In her examination of the problem of identity and its fractal character H. Arendt makes an important step towards a more profound understanding of the thinking Self. Her objective is evident - it is her effort to eliminate the negative effects of the so called identifying monism resulting in mass hysterias. Arendt argues, that the thinking Self is never identical with itself, therefore the definition of subjectivity itself should be seen from the perspective of the difference between the Self (...) and the statments about the Self. For Arendt it means opening the Self, so far shut off in his private cell of the only belief, that it is the Self, who produces the subjectivity. The fractal version of subjectivity represents an open universe of freely chosen responsibility ensuring that there is the only correspondence between the Self and the statments about the Self, namely the correspondence based on knowing the price, paid for it by every Self. And that means tolerating the differences, produced by mind as the thinking Self, differences by which we are constantly being negatively determined. Contrary to identifying monism, however, this determination is not a militant one. (shrink)

Formal systems are standardly envisaged in terms of a grammar specifying well-formed formulae together with a set of axioms and rules. Derivations are ordered lists of formulae each of which is either an axiom or is generated from earlier items on the list by means of the rules of the system; the theorems of a formal system are simply those formulae for which there are derivations. Here we outline a set of alternative and explicitly visual ways of envisaging and analyzing (...) at least simple formal systems using fractal patterns of infinite depth. Progressively deeper dimensions of such a fractal can be used to map increasingly complex wffs or increasingly complex 'value spaces', with tautologies, contradictions, and various forms of contingency coded in terms of color. This and related approaches, it turns out, offer not only visually immediate and geometrically intriguing representations of formal systems as a whole but also promising formal links (1) between standard systems and classical patterns in fractal geometry, (2) between quite different kinds of value spaces in classical and infinite-valued logics, and (3) between cellular automata and logic. It is hoped that pattern analysis of this kind may open possibilities for a geometrical approach to further questions within logic and metalogic. (shrink)

Classical fractal dimensions have recently been effectivized by characterizing them in terms of real-valued functions called gales, and imposing computability and complexity constraints on these gales. This paper surveys these developments and their applications in algorithmic information theory and computational complexity theory.

This paper is the third and final one in a sequence of three. All three papers emphasize that a proposition can be justified by an infinite regress, on condition that epistemic justification is interpreted probabilistically. The first two papers showed this for one-dimensional chains and for one-dimensional loops of propositions, each proposition being justified probabilistically by its precursor. In the present paper we consider the more complicated case of two-dimensional nets, where each "child" proposition is probabilistically justified by two "parent" (...) propositions. Surprisingly, it turns out that probabilistic justification in two dimensions takes on the form of Mandelbrot's iteration. Like so many patterns in nature, probabilistic reasoning might in the end be fractal in character. (shrink)

Natural forms, often characterized by irregularity and roughness, have a unique complexity that exhibit self-similarity across different spatial scales or levels of magnification. Our visual system is remarkably efficient in the processing of natural scenes and tuned to the multi-scale, fractal-like properties they possess. The fractal-like scaling characteristics are ubiquitous in many physical and biological domains, with recent research also highlighting their importance in aesthetic perception, particularly in the visual and, to some extent, auditory modalities. Given the multitude (...) of fractal-like scaling manifestations, we explore potential commonalities in the way they might affect aesthetic preference within and across different physical and sensory domains. We use a range of visual and tactile stimuli to explore the equivalence of fractal-scaling effects on aesthetic preferences within and across visual and tactile modalities. Our results suggest that, despite some superficial differences, the underlying dimensional structure mediating the preference across the two domains is remarkably similar. The qualitative nature of this dimensional structure as well as suggestions for future research are discussed. (shrink)

