Two-dimensional Fibonacci Research Articles

On the Construction of 3D Fibonacci Spirals

The paper aims to extend the classical two-dimensional (2D) Fibonacci spiral into three-dimensional (3D) space by using geometric constructions starting from cubic Fibonacci identities and relying on affine maps and parametrizations of the curves. We have already performed a comprehensive survey of cubic Fibonacci identities, which, to our surprise, uncovered only a handful of homogenous cubic identities. Obviously, the goal here is to show how one could use a particular homogenous cubic Fibonacci identity for generating 3D geometric designs similar in spirit to the way the classical Fibonacci spiral is built in 2D starting from a quadratic Fibonacci identity. This made us realize that for any cubic identity there are many different ways of packing cuboids, while only an insignificant fraction of those possible tilings might allow a smooth spiral-like curve to be drawn through them. After reviewing the state of the art, we present accurate details on ways to construct such 3D spirals using affine maps. We go on to prove the continuity and smoothness of such 3D spirals by giving a parametrization of the intersection of the surfaces that define the curves. Throughout the paper, we visualize the resulting 3D spirals by generating geometrically correct stereoscopic views. Finally, it is to be mentioned that the recursive 3D packing of cuboids tends to lead to fractal structures, which will need further investigations.

Surreal Birthdays and Their Arithmetic

Summary This paper is about the structure underlying surreal numbers, namely, the birthdays of surreal number forms. The results are intriguing because (i) addition of surreal forms results in simple addition of their birthdays, but (ii) multiplication results in a much more complicated structure of birthdays, given by a kind of two-dimensional Fibonacci sequence. The paper also provides many examples and illustrations designed to help a student become familiar with this interesting set of numbers.

Two-dimensional Fibonacci words: Tandem repeats and factor complexity

If x is a non-empty string then the repetition xx is called a tandem repeat. Similarly, a tandem in a two-dimensional array X is a configuration consisting of copies of a same primitive block W that touch each other with one side or a corner. In [6], Apostolico and Brimkov have proved various bounds for the number of tandems in a two-dimensional word of size m×n. Of the two types of tandems considered therein, they also proved that, for one type, the number of occurrences in an m×n Fibonacci array attained the general upper bound, O(m2nlogn). In this paper, we derive an expression for the exact number of tandems in a given finite Fibonacci array fm,n. As a required result, we derive the factor complexities of fm,n, m,n≥0 and that of the infinite Fibonacci word f∞,∞. Generations of f∞,∞ and fm,n, for any given m,n≥1 using a two-dimensional homomorphism are also achieved.

Multiple-image encryption algorithm based on Sarrus rule and 3D Fibonacci matrix

Using images to transmit information becomes more common in the new era, but the image information can easily be stolen and tampered with the malicious damage during transmission. In this paper, firstly, inspired by the famous Sarrus rule the theory of linear algebra theory, the Sarrus model is established to scramble images; secondly, the pixel positions of plain images are scrambled with the established Sarrus model to obtain the stereo scrambled matrix; thirdly, two-dimensional Fibonacci matrix is extended to three-dimensional (3D) space; finally, the image matrix is divided into cubes which are diffused by the 3D Fibonacci matrix. Experimental analyses indicate that the new algorithm has the characteristics of large key space, high key sensitivity, high efficiency and can resist to the plaintext attacks, statistical attacks, etc.

Application of Two-Dimensional Fibonacci Wavelets in Fractional Partial Differential Equations Arising in the Financial Market

Application of Two-Dimensional Fibonacci Wavelets in Fractional Partial Differential Equations Arising in the Financial Market

Bose-Einstein Condensation and Quasicrystals

We consider interacting Bose particles in an external local potential. It is shown that large class of external quasicrystal potentials cannot sustain any type of Bose-Einstein condensates. Accordingly, at spatial dimensions D ≤ 2 in such quasicrystal potentials a supersolid is not possible via Bose-Einstein condensates at finite temperatures. The latter also hold true for the two-dimensional Fibonacci tiling. However, supersolids do arise at D ≤ 2 via Bose-Einstein condensates from infinitely long-range, nonlocal interparticle potentials.

Fibonacci array-based focused acoustic camera for estimating multiple moving sound sources

Multiple moving sound sources’ aliasing problem occurs in many fields. The number and relative position of these sources provide critical information for further signal processing. However, estimating the number and relative position under aliasing of multiple moving sound sources is challenging. In this study, the Fibonacci sequence was applied to design a novel two-dimensional Fibonacci array, which has lower maximum sidelobe level than other typical array structures in a certain frequency scope. Conventional beamforming was utilized as an acoustic camera for stationary sound source localization by two-dimensional spatial scanning. Unlike the acoustic camera, a spatiotemporal map (STM) method was developed to estimate the number and relative position of multiple moving sound sources through spatial scanning and temporal focusing. STM is called focused acoustic camera because of the spatiotemporal information extraction of dynamic sources. A deconvolution approach called clean STM (CSTM) was created to estimate the spatiotemporal parameters for further signal processing and to overcome the influence of array pattern on STM. CSTM was combined with two-dimensional time-varying spatial filtering rearrangement and harmonic product spectrum-based sparse filtering enhancement. Consequently, a systematic technique solution was developed to be potentially applied in wayside acoustic bearing fault diagnosis systems. Simulation and experimental cases verified the effectiveness of the proposed method and the systematic technique solution.

