Fibonacci Numbers (abridged)


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Fibonacci Numbers (abridged)

  1. 1. Fibonacci Numbers 2014
  2. 2. Fibonacci Numbers ● The Fibonacci sequence is named after Fibonacci. His 1202 book Liber Abaci introduced the sequence to Western European mathematics,[4] although the sequence had been described earlier in Indian mathematics.[5][6][7] By modern convention, the sequence begins either with F0 = 0 or with F1 = 1. The Liber Abaci began the sequence with F1 = 1, without an initial 0. ● Fibonacci numbers are closely related to Lucas numbers in that they are a complementary pair of Lucas sequences. They are intimately connected with the golden ratio; for example, the closest rational approximations to the ratio are 2/1, 3/2, 5/3, 8/5, ... . Applications 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,[8] such as branching in trees, phyllotaxis (the arrangement of leaves on a stem), the fruit sprouts of a pineapple,[9] the flowering of artichoke, an uncurling fern and the arrangement of a pine cone.[10] ● The Fibonacci sequence appears in Indian mathematics, in connection with Sanskrit prosody.[6][11] In the Sanskrit oral tradition, there was much emphasis on how long (L) syllables mix with the short (S), and counting the different patterns of L and S within a given fixed length results in the Fibonacci numbers; the number of patterns that are mshort syllables long is the Fibonacci number Fm + 1 .[7]
  3. 3. The number of binary strings of length n without consecutive 1s is the Fibonacci number Fn+2 . For example, out of the 16 binary strings of length 4, there are F6 = 8 without consecutive 1s – they are 0000, 0001, 0010, 0100, 0101, 1000, 1001 and 1010. By symmetry, the number of strings of length n without consecutive 0s is also Fn+2 . Equivalently, Fn+2 is the number of subsets S ⊂ {1,...,n} without consecutive integers: {i, i+1} ⊄ S for every i. The symmetric statement is: Fn+2 is the number of subsets S ⊂ {1,...,n} without two consecutive skipped integers: that is, S = {a1 < ... < ak } with ai+1 ≤ ai + 2. Fibonacci Numbers The Fibonacci numbers occur in the sums of "shallow" diagonals in Pascal's triangle (see Binomial coefficient).[18] These numbers also give the solution to certain enumerative problems.[19] The most common such problem is that of counting the number of compositions of 1s and 2s that sum to a given total n: there are Fn+1 ways to do this. For example F6 = 8 counts the eight compositions: 1+1+1+1+1 = 1+1+1+2 = 1+1+2+1 = 1+2+1+1 = 2+1+1+1 = 2+2+1 = 2+1+2 = 1+2+2, all of which sum to 6−1 = 5. The Fibonacci numbers can be found in different ways among the set of binary strings, or equivalently, among the subsets of a given set.
  4. 4. Closed-form expression Like every sequence defined by a linear recurrence with constant coefficients, the Fibonacci numbers have a closed-form solution. It has become known as Binet's formula, even though it was already known by Abraham de Moivre:[20] Matrix form[edit] A 2-dimensional system of linear difference equations that describes the Fibonacci sequence is The eigenvalues of the matrix A are and , for the respective eigenvectors and . Since , and the above closed-form expression for the nth element in the Fibonacci series as an analytic function of n is now read off directly, The matrix 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: The Golden Ratio
  5. 5. Fibonacci Examples ● 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. ● Fibonacci numbers are used by some pseudorandom number generators. ● 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 φ. A tape-drive implementation of the polyphase merge sort was described in The Art of Computer Programming. ● Fibonacci numbers arise in the analysis of the Fibonacci heap data structure. ● Fibonacci sequences appear in biological settings,[8] in two consecutive Fibonacci numbers, such as branching in trees, arrangement of leaves on a stem, the fruitlets of a pineapple,[9] the flowering of artichoke, an uncurling fern and the arrangement of apine cone,[10] and the family tree of honeybees.[52]
  6. 6. Fibonacci in Music ● Ernő Lendvaï analyzes Béla Bartók's works as being based on two opposing systems, that of the golden ratio and theacoustic scale.[11] In the third movement of Bartok's Music for Strings, Percussion and Celesta, the opening xylophone passage uses Fibonacci rhythm as such: 1:1:2:3:5:8:5:3:2:1:1.[12] ● The Fibonacci numbers are also apparent in the organisation of the sections in the music of Debussy's Image, Reflections in Water, in which the sequence of keys is marked out by the intervals 34, 21, 13 and 8.[12] ● Polish composer Krzysztof Meyer structured the values in his Trio for clarinet, cello and piano according to the Fibonacci sequence.[13] ● Fibonacci's name was adopted by a Los Angeles-based art rock group The Fibonaccis, that recorded from 1981 to 1987. ● American musician BT also recorded a song titled "Fibonacci Sequence". The narrator in the song goes through all the numbers of the sequence from 1 to 21 (0 is not mentioned). The track appeared on a limited edition version of his 1999 album Movement in Still Life, and is also featured on the second disc of the Global Underground 013: Ibiza compilation mixed by Sasha.[14] ● Voiceover and recording artist Ken Nordine described Fibonacci numbers in a word jazz piece called "Fibonacci Numbers" on his album A Transparent Mask.[15] ● American musician Doctor Steel has a song titled "Fibonacci Sequence" on his album People of Earth.[16] ● Australian electronic group Angelspit uses the Fibonnaci in the song "Vermin." The lyrics start with, "1, 2 3 5 8, Who do we decapitate?" and continues through a few more iterations of the sequence.