We examine two connectionist networks—a fractal learning neural network (FLNN) and a Simple Recurrent Network (SRN)—that are trained to process center-embedded symbol sequences. Previous work provides evidence that connectionist networks trained on infinite-state languages tend to form fractal encodings. Most such work focuses on simple counting recursion cases (e.g., anbn), which are not comparable to the complex recursive patterns seen in natural language syntax. Here, we consider exponential state growth cases (including mirror recursion), describe a new training scheme (...) that seems to facilitate learning, and note that the connectionist learning of these cases has a continuous metamorphosis property that looks very different from what is achievable with symbolic encodings. We identify a property—ragged progressive generalization—which helps make this difference clearer. We suggest two conclusions. First, the fractal analysis of these more complex learning cases reveals the possibility of comparing connectionist networks and symbolic models of grammatical structure in a principled way—this helps remove the black box character of connectionist networks and indicates how the theory they support is different from symbolic approaches. Second, the findings indicate the value of future, linked mathematical and empirical work on these models—something that is more possible now than it was 10 years ago. (shrink)

It has long been appreciated that the brain is oscillatory 1. Early measurements of brain electrophysiology revealed rhythmic synchronization unifying large swaths of the brain. The study of neural oscillation has enveloped cognitive neuroscience and neural systems. The traditional belief that oscillations are epiphenomenal of neuron spiking is being challenged by intracellular oscillations and the theoretical backing that oscillatory activity is fundamental to physics. Subjective experience oscillates at three particular frequency bands in a cognitive triad: perception at 5 Hz, action (...) at 2 Hz, and attention at 0.1 Hz. This triad functions as a means of information flow across scales of magnitude in a biological fractal. The _Homunculus Solution _is proposed in which mental experience occurs at fixed scales of biology. The mind is composed of minds, perceived as "the voices in your head." Each voice has voices inside its head to increasingly microscopic scales, forming an interactive fractal of subjective experience. Normal 0 false false false EN-US X-NONE X-NONE. (shrink)

There is a whole discussion around the genuine/non genuine appurtenance of the Fractal Art to the Art (Ken Keller, Tad Boniecki, Noel Huntley a.o.). Fractal Art is a new way to manipulate shapes, colors and light. It is a subclass of the visual digital art that could describe as that art form produced using a computer (PC, Mac), fractal and graphical software and output devices (monitors, plotters, printers etc.) or using fractal rules and traditional painting techniques (...) (example: Pollock) as essential tools in the creative process. It is crucial to understand that the use of a computer is not a sine qua non condition, even most of the fractal artworks were digitally realized. Fractal Art is an experimental art, engaging in a new way the relations between Creator and Work. Fractal Art has a genuine equilibrium between “pragmatic” and “theoretic”. It is rather about “discovering” than “manufacturing”, rather “evocation” than “mimesis”. The conclusion has to be: Fractal Art is a genuine art form. (shrink)

Oblak, Boyadzhieva, and Bon claim that fractality is a modal property of the experience of perceptual presence. I will instead argue that the concept of fractality implies amodality and that modal ….

This paper seeks to build on the extensive connections that have arisen between automata theory, combinatorics on words, fractal geometry, and model theory. Results in this paper establish a characterization for the behavior of the fractal geometry of “k-automatic” sets, subsets of $[0,1]^d$ that are recognized by Büchi automata. The primary tools for building this characterization include the entropy of a regular language and the digraph structure of an automaton. Via an analysis of the strongly connected components of (...) such a structure, we give an algorithmic description of the box-counting dimension, Hausdorff dimension, and Hausdorff measure of the corresponding subset of the unit box. Applications to definability in model-theoretic expansions of the real additive group are laid out as well. (shrink)

Successful applications of hierarchical complexity to the behaviors of organisms, animals and humans, and social entities evidence the scaling properties of self-similarity, thus the bounded fractal characteristics of orders of hierarchical complexity. The theory specifies an identical sequence of discrete-state transition steps required from each stage of performance to the next. It repeats at all scales. Tasks nested within the step sequence evidence self-similarity with the orders of complexity. This model introduces questions about noise categories when system tasks are (...) fully accounted for, dependent, self-similar, and measurable. Ubiquitous transition steps are inherent dynamics of evolution. (shrink)

