Fractal

Receiving of remote sensed data's signals in urban space information reception centers is usually difficult, because of complex electromagnetic situation in cities and insufficient EMC. Traditional methods for digital reconstruction of images use smoothing and autoregressive forecasting. In this case anomalous spikes on the image are lost. Alternative method based on Kolmogorov-Wiener's filters and fractal properties of satellite images is

Fractal analysis of the morphology of ice crystal particles in frozen food was attempted for frozen soybean curd (tofu). From a microscopic image analysis of the ice crystal particles, it was found that the perimeter of the ice crystal particles could be recognized as a fractal. Effects of the storage time and storage temperature on the fractal dimension (d p ) of the perimeter of the ice crystal particles were also investigated. As storage time was increased, the d p value tended to decrease. When the storage temperature was increased, the value of d p decreased more rapidly. The changes corresponded to the visual observation of the shape change for the ice crystal particles during storage reported by many researchers, suggesting that the fractal dimension d p could be used as a quantitative indicator reflecting the surface roughness of ice crystal particles.

Many desirable properties make fractals a powerful mathematic model applyied in several image processing and pattern recognition tasks: image coding, segmentation, feature extraction and indexing, just to cite some of them. Unfortunately, they are based on a strong asymmetric scheme, so suffering from very high coding times. On the other side, linear transfoms are quite time balanced, allowing to be usefully integrated in real-time applications, but they do not provide comparable performances with respect to the image quality for high bit rates. Owning to their potential for preserving the original image energy in a few coefficients in the frequency domain, linear transforms also known a widespread diffusion in some side applications such as to select representative features or to define new image quality measures. In this paper, we investigate different levels of embedding linear trasforms in the fractal coding scheme. Experimental results have been organized as to point out what is the contribution of each embedding step to the objective quality of the decoded image.

Fractal measurements using the Cantor's dust method in a linear one-dimensional analysis mode were made on the fracture patterns revealed on two-dimensional, planar surfaces in four granites. This method allows one to conclude that: (1) The fracture systems seen on two-dimensional surfaces in granites are consistent with the part of fractal theory that predicts a repetition of patterns on different scales of observation, self similarity. Fractal analysis gives essentially the same values of D on the scale of kilometres, metres and centimetres (five orders of magnitude) using mapped, surface fracture patterns in a Sierra Nevada granite batholith (Mt. Abbot quadrangle, Calif.). (2) Fractures show the same fractal values at different depths in a given batholith. Mapped fractures (main stage ore veins) at three mining levels (over a 700 m depth interval) of the Boulder batholith, Butte, Mont. show the same fractal values although the fracture disposition appears to be different at different levels. (3) Different sets of fracture planes in a granite batholith, Central France, and in experimental deformation can have different fractal values. In these examples shear and tension modes have the same fractal values while compressional fractures follow a different fractal mode of failure. The composite fracture patterns are also fractal but with a different, median, fractal value compared to the individual values for the fracture plane sets. These observations indicate that the fractal method can possibly be used to distinguish fractures of different origins in a complex system. It is concluded that granites fracture in a fractal manner which can be followed at many scales. It appears that fracture planes of different origins can be characterized using linear fractal analysis.

The temporal structure of macroscopic brain activity displays both oscillatory and scale-free dynamics. While the functional relevance of neural oscillations has been largely investigated, both the nature and the role of scale-free dynamics in brain processing have been disputed. Here, we offer a novel method to rigorously enrich the characterization of scale-free brain activity using a robust wavelet-based assessment of self-similarity and multifractality. For this, we analyzed human brain activity recorded with magnetoencephalography (MEG) while participants were at rest or performing a visual motion discrimination task. First, we report consistent infraslow (from 0.1 to 1.5 Hz) scale-free dynamics (i.e., self-similarity and multifractality) in resting-state and task data. Second, we observed a fronto-occipital gradient of self-similarity reminiscent of the known hierarchy of temporal scales from sensory to higher-order cortices; the anatomical gradient was more pronounced in task...

