Fibonacci Pattern Research Articles

Algebraic Nexus of Fibonacci Forms and Two-Simplex Topology in Multicellular Morphogenesis

Background: Fibonacci patterns and tubular forms both arose early in the phylogeny of multicellular organisms. Tubular forms offer the advantage of a regulated internal milieu, and Fibonacci forms may offer packing efficiencies. The underlying mechanisms behind the cellular genesis of Fibonacci and tubular forms remain unknown. Methods: In a multicellular organism, cells adhere to form a macrostructure and to coordinate further replication. We propose and prove simple theorems connecting cell replication and adhesion to Fibonacci forms and simplicial topology. Results: We identify some cellular and molecular properties whereby the contact inhibition of replication by adhered cells may approximate Fibonacci growth patterns. We further identify how a component 2→3 cellular multiplication step may generate a multicellular structure with some properties of a two-simplex. Tracking the homotopy of a two-simplex to a circle and to a tube, we identify some molecular and cellular growth properties consistent with the morphogenesis of tubes. We further find that circular and tubular cellular aggregates may be combinatorially favored in multicellular adhesion over flat shapes. Conclusions: We propose a correspondence between the cellular and molecular mechanisms that generate Fibonacci cell counts and those that enable tubular forms. This implies molecular and cellular arrangements that are candidates for experimental testing and may provide guidance for the synthetic biology of hollow morphologies.

Evolutionary origins of Fibonacci phyllotaxis in land plants

Fibonacci phyllotaxis is commonly seen in all major groups of land plants. While a precise correlation is found between the internal pattern of the primary vascular system and the external pattern of appendages on the stem surface, it remains a big question how this regularity of Fibonacci phyllotaxis came into being in the course of evolution. Here I address this problem with a model describing phylogenetic and ontogenetic changes in vascular phyllotaxis based on two hypotheses. The first is that the Fibonacci pattern of vascular connection is uniquely determined by the primary arrangement of incipient primordia, the sources of the primary signal system in vascular tissue differentiation. The second is that the surface-area-to-volume ratio of primary vascular tissues serves as a measure of fitness in evolution. The model explains the empirical rule on the manner in which vascular connection is reconfigured during ontogeny, especially during juvenile development. Fossil and phylogenetic evidence suggests that Fibonacci phyllotaxis appeared shortly after the innovation of indefinite lateral organ initiation in a regular sequence.

Applications of Fibonacci Sequences and Golden Ratio

The study mainly focuses on the use of the Golden Ratio and the Fibonacci sequence. The connection between them can be clearly visible in nature. With the help of the Fibonacci sequence scientists have solved many mysteries related to nature. Everything that is around us somehow or other depends on Fibonacci numbers, the Golden Ratio, and the Fibonacci sequence. Some examples are –Flower petals- Lily, Rose, Daisy, Marigold, Sunflower, Iris, Buttercups, wild rose, larkspur Trillium, Bloodroot, Aster, and Susan; Seed heads-Sunflower; Snail; Fruit-Apple, Banana, Pineapple; Human Face; Tree Branches; Cyclone; Pinecones; Shells; Spiral Galaxies; Bees; Famous architecture design – Taj Mahal, in Hindu rituals, in decoding-coding the data, in providing security to the sensitive data and all over the world, in mother's womb (about her baby's position), etc. The current study reflects that there is no limitation to the Fibonacci pattern and Golden Ratio in our surroundings.

Theory, Design, and Implementation of a New Family of Ultra-Wideband Metamaterial Microstrip Array Antennas Based on Fractal and Fibonacci Geometric Patterns

The theory and design of a new family of ultra-wideband (UWB) metamaterial (MTM) microstrip array antennas based on fractal and Fibonacci geometric patterns are investigated. First, the UWB microstrip array antenna is presented with two radiating MTM elements. Then, using fractal and Fibonacci geometric patterns, the array antenna is expanded. Improvements in the antenna parameters is achieved by repeating the second and third iterations of the fractal and Fibonacci patterns. As the order of iteration of the fractal and Fibonacci geometric patterns increases, the impedance bandwidth of the MTM microstrip array antenna increases, and its radar cross-section (RCS) decreases. The impedance bandwidth of the array antenna with two MTM elements is 3.37–9.2 GHz, while the bandwidth of the third-iteration Fibonacci and fractal MTM array antennas are 3.5–10.1 GHz and 3.55–10.34 GHz, respectively. Furthermore, the proposed array antennas exhibit lower RCS due to metal area reduction, with respect to the array antenna with two MTM elements.