In recent years the magnificent world of fractals has been revealed. Some of the fractal images resemble natural forms so closely that Benoit Mandelbrot's hypothesis, that the fractal geometry is the geometry of natural objects, has been accepted by scientists and non-scientists alike. The present paper critically examines Mandelbrot's hypothesis. It first analyzes the concept of a fractal. The analysis reveals that fractals are endless geometrical processes, and not geometrical forms. A comparison between fractals and irrational numbers (...) shows that the former are ontologically and epistemologically even more problematic than the latter. Therefore, it is argued, a proper understanding of the concept of fractal is inconsistent with ascribing a fractal structure to natural objects. Moreover, it is shown that, empirically, the so-called fractal images disconfirm Mandelbrot's hypothesis. It is conceded that the fractal geometry can be used as a useful rough approximation, but this fact has no bearing on the physical theory of natural forms. (shrink)

This study investigates global properties of three categories of English text: canonical fiction, non-canonical fiction, and non-fictional texts. The central hypothesis of the study is that there are systematic differences with respect to structural design features between canonical and non-canonical fiction, and between fictional and non-fictional texts. To investigate these differences, we compiled a corpus containing texts of the three categories of interest, the Jena Corpus of Expository and Fictional Prose. Two aspects of global structure are investigated, variability and self-similar (...) patterns, which reflect long-range correlations along texts. We use four types of basic observations, the frequency of POS-tags per sentence, sentence length, lexical diversity, and the distribution of topic probabilities in segments of texts. These basic observations are grouped into two more general categories, the lower-level properties and, which are observed at the level of the sentence, and the higher-level properties and, which are observed at the textual level. The observations for each property are transformed into series, which are analyzed in terms of variance and subjected to Multi-Fractal Detrended Fluctuation Analysis, giving rise to three statistics: the degree of fractality () of the fractal spectrum. The statistics thus obtained are compared individually across text categories and jointly fed into a classification model. Our results show that there are in fact differences between the three text categories of interest. In general, lower-level text properties are better discriminators than higher-level text properties. Canonical fictional texts differ from non-canonical ones primarily in terms of variability in lower-level text properties. Fractality seems to be a universal feature of text, slightly more pronounced in non-fictional than in fictional texts. On the basis of our results obtained on the basis of corpus data we point out some avenues for future research leading toward a more comprehensive analysis of textual aesthetics, e.g., using experimental methodologies. (shrink)

Digital companies exhibit discontinuous growth in the process of shifting from their existing core business to a newer and less familiar business. This pattern of growth often ends in failure mainly because companies invest most of their resources in maintaining the value network of their existing core business, which ultimately results in a “lock-in” effect. The fractal theory assumes that there are similarities among fractals within companies. These similarities may reduce the threats posed by the value network lock-in effect (...) and increase the chances of successful discontinuous growth. In this study, we applied fractal theory to consider the following questions: in what aspects does the successful discontinuous growth of digital companies exhibit fractal characteristics? What strategy does digital companies use to ensure these fractal characteristics? We adopted an exploratory single-case study method and chose ByteDance as the case company to analyze its successful shift from Toutiao to Douyin. Our results show that a necessary condition for the successful discontinuous growth of digital companies is that similarities exist between the existing core business and the new business and entrepreneurial bricolage is a strategy used by digital companies to ensure the existence of fractal characteristics of similarities. We discuss the theoretical contributions and practical implications of this finding. (shrink)

