HOME

TheInfoList



OR:

A Fibonacci word is a specific sequence of binary digits (or symbols from any two-letter
alphabet An alphabet is a standard set of letter (alphabet), letters written to represent particular sounds in a spoken language. Specifically, letters largely correspond to phonemes as the smallest sound segments that can distinguish one word from a ...
). The Fibonacci word is formed by repeated
concatenation In formal language theory and computer programming, string concatenation is the operation of joining character strings end-to-end. For example, the concatenation of "snow" and "ball" is "snowball". In certain formalizations of concatenati ...
in the same way that the
Fibonacci number In mathematics, the Fibonacci sequence is a Integer sequence, sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted . Many w ...
s are formed by repeated addition. It is a paradigmatic example of a Sturmian word and specifically, a morphic word. The name "Fibonacci word" has also been used to refer to the members of a
formal language In logic, mathematics, computer science, and linguistics, a formal language is a set of strings whose symbols are taken from a set called "alphabet". The alphabet of a formal language consists of symbols that concatenate into strings (also c ...
''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 S_0 be "0" and S_1 be "01". Now S_n = S_S_ (the concatenation of the previous sequence and the one before that). The infinite Fibonacci word is the limit S_, that is, the (unique) infinite sequence that contains each S_n, for finite n, as a prefix. Enumerating items from the above definition produces: : S_0    0 : S_1    01 : S_2    010 : S_3    01001 : S_4    01001010 : 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, ...


Closed-form expression for individual digits

The ''n''th digit of the word is 2 + \lfloor n \varphi \rfloor - \lfloor (n+1)\varphi \rfloor where \varphi is the
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if \fr ...
and \lfloor \,\ \rfloor is the
floor function In mathematics, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least integer greater than or eq ...
. As a consequence, the infinite Fibonacci word can be characterized by a cutting sequence of a line of slope 1/\varphi or \varphi-1. See the figure above.


Substitution rules

Another way of going from ''S''''n'' to ''S''''n''+1 is to replace each symbol 0 in ''S''''n'' with the pair of consecutive symbols 0, 1 in ''S''''n''+1, and to replace each symbol 1 in ''S''''n'' with the single symbol 0 in ''S''''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 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''th digit of the word is \lfloor n \varphi \rfloor - \lfloor (n-1)\varphi \rfloor - 1.


Discussion

The word is related to the famous sequence of the same name (the
Fibonacci sequence In mathematics, the Fibonacci sequence is a Integer sequence, sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted . Many w ...
) in the sense that addition of integers in the
inductive definition In mathematics and computer science, a recursive definition, or inductive definition, is used to define the elements in a set in terms of other elements in the set ( Aczel 1977:740ff). Some examples of recursively definable objects include fact ...
is replaced with string concatenation. This causes the length of ''S''''n'' to be ''F''''n''+2, the (''n''+2)nd Fibonacci number. Also the number of 1s in ''S''''n'' is ''F''''n'' and the number of 0s in ''S''''n'' is ''F''''n''+1.


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 A palindrome (Help:IPA/English, /ˈpæl.ɪn.droʊm/) is a word, palindromic number, number, phrase, or other sequence of symbols that reads the same backwards as forwards, such as ''madam'' or ''racecar'', the date "Twosday, 02/02/2020" and th ...
. Example: 01''S''4 = 0101001010 is a palindrome. The palindromic density of the infinite Fibonacci word is thus 1/φ, where φ is the
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if \fr ...
: this is the largest possible value for aperiodic words. * 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 factors of the same length anywhere in the Fibonacci word. The difference between their
Hamming weight The Hamming weight of a string (computer science), string is the number of symbols that are different from the zero-symbol of the alphabet used. It is thus equivalent to the Hamming distance from the all-zero string of the same length. For the mo ...
s (the number of occurrences of "1") never exceeds 1. * The subwords 11 and 000 never occur.For the subwords that do occur, see , pp. 14 and 18 (using the letters a and b in place of the digits 0 and 1) * 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, with slope 1/\varphi. The infinite Fibonacci word is the standard word generated by the directive sequence (1,1,1,....). * The infinite Fibonacci word is recurrent; that is, every subword occurs infinitely often. * If u is a subword of the infinite Fibonacci word, then so is its reversal, denoted u^R. * If u is a subword of the infinite Fibonacci word, then the least period of u is a Fibonacci number. * The concatenation of two successive Fibonacci words is "almost commutative". S_=S_nS_ and S_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, is transcendental. * The letters "1" can be found at the positions given by the successive values of the Upper Wythoff sequence : \lfloor n\varphi^2 \rfloor * The letters "0" can be found at the positions given by the successive values of the Lower Wythoff sequence : \lfloor n\varphi \rfloor * The distribution of n=F_k points on the
unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
, placed consecutively clockwise by the golden angle \frac, generates a pattern of two lengths \frac,\frac 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 S_ 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. The critical exponent for the infinite Fibonacci word is 2+\varphi \approx 3.618. It is the smallest index (or critical exponent) among all Sturmian words. * The infinite Fibonacci word is often cited as the
worst case In computer science, best, worst, and average cases of a given algorithm express what the resource usage is ''at least'', ''at most'' and ''on average'', respectively. Usually the resource being considered is running time, i.e. time complexity, b ...
for algorithms detecting repetitions in a string. * The infinite Fibonacci word is a morphic word, generated in * by the endomorphism 0 → 01, 1 → 0. * The ''n''th element of a Fibonacci word, s_n, is 1 if the Zeckendorf representation (the sum of a specific set of Fibonacci numbers) of ''n'' 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 of fibbinary numbers
modulo In computing and mathematics, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, the latter being called the '' modulus'' of the operation. Given two positive numbers and , mo ...
2.


Applications

Fibonacci based constructions are currently used to model physical systems with aperiodic order such as
quasicrystal A quasiperiodicity, quasiperiodic crystal, or quasicrystal, is a structure that is Order and disorder (physics), ordered but not Bravais lattice, periodic. A quasicrystalline pattern can continuously fill all available space, but it lacks trans ...
s, and in this context the Fibonacci word is also called the Fibonacci quasicrystal. Crystal growth techniques have been used to grow Fibonacci layered crystals and study their light scattering properties.


See also

*
Mathematics and art Mathematics and art are related in a variety of ways. Mathematics has itself been described as an art mathematical beauty, motivated by beauty. Mathematics can be discerned in arts such as Music and mathematics, music, dance, painting, Mathema ...
* Tribonacci word


Notes


References

*. *. * *. *. *. *. *. Reprint of the 2002 hardback. *. *. *.


External links


A detailed and accessible description, on Ron Knott's site
* * {{DEFAULTSORT:Fibonacci Word Binary sequences Fibonacci numbers