${\displaystyle S_{10}}$

${\displaystyle S_{17}}$

Fibonacci curves made from the 10th and 17th Fibonacci words^[1]

AFibonacci word is a specific sequence ofbinary digits (or symbols from any two-letteralphabet). The Fibonacci word is formed by repeatedconcatenation in the same way that theFibonacci numbers are formed by repeated addition.

It is a paradigmatic example of aSturmian word and specifically, amorphic word.

The name "Fibonacci word" has also been used to refer to the members of aformal languageL consisting of strings of zeros and ones with no two repeated ones. Any prefix of the specific Fibonacci word belongs toL, but so do many other strings.L has a Fibonacci number of members of each possible length.

Let${\displaystyle S_0}$ be "0" and${\displaystyle S_1}$ be "01". Now${\displaystyle S_n=S_{n-1}{S}_{n-2}}$ (the concatenation of the previous sequence and the one before that).

The infinite Fibonacci word is the limit${\displaystyle S_{\mathrm{\infty }}}$, that is, the (unique) infinite sequence that contains each${\displaystyle S_n}$, for finite${\displaystyle n}$, as a prefix.

Enumerating items from the above definition produces:

${\displaystyle S_0}$    0

${\displaystyle S_1}$    01

${\displaystyle S_2}$    010

${\displaystyle S_3}$    01001

${\displaystyle S_4}$    01001010

${\displaystyle S_5}$    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, ... (sequenceA003849 in theOEIS)

Then^th digit of the word is${\displaystyle 2+\lfloor n\phi \rfloor -\lfloor (n+1)\phi \rfloor }$ where${\displaystyle \phi }$ is thegolden ratio and${\displaystyle \lfloor \rfloor }$ is thefloor function (sequenceA003849 in theOEIS). As a consequence, the infinite Fibonacci word can be characterized by a cutting sequence of a line of slope${\displaystyle 1/\phi }$ or${\displaystyle \phi -1}$. See the figure above.

Another way of going fromS_n toS_n +1 is to replace each symbol 0 inS_n with the pair of consecutive symbols 0, 1 inS_n +1, and to replace each symbol 1 inS_n with the single symbol 0 inS_n +1.

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 therabbit 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:

Then^th digit of the word is${\displaystyle \lfloor n\phi \rfloor -\lfloor (n-1)\phi \rfloor -1.}$

The word is related to the famous sequence of the same name (theFibonacci sequence) in the sense that addition of integers in theinductive definition is replaced with string concatenation. This causes the length ofS_n to beF_n +2, the (n +2)nd Fibonacci number. Also the number of 1s inS_n isF_n and the number of 0s inS_n isF_n +1.

• The infinite Fibonacci word is not periodic and not ultimately periodic.^[2]
• 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 apalindrome. Example: 01S₄ = 0101001010 is a palindrome. Thepalindromic density of the infinite Fibonacci word is thus 1/φ, where φ is thegolden ratio: this is the largest possible value for aperiodic words.^[3]
• In the infinite Fibonacci word, the ratio (number of letters)/(number of zeroes) is φ, as is the ratio of zeroes to ones.^[4]
• The infinite Fibonacci word is abalanced sequence: Take twofactors of the same length anywhere in the Fibonacci word. The difference between theirHamming weights (the number of occurrences of "1") never exceeds 1.^[5]
• The subwords 11 and 000 never occur.^[6]
• Thecomplexity function of the infinite Fibonacci word isn + 1: it containsn + 1 distinct subwords of lengthn. 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 aSturmian word,^[7] with slope${\displaystyle 1/\phi }$. The infinite Fibonacci word is thestandard word generated by thedirective sequence (1,1,1,....).
• The infinite Fibonacci word is recurrent; that is, every subword occurs infinitely often.
• If${\displaystyle u}$ is a subword of the infinite Fibonacci word, then so is its reversal, denoted${\displaystyle u^R}$.
• If${\displaystyle u}$ is a subword of the infinite Fibonacci word, then the least period of${\displaystyle u}$ is a Fibonacci number.
• The concatenation of two successive Fibonacci words is "almost commutative".${\displaystyle S_{n+1}=S_n{S}_{n-1}}$ and${\displaystyle S_{n-1}{S}_n}$ differ only by their last two letters.
• The number 0.010010100..., whose digits are built with the digits of the infinite Fibonacci word, istranscendental.
• The letters "1" can be found at the positions given by the successive values of the Upper Wythoff sequence (sequenceA001950 in theOEIS):${\displaystyle \lfloor n{\phi }^2\rfloor }$
• The letters "0" can be found at the positions given by the successive values of the Lower Wythoff sequence (sequenceA000201 in theOEIS):${\displaystyle \lfloor n\phi \rfloor }$
• The distribution of${\displaystyle n=F_k}$ points on theunit circle, placed consecutively clockwise by the golden angle${\displaystyle \frac{2\pi }{{\phi }^2}}$, generates a pattern of two lengths${\displaystyle \frac{2\pi }{{\phi }^{k-1}},\frac{2\pi }{{\phi }^k}}$ on the unit circle. Although the above generating process of the Fibonacci word does not correspond directly to the successive division of circle segments, this pattern is${\displaystyle S_{k-1}}$ if the pattern starts at the point nearest to the first point in clockwise direction, whereupon 0 corresponds to the long distance and 1 to the short distance.
• The infinite Fibonacci word contains repetitions of 3 successive identical subwords, but none of 4.^[2] Thecritical exponent for the infinite Fibonacci word is${\displaystyle 2+\phi \approx 3.618}$.^[8] It is the smallest index (or critical exponent) among all Sturmian words.
• The infinite Fibonacci word is often cited as theworst case for algorithms detecting repetitions in a string.
• The infinite Fibonacci word is amorphic word, generated in {0,1}* by the endomorphism 0 → 01, 1 → 0.^[9]
• Thenth element of a Fibonacci word,${\displaystyle s_n}$, is 1 if theZeckendorf representation (the sum of a specific set of Fibonacci numbers) ofn includes a 1, and 0 if it does not include a 1.
• The digits of the Fibonacci word may be obtained by taking the sequence offibbinary numbersmodulo 2.^[10]

