Natural Fractals

El contenido de esta comunicación forma parte, y hace referencia, a la Unidad Didáctica titulada "GEOMETRÍA FRACTAL ¿Qué son y cómo pueden los fractales ayudar a representar y a organizar el espacio?", que incluye el desarrollo completo del
resto de coordenadas curriculares (contextualización, objetivos didácticos, metodología, contenidos, evaluación y recursos) y de una serie de actividades propuestas para las materias de Matemáticas e Informática de ESO y Bachillerato. Este material está fue experimentado en 1996 con alumnos de niveles equivalentes a Tercero de ESO y Primero de Bachillerato.

Regarding the foundations of matter as energy fields in a fractal gearing structure can redefine the foundations of quantum physics in a unique new way, while remaining consistent with modern probabilisitic theories.  Further, working experimental models of fractal gear arrays have created effects that cannot be explained with current theories, and they are described here in detail.  This theory is the only way to explain the experimental phenomenon, and it is an explanation to phenomenon which I have been able to create in the lab for the first time in physics history.  Fractal gear theory is a deterministic model of quantum fields that can be used simultaneously with the current standard model and they both can borrow concepts from each other without breaking the logical consistency.  An example is the double-slit experiment; fractal gear theory states that a harmonic field structure is created by the geometry of the matter (the slits) and the electron or photon rides on it in an oscillatory manner so it does not matter if one or many particles enter simultaneously through the slits.  Many of the modern theories provide quantitative answers to the questions that arise from this new deterministic unified field theory. A prime example is that strings from string theory can be understood as a line of contact thru time within a fractal gear field array.  Physicists today have obfuscated theories with statistical data , whereas most of the real advances in physics have arisen from thought experiments and theoretical and geometric analysis. This theory was devised with very little mathematics and a whole lot of 3D geometry and analysis with almost zero experimental data or references beyond the foundations of modern quantum physics.  It is a grass roots endeavor to start from scratch and rebuild quantum physics with a correct picture that almost any mechanically inclined person can understand; a picture that has eluded physicists for over 100 years.  I believe that if a picture is worth 1000 words, then a geometry is worth 1000 formulas, but geometry is much easier to work with.

Fractal gear fields are a quantum foam theory and provides means for transmitting information and by its explanation of emergent space time, it is directly implied that information is able to travel faster than the speed of light within a bound domain.  The speed of transmission dissipates at a rate proportional to the total energy within the domain.  Fractal gear theory is a geometrical foundation for a pilot-wave theory in the spirit of DeBroglie -Bohm theory.  Further, the 3D geometric form of fractal gear fields shows clearly that all bell inequality measurements using photon polarization is biased and in error, due to angular variations in density of the field structure.

This paper investigates an approach to the use of fractals in architectural design. Two major aspects are discussed. First, the effect of the direction of the line segments on the generated fractal, as well as, the proportions between the generator and the initiator. Secondly, the meanings attached to the lines segments that constitute the fractal are based on some known architectural organization. The goal is to develop a basis for using fractals to suggest architectural forms.

In this paper I will present the use of fractal geometry to design tile motifs. A fractal is a geometric figure that combines the several characteristics among others: its parts have the same form as the whole, fragmented, and formation by iteration. The concept of fractals has been spread over all fields of sciences, technology, and art. This paper aims to provide an algorithm to creating motifs of tile algorithm for create the tile motif consists of base, iteration, coloration and duplication. In order to help the reader better understand the algorithm, I will present some script using Matlab. We describe a mathematically based algorithm that can fill a spatial region with sequence of randomly placed which may be transformed copies of one motif or several motifs. By using this algorithm, I can produce thousand variety of aesthetically pleasing tile motifs, of which we show a number of examples.