Feminists have long been aware that the notion of women is problematic and using it uncritically without further qualifications leads to exclusions. In the article, I argue that the source of these problems lies in the understanding of concepts as static and clearly defined. I deploy Hegel’s idea of syllogism to define dynamic concepts, which I term ‘fractal concepts’ because of their complexity and constant development. In such structures the balance between the universal, the particular and the individual is (...) maintained and, consequently, every woman is essential for the definition of such a non-essentialist notion of women. I propose that an epistemological interpretation of the Hegelian idea of recognition can be used to describe the development of fractal concepts in the confrontation of different perspectives. It also explains the obstruction of this process by the desire for domination and control, which results in the appropriation of group identities by their privileged members. I situate the logical and political framework developed in the paper in the context of feminist controversies around the concept of woman and argue that it provides us with a comprehensive account of intersectionality as well as an explicative solution to the regress problem. (shrink)

Abstract The first fractal constructions appeared in mathematics in the second half of the 19th century. Their history is divided into two periods. The first period lasted 100 years and is a good example of the method of proofs and refutations discovered by Lakatos. The modern history of these objects started 20 years ago, when Mandelbrot decided to create fractal geometry, a general theory concentrated on specific properties of fractals. His approach has been surprisingly effective. The aim of (...) this paper is to examine the reasons for Mandelbrot's success and for the present popularity of fractals. They are now known not only in mathematics but also in many fields of natural, social and applied sciences, and even in the arts. (shrink)

The principle of Identity of Indiscernibles has been challenged with various thought experiments involving symmetric universes. In this paper, I describe a fractal universe and argue that, while it is not a symmetric universe in the classical sense, under the assumption of a relational theory of space it nonetheless contains a set of objects indiscernible by pure properties alone. I then argue that the argument against the principle from this new thought experiment resists better than those from classical symmetric (...) universes three main objections put forth against this kind of arguments. (shrink)

One of the most classic ontological problems of philosophy is the question of why there exists something instead of nothing. In John L. Culliney and David Jones’s The Fractal Self: Science, Philoso...

Our universe, science reveals, began in utter simplicity, then evolved into burgeoning complexity. Starting with subatomic particles, dissimilar entities formed associations—binding, bonding, growing, branching, catalyzing, cooperating—as “self” joined “other” following universal laws with names such as gravity, chemical attraction, and natural selection. Ultimately life arose in a world of dynamic organic chemistry, and complexity exploded with wondrous new potential. Fast forward to human evolution, and a tension that had existed for billions of years now played out in an unprecedented arena (...) of conscious calculation and cultural diversity. Cooperation interleaving with competition; intimacy oscillating with integrity—we dwell in a world where yin meets yang in human affairs on many levels. In The Fractal Self, John Culliney and David Jones uncover surprising intersections between science and philosophy. Connecting evidence from evolutionary science with early insights of Daoist and Buddhist thinkers, among others, they maintain that sagely behavior, envisioned in these ancient traditions, represents a pinnacle of human achievement emerging out of our evolutionary heritage. They identify an archetype, “the fractal self,” a person in any walk of life who cultivates a cooperative spirit. A fractal self is a sage in training, who joins others in common cause, leads from within, and achieves personal satisfaction in coordinating smooth performance of the group, team, or institution in which he or she is embedded. Fractal selves commonly operate with dedication and compassionate practice in the service of human society or in conserving our planet. But the competitive side of human nature is susceptible to greed and aggression. Self-aggrandizement, dictatorial power, and ego-driven enforcement of will are the goals of those following a self-serving path—individuals the authors identify as antisages. Terrorist leaders are an especially murderous breed, but aggrandizers can be found throughout business, religion, educational institutions, and governments. Humanity has reached an existential tipping point: will the horizon already in view expand with cooperative progress toward godlike emergent opportunities or contract in the thrall of corrupt oligarchs and tribal animosities? We have brought ourselves to a chaotic edge between immense promise and existential danger and are even now making our greatest choice. (shrink)

Primal emergence -- Out of the dreamtime -- The quickening of chemistry -- Ecology emergent -- Intimate ark : sex and emergence -- Social order in nature : between conflict and cooperation -- Self within world -- From self to sage -- From self to no-self to all-self -- Anti-sage : from cult to empire -- Into Indra's net.

