roll as series derived from unit fractions (Gillings).
The Fibonacci numbers are a series where you add the last two to get the next.; 0, 1, 1, 2, 3 ...
The golden section numbers are ±0·61803 39887... and ±1·61803 39887... the only number which is its own reciprocal. They are commonly derived in architectural proportions from geometry rather than the complicated general formulas in this article.
I think Milo Gardiner is knowledgable on this. You could put questions on his talk page. There is also some mention in
Eves "In Mathematical Circles" 1969, Prindle Weber and Schmidt, pp 10-11
Niven "Numbers, Rational and Irrational"
C. D. Olds "Continued Fractions"
Gillings, "Mathematics in the Time of the Pharoahs"

Fibonacci Series in Architectural Proportions or Orders

The Greek Orders of Architecture were derived from architectural proportions whose Golden ratios and other mathematical series were noted in the inscription grids, sekeds etc; the Egyptian,and other weights and measures and themselves formed the basis of later architectural ideals found in the writings of Vitruvius about Roman systemas used by architects and later Palladio

Length

Fibonacci series can be found amongMesopotamian units, where the cubit or ku was divided into different digits (shusi of about 17 mm uban of about 20 mm) and hands (qat). Similarly the Greek units also have long, median and short forms reflecting both arithmetric and geometric series.

The proportions continue through the length and width of boards to the diameters of columns and the proportions of rooms such that a room will be twice its height in width and 3 times its height in length.

Column diameters may be related to circumference as 7:22 and their height as 1:8 or with a capital and base 1:13

  • The Ionian pous is a typical short pous measuring 296 mm
  • The Attic pous measures 309.4 mm
  • The Athenian pous measures 316 mm and is considered of median length
  • Long pous are actually remen (4 hands) and pygons
  • See cubit for the discussion of the choice of division into hands or palms

Generally the sexagesimal (base sixty) or decimal (base-ten) multiples have Mesopotamian origins while the septenary (base-seven) multiples have Egyptian origins. The following table compares several related series based on the module of the Greek daktylos of finger

One example of fibonacci series in architectural proportion would be in found in the board measures used when boards and beams were split, sawed and planed by hand or worked by adze before board measure was modularized for saw mills...its still similar but not exact.

1 finger, 18.5 mm (1x)
2 fingers, 37 mm (2x)
3 fingers, 55.5 mm strapping
5 fingers, 92.5 mm (2x4)
8 fingers, 148 mm (2x6)
13 fingers 240.5 mm (2x10)

The proportions continue through the length and width of boards to the diameters of columns and the proportions of rooms such that a room will be twice its height in width and 3 times its height in length. Column diameters may be related to circumference as 7:22 and their height as 1:8 or with a capital and base 1:13

diameter 0, 3.81", 3.81", 7.64", 11.45" 19" 30.6"
circumference 0, 1', 1', 2', 3', 5', 8',
height 0, 3.14, 3.14, 6.28, 9.42, 15.7, 25.1,

—Preceding unsigned comment added by Rktect (talk • contribs)

There is nothing about Fibonacci numbers in our current articles on the Rhind Mathematical Papyrus and Egyptian Mathematical Leather Roll. I am skeptical of such claims and would want to see solid scholarly sources before including anything like that here. —David Eppstein 05:45, 24 July 2007 (UTC)