Quasiperiodicities in Fibonacci strings

We consider the problem of finding quasiperiodicities in a Fibonacci string. A factor u of a string y is a cover of y if every letter of y falls within some occurrence of u in y. A string v is a seed of y, if it is a cover of a superstring of y. A left seed of a string y is a prefix of y that it is a cover of a superstring of y. Similarly a right seed of a string y is a suffix of y that it is a cover of a superstring of y. In this paper, we present some interesting results regarding quasiperiodicities in Fibonacci strings, we identify all covers, left/right seeds and seeds of a Fibonacci string and all covers of a circular Fibonacci string.Comment: In Local Proceedings of "The 38th International Conference on Current Trends in Theory and Practice of Computer Science" (SOFSEM 2012

Fibonacci words in hyperbolic Pascal triangles

The hyperbolic Pascal triangle ${\cal HPT}_{4,q}$ $(q\ge5)$ is a new mathematical construction, which is a geometrical generalization of Pascal's arithmetical triangle. In the present study we show that a natural pattern of rows of ${\cal HPT}_{4,5}$ is almost the same as the sequence consisting of every second term of the well-known Fibonacci words. Further, we give a generalization of the Fibonacci words using the hyperbolic Pascal triangles. The geometrical properties of a ${\cal HPT}_{4,q}$ imply a graph structure between the finite Fibonacci words.Comment: 10 pages, 4 figures, Acta Univ. Sapientiae, Mathematica, 201

Local symmetry dynamics in one-dimensional aperiodic lattices

A unifying description of lattice potentials generated by aperiodic one-dimensional sequences is proposed in terms of their local reflection or parity symmetry properties. We demonstrate that the ranges and axes of local reflection symmetry possess characteristic distributional and dynamical properties which can be determined for every aperiodic binary lattice. A striking aspect of such a property is given by the return maps of sequential spacings of local symmetry axes, which typically traverse few-point symmetry orbits. This local symmetry dynamics allows for a classification of inherently different aperiodic lattices according to fundamental symmetry principles. Illustrating the local symmetry distributional and dynamical properties for several representative binary lattices, we further show that the renormalized axis spacing sequences follow precisely the particular type of underlying aperiodic order. Our analysis thus reveals that the long-range order of aperiodic lattices is characterized in a compellingly simple way by its local symmetry dynamics.Comment: 15 pages, 12 figure