  7. 7. Fibonacci References ● Huylebrouck, Dirk; Gyllenberg, Mats; Sigmund, Karl (2000). "The Fibonacci Chimney". The Mathematical Intelligencer 22 (4): 46.doi:10.1007/BF03026769. ISSN 0343-6993. Retrieved 2009-06-23. ● Smith, Peter (2007). Sustainability at the Cutting Edge, Second Edition: Emerging Technologies for low energy buildings(date=December 2007). Elsevier. p. 151. ISBN 0-7506-8300-7. ● The Engineer, "Eden Project gets into flower power". ● Munroe, Randall. "Alone". xkcd. Retrieved 2011-04-14. ● The Educational Forum, 55, 3 (Spring 1991), 243-259 ● Journal of Moral Education, 21, 1 (Winter, 1992), 29-40 ● Di Carlo, Christopher (2001). "Interview with Maynard James Keenan". Retrieved 2007-05-22. ● An exposition of how the fibonacci sequence appears in Lateralus set to pictures from the Hubble telescope: ● Norris, Chris (2001). "Hammer Of The Gods". Retrieved 2007-04-25. ● Maconie, Robin (2005). Other Planets, 26 & 28. ISBN 0-8108-5356-6. Citing Lendvai (1972). "Einführung in die Formen- und Harmonienwelt Bartóks" (1953), Béla Bartók: Weg und Werk, p.105-49. Bence Szabolcsi, ed. ● *Lendvaï, Ernő (1971). Béla Bartók: An Analysis of his Music. introd. by Alan Bush. London: Kahn & Averill. ISBN 0-900707-04-6.OCLC 240301. ● ^ Jump up to:a b Smith, Peter F. The Dynamics of Delight: Architecture and Aesthetics (New York: Routledge, 2003) p. 83, ISBN 0-415-30010-X ● Weselmann, Thomas (2003) Musica incrostata. Poznan ● BT - Fibonacci Sequence on YouTube ● Fibonacci Numbers: Ken Nordine at ● People of Earth track list ● "Obituary: Mario Merz". The Guardian (London). 2003-11-13. Retrieved 2008-09-14. ● "Fibonacci Accessories: Scarf". Retrieved 2007-12-31. ● Ingmar Lehman: „Fibonacci-numbers in visual arts and literature" (German)(last called on November 7, 2009) ● 2009: Martina Schettina:Mathemagische Bilder - Bilder und Texte. Vernissage Verlag Brod Media, Wien 2009, ISBN 978-3-200-01743-6 (German) ● About the exhibition, interview on Radio Ö1(recalled at February 28, 2010)
  8. 8. Fibonacci References ● Ball, Keith M (2003), "8: Fibonacci's Rabbits Revisited", Strange Curves, Counting Rabbits, and Other Mathematical Explorations, Princeton, NJ: Princeton University Press, ISBN 0-691-11321-1. ● Beck, Matthias; Geoghegan, Ross (2010), The Art of Proof: Basic Training for Deeper Mathematics, New York: Springer. ● Bóna, Miklós (2011), A Walk Through Combinatorics (3rd ed.), New Jersey: World Scientific. ● Lemmermeyer, Franz (2000), Reciprocity Laws, New York: Springer, ISBN 3-540-66957-4. ● Lucas, Édouard (1891), Théorie des nombres (in French) 1, Gauthier-Villars. ● Pisano, Leonardo (2002), Fibonacci's Liber Abaci: A Translation into Modern English of the Book of Calculation, Sources and Studies in the History of Mathematics and Physical Sciences, Sigler, Laurence E, trans, Springer, ISBN 0-387-95419-8