|
|
| Line 1: |
Line 1: |
| [[File:Fibonacci word cutting sequence.png|thumb|350px|Characterization by a [[cutting sequence]] with a line of slope <math>1/\varphi</math> or <math>\varphi-1</math>, with <math>\varphi</math> the [[golden ratio]].]]
| | I'm Magnolia and I live with my husband and our two children in Burbank, in the CA south area. My hobbies are Sand castle building, Color Guard and Aircraft spotting.<br><br>My site: [http://www.trust-buyer.com/plus/guestbook.php Fifa 15 Coin Generator] |
| A '''Fibonacci word''' is a specific sequence of [[Binary numeral system|binary]] digits (or symbols from any two-letter [[alphabet]]). The Fibonacci word is formed by repeated [[concatenation]] in the same way that the [[Fibonacci number]]s are formed by repeated addition.
| |
| | |
| It is a paradigmatic example of a [[Sturmian word]].
| |
| | |
| The name “Fibonacci word” has also been used to refer to the members of a [[formal language]] ''L'' consisting of strings of zeros and ones with no two repeated ones. Any prefix of the specific Fibonacci word belongs to ''L'', but so do many other strings. ''L'' has a Fibonacci number of members of each possible length.
| |
| | |
| == Definition ==
| |
| | |
| Let <math>S_0</math> be "0" and <math>S_1</math> be "01". Now <math>S_n = S_{n-1}S_{n-2}</math> (the concatenation of the previous sequence and the one before that).
| |
| | |
| The infinite Fibonacci word is the limit <math>S_{\infty}</math>.
| |
| | |
| == The Fibonacci words ==
| |
| | |
| We have:
| |
| | |
| <math>S_0</math> 0
| |
| | |
| <math>S_1</math> 01
| |
| | |
| <math>S_2</math> 010
| |
| | |
| <math>S_3</math> 01001
| |
| | |
| <math>S_4</math> 01001010
| |
| | |
| <math>S_5</math> 0100101001001
| |
| | |
| ...
| |
| | |
| The first few elements of the infinite Fibonacci word are:
| |
| | |
| 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, ... {{OEIS|A003849}}
| |
| | |
| == Closed-form expression for individual digits ==
| |
| The n<sup>th</sup> digit of the word is <math>2+\left\lfloor { {n} \,\varphi} \right\rfloor - \left\lfloor {\left( {n + 1} \right)\,\varphi } \right\rfloor</math> where <math>\varphi</math> is the [[golden ratio]] and <math>\left\lfloor x \right\rfloor</math> is the [[floor function]] {{OEIS|A003849}}.
| |
| | |
| == Substitution rules ==
| |
| | |
| Another way of going from ''S''<sub>''n''</sub> to ''S''<sub>''n'' + 1</sub> is to replace each symbol 0 in ''S''<sub>''n''</sub> with the pair of consecutive symbols 0, 1 in ''S''<sub>''n'' + 1</sub>, and to replace each symbol 1 in ''S''<sub>''n''</sub> with the single symbol 0 in ''S''<sub>''n'' + 1</sub>.
| |
| | |
| Alternatively, one can imagine directly generating the entire infinite Fibonacci word by the following process: start with a cursor pointing to the single digit 0. Then, at each step, if the cursor is pointing to a 0, append 1, 0 to the end of the word, and if the cursor is pointing to a 1, append 0 to the end of the word. In either case, complete the step by moving the cursor one position to the right.
| |
| | |
| A similar infinite word, sometimes called the '''rabbit sequence''', is generated by a similar infinite process with a different replacement rule: whenever the cursor is pointing to a 0, append 1, and whenever the cursor is pointing to a 1, append 0, 1. The resulting sequence begins
| |
| :0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, ...
| |
| However this sequence differs from the Fibonacci word only trivially, by swapping 0s for 1s and shifting the positions by one.
| |
| | |
| A closed form expression for the so-called rabbit sequence:
| |
| | |
| The n<sup>th</sup> digit of the word is <math> \left\lfloor { {n} \,\varphi} \right\rfloor - \left\lfloor {\left( {n - 1} \right)\,\varphi } \right\rfloor -1 </math> where <math>\varphi</math> is the [[golden ratio]] and <math>\left\lfloor x \right\rfloor</math> is the [[floor function]].