In recent years there has been deep development in the use of techniques to analyse the statistical properties of point process time series. Furthermore many efforts have been made to establish robust methods to identify time clustering structures in the temporal distribution of seismic events. In this paper we intend to give a systematic review of some common used techniques to characterise the temporal properties of an earthquake sequence; we also present new methods, commonly used in other scientific fields, to reveal time clustering structures in a seismic sequence and to clearly identify non-poissonian behaviours at different time scales. An application of these methods has been performed on the seismicity of Irpinia-Basilicata area (Southern Italy).

В статье рассмотрены основные положения теорий динамики рынков, в частности, гипотезы эффективного рынка, гипотезы когерентного рынка, гипотезы фрактального рынка и гипотезы синергетического рынка. Проанализированы инструменты фрактального рынка для принятия на их основе обоснованных инвестиционных решений. Приведены и обоснованы инструменты технического анализа, основанного на игре за трендом для прогнозирования динамики рынка, согласно гипотезе фрактального рынка, в частности, волновую модель Эллиотта и применения нейро-нечетких моделей.
Ключевые слова: эффективный рынок, когерентный рынок, фрактальный рынок, синергетический рынок, технический анализ

A wide band iris excited fractal slotted array has been realized and experimentally verified at X band. The design starts with a centered plus shaped slot in the broadwall of a rectangular waveguide and the same has been used in second iteration for fractal configuration, which exhibits dual resonance. The fractal unit cell is optimized for bandwidth by carrying out parametric study of slot-iris combinations and 2.4 GHz bandwidth is achieved for both radiation pattern and impedance response. An empirical expression for conductance of the unit cell has been developed to facilitate design of antenna array using the unit cell as an element. The optimized unit cell has been used for designing a four element array, which exhibits a pattern bandwidth of 1.4 GHz and impedance bandwidth of 2 GHz. The array is simulated in commercial fullwave 3D FEM simulator. The designed array has been fabricated and measured response has been compared with simulated data.

Fractal properties of surfaces have been explored by many investigators. Most have concluded that fractal characterisation is useful. This note questions the philosophy of using fractals to describe and control engineering surfaces. It concludes that the benefits are more virtual than real. The functional significance of fractal parameters is also examined and the overall question arises as to whether scale invariant parameters are appropriate.

A study of the quality and accuracy of the methods based on frequency analysis for the fractal characterization of surfaces as measured by scanning tunnelling microscopy (or profilometry) is made. The study is based on computer simulation of images of fractal surfaces. A discussion of the mathematical algorithms used for computer generation of fractal surfaces then follows. The main conclusion is that studies of fractal characterization by frequency analysis reported in previous papers in the STM field, as well as conclusions about the performance of the various methods, are doubtful. New methods for frequency analysis that in some cases produce more reliable results are proposed.

A novel methodology ''fractal browning indicator" (FBI) is presented, that describes the enzymatic browning kinetic based on the use of irregular color patterns from banana slice images. It uses the fractal Fourier texture image value in a selected area, to calculate a fractal dimension (FD), which represents the complexity of color distribution. During the procedure, colors from digital images were first transformed to L à a à b à space color using a transformation function (quadratic model), in order to derivate three color channels, lightness (L à ), redness (a à ), and yellowness (b à ). In the results, lightness and yellowness parameters decreased during the browning kinetic, when their respective FD values increased, indicating major color distribution complexity in a selected area analyzed during the kinetic. The redness color (a à ) did not show any statistical variation. The empirical power law model was suitable to correlate enzymatic browning kinetic data both for FBI and for the traditional method (when an L à mean was used). However, enzymatic browning rates using the FBI method, were between 8.5 and 35 times higher than rates calculated with the traditional method.

It has been known for a century that small particles dispersed in liquids and gases can form aggregates with extraordinarily low densities, and a variety of studies in the 1960s and 1970s, based on conceptual models and computer models, suggested that “anomalous” scaling relationships were associated with the structures of these aggregates. However, the lack of a suitable theoretical framework and the limits of computer technology inhibited the development of a coherent understanding of the structures of these aggregates and the kinetics of their formation. In the 1980s, the popularization of fractal geometry and rapid advances in computer technology removed these barriers to progress. In this review, the early work on fractal aggregates is discussed and the basic particle-cluster and cluster-cluster aggregation models introduced in the 1980s are described. During the 1990s, interest has been focused on the subtle relationships between aggregation, gelation and spinodal decomposition and on the physical behavior of systems containing fractal aggregates. The following papers in this special issue of the Journal of Sol-Gel Science and Technology on “Computer simulations of aggregation and sol-gel processes” describe recent advances in these directions. They are previewed in this introductory contribution.