Los cursos cubren los principios básicos de ecología de poblaciones, fractales, oceanología, vida en el medio fluido, donde puse lo hallado, ecología marina, ecología de barrancas y arroyos, nadar sin turbulencia. Se énfasis en los conceptos básicos y en los procesos que ocurren en la naturaleza, así como la conectividad entre la tierra y el océano. Permite a los alumnos familiarizarse con el conocimiento multidisciplinario y las habilidades de trabajo interdisciplinario necesarias en el campo de las ciencias del mar, limnología y recursos acuáticos. Dirigido a estudiantes de licenciatura en cualquier área o disciplina.

Tumour growth can be described in terms of mathematical models from different points of view due to its multiscale nature. Dynamic scaling is a heuristic discipline that exploits the geometrical features of growing fronts using different concepts from the theory of stochastic processes and fractal geometry. This work is concerned with some problems that arise in the study of tumour–host interfaces. The behaviour of their fuctuations leads to some stochastic evolution equations, which are studied here in the radial symmetry case. Some questions concerning the dynamic scaling of these models and their comparison with experimental results are addressed.

B I O M I M I C R Y
" Doing it nature's way has the potential to change the way we grow food, make materials, harness energy, heal ourselves, store information, and conduct business. " — ​ Janine Benyus

The Biomimicry Institute defines biomimicry as the art and science of seeking sustainable innovation by emulating nature's time­tested patterns and strategies. It's a new approach to product design that gives rise to what Janine Benyus calls the Biomimicry Revolution: " ... ​ an era based not on what we can extract from nature, but on what we can learn from her. " ​

Whether Fractal Entrainment (FE) can be considered applied biomimicry or not, natural patterns are certainly at the core of this bio­inspired and novel technology, conceived to further harness the transformative power of sound energy.

From the air we breathe, to the patterns of light in the screen that enable us to write and read this, everything in the universe—including the wind, the foliage that reveals its shape­shifting patterns, the image of a starry night and all it might contain and convey, even a heart, pumping in reverence at the wonder of being alive—is said to be composed of vibrating filaments and membranes of energy. Life is an ongoing succession of ups and downs, comings and goings, twists and turns, yet embracing this is not always easy. We are constantly drifting between states of flow and resistance and experiencing the many possible degrees of balance between those two poles. If consciousness unfolds through the countless intercommunications that our inner and outer " environments " establish, moment by moment, it is quite probable that efficiency in information flow might have biological advantages.

This is why Fractal Entrainment aims at replicating some of the ways in which nature favors the flow of energy, and the seemingly endless succession of

1/f serial correlations and statistical self-similarity (fractal structure) have been measured in various dimensions of musical compositions. Musical performances also display 1/f properties in expressive tempo fluctuations, and listeners predict tempo changes when synchronizing. Here the authors show that the 1/f structure is sufficient for listeners to predict the onset times of upcoming musical events. These results reveal what information listeners use to anticipate events in complex, non-isochronous acoustic rhythms, and this will entail innovative models of temporal synchronization. This finding could improve therapies for Parkinson's and related disorders and inform deeper understanding of how endogenous neural rhythms anticipate events in complex, temporally structured communication signals.

Just as religion is impassable to its own dogmatic philosophies and ideas, present day science also acts in the same manner. Embracing its reductionist approach, it loses its sight to the most beautiful and insightful possibilities associated with the cosmos and its fundamental constituents around us. The present paper is a holistic ride into few of life’s beautiful constructs that occur as a playful act of the cosmos, and in part hypothesize that the dynamic biological structures or forms could be thought of as frozen energy patterns. Here, the supposition is that the electromagnetic (EM) fields that envelop our ecosystems play a central role in its functioning and evolution acting as a unified network interconnecting life at various scales. Supporting a synergetic and holistic view, the present paper is skeptical about the present school of thought, and to other investigations carried out under the sheath of modern science by raising a few fundamental questions.

