Fractal Geometry Of Nature Research Articles

Fractal patterns in music

If aesthetic preferences are affected by the fractal geometry of nature, scaling regularities would be expected to appear in all art forms, including music. While a variety of statistical tools have been proposed to analyze time series in sound, no consensus has yet emerged regarding the most meaningful measure of complexity in music or how to discern fractal patterns in compositions in the first place. Here, we offer a new approach based on the self-similarity of melodic lines recurring at various temporal scales. In contrast to the statistical analyses advanced in recent literature, the proposed method does not depend on averaging within time-windows and is distinctively local. The corresponding definition of the fractal dimension is based on the temporal scaling hierarchy and depends on the tonal contours of musical motifs. The new concepts are tested on musical “renditions” of the Cantor Set and the Koch Curve, and then applied to a number of carefully selected masterful compositions spanning five centuries of music making.

Fractals: The Secret Code of Creation

Fractals: The Secret Code of Creation

Fractal Impedance Element Synthesis Based on Multilayer Resistance-Capacitance Environment with C-R-Nc Layer Structure

The idea of fractural calculus appeared almost at the same time as the development of conventional differential calculus began. However, modern scientists became familiar with it only from pioneering works of B. Mandelbrot on fractal geometry of nature in 1980s. Since then, numerous scientific works were published on various aspects of fractural calculus and its application in science and engineering, and the interest to this area remains high. However, for physical implementation of fractural integration operators two-terminal elements with fractional power and frequency law (the fractural impedance elements - FIE) are required. Commercial FIE that could be produced industrially have not been developed so far. The greatest potential, in this case, have FIE built on the basis of multilayer resistance-capacitance element with distributed parameters (RC-distributed elements). One of the baseline designs of such FIEs with R-C-NR layer structure was manufactured as a prototype model. However, when manufacturing and testing of this element, flaws that may prevent its application were detected. The article suggests making FIE based on RC-distributed elements with dual C-R-NC layer structure where, as it is assumed, it is possible to avoid flaws found in FIE with R-C-NR layer structure. However, to solve this problem it is necessary to develop an algorithm and a program of structure synthesis of new type FIE enabling production elements with parameters not worse than those of the existing FIE with R-C-NR layer structure. For this purpose, an algorithm of frequency characteristic analysis of equivalent circuit FIE based on RC-distributed parameters having C-R-NC layer structure and being the component of synthesis algorithm, was developed. Principal stages of synthesis algorithm based on search optimization genetic algorithm were formulated. Ways of information coding of internal structure and parameter C-R-NC FIE were defined. Method of function calculation of individual fitness within population of object representation and ways of genetic operators (selection, crossing-over and mutation) realization in the process of genetic algorithm performance was described. Thereupon the synthesis program was developed and the results of its operation are given in the article. Synthesis result validity was verified by means of circuit simulation software OrCAD by application of Pspice-based C-R-NC-line model. The obtained FIE equivalent circuits integral C-R-NC structures, taking into account chosen materials and layer parameters, were designed. Comparative analysis results of synthesis program operation of C-R-NC FIE and R-C-NR FIE for synthesis FIE with the same requirements to frequency impedance FIE characteristics are given, showing that synthesizing C-R-NC FIE; the FIE with phase uniformity phase-frequency impedance characteristics from - 45° to about - 80° without galvanic coupling between poles and synthesizing R-C-NR FIE, the FIE with phase uniformity of phase-frequency impedance from -10° to about -45° with galvanic coupling between the pole are more likely to be realized.

The Fractal Geometry of Mathematics Classrooms: Navigating Classroom Environments in Factory-Model Schools

Mathematics education has a diverse and complex history. After decades of national reform, mathematics classrooms today are still in desperate need of change. This article examines the current state of factory-model schooling through the lens of Mandelbrot's The Fractal Geometry of Nature in order to rethink pedagogic practices.

