I know that the Fibonacci numbers converge to a ratio of .618, and that this ratio is found all throughout nature, etc. I suppose the best way to ask my question is: where was this .618 value first found? And what is the...significance?
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8$\begingroup$ The ratio doesn't converge to $.618$, it converges to $\frac{\sqrt5-1}2$. $\endgroup$– Kevin O'BryantJun 30, 2010 at 16:16
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$\begingroup$ I don't understand why this question has so many down-votes. Any down-voters care to explain themselves? $\endgroup$– Kevin O'BryantJun 30, 2010 at 16:17
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8$\begingroup$ @Kevin: ...or, depending upon exactly what you mean, the successive ratios converge to $\varphi = \frac{\sqrt{5}+1}{2}$. (FWIW, the latter is usually defined to be the golden ratio, not its reciprocal.) I didn't downvote, but I think that at least the second question asking about the significance of the golden ratio is not a good one for our site. I expect that most research mathematicians have heard more than enough about $\varphi$. That pretty much goes for me, although I wouldn't mind watching Donald Duck in Mathemagic Land once more for old times' sake. $\endgroup$– Pete L. ClarkJun 30, 2010 at 17:28
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3$\begingroup$ The chicken or the egg: that is the question. $\endgroup$– Victor ProtsakJul 1, 2010 at 5:46
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3$\begingroup$ My daughter (aged something like 6 at the time - a long time ago) told me: "well God didn't say 'let there be eggs'". The question becomes 'which is the chicken?' $\endgroup$– Mark BennetMar 16, 2011 at 19:41
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8 Answers
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The golden ratio in mathematics dates back to the Pythagoreans, circa 500 BC, it's true. But the Fibonacci numbers also have a long heritage going back to Pingala in India circa 200 BC.
However, the mystical claims about the golden ratio and Fibonacci numbers going back hundreds of millions of years in biology and showing up in every piece of ancient art and architecture seem to date back only to Pacioli in the 16th century AD.
answered Oct 22, 2009 at 20:41
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3$\begingroup$ David, I agree with your criticism about the Fibonnaci numbers not "...showing up in every piece of ancient art and architecture...". But why is it 'mystical' that patterns related to the Fibonacci numbers that have certain useful phenomenological properties (with respect to things like leaf/petal arrangements/staggerings for example), were discovered by evolutionary algorithms? My 'hundreds of millions of years' number simply comes from the divergence of angiospermae (flowering plants) from gymnospermae approximately ~200-250 million years ago. $\endgroup$– MensenNov 4, 2009 at 20:54
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As previous answers have pointed out, both the golden ratio and the Fibonacci numbers go back thousands of years. However, I believe the connection between the two was discovered around 1730. At that time, Daniel Bernoulli and Abraham de Moivre independently came up with the generating function for the Fibonacci numbers, and the resulting formula for the $n$th Fibonacci number in terms of the golden ratio.
answered Jun 30, 2010 at 23:20
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1$\begingroup$ What about Kepler? He clearly states the relation in "Hexagonal snow" (1611). $\endgroup$ Jul 1, 2010 at 5:08
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$\begingroup$ Thanks for drawing my attention to this, Victor. It seems that Kepler should get credit for guessing that the golden ratio is the limiting ratio of consecutive Fibonacci numbers, but apparently this was from numerical experiments (I can't find his "Snowflake" book right now). I still think that Daniel Bernoulli and Abraham de Moivre found the first proofs. $\endgroup$ Jul 1, 2010 at 5:32
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1$\begingroup$ Translation from the Russian edition, at the end of the section "Regular solids based on number five and their genesis from divine proportions", after stating the relation, Kepler says: "I omit all further arguments that I could give in confirmation of these most pleasant reasonings. A special place would be required for them". I don't know whether the notion of limit Kepler possessed was robust enough to allow for a possibility of a proof, but his manner of expression leaves little doubt that he understood the precise relationship between the "extreme and mean ratio" and $F_{n+1}/F_n.$ $\endgroup$ Jul 1, 2010 at 5:57
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Golden ratio came first. Wikipedia has a rather thorough article on it. It's not nearly as pervasive in nature or architecture as people like to say it is. It will show up in anything with regular pentagons, though.
answered Oct 11, 2009 at 19:47
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$\begingroup$ Yeah, Golden ratio been here from ancient times. $\endgroup$ Oct 11, 2009 at 19:53
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The book A Mathematical History of the Golden Number by Roger Herz-Fischler is an exhaustive study of nearly all references to the golden ratio, from the earliest times, and is available as a free e-book. As has been pointed out by others, the golden ratio is older than the Fibonacci numbers. On page 53, Herz-Fischler notes that a pentagram appears as "a pot mark on a jar" dating from 3100 BC in Egypt.