In this paper I will present the use of fractal geometry to design Indonesian batik patterns. Fractals are geometric figures consisting of several different characteristics: the parts have the same shape as a whole, are fragmented, and are made by iteration. The concept of fractals has spread across all fields of knowledge, technology, and art. This paper aims to provide a method for making geometric patterns called Parang. The generator algorithm consists of base, iteration, coloring, and duplication. To help talk more about learning algorithms, I will present a script program using Matlab. We consider a mathematical-based algorithm that can fill a spatial region with a finished placement sequence of one motif or several motifs. Using these algorithms, I can produce thousands of variations of aesthetically beautiful Parang patterns, which we present a number of examples.

Two indicators of fractality, related to Hausdorff dimension, are identified. The first indicator of fractality of objects and processes, implied in the etymological meaning of the stem of the word “fractal”, is the fractional value of the Hausdorff dimension of an object or process (hereinafter, the indicator “fractionality”). The second indicator, represented in the definitions of fractal based on Hausdorff dimension, is that the value of the Hausdorff dimension (D) strictly exceeds the value of the topological dimension (DT), or D > DT (hereinafter, the indicator “exceedance”). Objects and processes, recognized in contemporary science as fractals, have been divided into three groups on the basis of the criterion of the relationship between the values of their Hausdorff and topological dimensions: 1) fractals with an integer value of Hausdorff dimension equal to the value of their topological dimension; hence, they do not have indicators “fractionality” and “exceedance”; 2) fractals with a fractional value of Hausdorff dimension exceeding the value of topological dimension by less than one, consequently, they have indicators “fractionality” and “exceedance”; 3) fractals with an integer value of Hausdorff dimension exceeding the value of their topological dimension by one, hence, they do not have indicator “fractionality”. All three groups of fractals and their underlying variants of the relationship between the values of Hausdorff and topological dimensions satisfy the Szpilrajn inequality: (D ≥ DT). If we ascribe to the indicator “exceedance” and, in particular, to the indicator “fractionality” the status of the attributive (necessary) indicator, it means a deviation from the Szpilrajn inequality, and such a deviation unjustifiably narrows the class of fractals, since the indicator “exceedance” allows to consider as fractals only the objects of the second and third groups, and the indicator “fractionality” (which is a particular case of the indicator “exceedance”) allows to classify as fractals only the objects of the second group. Despite the absence of one or both proposed by Mandelbrot indicators of fractality, related to Hausdorff dimension, some objects or processes are recognized by the scientific community as fractals because of the presence of another indicator of fractality – widely interpreted self-similarity. Thus, the indicators “exceedance” and “fractionality” are not attributive indicators of fractality. The search for a fundamental characteristic of fractals, the characteristic, which connects self-similarity and Hausdorff dimension, and, presumably, provides a correlation between both of values – the self-similarity dimension and the Hausdorff dimension of fractal, is one of the most important and promising directions in the study of fractals. Keywords: fractal, Hausdorff dimension (Hausdorff-Besicovitch dimension), topological dimension, indicator of fractality, the Szpilrajn inequality, self-similarity (auto-similarity), Benoît Mandelbrot.

Over the years, a number of measures have been defined for the purpose of determining the number of independent dimensions contained in a space. The most common dimensionality measures are the topological dimensionality and various kinds of fractal dimensionalities. While each of these dimensionality measures is useful in its own right, none of them accurately quantifies the effective number of independent directions passing through locations contained in a local region of a space. This article introduces a new dimensionality measure, called the connectivity dimensionality field, which is the true measure for the effective number of independent directions passing through locations in a space. In contrast to the fractal dimensionality, the connectivity dimensionality field is a topological property because its value at each material location is invariant to deformations of the space preserving connectivity. The connectivity dimensionality field is a fundamental concept that applies to many different kinds of discrete spaces, continuous spaces, and discrete-continuous dual spaces. A discrete space is a space in which positions cannot be varied differentially, and a continuous space is a space in which positions can be varied differentially. A discrete-continuous dual space has complementary discrete and continuous representations, and a process called discrete-continuous dual matching relates the discrete and continuous representations to each other. This article formally defines two basic types of discrete-continuous duality: (a) asymptotic and (b) strict. A rigorous method is provided for computing the connectivity dimensionality field in edge-vertex graphs, continuous spaces, and discrete-continuous dual spaces. Many examples are given to illustrate the key concepts. Single points, unbranched lines, and periodic lattices are examples of discrete-continuous dual spaces in which the connectivity dimensionality field is a constant nonnegative integer. In other types of discrete-continuous dual spaces, the connectivity dimensionality field contains inherent uncertainty. For the first time, a comprehensive theory is derived that predicts the inherent uncertainty associated with the connectivity dimensionality field in discrete-continuous dual spaces. The study of discrete-continuous dual spaces with variable connectivity dimensionality fields transcends variable-based mathematics.