Fraktale – powrót do źródeł

Fractals have become fashionable. Therefore, in recent years there have been many articles in which the authors support something that they call a fractal account, including, for example, the sum of fractal dimensions. This paper is a recapitulation of what a fractal account is in the Earth sciences, what are its uses and boundaries. The definition of fractals is: it has a non trite structure in all scales, it is very hard to describe fractal structure in the Euclidean geometry, it is self-similar (directly or statistical), its Hausdorf dimension is greater than its topological dimension, it is described by recurrent formula, its dimension is not an integral number. In the face of such a wide and imprecise formula, various fields of science have introduced their definitions of fractal. It only has to meet most of the conditions included in the definition. In the analysis of geological objects in Earth sciences and in oil and gas industry, fractals are defined by the recurrence formula with its range of applicability, fractal dimension share a part of the space occupied by the fractal object, so the highest value of fractal dimension is equal to 3. Fundamental work in which the name of fractals for self- similar objects were introduced was The Fractal Geometry of Nature by Mandelbrot (1977). In the Earth sciences, statistical fractals (pseudofractals) are used. The straight line in the log-log plot is the indicator of fractal structure. In other words, the fractal structures are associated with the power patterns obtained during the analysis of geological objects. Generally, in the analysis of geological objects the Menger sponge and box methods of fractal dimension calculations are used. Fractals provide a unique opportunity to characterize complicated objects with the use of a single number, nevertheless, in order for the obtained results to be applicable and comparable with the results of other analyzes, both the model of the analyzed object and the method of calculation of the fractal dimension should be given, as well as the scope of applicability of this dimension.

Bionic architecture: back to the origins and a step forward

The article reviews research in the field of architectural bionics and its historical background. The article considers the development of bionic architecture from its origin till the present day. It was established how various functional elements of live forms, nature as a whole, and architectural shape formation interact and correlate. The basic factors, influencing the shape forming elements of the architectural composition, were identified. The main terms, associated with bionic shape formation, were indicated. Modern trends in bionic architecture were considered. The examples of outstanding architectural specimens and their creators were given. The concepts of architectural bionics and bionic architecture were defined and differentiated. The article considers the development of the bionic architectural style and determines the historical background of its development. The issues of fractal geometry are also addressed by reviewing the French scientist B. Mandelbrot's treatise "The Fractal Geometry of Nature". The conclusions were made about the advisability of using fractal algorithms in architectural morphogenesis. The article considers the principles of bionic architectural design that contribute to the development of architects' imaginative and analytical thinking and allow them to apply innovative methods of shape formation in town planning. It was concluded that one of the most important conditions for the optimal functioning of an artificial environment is its organic unity with natural environment, as well as mimesis (deliberate imitation), which is an objective prerequisite of the bionic trend in architecture.

Bionic architecture: back to the origins and a step forward

The article reviews research in the field of architectural bionics and its historical background. The article considers the development of bionic architecture from its origin till the present day. It was established how various functional elements of live forms, nature as a whole, and architectural shape formation interact and correlate. The basic factors, influencing the shape forming elements of the architectural composition, were identified. The main terms, associated with bionic shape formation, were indicated. Modern trends in bionic architecture were considered. The examples of outstanding architectural specimens and their creators were given. The concepts of architectural bionics and bionic architecture were defined and differentiated. The article considers the development of the bionic architectural style and determines the historical background of its development. The issues of fractal geometry are also addressed by reviewing the French scientist B. Mandelbrot’s treatise “The Fractal Geometry of Nature”. The conclusions were made about the advisability of using fractal algorithms in architectural morphogenesis. The article considers the principles of bionic architectural design that contribute to the development of architects’ imaginative and analytical thinking and allow them to apply innovative methods of shape formation in town planning. It was concluded that one of the most important conditions for the optimal functioning of an artificial environment is its organic unity with natural environment, as well as mimesis (deliberate imitation), which is an objective prerequisite of the bionic trend in architecture.

From Fractal Geometry to Fractal Analysis

Euclidian geometry pertained only to the artificial realities of the first, second and third dimensions. Fractal geometry is a new branch of mathematics that proves useful in representing natural phenomena whose dimensions (fractal dimensions) are non-integer values. Fractal geometry was conceived in the 1970s, and mainly developed by Benoit Mandelbrot. In fractal geometry fractals are normally the results of an iterative or recursive construction using corresponding algorithm. Fractal analysis is a nontraditional mathematical and experimental method derived from Mandelbrot’s Fractal Geometry of Nature, Euclidean geometry and calculus. The main aims of the present study are: 1) to address the dimensional imbalances in some texts on fractal geometry, proving that logarithm of a physical quantity (e.g. length of a segment) is senseless; 2) to define the modified capacity dimension, calculate its value for Koch fractal set and show that such definition satisfies basic demands of physics, before all the dimensional balance; and 3) to calculate theoretically the fractal dimension of a circle of unit radius. A quantitative determination of the similarity using the set of Koch fractals is carried out. An important result is the relationship between the modified capacity dimension and fractal dimension obtained using the log-log method. The text includes some important modifications and advances in fractal theory. It is important to notice that these modifications and quantifications do not affect already known facts in fractal geometry and fractal analysis.

