Monday, November 23, 2009

Fibonacci numbers and the extreme and mean ratio - some history

Ruth Tatlow's article The Use and Abuse of Fibonacci Numbers and the Golden Section in Musicology Today (Understanding Bach, 1, 69-85, 2006) besides being very interesting in itself as a documented criticism of the "Golden numberism [that] has thoroughly infected musicology", incidentally provides useful references for those interested in the history of the representation of the EMR ("extreme and mean ratio" or "golden section") as the limit of the sequence of quotients Fn+1/Fn of consecutive Fibonacci numbers [1]. One may wonder who noticed this first? Leonardo Pisano (c. 1170 – c. 1250, also known as Fibonacci)? Not even close!

Evidence that this fact was noticed as early as the beginning of the 16th century was discovered by Leonard Curchin and Roger Herz-Fischler (handwritten annotation in the 1509 Luca Pacioli's edition of Elements [2]). When it comes to published work which associates the EMR with the limit of the sequence of quotients of consecutive Fibonacci numbers, Johannes Kepler wrote [3] the following, in 1611, about "this proportion that the geometers of today call divine":
It is impossible to provide a perfect example in round numbers. However, the further we advance from the number one, the more perfect the example becomes. Let the smallest numbers be 1 and 1... Add them, and the sum will be 2; add to this the greater of the 1s, result 3; add 2 to this, and get 5; add 3, get 8; 5 to 8, 13; 8 to 13, 21. As 5 is to 8, so 8 is to 13 approximately, and as 8 to 13, so 13 is to 21, approximately.
However, as communicated in 1995 by Peter Schreiber [4], in a rare book by the German reckoning master Simon Jacob (d. 1564) one can find a remark that the sequence of quotients of consecutive Fibonacci numbers approaches the EMR.


[1] The term "extreme and mean ratio" goes back to Euclid's "Elements". Luca Pacioli introduced the "divine proportion" term in 1509, while the term "golden section" was introduced in 1835 by Martin Ohm. The exact value for the EMR is (1+sqrt(5))/2. The phrase "Fibonacci sequence" was coined by Edouard Lucas (1842-1891).
[2] Leonard Curchin and Roger Herz-Fischler, "De quand date le premier rapprochement entre la suite de Fibonacci et la division en extrême et moyenne raison?" (French) ["When were the first parallels drawn between the Fibonacci sequence and the golden section?"], Centaurus 28 (1985), no. 2, 129-138.
[3] Kepler, Johannes. Vom sechseckigen Schnee. (German) [On hexagonal snowflakes]. Translated from the Latin and with an introduction and notes by Dorothea Goetz. Ostwald's Classics of the Exact Sciences, 273. Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1987.
[4] Peter Schreiber, "A Supplement to J. Shallit’s Paper "Origins of the analysis of the Euclidean algorithm"", Historia Mathematica, Vol. 22, Issue 4, 422-424 (1995).

Friday, September 18, 2009

Bernhard Riemann

Georg Friedrich Bernhard Riemann (September 17, 1826 – July 20, 1866) - one of the greatest mathematicians of all time.
Dedekind has a deeply touching account of Riemann's final moments, his passing away being marked by the words of Lord's prayer. Riemann's tombstone (see here and here) in Biganzolo (Italy) refers to Romans 8:28 ("And we know that all things work together for good to them that love God, to them who are the called according to his purpose"):
Here rests in God
Georg Friedrich Bernhard Riemann
Professor in Göttingen
born in Breselenz, September 17th, 1826
died in Selasca, Juli 20th, 1866
Those, who love God, all things
must serve to its best manner.

Friday, February 20, 2009

Fibonacci modulo m

The general problem of the periods of the Fibonacci sequence modulo m is definitely non-trivial (with the case m = p - prime - playing a very important role). An important reference can be found here ("The Fibonacci Sequence Under Various Moduli" - M.Sc. Thesis by Marc Renault, 1996). Also see the article (PDF) "The Fibonacci sequence modulo p^2...".

An example that teachers use relatively often as a middle-school problem: "Find the period of the sequence of the last digits of the Fibonacci numbers"! That will correspond to the modulus m=10, the answer is 60, and the elements of the period are
9,9,8,7,5,2,7,9,6,5,1,6,7,3,0,3,3,6,9,5,4,9,3,2 ,5,7,2,9,1
If the modulus m is 2011 (that is the 305-th prime), the period of the Fibonacci sequence modulo m is 2010.