Sobre la posibilidad del Arte Fractal Parte I

Benoit Mandelbrot había descubierto la estructura que regía a la naturaleza, expuesta en su libro La geometría fractal de la naturaleza (The Fractal Geometry of Nature).[i] Descubrió en resumidas cuentas, que los contornos de nuestra realidad están regidos por ciertas ecuaciones, que se repiten en diversos niveles. Imágenes con patrones muy agradables, con armonías y secuencias perfectas. Después descubrieron que, en efecto, aquello que solo se pensaba ser algo bello (como un tipo de arte matemático) tenía una repercusión en la ciencia, las compañías empezaron a realizar antenas con diseños fractales al darse cuenta que su receptividad aumentaba de una forma bastante considerable, adicional a eso, se empezó a aplicar este conocimiento a la tecnología de los ordenadores; sus microprocesadores; provocando un crecimiento exponencial. En la actualidad la Geometría Fractal ha revolucionado la forma de ver lo que nos rodea. Al descubrir esto, empiezo a gestar en mi mente un sinnúmero de posibilidades, el simple hecho de entender la posibilidad de que, de la misma forma en como una obra de arte puede ser solo contemplativa, también es posible que encierre mucho más, como en el caso de la Geometría Fractal. El arte en general tiene un propósito mucho

the ever decreasing area of forests has lead to environmental and economical challenges and has brought with it a renewed interest in developing methodologies that quantify the extent of deforestation and reforestation. in this study we analyzed the deforested areas of the Apuseni Mountains, which has been under economic pressure in recent years and resulted in widespread deforestation as a means of income. Deforested surface dynamics modeling was based on images contained in the Global forest Database, provided by the Department of Geographical Sciences at Maryland University between 2000 and 2014. The results of the image particle analysis and modelling were based on total Area (ha), count of patches and Average Size whereas deforested area distribution was based on the Local connected fractal Dimension, fractal fragmentation index and tug-of-War Lacunarity as indicators of forest fragmentation or heterogeneity. The major findings of the study indicated a reduction of the tree cover area by 3.8%, an increase in fragmentation of 17.7% and an increase in heterogeneity by 29%, while fractal connectivity decreased only by 0.1%. The fractal and particle analysis showed a clustering of forest loss areas with an average increase from 1.1 to 3.0 ha per loss site per year. in conclusion, the fractal and particle analysis provide a relevant methodological framework to further our understanding of the spatial effects of economic pressure on forestry. In view of its effect on climate, deforestation is athe threatening challenge for contemporary society. The major factors leading to the decline of forested areas include an expansion of agricultural land 1-4 , human settlements 5 , timber demand 6 , mining exploitations and population growth 7 leading to environmental and economic challenges in the long term. Therefore the aim of the local or regional authorities is to reduce deforestation and to regenerate the existing forests wherever possible due to their importance in supporting the biotic system, regional and global climate change, the preservation of atmospheric oxygen and carbon, conservation of biodiversity and benefits to local communities 6,8-12. Forest fragmentation is a major result of deforestation 13,14. It leads to habitat modification 15 , and subdivision of plant and animal populations. Thus, changes in species interactions occur 13,14 , leading to further tree mortality

This research is concerned chiefly with how global media icons as iconics assigned to terrorism are motivated, engineered, choreographed, and represented by all interested parties, and to what psycho-socio-political ends. Initially motivated largely by Marshall McLuhan from an essay on weapons called “War of the Icons,” where I actually enter the conversation today is at the final section of Joseph Tuman’s Communicating Terror: The Rhetorical Dimensions of Terrorism, where Tuman addresses terrorists’ use of codes and cues in their iconic media releases.