On the Fractal Design in Human Brain and Nervous Tissue

Digital imaging techniques have enabled to gain insight into complex structure-functional processes involved in the neo-cortex maturation and in brain development, already recognized in anatomical and histological preparations. Despite such a refined technical progress most diagnostic records sound still elusive and unreliable because of use of conventional morphometric approaches based on a unique scale of measure, inadequate for investigating irregular cellular components and structures which shape nervous and brain tissues. Instead, these could be efficiently analyzed by adopting principles and methodologies derived from the Fractal Geometry. Through his masterpiece, The Fractal Geometry of Nature [1], Benoît Mandelbrot has provided a novel epistemological framework for interpreting the real life and the natural world as they are, preventing whatever approximation or subjective sight. Founded upon a body of well-defined laws and coherent principles, the Fractal Geometry is a powerful tool for recognizing and quantitatively describing a good many kinds of complex shapes, living forms, organized patterns, and morphologic features long range correlated with a broad network of functional interactions and metabolic processes that contribute to building up adaptive responses making life sustainable. Scale free dynamics characterized biological systems which develop through the iteration of single generators on different scales thus preserving proper self-similar traits. In the last decades several studies have contributed to showing how relevant may be the recognition of fractal properties for a better understanding of brain and nervous tissues either in healthy conditions or in altered and pathological states.

Benoit B. Mandelbrot (1924–2010)

Benoit B. Mandelbrot, who advanced the concept of power law scaling as a fundamental property of a broad range of natural processes and patterns in geophysics, economics, mathematics, and virtually all of science, died on 14 October 2010 in Cambridge, Mass., at the age of 85. Mandelbrot, known as the “father of fractal geometry,” was a mathematician who developed the scaling concepts of self‐similarity and self‐affinity and found examples in spatial, temporal, and size patterns across a broad spectrum of disciplines. He coined the term “fractal” (from the Latin noun “fractus,” meaning fragmented) for shapes and patterns that exhibit self‐similarity, meaning that they are statistically scale independent. Such shapes are characterized by fractional power law exponents, between the integer (Euclidean) dimensions. He is best known through his books, including Les Objets Fractals: Forme, Hasard et Dimension; Fractals: Form, Chance and Dimension; The Fractal Geometry of Nature; andMultifractals and 1/f Noise: Wild Self‐Affinity in Physics [Mandelbrot, 1975, Mandelbrot 1977, Mandelbrot 1982, Mandelbrot 1999].

Fractal geometry of nature (bone) may inspire medical devices shape

AbstractMedical devices, as orthopaedics prostheses and dental implants, have been designed over years on the strength of mechanical, clinical and biological indications. This sequence is the commonly accepted cognitive and research process: adapting the device to the surrounding environment (host tissue). Inverting this traditional logical approach, we started from bone microarchitecture analysis. Here we show that a unique geometric rule seems to underlie different morphologic and functional aspects of human jaw bone tissue: fractal properties of white trabeculae in low quality bone are similar to fractal properties of black spaces in high quality bone and vice versa. These data inspired the fractal bone quality classification and they were the starting point for reverse engineering to design specific dental implants threads. We introduce a new philosophy: bone decoding and with these data devices encoding. In the future, the method will be implemented for the analysis of other human or animal tissues in order to project medical devices and biomaterials with a microarchitecture driven by nature.

A nonlinear dynamics model for simulating long range correlations of cognitive bistability

Simulation results of bistable perception due to ambiguous visual stimuli are presented which are obtained with a behavioral nonlinear dynamics model using perception-attention-memory coupling. This model provides an explanation of recent experimental results of Gao et al. (Cogn Process 7:105-112, 2006a) and it supports their speculation that the fractal character of perceptual dominance time series may be understood in terms of nonlinear and reentrant dynamics of brain processing. Percept reversals are induced by attention fatigue and noise, with an attention bias which balances the relative percept duration. Dynamical coupling of the attention bias to the perception state introduces memory effects leading to significant long range correlations of perceptual duration times as quantified by the Hurst parameter H > 0.5 (Mandelbrot, The fractal geometry of nature, 1991), in agreement with Gao et al. (Cogn Process 7:105-112, 2006a).