Mineralized scale patterns on the cell periphery of the chrysophyte Mallomonas determined by comparative 3D Cryo-FIB SEM data processing

Unicellular protists can biomineralize spatially complex and functional shells. A typical cell of the photosynthetic synurophyte Mallomonas is covered by about 60–100 silica scales. Their geometric arrangement, the so-called scale case, mainly depends on the species and on the cell cycle. In this study, the scale case of the synurophyte Mallomonas was preserved in aqueous suspension using high-pressure freezing (HPF). From this specimen, a three-dimensional (3D) data set spanning a volume of about 25.6 μm × 19.2 μm × 4.2 μm with a voxel size of 12.5 nm × 12.5 nm × 25.0 nm was collected by Cryo-FIB SEM in 3 h and 24 min. SEM imaging using In-lens SE detection allowed to clearly differentiate between mineralized, curved scales of less than 0.2 μm thickness and organic cellular ultrastructure or vitrified ice. The three-dimensional spatial orientations and shapes of a minimum set of scales (N = 13) were identified by visual inspection, and manually segmented. Manual and automated segmentation approaches were comparatively applied to one arbitrarily selected reference scale using the differences in grey level between scales and other constituents. Computational automated routines and principal component analysis of the experimentally extracted data created a realistic mathematical model based on the Fibonacci pattern theory. A complete in silico scale case of Mallomonas was reconstructed showing an optimized scale coverage on the cell surface, similarly as it was observed experimentally. The minimum time requirements from harvesting the living cells to the final scale case determination by Cryo-FIB SEM and computational image processing are discussed.

Quasiperiodic magnetic chain as a spin filter for arbitrary spin states

We show that a quasiperiodic magnetic chain comprising magnetic atomic sites sequenced in Fibonacci pattern can act as a prospective candidate for spin filters for particles with arbitrary spin states. This can be achieved by tuning a suitable correlation between the amplitude of the substrate magnetic field and the on-site potential of the magnetic sites, which can be controlled by an external gate voltage. Such correlation leads to a spin filtering effect in the system, allowing one of the spin components to completely pass through the system while blocking the others over the allowed range of energies. The underlying mechanism behind this phenomena holds true for particles with any arbitrary spin states S = 1, 3/2, 2, . . ., in addition to the canonical case of spin-half particles. Our results open up the interesting possibility of designing a spin demultiplexer using a simple quasiperiodic magnetic chain system. Experimental realization of this theoretical study might be possible by using ultracold quantum gases, and can be useful in engineering new spintronic devices.

Leaf arrangements are invalid in the taxonomy of orchid species.

The selection and validation of proper distinguishing characters are of crucial importance in taxonomic revisions. The modern classifications of orchids utilize the molecular tools, but still the selection and identification of the material used in these studies is for the most part related to general species morphology. One of the vegetative characters quoted in orchid manuals is leaf arrangement. However, phyllotactic diversity and ontogenetic changeability have not been analysed in detail in reference to particular taxonomic groups. Therefore, we evaluated the usefulness of leaf arrangements in the taxonomy of the genus Epipactis Zinn, 1757. Typical leaf arrangements in shoots of this genus are described as distichous or spiral. However, in the course of field research and screening of herbarium materials, we indisputably disproved the presence of distichous phyllotaxis in the species Epipactis purpurata Sm. and confirmed the spiral Fibonacci pattern as the dominant leaf arrangement. In addition, detailed analyses revealed the presence of atypical decussate phyllotaxis in this species, as well as demonstrated the ontogenetic formation of pseudowhorls. These findings confirm ontogenetic variability and plasticity in E. purpurata. Our results are discussed in the context of their significance in delimitations of complex taxa within the genus Epipactis.