I have been working closely with rhetorical analysis of select media events over the past three years and am especially interested in currently available means of establishment for pre-emptive analysis of what underlies the iconic media content that will be indiscriminately channeled across the globe; e.g., motive, message, medium, responsibility, ambiguity of meaning, and possible consequence. With this perspective comes an immediacy that questions the controls and variables at play in the rhetorical processes connecting terrorism with global media entities and channels.

Analysis of the symbiotic, parasitic relationship — between terrorists/terrorism and the media, where they collude and collide — informs much of my research. Their required relationship is best realized today across new media channels in resulting global representations or media moments that, especially because of digital technologies, are iconic: bigger than life, bigger than the moment, dangerous in global scope and impact, acting as virtual subscripts of artificial intelligence with agency in cyberspace..

Fractals are mathematical structures under exploration and development in virtually every academic discipline and beyond: “besides mathematics and science, fractals have direct applications in many fields, including music, literature, visual art, architecture, sculpture, dance, technology, business, finance, economics, psychology, and sociology” (Frame & Mandelbrot, 2002, p. 4). Since fractal geometry today is “the only available language for the study of roughness, a concept that is basic and related to our senses, but has been the last to give rise to a science,” (Ibid., p. 11) and ‘roughness’ or unevenness or irregularity is a natural, universal tenet of thought and matter, the science behind fractal geometry unifying it across disciplines holds much promise.

The rise of a multidisciplinary discourse containing measurable -- and predictable – patterns of irregularity is attributed to fractal geometry with the seriousness and playfulness of mathematics at its root. Because of these fundamentals, the newness of the study of fractal geometry lends itself to extreme unpredictability. Since formulas and algorithms create fractal images and patterns, much of the fun and exploration of fractals lies in ability to experience new mathematical conjectures in anticipation of new discoveries.

Such an exploratory method has been developing across the disciplines with diverse jargon but one common language uniting them all, that being this staunchly predictable yet wildly unpredictable mathematical integrity of the fractal, dependent upon the available formula or algorithm used to suggest its creation and level of complexity.

We have yet to discover or unveil a serious application of fractal geometries related to patterns of irregularity in consideration of the work of Irwin Panofsky and the positing of iconography in art history, e.g., or in philosophy with study of Michel Foucault’s architecture of knowledge or history of insanity, which appear to naturally lend themselves to such interrogations.

the ever decreasing area of forests has lead to environmental and economical challenges and has brought with it a renewed interest in developing methodologies that quantify the extent of deforestation and reforestation. in this study we analyzed the deforested areas of the Apuseni Mountains, which has been under economic pressure in recent years and resulted in widespread deforestation as a means of income. Deforested surface dynamics modeling was based on images contained in the Global forest Database, provided by the Department of Geographical Sciences at Maryland University between 2000 and 2014. The results of the image particle analysis and modelling were based on total Area (ha), count of patches and Average Size whereas deforested area distribution was based on the Local connected fractal Dimension, fractal fragmentation index and tug-of-War Lacunarity as indicators of forest fragmentation or heterogeneity. The major findings of the study indicated a reduction of the tree cover area by 3.8%, an increase in fragmentation of 17.7% and an increase in heterogeneity by 29%, while fractal connectivity decreased only by 0.1%. The fractal and particle analysis showed a clustering of forest loss areas with an average increase from 1.1 to 3.0 ha per loss site per year. in conclusion, the fractal and particle analysis provide a relevant methodological framework to further our understanding of the spatial effects of economic pressure on forestry. In view of its effect on climate, deforestation is athe threatening challenge for contemporary society. The major factors leading to the decline of forested areas include an expansion of agricultural land 1-4 , human settlements 5 , timber demand 6 , mining exploitations and population growth 7 leading to environmental and economic challenges in the long term. Therefore the aim of the local or regional authorities is to reduce deforestation and to regenerate the existing forests wherever possible due to their importance in supporting the biotic system, regional and global climate change, the preservation of atmospheric oxygen and carbon, conservation of biodiversity and benefits to local communities 6,8-12. Forest fragmentation is a major result of deforestation 13,14. It leads to habitat modification 15 , and subdivision of plant and animal populations. Thus, changes in species interactions occur 13,14 , leading to further tree mortality