The Time Fractals

Ralph N. Elliott wrote the wave principle in 1938. In 1975 Benoit B. Mandelbrot coined the term fractal and in 1982 published his ideas in Fractal Geometry of Nature. The book brought fractals into the mainstream of professional and popular mathematics. In February 1999, Benoit Mandelbrot submitted an article to Scientific American called ”A Multifractal Walk down Wall Street.” In the article, he discussed how fractal geometry can be used to model the stock market curves. The enclosed research reworks the Mandelbrot Multifractal from a time cycle rather than trend perspective to prove that time fractal is more proportionate than the price fractal and is the real law of nature, which drives everything in nature. The case is validated by illustrating power law curves in time cycle periodicities. Power law is seen across nature and in diverse social trends. The power law in prices is a subject of extended study, but there has been no research attempt made to prove power law in time cycle periodicities. Testing cycle periodicity needs large historical data. Long term time series are difficult to obtain and many emerging markets have seen stock market trading activity only start a decade back. The continued prosperity after 1980’s was a reason why time fractals did not get researchers attention, unlike price fractal which was actively studied and researched. The fact that what we can see is what we can relate too more also made researchers focus more on price than time, which is less visible. Cycles are not conventionally believed to be patterns. Patterns are understood either conventionally or as Elliott wave fractals. Even few Elliott wave practitioners have admitted the limitation of the Elliott Wave structure as being more sharp on form than on time. These were few reasons why time time fractals remained unproven. This study further connects its findings with the existing research on various economic cycles finally extending the proof to a long – short intermarket strategy on an asset pair.

Fractal geometry in the nucleus

In his groundbreaking work, ‘The Fractal Geometry of Nature’ (Mandelbrot, 1982), Benoit Mandelbrot described how fractals could account for a long list of natural forms ranging from trees to coastlines. Two recent studies in Science and The EMBO Journal have suggested that chromatin and the nucleoplasmic space surrounding it should be added to the fractal list.

Critical-like self-organization and natural selection: Two facets of a single evolutionary process?

We argue that critical-like dynamics self-organize relatively easily in non-equilibrium systems, and that in biological systems such dynamics serve as templates upon which natural selection builds further elaborations. These critical-like states can be modified by natural selection in two fundamental ways, reflecting the selective advantage (if any) of heritable variations either among avalanche participants or among whole systems. First, reproducing (avalanching) units can differentiate, as units adopt systematic behavioural variations. Second, whole systems that are exposed to natural selection can become increasingly or decreasingly critical. We suggest that these interactions between SOC-like dynamics and natural selection have profound consequences for biological systems because they could have facilitated the evolution of division of labour, compartmentalization and computation, key features of biological systems. The logical conclusion of these ideas is that the fractal geometry of nature is anything but coincidental, and that natural selection is itself a fractal process, occurring on many temporal and spatial scales.

An adhesion model for elastic-plastic fractal surfaces

The strong adherence (stiction) of adjacent surfaces is a major design concern in microelectromechanical system. The present research concerns the elastic-plastic adhesion of a rough surface contacting with a rigid flat. Rough surface is characterized by fractal geometry using a two-variable Weierstrass-Mandelbrot function [The Fractal Geometry of Nature (Freeman, New York, 1982)]. The microcontact model of single asperity is established in terms of fractal parameters, and the plastic adhesion model is also developed with the Dugdale approximation to consider the adhesive interaction within and outside the contact area. Then according to the plastic flow criterion, the Maguis-Dugdale transition theory [Adhesion of Adhesion and Rapture of Elastic Solids (Springer-Verlag, New York, 2005)] and the plastic adhesion model are used to solve the elastic-plastic adhesive interaction for the two approaching surfaces by incorporating the fractal surface model. Simulations of the adherence force versus the approaching distance of surfaces are performed. The necessity of considering the plastic deformation in microsized interface adhesion is also validated by comparing the result of the presented model with that of the Morrow model [J. Tribol 127, 206 (2005)]. The influence of fractal parameters on adherence force shows that controlling the surface topography is a readily means to control the adherence force for the elastic-plastic adhesion of rough surfaces, which is due to the intermolecular adhesive interactions.