Engineering electronic states of periodic and quasiperiodic chains by buckling

The spectrum of spinless, non-interacting electrons on a linear chain that is buckled in a non-uniform, quasiperiodic manner is investigated within a tight binding formalism. We have addressed two specific cases, viz., a perfectly periodic chain wrinkled in a quasiperiodic Fibonacci pattern, and a quasiperiodic Fibonacci chain, where the buckling also takes place in a Fibonacci pattern. The buckling brings distant neighbors in the parent chain to close proximity, which is simulated by a tunnel hopping amplitude. It is seen that, in the perfectly ordered case, increasing the strength of the tunnel hopping (that is, bending the segments more) absolutely continuous density of states is retained towards the edges of the band, while the central portion becomes fragmented and host subbands of narrowing widths containing extended, current carrying states, and multiple isolated bound states formed as a result of the bending. A switching “on” and “off” of the electronic transmission can thus be engineered by buckling. On the other hand, in the second example of a quasiperiodic Fibonacci chain, imparting a quasiperiodic buckling is found to generate continuous subband(s) destroying the usual multifractality of the energy spectrum. We present exact results based on a real space renormalization group analysis, that is corroborated by explicit calculation of the two terminal electronic transport.

The ontogeny, phyllotactic diversity, and discontinuous transitions of Diphasiastrum digitatum (Lycopodiaceae)

Fibonacci phyllotactic patterns in seed plants are well documented, but whether such predominance holds true for lower vascular plants is relatively unknown. We investigated Diphasiastrum digitatum (Lycopodiaceae) phyllotaxis throughout its ontogeny to extend our knowledge of pattern frequency of lower vascular plants and to measure quantitative variables associated with discontinuous phyllotactic transitions. These investigations allowed us to test whether the same mechanisms inherent in shoot apical meristem (SAM) development of seed plants are applicable to early-diverged lower vascular plants SAM development. Divergence angle, plastochron ratio, leaf insertion angle, circumferential ratio, radial ratio, half conic angle, area, circumference, and circularity of the shoot apical meristem were compared among different phyllotactic patterns and different meristem types observed throughout D. digitatum ontogeny, using scanning electron microscopy. Fibonacci patterns were not predominant during six stages of D. digitatum ontogeny. In all five cases of discontinuous transition associated with strobili formation, divergence angle was the only variable that has changed consistently. The predominance of non-Fibonacci patterns/series in D. digitatum is inconsistent with the prediction of interpretive model of phyllotaxis. We hypothesize this is because its SAM, due to its frequent dichotomy, is not circular and primordia initiation is restricted spatially and temporally at the beginning of pattern formation. Change in divergence angle associated with discontinuous transitions is most likely due to the change of the location of new auxin maxima, due to the change of SAM shape and size.

Printed Aperiodic Cavity for Computational and Microwave Imaging

We demonstrate a frequency-diverse aperture for microwave imaging based on a planar cavity at K-band frequencies (18–26.5 GHz). The structure consists of an array of radiating circular irises patterned into the front surface of a double-sided printed circuit board. The irises are distributed in a Fibonacci pattern to maximize spatial diversity at the scene. The printed cavity is a phase-diverse system and encodes imaged scene information onto a set of frequencies that span the K-band. Similar to recently reported metamaterial apertures, the printed cavity imager does not require any mechanically moving parts or complex phase shifting networks. Imaging of a number of targets is shown; these reconstructed images demonstrate the ability of the system to perform imaging at the diffraction limit. The proposed printed cavity imager possesses a relatively large quality factor that can be traded off to achieve higher radiation efficiency. The general mode characteristics of the printed cavity suggest advantages when used in computational imaging scenarios.