Two-dimensional Fibonacci grating for far-field super-resolution imaging

A two-dimensional (2D) Fibonacci grating is used to transform evanescent waves into propagating waves for far-field super-resolution imaging. By detecting far-field intensity distributions of light field through objects in front of the 2D Fibonacci grating in free space at once, we can retrieve the image of objects with beyond λ/7 spatial resolution. We also find that the coherent illumination case can give a better resolution than incoherent illumination case by such 2D grating-assisted imaging system. The analytical results are verified by numerical simulation.

Light localization in optically induced deterministic aperiodic Fibonacci lattices

As light localization becomes increasingly pronounced in photonic systems with less order, we investigate optically induced two-dimensional Fibonacci structures that are supposed to be among the most ordered realizations of deterministic aperiodic patterns. For the generation of corresponding refractive index structures, we implement a recently developed incremental induction method using nondiffracting Bessel beams as waveguide formation entities. Even though Fibonacci structures present slightly reduced order, we show that transverse light transport here is significantly hampered in comparison with discrete diffraction in a periodic lattice. Numerical simulations that support our experimental findings help to identify three cases of input waveguide configurations that significantly determine the initial propagation in a Fibonacci structure. These crucial starting conditions determine the character of light transport, yielding either localization or enhanced expansion. A diverse set of light transport scenarios is identified therein.

Scattering performance of plasmonic nanorod antennas in randomly tilted disordered and Fibonacci configurations

In this letter, we formulate and investigate the optical performance of finite arrays of plasmonic nanorod antennas in randomly tilted disordered and Fibonacci configurations. To efficiently model the scattering performance of a nanorod, we take advantage of characteristic basis function method in conjunction with macro basis functions. We study how random rotations of the nanorods, as a disorder, may affect their electromagnetic response. Also, the impact of quasi-periodic order on the performance of the array in the form of two-dimensional Fibonacci quasi-lattice comprising of plasmonic nanorods with two different lengths is studied.

Measurement of surface plasmon correlation length differences using Fibonacci deterministic hole arrays

Using terahertz (THz) transmission measurements through two-dimensional Fibonacci deterministic subwavelength hole arrays fabricated in metal foils, we find that the surface plasmon-polariton (SPP) correlation lengths for aperiodic resonances are smaller than those associated with the underlying grid. The enhanced transmission spectra associated with these arrays contain two groups of Fano-type resonances: those related to the two-dimensional Fibonacci structure and those related to the underlying hole grid array upon which the aperiodic Fibonacci array is built. For both groups the destructive interference frequencies at which transmission minima occur closely match prominent reciprocal vectors in the hole array (HA) structure-factor in reciprocal space. However the Fibonacci-related transmission resonances are much weaker than both their calculated Fourier intensity in k space and the grid-related resonances. These differences may arise from the complex, multi-fractal dispersion relations and scattering from the underlying grid arrays. We also systematically studied and compared the transmission resonance strength of Fibonacci HA and periodic HA lattices as a function of the number of holes in the array structure. We found that the Fibonacci-related resonance strengths are an order of magnitude weaker than that of the periodic HA, consistent with the smaller SPP correlation length for the aperiodic structure.

Hyperbolas over two-dimensional Fibonacci quasilattices

For the number ns(α, β; X) of points (x1, x2) in the two-dimensional Fibonacci quasilattices \( \mathcal{F}_m^2 \) of level m = 0, 1, 2,… lying on the hyperbola x12 − αx22 = β and such that 0 ≤ x1 ≤ X, x2 ≥ 0, the asymptotic formula $$ {n_s}\left( {\alpha, \beta; X} \right)\sim {c_s}\left( {\alpha, \beta } \right)\ln X\,\,\,\,{\text{as}}\,\,\,\,X \to \infty $$ is established, and the coefficient cs(α, β) is calculated exactly. Using this, we obtain the following result. Let Fm be the Fibonacci numbers, Ai ∈ \( \mathbb{N} \), i = 1, 2, and let \( \overleftarrow {{A_i}} \) be the shift of Ai in the Fibonacci numeral system. Then the number ns(X) of all solutions (A1, A2) of the Diophantine system $$ \left\{ {\begin{array}{*{20}{c}} {A_1^2 + \overleftarrow {A_1^2} - 2{A_2}{{\overleftarrow A }_2} + \overleftarrow {A_2^2} = {F_{2s}},} \\ {\overleftarrow {A_1^2} - 2{A_1}{{\overleftarrow A }_1} + A_2^2 - 2{A_2}{{\overleftarrow A }_2} + 2\overleftarrow {A_2^2} = {F_{2s - 1}},} \\ \end{array} } \right. $$ 0 ≤ A1 ≤ X, A2 ≥ 0, satisfies the asymptotic formula $$ {n_s}(X)\sim \frac{{{c_s}}}{{{\text{ar}}\cosh \left( {{{1} \left/ {\tau } \right.}} \right)}}\ln X\,\,\,\,{\text{as}}\,\,\,\,X \to \infty . $$ Here τ = (−1 + √5)/2 is the golden ratio, and cs = 1/2 or 1 for s = 0 or s ≥ 1, respectively.