| |
| | |
| == Discussion ==
| |
| The word is related to the famous sequence of the same name (the [[Fibonacci number|Fibonacci sequence]]) in the sense that addition of integers in the [[inductive definition]] is replaced with string concatenation. This causes the length of ''S''<sub>''n''</sub> to be ''F''<sub>''n'' + 2</sub>, the (''n'' + 2)th Fibonacci number. Also the number of 1s in ''S''<sub>''n''</sub> is ''F''<sub>''n''</sub> and the number of 0s in ''S''<sub>''n''</sub> is ''F''<sub>''n'' + 1</sub>.
| |
| | |
| == Other properties ==
| |
| * The infinite Fibonacci word is not periodic and not ultimately periodic.
| |
| * The last two letters of a Fibonacci word are alternately "01" and "10".
| |
| * Suppressing the last two letters of a Fibonacci word, or prefixing the complement of the last two letters, creates a [[palindrome]]. Example: 01<math>S_4</math> = 0101001010 is a palindrome. The [[palindromic density]] of the infinite Fibonacci word is thus 1/φ, where φ is the [[Golden ratio]]: this is the largest possible value for aperiodic words.<ref name=AB443>{{citation | last1=Adamczewski | first1=Boris | last2=Bugeaud | first2=Yann | chapter=8. Transcendence and diophantine approximation | editor1-last=Berthé | editor1-first=Valérie | editor2-last=Rigo | editor2-first=Michael | title=Combinatorics, automata, and number theory | location=Cambridge | publisher=[[Cambridge University Press]] | series=Encyclopedia of Mathematics and its Applications | volume=135 | page=443 | year=2010 | isbn=978-0-521-51597-9 | zbl=pre05879512 }}</ref>
| |
| * In the infinite Fibonacci word, the ratio (number of letters)/(number of zeroes) is φ, as is the ratio of zeroes to ones.
| |
| * The infinite Fibonacci word is a [[balanced sequence]]: Take two [[Substring|factors]] of the same length anywhere in the Fibonacci word. The difference between their [[Hamming weight]]s (the number of occurrences of "1") never exceeds 1.<ref name=Lot47>{{harvtxt|Lothaire|2011}}, p. 47.</ref>
| |
| * The subwords ''11'' and ''000'' never occur.
| |
| * The [[complexity function]] of the infinite Fibonacci word is ''n''+1: it contains ''n''+1 distinct subwords of length ''n''. Example: There are 4 distinct subwords of length 3 : "001", "010", "100" and "101". Being also non-periodic, it is then of "minimal complexity", and hence a [[Sturmian word]],{{sfnp|de Luca|1995}} with slope <math>1/\phi^2</math>. The infinite Fibonacci word is the [[Sturmian word#Discussion|standard word]] generated by the [[Sturmian word#Discussion|directive sequence]] (1,1,1,....).
| |
| * The infinite Fibonacci word is recurrent; that is, every subword occurs infinitely often.
| |
| * If <math>u</math> is a subword of the infinite Fibonacci word, then so is its reversal, denoted <math>u^R</math>.
| |
| * If <math>u</math> is a subword of the infinite Fibonacci word, then the least period of <math>u</math> is a Fibonacci number.
| |
| * The concatenation of two successive Fibonacci words is "almost commutative". <math>S_{n+1}=S_nS_{n-1}</math> and <math>S_{n-1}S_n</math> differ only by their last two letters.
| |
| * As a consequence, the infinite Fibonacci word can be characterized by a cutting sequence of a line of slope <math>\phi</math> or <math>\phi-1</math>. See figure above.
| |
| * The number 0.010010100..., whose decimals are built with the digits of the infinite Fibonacci word, is [[Transcendental number|transcendental]].
| |
| * The letters "1" can be found at the positions given by the successive values of the Upper Wythoff sequence (OEIS A001950): <math>\lfloor n\phi^2\rfloor</math>
| |
| * The letters "0" can be found at the positions given by the successive values of the Lower Wythoff sequence (OEIS A000201): <math>\lfloor n\phi\rfloor</math>
| |
| * The infinite Fibonacci word can contain repetitions of 3 successive identical subwords, but never 4. The [[Critical exponent of a word|critical exponent]] for the infinite Fibonacci word is <math>2+\phi=3.618</math> repetitions.<ref name=AS37>{{harvtxt|Allouche|Shallit|2003}}, p. 37.</ref> It is the smallest index (or critical exponent) among all Sturmian words.
| |
| * The infinite Fibonacci word is often cited as the [[worst case]] for algorithms detecting repetitions in a string.