Within the expanded field of drawing, my Master of Fine Art project explores this liferidden planet's journey through time and space by investigating the role played by cataclysm and renewal in planetary deep time. Instead of binaries, this exegesis asks, can destruction and creation be seen as a single evolutionary force and, if so, how does an artist interpret such a force? Using Earth's evolutionary development as its foundation, my project aims to suggest a visual story of the interconnectedness of matter through deep time, viewing a brief but impactful human presence in perspective amidst an unfolding epic. To reinforce this aim my practice materially apposes the domestic and the numinous. Answers and inspiration are sought within three main research platforms: art history and practice, the natural sciences, and posthuman critical theory. Incomprehensible reaches of deep time and the interconnectedness of matter within the cosmos, as revealed by recent science, inspire ineffable awe, a fundamental principle of the sublime. This is where I trace my art practice's field and history. Produced in tandem with research, my artwork features domestic materials, directed chance and recycled elements using, as the Earth does, products from earlier experiments in revised ways.

O que se apresenta em sala de aula em relação à geometria sempre é o básico, mas não percebemos que ao nosso redor existe uma geometria de figuras não regulares, que apresenta auto similaridade fragmentada, essas figuras chamadas de fractais podem entrar no âmbito escolar como uma forma de aplicar a matemática. E é com o objetivo de trabalhar a geometria fractal associando ao conteúdo de Progressão Geométrica (PG) que se propôs este trabalho. Para a realização deste estudo foi feito um levantamento bibliográfico com o intuito de colher material necessário para apresentar adequadamente aos discentes, por meio de uma oficina para os estudantes da Escola de Ensino Médio Presidente Geisel (Polivalente) parceira do Programa Institucional de Bolsa de Iniciação à Docência (PIBID), trabalhando a relação do Fractal com as PG, permitindo que eles possam usar a imaginação e construir seus próprios fractais. Concluindo que a geometria fractal é de fato um ramo da Matemática com grande atuação e importância, além de atrair a curiosidade e interesse dos estudantes quando apresentados em sala de aula.

This paper should be interpreted as an introduction to the concept of space filling curves. Looking at examples created by Peano, Hilbert and Lebesgue. Such curves have motivation in computer science as an efficient way to traverse multi-dimensional data. The contents throughout is heavily influenced by what is one of the only books to really study the subject, Han's Sagan's Space Filling Curves [1].

In this paper, we discuss some applications of exciting fields like Mathematics, Statistics, Medical, Biological Sciences, Engineering, Economics etc. We show their real-life applications both theoretically and analytically considering some phenomena of the nature. We mainly focused on some applications of dynamical system in real life cases known as chaos, iteration, fractal geometry, Mandelbrot and Julia sets. We derive some mathematical formula concerning dynamical systems. Necessary programs are considered for all cases. We use Mathematica and MATLAB to perform programming.

Grafik tasarım içerisinde dolaylı ya da doğrudan matematik ile ilişkisi bulunmaktadır. Oran, sembol, renk değerleri gibi konular içerisinde sayısal olarak da ölçülebilinen estetik kurallar grafik tasarımı etkilemiş ve değiştirmiştir. Matematiksel oranlar içerisinde en bilinen olanı ise altın orandır. Altın oranın sanata olan etkisi gibi ayrıca grafik tasarıma da etkisi bulunmaktadır. Birçok grafik tasarım alanında altın oran gözlenmektedir. Aynı şekilde bilgisayarların günlük kullanımlarının artmasıyla yeni tasarım alanları ortaya çıkmıştır. Fraktallar ise 1904 yılında Mandelbrot tarafından ortaya atıldığında bir geometrik açıklama aracı olarak görülmekteydi. Fraktallar çok fazla karmaşık bir yapıya sahip oldukları için görselleştirilebilmeleri ancak bilgisayar kullanımının yaygınlaşmasından sonra olabilmiştir. Görselleştirilmelerinden ortaya çıkan imgeler sonucunda tasarımcılar fraktallara görsel bir alan olarak da bakmaya başlamışlardır.