Band structure of phononic crystal constructed by Fibonacci super-cell on square lattice

Using the plane-wave expansion method, we study the band structure of a phononic crystal with water rods embedded in mercury to form a super-cell according to the two-dimensional Fibonacci array. It is found that each band splits into three sub-bands with the change of quasilattice constants and there exist intermediate states, which reflect the peculiar properties of quasiperiodic systems.

Polynomial Fibonacci–Hessenberg matrices

A Fibonacci–Hessenberg matrix with Fibonacci polynomial determinant is referred to as a polynomial Fibonacci–Hessenberg matrix. Several classes of polynomial Fibonacci–Hessenberg matrices are introduced. The notion of two-dimensional Fibonacci polynomial array is introduced and three classes of polynomial Fibonacci–Hessenberg matrices satisfying this property are given.

Quasi-periodic distribution of plasmon modes in two-dimensional Fibonacci arrays of metal nanoparticles

In this paper we investigate for the first time the near-field optical behavior of two-dimensional Fibonacci plasmonic lattices fabricated by electron-beam lithography on transparent quartz substrates. In particular, by performing near-field optical microscopy measurements and three dimensional Finite Difference Time Domain simulations we demonstrate that near-field coupling of nanoparticle dimers in Fibonacci arrays results in a quasi-periodic lattice of localized nanoparticle plasmons. The possibility to accurately predict the spatial distribution of enhanced localized plasmon modes in quasi-periodic Fibonacci arrays can have a significant impact for the design and fabrication of novel nano-plasmonics devices.

Electronic transport in multidimensional Fibonacci lattices

In this article, the Kubo-Greenwood formula is used to investigate the electronic transport behaviour in macroscopic systems by means of an exact renormalization method. The convolution technique is employed in the analysis of two-dimensional Fibonacci lattices. The dc electrical conductance spectra of multidimensional systems exhibit a quantized behaviour when the electric field is applied along a periodically arranged atomic direction, and it becomes a devil's stair if the perpendicular subspace of the system is quasiperiodic. The spectrally averaged conductance shows a power-law decay as the system length grows, neither constant as in periodic systems nor exponential decays occurred in randomly disordered lattices, revealing the critical localization nature of the eigenstates in quasicrystals. Finally, the ac conductance along periodic and quasiperiodic directions is compared with the optical conductivity measured in decagonal quasicrystals.

Kubo conductivity in two-dimensional Fibonacci lattices

The electronic transport at zero degrees in quasiperiodic systems is investigated by using the Kubo–Greenwood formula and a novel renormalization method, which allows an evaluation in an exact way of the products of the Green’s function in macroscopic Fibonacci chains. The analysis of transport properties in two-dimensional Fibonacci superlattices and in double quasiperiodic lattices is carried out by means of the convolution technique. The spectrally averaged conductance shows a linear dependence with the width of the system and a power-law decay as its length increases along the applied electric field.

Exact results of the Kubo conductivity in macroscopic Fibonacci systems: a renormalization approach

In this work, the Kubo–Greenwood formula is used to investigate the electrical conduction in macroscopic Fibonacci lattices within a single-band tight-binding model. This investigation is carried out by means of a renormalization method, which allows the iterative evaluation of the products of the Green’s function in an exact way. The results of d.c. conductivity show an extremely fine band structure and a periodic oscillating pattern in the neighborhood of the transparent state. The a.c. conductivity of these transparent states as a function of the frequency shows a regular oscillating behavior, whose maximums decay following an inverse power law. Furthermore, the d.c. conduction in two-dimensional Fibonacci superlattices reveals a smooth dependence on the Fermi energy location and finally the transition from one- into two-dimensional conductivity is also analyzed.

Electronic spectrum of the two-dimensional Fibonacci lattice

The electron spectrum and wave functions of the two-dimensional Fibonacci lattice with “central” and “vertex” decorations are considered. The energy spectrum contains a singular part but does not contain a hierarchical gap structure. The measure of permittable states of the spectrum (the Lebesgue measure) is not equal to zero and in the case of central decoration is close to unity. Most of the wave functions are critical.

Electronic spectrum of a two-dimensional Fibonacci lattice

The electronic spectrum and wave functions of a new quasicrystal structure—a two-dimensional Fibonacci lattice—are investigated in the tight-binding approximation using the method of the level statistics. This is a self-similar structure consisting of three elementary structural units. The “central” and “nodal” decoration of this structure are examined. It is shown that the electronic energy spectrum of a two-dimensional Fibonacci lattice contains a singular part, but in contrast to a one-dimensional Fibonacci lattice the spectrum does not contain a hierarchical gap structure. The measure of allowed states (Lebesgue measure) of the spectrum is different from zero, and for “central” decoration it is close to 1. The character of the localization of the wave functions is investigated, and it is found that the wave functions are “critical.”