Fibonacci based constructions are currently used to model physical systems with aperiodic order such asquasicrystals, and in this context the Fibonacci word is also called theFibonacci quasicrystal.^[11] Crystal growth techniques have been used to grow Fibonacci layered crystals and study their light scattering properties.^[12]

• Mathematics and art
• Tribonacci word

• Adamczewski, Boris; Bugeaud, Yann (2010), "8. Transcendence and diophantine approximation", inBerthé, Valérie; Rigo, Michael (eds.),Combinatorics, automata, and number theory, Encyclopedia of Mathematics and its Applications, vol. 135, Cambridge:Cambridge University Press, p. 443,ISBN 978-0-521-51597-9,Zbl 1271.11073.
• Allouche, Jean-Paul;Shallit, Jeffrey (2003),Automatic Sequences: Theory, Applications, Generalizations,Cambridge University Press,ISBN 978-0-521-82332-6.
• Berstel, Jean (1986),"Fibonacci Words – A Survey"(PDF), in Rozenberg, G.; Salomaa, A. (eds.),The Book of L, Springer, pp. 13–27,doi:10.1007/978-3-642-95486-3_2,ISBN 9783642954863
• Bombieri, E.;Taylor, J. E. (1986),"Which distributions of matter diffract? An initial investigation"(PDF),Le Journal de Physique,47 (C3):19–28,doi:10.1051/jphyscol:1986303,MR 0866320,S2CID 54194304.
• Dharma-wardana, M. W. C.; MacDonald, A. H.; Lockwood, D. J.; Baribeau, J.-M.; Houghton, D. C. (1987), "Raman scattering in Fibonacci superlattices",Physical Review Letters,58 (17):1761–1765,Bibcode:1987PhRvL..58.1761D,doi:10.1103/physrevlett.58.1761,PMID 10034529.
• Kimberling, Clark (2004), "Ordering words and sets of numbers: the Fibonacci case", in Howard, Frederic T. (ed.),Applications of Fibonacci Numbers, Volume 9: Proceedings of The Tenth International Research Conference on Fibonacci Numbers and Their Applications, Dordrecht: Kluwer Academic Publishers, pp. 137–144,doi:10.1007/978-0-306-48517-6_14,ISBN 978-90-481-6545-2,MR 2076798.
• Lothaire, M. (1997),Combinatorics on Words, Encyclopedia of Mathematics and Its Applications, vol. 17 (2nd ed.),Cambridge University Press,ISBN 0-521-59924-5.
• Lothaire, M. (2011),Algebraic Combinatorics on Words, Encyclopedia of Mathematics and Its Applications, vol. 90,Cambridge University Press,ISBN 978-0-521-18071-9. Reprint of the 2002 hardback.
• de Luca, Aldo (1995), "A division property of the Fibonacci word",Information Processing Letters,54 (6):307–312,doi:10.1016/0020-0190(95)00067-M.
• Mignosi, F.; Pirillo, G. (1992),"Repetitions in the Fibonacci infinite word",Informatique Théorique et Application,26 (3):199–204,doi:10.1051/ita/1992260301991.
• Ramírez, José L.; Rubiano, Gustavo N.; De Castro, Rodrigo (2014), "A generalization of the Fibonacci word fractal and the Fibonacci snowflake",Theoretical Computer Science,528:40–56,arXiv:1212.1368,doi:10.1016/j.tcs.2014.02.003,MR 3175078,S2CID 17193119.

• A detailed and accessible description, on Ron Knott's site
• Weisstein, Eric W.,"Rabbit Sequence",MathWorld
• Fibonacci Word (First 200,000 bits) onYouTube

• Binary sequences
• Fibonacci numbers

• Articles with short description
• Short description is different from Wikidata