Formüllerin görselleştirmesine ek olarak ise, oyun tasarımları söylenebilir, çeşitli formüller ve kodlamalar sayesinde görsel olarak ortaya çıkan veri bütünleri ve görsel ögeler tasarım açısından zengin bir alan ortaya çıkartmışlardır. Oyun tasarımı tarihsel süreci boyunca hızlı gelişen ve hala gelişmeye devam eden bir sektördür. Oyun yapımcıları sürekli olarak yeni tarz oyunlar çıkartmaya çalışmakta, oyuncuya ise en iyi deneyimi hangi şekilde sunmaları gerektikleri arasında araştırmalar yapmaktadırlar. Oyuncular sadece artık sadece oyunun içerisindeki görsellik ya da oyunun sadece içerik kısmına bakmamaktadırlar. Oyuncuyu oyun içerisinde devamlılığını sağlayabilmesi için oyun içi görsellerin yanı sıra, senaryo yapısı, oyun dışında oyun ile ilgili ürünlerin tasarımı, gerçek dünya da oyun ile etkileşim gibi konular ortaya çıkmaktadır.

Oyun içerisindeki veri görselleştirilmesi kavramları da incelendiğinde görülmektedir ki, oyunların oluşturduğu veri setlerinin ortaya çıkması bunların işlenerek oyunculara iletilmesi gerekmektedir. Veri setlerinin görselleştirilmesi de belirli görsel kurallar ve yasalara dayanmaktadır. Oyun tasarımı içerisinde ise  görsel zenginliğin yanı sıra, oyun içerisindeki oyuncuyla oyun arasındaki veri iletiminin başarı şekilde sağlanmasından geçtiği görülmektedir. Bu ise başarılı bir görselleştirme sonucunda ancak sağlanabilmektedir.

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In graphic design, there is direct or indirect relationship with mathematics. The aesthetic rules that can be measured numerically in subjects such as ratio, symbol, color values have affected and changed the graphic design. Among all the mathematical ratios, golden ratio is the most common and popular one. The golden ratio has the effect on graphic design such as its effect on art and it is also be observed in many graphic
design areas. Likewise, with the increase of the amount of daily compute usage, new design fields have emerged. On the other hand, in 1904, when the fractals were laid out by Mandelbrot, they were seen as geometric description format. However, due to fractals’ complex structure, they can only be visualized after the extensive usage of computers. As a consequence of the visualizations, designers had began to see the fractals as a visual field.

Game designs can be the example in addition to visualization of formulas. Due to various formulas and codings, visually occurring data integrations and images have revealed enriched design area. The game design is a sector that develops rapidly throughout its historical process and is still evolving. Game designers are constantly trying to publish new styles and offering the best experience to the player. Gamers are not interested in only the visual quality or content within the game anymore. In order to ensure the player’s continuity in the game, in addition to visuals, the subjects such as scenario, design of the products related to game and the interaction between real world and the game also gained importance.

If the concepts of data visualization within the game are examined, datasets made by the games should be come up and transmitted to players by processing them. The visualization of datasets also based on certain visual rules and laws. In addition to the visual richness in the game design, providing the data transmission between the game and the player also important and those can be managed by the result of the successful visualization.

La inmensa plétora de obras narrativas escritas en español en los últimos treinta años con técnicas fragmentarias, reticulares o “fractales”, no pocas de ellas creadas por algunos de los más importantes autores y novelistas, obliga a buscar una explicación de esta proliferación y diversidad de estrategias literarias. Aunque la estructura fragmentaria no es una novedad, sino una tradición, es obvio y demostrable que el número de obras fragmentarias se ha incrementado exponencialmente en las últimas décadas, y ese hecho precisa de un profundo análisis. Criticada esta práctica a finales de los años 90 por algunos críticos españoles relevantes, y con renuentes contemporáneos, tras el análisis que va a realizarse en este texto parece razonable pensar que el número e importancia de los escritores fragmentarios -ocasional o sistemáticamente- invita a una validación más general y estable de estas estrategias narrativas.
Artículo “La estrategia de Chopin. Modos fractales, reticulares y fragmentarios de escritura narrativa en español entre finales del XX y principios del XXI”, Rilce. Revista de Filología Hispánica, 37.3 (2021), pp. 1002-1023, ISSN: 0213-2370.