| |
| * The infinite Fibonacci word is a [[morphic word]], generated in {0,1}∗ by the endomorphism 0 → 01, 1 → 0.<ref name=LotII11>{{harvtxt|Lothaire|2011}}, p. 11.</ref>
| |
| | |
| ==Applications==
| |
| Fibonacci based constructions are currently used to model physical systems with aperiodic order such as [[Fibonacci quasicrystal|quasicrystal]]s. Crystal growth techniques have been used to grow Fibonacci layered crystals and study their light scattering properties.{{sfnp|Dharma-wardana|MacDonald|Lockwood|Baribeau|1987}}
| |
| | |
| ==See also==
| |
| * [[Tribonacci word]]
| |
| | |
| ==Notes==
| |
| {{Reflist|colwidth=30em}}
| |
| | |
| ==References==
| |
| *{{citation
| |
| | last1 = Allouche | first1 = Jean-Paul
| |
| | last2 = Shallit | first2 = Jeffrey | author2-link = Jeffrey Shallit
| |
| | isbn = 978-0-521-82332-6
| |
| | publisher = [[Cambridge University Press]]
| |
| | title = Automatic Sequences: Theory, Applications, Generalizations
| |
| | year = 2003}}.
| |
| *{{citation
| |
| | last1 = Dharma-wardana | first1 = M. W. C.
| |
| | last2 = MacDonald | first2 = A. H.
| |
| | last3 = Lockwood | first3 = D. J.
| |
| | last4 = Baribeau | first4 = J.-M.
| |
| | last5 = Houghton | first5 = D. C.
| |
| | journal = Physical Review Letters
| |
| | pages = 1761–1765
| |
| | title = Raman scattering in Fibonacci superlattices
| |
| | volume = 58
| |
| | year = 1987}}.
| |
| *{{citation
| |
| | last1 = Lothaire | first1 = M. | author1-link = M. Lothaire
| |
| | last2 = Perrin | first2 = D.
| |
| | last3 = Reutenauer | first3 = C.
| |
| | last4 = Berstel | first4 = J.
| |
| | last5 = Pin | first5 = J. E.
| |
| | last6 = Pirillo | first6 = G.
| |
| | last7 = Foata | first7 = D.
| |
| | last8 = Sakarovitch | first8 = J.
| |
| | last9 = Simon | first9 = I.
| |
| | last10 = Schützenberger | first10 = M. P.
| |
| | last11 = Choffrut | first11 = C.
| |
| | last12 = Cori | first12 = R.
| |
| | edition = 2nd
| |
| | isbn = 0-521-59924-5
| |
| | publisher = [[Cambridge University Press]]
| |
| | series = Encyclopedia of Mathematics and Its Applications
| |
| | title = Combinatorics on Words
| |
| | volume = 17
| |
| | year = 1997}}.
| |
| *{{citation
| |
| | last = Lothaire | first = M. | author-link = M. Lothaire
| |
| | isbn = 978-0-521-18071-9
| |
| | publisher = [[Cambridge University Press]]
| |
| | series = Encyclopedia of Mathematics and Its Applications
| |
| | title = Algebraic Combinatorics on Words
| |
| | volume = 90
| |
| | year = 2011}}. Reprint of the 2002 hardback.
| |
| *{{citation
| |
| | last = de Luca | first = Aldo
| |
| | doi = 10.1016/0020-0190(95)00067-M
| |
| | issue = 6
| |
| | journal = [[Information Processing Letters]]
| |
| | pages = 307–312
| |
| | title = A division property of the Fibonacci word
| |
| | volume = 54
| |
| | year = 1995}}.
| |
| *{{citation
| |
| | last1 = Mignosi | first1 = F.
| |
| | last2 = Pirolli | first2 = G.
| |
| | issue = 3
| |
| | journal = Informatique théorique et application
| |
| | pages = 199–204
| |
| | title = Repetitions in the Fibonacci infinite word
| |
| | url = http://cat.inist.fr/?aModele=afficheN&cpsidt=5478956
| |
| | volume = 26
| |
| | year = 1992}}.
| |
| | |
| == External links ==
| |
| *[http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibrab.html A detailed and accessible description, on Ron Knott's site]
| |
| *{{mathworld|title=Rabbit Sequence|urlname=RabbitSequence}}
| |
| *{{youtube|id=ZDGGEQqSXew|title=Fibonacci Word (First 200,000 bits)}}
| |
| | |
| {{DEFAULTSORT:Fibonacci Word}}
| |
| [[Category:Binary sequences]]
| |
| [[Category:Fibonacci numbers]]
| |