We present evidence based theory for the emergence of plant structure in which CO 2 is not only the source of carbon for plant growth, but also plays a critical role as a source of charge (ionization), with charge density dictating plant structures at a wide range of scales. As levels of charge density increase beyond a critical point, dis-sipative systems lead to the emergence of macroscopic quantum processes analogous with high temperature super conductivity and coherent random lasing. The assembly of molecules into larger, ordered structures operates within charge-induced coherent bosonic fields acting as a structuring force in competition with exterior potentials. Within these processes many of the phenomena associated with standard quantum theory are recovered, including quantization, non-dissipation, self-organization, confinement , structuration conditioned by the environment, environmental fluctuations leading to macroscopic quantum decoherence and evolutionary time described by a time dependent Schrödinger-like equation, which describes models of bifurcation and duplication. The work provides a strong case for the existence of quintessence-like behaviour, with macroscopic quantum potentials and associated forces having their equivalence in standard quantum mechanics and gravitational forces in general relativity. The theory offers new insight into evolutionary processes in structural biology, with selection at any point in time, being made from a wide range of spontaneously emerging potential structures (dependent on conditions), which offer advantage for a specific organism. This is valid for both the emergence of structures from a prebiotic medium and the wide range of different plant structures we see today.

A series of experiments examined auditory contour formation, investigating listeners' sensitivities to a family of random fractals known as fractional Brownian noises. Experiments 1A and IB looked at identification of contours when 3 different noises were portrayed using variations in the pitch, duration, or loudness of successive notes of a sequence. Listeners could categorize pitch and loudness encodings, but not duration mappings. Experiment 2 looked at the effect of simultaneous presentation of pitch and loudness information, finding that these dimensions combined additively to increase identification of the noise distributions. Experiment 3 looked at discrimination of pitch contours as a function of changing fractal dimension. Discrimination curves approximated an inverted U shape, a finding that is not understandable in terms of sensitivity to differences in fractal dimension per se, nor in terms of "tuned" perceptual sensitivity to statistical regularities of the environment.

Mathematical modeling of tumor development has become a real hype within the last decade. The abundance of mathematical models has created a great need for the validation of their biological relevance. Recently, in order to characterize the tumor growth dynamics, Brú et al. have determined some statistical properties of both in vitro and in vivo solid tumor-surfaces by using fractal scaling analysis. Surprisingly, for all tumor surfaces, the statistical observables converged to a unique set of critical exponents which indicates some common features of tumor growth dynamics (linear growth rate, growth activity limited to the outer rim of the tumor mass and diffusion of newborn tumor cells on the surface from lower to higher curvature regions, typical of Molecular Beam Epitaxy (MBE) Universality). Here, we develop and analyze a lattice-gas cellular automaton (LGCA) model of solid tumor growth. Random walk dynamics are assumed for tumor cell migration and a density-dependent birth process describes the cell mitotic dynamics. Fractal scaling analysis shows that for any parameter variation the model interface dynamic follows Edward–Wilkinson (EW) Universality, which differs from experimental findings. However, the model recovers some features, i.e. linear growth rate for tumor size and proliferative activity restricted to the outer layer, observed in experiments.