This paper presents a review of the fractal approach for describing the large scale distribution of galaxies. We start by presenting a brief, but general, introduction to fractals, which emphasizes their empirical side and applications rather than their mathematical side. Then we discuss the standard correlation function analysis of galaxy catalogues and many observational facts that brought increasing doubts about the reliability of this method, paying special attention to the standard analysis implicit assumption of an eventual homogeneity of the distribution of galaxies. Some new statistical concepts for analysing this distribution is presented, and without the implicit assumption of homogeneity they bring support to the hypothesis that the distribution of galaxies does form a fractal system. The Pietronero-Wertz's single fractal (hierarchical) model is presented and discussed, together with the implications of this new approach for understanding galaxy clustering.

Durante un periodo più o meno lungo di relativa stabilità, il sistema riesce a neutralizzare le perturbazioni, ma ad un certo punto elementi pregressi provocano una crisi di instabilità durante la quale i vari elementi del sistema sono sottoposti a tensione, come un gas surriscaldato in cui aumenti progressivamente il movimento delle molecole. La rottura, o biforcazione catastrofica, è preceduta da uno stato caotico in cui ogni minima fluttuazione può essere estremamente amplificata da fenomeni di feedback positivo. Il futuro del sistema diventa imprevedibile se non si conosce la storia delle condizioni al contorno che hanno provocato lo stato attuale (René Thom, determinista; gli indeterministi sostengono invece che il sistema diventa imprevedibile e basta). In tale stato, una fluttuazione più ampia o una sincronia di condizioni catapultano il sistema ad uno stadio superiore il quale procede in un nuovo stato stabile.

Un sistema come quello capitalistico può rimanere stabile, cioè in equilibrio omeostatico, mediante l'autoregolazione dei propri flussi di energia (retroazione negativa, come nel termostato: keynesismo, patto sociale) per un certo periodo, fino al momento in cui i cicli stabilizzanti non vanno in crisi; la retroazione, da negativa diventa positiva (ad esempio sovrapproduzione e speculazione), le perturbazioni si accumulano, ed il sistema esplode in una biforcazione catastrofica verso un nuovo livello di stabilità. L'importante è ciò che succede nel periodo di fluttuazione caotica: in esso si osserva un fenomeno detto di nucleazione, vale a dire che una regione del sistema improvvisamente si auto-organizza in un nuovo ordine fino a raggiungere, via catastrofe, il nuovo assetto, il quale viene di nuovo stabilizzato da una serie di cicli di retroazione.

There is always another way to say the same thing that doesn't look at all like the way it was said before." Richard Feynman. In this essay, a novel approach to cosmology is presented that mathematically models the Universe as an iterated function system (IFS) analogous to the famous Mandelbrot Set IFS (M): z=z 2 +c, where z and c are complex numbers. In theoretical physics, wavefunctions are functions of a complex space that are commonly used to model the dynamics of particles and waves. In the IFS framework presented herein, complex dynamical systems are generated via the iteration process, where the act of iteration corresponds to 1) a change in the state of the system and 2) a change to the wavefunction itself. In this manner, M can be considered a wavefunction generator. In this framework, all observables, including gravity and time, are thought to be generated by the iteration process. Feynman understood that there are many ways of looking at the Universe that are equivalent in nature but different psychologically. Understanding cosmology in terms of fractals and iterated function systems requires a paradigm shift in the way we approach cosmology. This is an evidence based dissertation and does not contradict the standard model; rather, it attempts to reconstruct it using the principles of the fractal paradigm as outlined in this essay. It is the contention of the author that in order to understand the true nature of light, the universe and everything, we must first understand the important role that fractal cosmology plays in the study of our complex dynamical universe.

This article engages with the works of Sudanese American poet Safia Elhillo, a rising star in Arab, African, and diasporic literary circles. It explores the chronotopic use of metre, repetition, keywords, and mise-en-abyme in Elhillo’s works to think through the ontological stakes of postcolonial verse. It consequently proposes fractals as a non-binary way of conceptualizing hybridity, linking these to a polychronic logic, where the poems’ form and content foreground temporality as a recursive, self-repetitive contemporaneity. Further, through a sustained focus on how Elhillo creates a network of intertextual links between her poems, the article shows how recursive structures that highlight time-space dialectics cast scale as the main link between the various building blocks of the poems. By linking these chronotopic fractals to patterns of orature that draw on both Arabic and African traditions, the article gives an aesthetic reading of the betweenness of Elhillo’s American Sudanese positionality.