The Complexity Revolution

The Complexity Revolution

Estimation of intrinsic processes affected by additive fractal noise

Fractal Gaussian models have been widely used to represent the singular behavior of phenomena arising in different applied fields; for example, fractional Brownian motion and fractional Gaussian noise are considered as monofractal models in subsurface hydrology and geophysical studies Mandelbrot [The Fractal Geometry of Nature, Freeman Press, San Francisco, 1982 [13]]. In this paper, we address the problem of least-squares linear estimation of an intrinsic fractal input random field from the observation of an output random field affected by fractal noise (see Angulo et al. [Estimation and filtering of fractional generalised random fields, J. Austral. Math. Soc. A 69 (2000) 1–26 [2]], Ruiz-Medina et al. [Fractional generalized random fields on bounded domains, Stochastic Anal. Appl. 21 (2003a) 465–492], Ruiz-Medina et al. [Fractional-order regularization and wavelet approximation to the inverse estimation problem for random fields, J. Multivariate Anal. 85 (2003b) 192–216]. Conditions on the fractality order of the additive noise are studied to obtain a bounded inversion of the associated Wiener–Hopf equation. A stable solution is then obtained in terms of orthogonal bases of the reproducing kernel Hilbert spaces associated with the random fields involved. Such bases are constructed from orthonormal wavelet bases (see Angulo and Ruiz-Medina [Multiresolution approximation to the stochastic inverse problem, Adv. in Appl. Probab. 31 (1999) 1039–1057], Angulo et al. [Wavelet-based orthogonal expansions of fractional generalized random fields on bounded domains, Theoret. Probab. Math. Stat. (2004), in press]). A simulation study is carried out to illustrate the influence of the fractality orders of the output random field and the fractal additive noise on the stability of the solution derived.

> Mandelbrot BB: Fractals and Chaos

Benoit Mandelbrot has produced a comprehensive, well-presented review of essential topics related to Mandelbrot set theory and applications. last part of the title The Mandelbrot set and beyond fully describes its potential allowing the reader to navigate through pictures, hard-to-find early papers and important and effective chapters on the historical background. All chapters are assembled in a way that the overall mix becomes a very well integrated source of know-how and knowledge bringing the readers into the Mandelbrot set world. spirit of the book is well summarized in a sentence on page 34: When seeking new insights, I look, look, look, and play with many pictures. (One picture is never enough). It is certainly true that in the last twenty years, mathematics has changed so deeply that to younger persons some chapter's stories might be simply incredible (p.36), as well, one should admit that after Mandelbrot's sets, initially describing trees, coastlines' shapes or allowing measuring the length of the Britain coast, and after the seminal book on The Fractal Geometry of Nature our way of looking at the world changed. Mandelbrot wrote: Why is geometry often described as 'cold' and 'dry'? One reason lies in its inability to describe the shape of a cloud, a mountain, a coastline or a tree. Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line. I think our vision of the world, from the atom to the higher length scales, is still changing using those concepts clearly illustrated in the current Mandelbrot's book. Selected notes and papers make this book unique within the several books published on this topic. It is clear the touch of the author under all aspects: a touch of pure genius. There are five main topics dominating the book, namely: Quadratic iteration and its Mandelbrot set – Quadratic Julia and Mandelbrot sets; Nonquadratic iterations – Nonquadratic rational dynamics; Kleinian groups' limit set – Iterated nonlinear function systems and the fractal limit sets of Kleinian groups; Multifractal invariant measures – Exponentially vanishing multifractal measures; Background and History. Cumulative bibliography is impressive and well done. It is clearly pointed out, following the pathway through the book, how fractal geometry played an important role in offering a quantitative tool in several areas. Circumstances and facts are put together also to bring important lessons for young scientists. author made a serious and effective effort to realize a book that contains more than history, more than mathematics... it is a sort of ideal book for stimulating new ideas, new concepts, and new discoveries. So far, it is an excellent book also for supporting courses at University, PhD and Post doc level. Moreover, it is indispensable for scientists not only as a lesson of a pathway in science but also as an important source for science of tomorrow. This is a valuable reference source to researchers from these and related areas including bioengineering, biophysics, nanobiosciences and, of course, applied mathematics.

Fractal-based scaling and scale-invariant dispersion of peak concentrations of crop protection chemicals in rivers.

A new regulatory approach is needed to characterize peak pesticide concentrations in surface waters over a range of watershed scales. Methods now in common use rely upon idealized edge-of-field scenarios that ignore scaling effects. Although some watershed-scale regulatory models are available, their complexity generally prevents them from being used duringthe pesticide registration decision process, even though nearly all exposure to both humans and aquatic organisms can occur only at this scale. The theory of fractal geometry offers a simpler method for addressing this regulatory need. Mandelbrot described rivers as "space-filling curves" (Mandelbrot, B. B. The Fractal Geometry of Nature; Freeman: New York, 1983), a class of fractal objects implying two useful properties we exploit in this work. The first is a simple power-law relationship in which log-log plots of maximum daily concentrations as a function of watershed area tend to be linear with a negative slope. We demonstrate that the extrapolation of such plots down to smaller watersheds agrees with edge-of-field concentrations predicted using the Pesticide Root Zone Model, but only when the modeling results are properly adjusted for use intensity within the watershed. We also define a second useful property, "scale-invariant dispersion", in which concentrations are well described by a single analytical solution to the convective--dispersion equation, regardless of scale. Both of these findings make it possible to incorporate the effect of watershed scale directly into regulatory assessments.