Generalized Fibonacci – Like Sequence Associated with Fibonacci and Lucas Sequences

The Fibonacci sequence, Lucas numbers and their generalization have many interesting properties and applications to almost every field. Fibonacci sequence is defined by the recurrence formula Fn=Fn-1+Fn-2, , and F0=0, F1=1, where Fn is a nth number of sequence. Many authors have been defined Fibonacci pattern based sequences which are popularized and known as Fibonacci-Like sequences. In this paper, Generalized Fibonacci-Like sequence is introduced and defined by the recurrence relation Bn=Bn-1+Bn-2, with B0=2s, B1=s+1, where s being a fixed integers. Some identities of Generalized Fibonacci-Like sequence associated with Fibonacci and Lucas sequences are presented by Binet’s formula. Also some determinant identities are discussed.

Aberrant phyllotactic patterns in cones of some conifers: a quantitative study

The scale patterns of 6000 cones from one single tree of <em>Pinus nigra</em> Arn. have been examined. Apart from the main Fibonacci pattern with 8 and 13 parastichies, nine aberrant spiral patterns with Fibonacci-type sequences have been found. They are quite rare and occur with different frequencies. The parastichy quotient 8/13 of the prevalent pattern is very close to the golden ratio 0.618. In case of the black pine it appeared that the greater the deviation of the parastichy quotient <em>m</em>/<em>n</em> from 0.618, the rarer the pattern. Similar results obtained for the sample of 1506 cones collected from three individual trees of larch (<em>Larix decidua</em> Mill.) suggest a true correlation between the frequency of a pattern and the deviation of its parastichy quotient from the golden ratio.

Big Solar Spiral of Stirring Up Global Systemic Conflicts

The authors are based on the Fibonacci pattern of global systemic conflicts (C-waves) and formulate the hypotheses about a metric relation between two global periodic processes, namely, between the sequence of 11-year cycles of solar activity and the process of evolutionary structurization of the family of C-waves of global systemic conflicts covering large and super-large time intervals and having a variable structural configuration.

Identities of Generalized Fibonacci-Like Sequence

The Fibonacci and Lucas sequences are well-known examples of second order recurrence sequences. The Fibonacci sequence, Lucas numbers and their generalization have many interesting properties and applications to almost every field. Fibonacci sequence is defined by the recurrence formula Fn=Fn-1+Fn-2, n≥2 and F0=0, F1=1, where Fn is a nth number of sequence. Many authors have defined Fibonacci pattern based sequences which are popularized and known as Fibonacci-Like sequences. In this paper, Generalized Fibonacci-Like sequence is introduced and defined by the recurrence relation Mn=Mn-1+Mn-2, n≥2, with M0=2, M1=s+1, where s being a fixed integers. Some identities of Generalized Fibonacci-Like sequence are presented by Binet’s formula. Also some determinant identities are discussed.

On a multisection style binomial summation identity for Fibonacci numbers

The Fibonacci sequence is obtained as weighted sum along the rows in the Pascal triangle by choosing a periodic up-and-down pattern of weights from the set $\{-1,-\frac{1}{2},0, \frac{1}{2}, 1\}$. A graphical illustration of this identity shows a novel "`beautiful"' Fibonacci pattern.

Phyllotaxic diversity in Magnolia flowers

Impermanent initials and thus the shift of the axis in the stochastic meristems are postulated to be responsible for ontogenetic phyllotactic diversity in plants. In the gynoecium of <i>Magnolia acuminata</i> the main Fibonacci pattern develops in much less than 50% of individual generative shoots. There is also an extremely wide spectrum of other patterns, among them even the rarest I, 3, 8, 11 ... pattern is present. Regarded sometimes as "impossible", the pattern has been documented in SEM for the first time. Beside the presence of various patterns, frequent ontogenetic transformations of phyllotaxis have been found in <i>Magnolia</i>. These are indicated by dislocations in the periodic distribution of carpel primordia. In other magnolias, exemplified by <i>M. soulangeana</i>, the Fibonacci pattern prevails, but not as much as in coniferous vegetative shoots, where, as demonstrated earlier, it reaches 95%. Other pattern numbers are also different. This suggests the involvement of the genetic factor and may be attributed to the higher frequency of discontinuous phyllotactic transformations in some species. The stochastic character of the meristem is perhaps more pronounced in some plants, which leads in turn to more frequent transitions and greater pattern diversity.