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2$\begingroup$ I cannot see the page in question in google, but unless there is evidence that the golden ratio was used in the construction, it might be a bit of a stretch. The fact that the golden ratio appears in a figure does not mean it was discovered. $\endgroup$ Aug 15, 2010 at 12:33
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$\begingroup$ The section I quoted is at the bottom of page 53; the link I gave should direct you to that page. You have to scroll down the page to see it. You are right that the mere use of a pentagram does not prove that the ancient Egyptians were aware of the golden ratio and its significance. But if you read further in that section, entitled "Examples Before Pythagoras (before c. -550)," the author lists many examples from various times and geographical locations, that show at least some understanding of the golden ratio. I think the earliest evidence he cites is from 4,500 B.C. in Palestine (page 57), $\endgroup$ Aug 15, 2010 at 16:52
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$\begingroup$ (continued): which suggests that prehistoric cultures may have had some familiarity with the golden ratio. In any case, the question was just about which came first, the golden ratio or the Fibonacci numbers. The golden ratio was definitely understood in the ancient world. The first unequivocal mention of it appears to be by Euclid in The Elements. There is evidence that the Fibonacci numbers were understood in ancient India, with Wikipedia citing a date as early as 200 B.C. That is 100 years after Euclid, but close enough that one could claim the question is not settled. $\endgroup$ Aug 15, 2010 at 17:05
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What is the significance? Most of the nice properties of the golden mean can be attributed to the fact that its continued fraction coefficients are uniformly bounded, as will be true in particular for any periodic continued fraction, which is to say any quadratic irrational, such as arises as the spectral radius of an indecomposable two-term linear recurrence relation. Among such continued fractions, the unique one with the minimum possible upper bound of 1 naturally exhibits these effects most prominantly, and it arises from (arguably) the simplest such recurrence.
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The golden ratio was used extensively in ancient art, but the man named Fibonacci (Leonardo of Pisa) lived around 1200 AD. It's possible that the Fibonacci series was known before Fibonacci but I'm not aware of this. So I think it's safe to assume the golden ratio is older.
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3$\begingroup$ The Fibonacci sequence (not series!) was known in ancient India, long before Fibonacci. $\endgroup$ Oct 22, 2009 at 20:43
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$\begingroup$ We've have Fibonacci numbers as long as we've had rabbits.. :) $\endgroup$– user1504Jun 30, 2010 at 18:42
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2$\begingroup$ Somewhat more accurately, we've had Fibonacci numbers as long as we've had bees. $\endgroup$ Jun 30, 2010 at 20:11
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6$\begingroup$ The golden ratio was used extensively in ancient art ... try this ... Misconceptions about the Golden Ratio, George Markowsky, College Math Journal: Volume 23, Number 1, Pages: 2-19 1992 $\endgroup$ Jul 1, 2010 at 1:13
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1$\begingroup$ We've had Fibonacci numbers as long as we've had pineapples. Or sunflowers. $\endgroup$– Todd Trimble ♦Feb 11, 2013 at 12:29
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The answer for either of these is "hundreds of millions of years" due to their emergence/use in biological development programs, the self-assembly of symmetrical viral capsids (the adenovirus for example), and maybe even protein structure. Because of their close relationship I'd be hard pressed to say which 'came first'.
If you google for it, you'll find plenty of books and papers. However, be extremely careful about examples without a well-explained functional role... there are an arbitrarily large number of coincidences out there if you're looking for them, and humans excel at numerology.
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9$\begingroup$ But I suspect the question was about human, conscious discovery of these concepts, insofar as existing records can show us. Otherwise this becomes partly a philosophical question; I may as a platonist argue that both concepts are eternal. $\endgroup$ Dec 25, 2009 at 1:14
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Golden ration came first in nature long before humans evolved to think about Fibonacci numbers.
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2$\begingroup$ Are you sure that the Fibonacci numbers didn't appear in nature? Please see the comments attached to other answers. $\endgroup$– S. Carnahan ♦Feb 11, 2013 at 8:08