The Fibonacci word is an example of a Sturmian word. The start of the cutting sequence shown here illustrates the start of the word 0100101001.
In mathematics, a Sturmian word (Sturmian sequence or billiard sequence[1]), named after Jacques Charles François Sturm, is a certain kind of infinitely long sequence of characters. Such a sequence can be generated by considering a game of English billiards on a square table. The struck ball will successively hit the vertical and horizontal edges labelled 0 and 1 generating a sequence of letters.[2] This sequence is a Sturmian word.
^Hordijk, A.; Laan, D. A. (2001). "Bounds for Deterministic Periodic Routing sequences". Integer Programming and Combinatorial Optimization. Lecture Notes in Computer Science. Vol. 2081. p. 236. doi:10.1007/3-540-45535-3_19. ISBN 978-3-540-42225-9.
^Győri, Ervin; Sós, Vera (2009). Recent Trends in Combinatorics: The Legacy of Paul Erdős. Cambridge University Press. p. 117. ISBN 978-0-521-12004-3.
In mathematics, a Sturmianword (Sturmian sequence or billiard sequence), named after Jacques Charles François Sturm, is a certain kind of infinitely...
addition. It is a paradigmatic example of a Sturmianword and specifically, a morphic word. The name "Fibonacci word" has also been used to refer to the members...
another name for eigenvector of a matrix Characteristic word, a subclass of Sturmianword Euler characteristic, a topological invariant Method of characteristics...
In mathematics, a Sturmian sequence may refer to: A Sturmianword: a sequence with minimal complexity function A sequence used to determine the number...
different irrational number. Beatty sequences can also be used to generate Sturmian words. Any irrational number r {\displaystyle r} that is greater than one...
"compact variant" It is conjectured that the Fibonacci word fractal appears for every sturmianword for which the slope, written in continued fraction expansion...
word, a sequence such that any factor (consecutive subsequence) that appears does so infinitely often, such as the Thue–Morse sequence or a Sturmian word...
number for which the digits with respect to some fixed base form a Sturmianword. The paperfolding constant (also named as "Gaussian Liouville number")...
complexity function at most n+1. A Sturmianword over a binary alphabet is one with complexity function n + 1. A sequence is Sturmian if and only if it is balanced...
where z is a letter not in A. Cutting sequence Lyndon word Hall wordSturmianword Lothaire (2005) p.524 Lothaire (2011) p. 10 Honkala (2010) p.505 Lothaire...
al. (2004) Synchronisation of a resonant tunneling diode oscillators Sturmianword Arnol'd, V.I. (1961). "Small denominators. I. Mapping the circle onto...
equivalent definitions of Sturmian words. For example, an infinite word is Sturmian if and only if it has n + 1 {\displaystyle n+1} distinct factors of...
individual grid lines crossed ("cut") as a curve crosses a square grid. Sturmian words are a special case of cutting sequences where the curves are straight...
periodic (meaning periodic after some nonperiodic initial segment). All Sturmian words are uniformly recurrent. Lothaire (2011) p. 30 Allouche & Shallit...
called the alphabet) is considered. A finite sequence of symbols is called a "word over A", and the free monoid A∗ is called the "Kleene star of A". Thus, the...
of real numbers encoded by lines of tiles (the encoding is related to Sturmian sequences made as the differences of consecutive elements of Beatty sequences)...
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