Link to full paper: https://www.tandfonline.com/eprint/RRZSCN4CGPZ5SFVYKXVY/full?target=10.1080/17449855.2021.1888150

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.

In this study the general theoretical framework of Complexity theory is presented, viewed through non-extensive statistical theory introduced by Constantino Tsallis in 1988. Nonlinear dynamics, fractional dynamics, thermodynamics and statistical physics constitute the basic elements of Complexity theory which can be applied to processes near, as well as, far from equilibrium. Moreover, the nonextensive statistics can be the fundamental linchpin of Complexity theory from microscopical to macroscopical level. Finally, a brief review of results concerning nonlinear analysis of time series corresponding to different complex systems is given.

We investigate Tsallis qq-statistics for various complex physical systems. In particular, the Tsallis qq-triplet is estimated for space plasmas, atmospheric dynamics, seismicity, and brain and cardiac activity. Remarkable agreement between theoretical Tsallis predictions and experimental estimations was observed in all cases. We also present theoretical extensions of Tsallis qq-statistics for intermittent turbulence theory, non-equilibrium statistics, and fractal extension of dynamics.

This article presents a summary of applications of chaos and fractals in robotics. Firstly, basic concepts of determin‐ istic chaos and fractals are discussed. Then, fundamental tools of chaos theory used for identifying and quantifying chaotic dynamics will be shared. Principal applications of chaos and fractal structures in robotics research, such as chaotic mobile robots, chaotic behaviour exhibited by mobile robots interacting with the environment, chaotic optimization algorithms, chaotic dynamics in bipedal locomotion and fractal mechanisms in modular robots will be presented. A brief survey is reported and an analysis of the reviewed publications is also presented.

Do literary texts have fractal pattern distribution of words? Perhaps. In this chapter from my dissertation I provide some preliminaary evidence that theme words in texts may in fact have a fractal distribution. More work needs to be done to investigate this possiblity.

O objetivo deste texto e ilustrar uma mudanca de foco das simetrias classicas em musica (fundada em espelhamentos) as simetrias complexas (fundadas em similaridades hierarquicas), oferecendo perspectivas importantes para a exploracao de novos caminhos criativos e investigativos em musica. Primeiro, sao apresentadas as simetrias classicas. Em seguida, as simetrias complexas, centradas em similaridades hierarquicas. Finalmente, realizo uma discussao sobre essa mudanca de foco, comentando a nao pertinencia de certos espelhamentos classicos, a traducao entre os suportes envolvidos para a realizacao de simetrias (uma notacao em uma superficie como uma folha de papel e sua traducao em sons), e a associacao entre simetrias fundadas em hierarquias e algumas hierarquias encontradas no mundo dos sons, mencionando brevemente alguns campos de especulacao sobre outros tipos e formas de simetria em musica.

Nearly all nontrivial real-world systems are nonlinear dynamical systems. Chaos describes certain nonlinear dynamical systems that have a very sensitive dependence on initial conditions. Chaotic systems are always deterministic and may be very simple, yet they produce completely unpredictable and divergent behavior. Systems of nonlinear equations are difficult to solve analytically, and scientists have relied heavily on visual and qualitative approaches to discover and analyze the dynamics of nonlinearity. Indeed, few fields have drawn as heavily from visualization methods for their seminal innovations: from strange attractors, to bifurcation diagrams, to cobweb plots, to phase diagrams and embedding. Although the social sciences are increasingly studying these types of systems, seminal concepts remain murky or loosely adopted. This article has three aims. First, it argues for several visualization methods to critically analyze and understand the behavior of nonlinear dynamical systems. Second, it uses these visualizations to introduce the foundations of nonlinear dynamics, chaos, fractals, self-similarity and the limits of prediction. Finally, it presents Pynamical, an open-source Python package to easily visualize and explore nonlinear dynamical systems' behavior.