Ontogenetic changes of phyllotaxis in Anagallis arvensis L.

During the ontogeny of <i>Anagallis</i> spontaneous changes of phyllotaxis appear in a regular sequence. The initial decussate pattern is followed by spiral Fibonacci phyllotaxis, this in turn, by a trimerous pattern, and finally Lucas spiral phyllotaxis is formed. In the course of the first and most common phyllotactic transition, from the decussate to spiral Fibonacci pattern, changes in primordia arrangement occur only within a limited sector of the apex circumference. In the complementary sector, primordia emerge as if the decussate phyllotaxis continued. It is likely that similar circumferential discontinuity accounts for further transitions. The common ontogenetic sequence of patterns in <i>Anagallis</i> is such that, theoretically, each transition requires minimal changes in shoot apex geometry. Although the meristem in <i>Anagallis</i> is able to produce primordia either in whorls or spirally, the elongated shoots of this plant seem to have leaves exclusively in whorls. It appeared that in shoots with an initially spiral pattern, leaves can be clustered in pseudo-whorls due to the uneven internode elongation. Pseudowhorls are composed usually of three (Fibonacci) or four (Lucas) leaves of successive nodes. The number of leaves in a pseudo-whorl equals the number of leaves positioned on one revolution of the ontogenetic helix, which is different in these two spiral patterns. In shoot apices with whorled phyllotaxis, the leaf and flower primordia of a whorl are of different size. On elongated shoots, flower buds emerging in the axils of leaves of one whorl also differ in size.

Phyllotactic spectra in cacti: Mammilaria species and some genera from Rebutia group

Phyllotaxis of globular or hemispheric cacti from <em>Mammillaria</em> genus and from <em>Rebutia</em> group shows the significant differences between the two groups. An extremely wide spectrum of phyllotaxis with many infrequent patterns has been found in <em>Rebutia</em> group while in <em>Mammilluria</em> genus it appears to be quite narrow with the main Fibonacci pattern dominant. Therefore one of possible factors responsible for these differences might be a genetic one. Ontogenetic transformations of phyllotaxis both continuous (quantitative) and discontinuous (qualitative) have been found in studied cacti. These developmental events together with dichotomy frequently occurring in some taxa increase the level of phyllotaxis diversity. As more than one expression of the pattern occurred due to continuous transformations in many studied cacti - the numbers of opposite spirals can no longer be treated as a feature of the diagnostic value in cactus taxonomy. Because of intense developmental changes significant is rather a range of phyllotaxis diversity (phyllotactic spectrum) than a single pattern or its expression in a given taxon.

A Bio-Inspired Patch Antenna Array Using Fibonacci Sequences in Trees [Education Column

This article presents the design, simulation, implementation, and test procedures for a bio-inspired patch antenna array and transmission system. The array structure was based on the Fibonacci pattern found in oak trees, with the antenna elements in the positions of the leaves. We present the antenna array's validation in an anechoic chamber, and tests performed on the transmission system under non-line-of-sight (NLOS) conditions.

An Efficient Algorithm based on Fibonacci pattern Hound packets for Detection of Wormhole Attack in DSR for Mobile Ad Hoc Networks

networks are playing very important role in the present world. Mobile Ad hoc Networks (MANET) are the extension of the wireless networks. These networks are playing crucial role in the each and every field of the human life. They are used in those places where a simple wireless network cannot use. They play a significant role in real tile applications such as military applications, home applications wireless sensor applications etc. Due to their adaptive nature they are threatened by number of attacks such as Modification, Black Hole attack, Wormhole attack etc. Wormhole attack is one of the dangerous active attacks in the mobile Ad hoc Networks (MANET). In this paper a secure and efficient approach for the detection of the wormhole attack in the Mobile Ad Hoc Networks (MANET) is described. The algorithm is implemented in a very popular on demand routing protocol, called DSR (Dynamic Source Routing) protocol. In this approach the solution is provided by sending the Hound Packet in the order of Fibonacci series and also modifies the table entries in terms of storing them in sorted order, so that the processing of Route Discovery becomes faster.