The interconnections between self and surroundings in Daoist thought have been explored in the past in a variety of contexts. This paper, however, explores the Daoist version of the relational self from the general perspective of chaos theory and, more specifically, examines the role of the self in creating emergent form and dynamism in nature and society. The fractal self and world merge through the disciplined effort of the sage until the effort becomes effortless. Both self and world are (...) transformed and become one through the emergent moment. This moment represents a new opportunity for subsequent patterns, or attractors, to emerge. We suggest that the self folds itself into the world, creating and being created by a new attractor, or pattern, in the organization of nature, which we argue is Dao. Our current investigation addresses the following aspects of chaos theory and its relation to Daoism: (1) the Daoist notion of wuwei as perfect congruence; (2) yin and yang at the edge of chaos; (3) the emergent nature of the myriad things; and (4) a Daoist warning against fractal disconnections in the world. Finally, we conclude that the self has the potential to become the world. To approach this amplified condition, the self must dedicate itself to and risk open engagement with events across the complexity of nature. Through openly engaging the world, the self is transformed, leaving the world forever changed. (shrink)

Benoit B. Mandelbrot, when discussing the global appeal of fractal patterns and designs, draws upon examples from across numerous world cultures. What may be missed in Mandelbrot's presentation is Immanuel Kant’s precedence in recognizing this sort of widespread beauty in art and nature, fractals avant la lettre. More importantly, the idea of the fractal may itself assist the aesthetic attitude which Kantian beauty requires. In addition, from a Kantian perspective, fractal patterns may offer a source for a (...) sense of community with humanity. I close with an excursus on the more sombre note of Kantian sublimity which fractals can also present. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Chaos, Fractals, and the Pedagogical Challenge of Jackson Pollock's "All-Over" PaintingsFrancis Halsall (bio)IntroductionThe "all-over" abstract canvases that Jackson Pollock produced between 1943 and 1951 present a pedagogical challenge in how to account for their apparently chaotic structure. One reason that they are difficult to teach about is that they have proved notoriously difficult for art historians to come to terms with. This is undoubtedly a consequence of their abstraction. (...) In the face of an apparent disintegration of the traditional pictorial distinction between figure and ground, multiplicities of (sometimes contradictory) readings present themselves. The question, posed to students, as to what they represent is an open one. To many, recalling the opinions of the baffled audience of the 1950s, they are nothing but an inchoate mess. In 1950 Time magazine referred to them as "chaos," prompting Pollock to wire a heated reply, "NO CHAOS DAMN IT."1Pollock's claim of "no chaos" can, however, be unpacked. This article looks at how a scientific analysis based on Chaos Theory and fractal patterns can be used to demonstrate to students that the paintings are indeed chaotic, but that this in turn provides evidence of an internal structure, an order within the chaos. This order is mimetically similar to other chaotic patterns and systems, from coastlines to economic systems. My overall position in this article is one of skepticism as to what such analysis ultimately has to offer in art historical, aesthetic, or pedagogical terms.The ProblemFor a student encountering Pollock's work for the first time, especially after a standard first-year survey of Western art, it can seem incomprehensible. [End Page 1] The challenge of Pollock's work is that none of the multiple interpretations that have been brought to bear on the paintings have any special claim to validity. Given the suggestive ambiguity and "chaos" of the images, there is little in the work itself that challenges the numerous, often contradictory, readings that have been posited. In fact, how art historians approach such images tells students something about the way art history itself works and how art historians are faced with particular choices in applying their discursive models. They must choose whether these accounts "fit" or satisfy the conditions of "rightness," to evoke Nelson Goodman's terms.2The numerous questions that have been asked of what the paintings actually represent are themselves demonstrative of the wide variety of techniques that modern art historians have at their disposal. Analyses have been based upon observations that are couched in the vocabularies of the political (often Marxist theory), feminism, psychoanalysis, iconography, formalism, and so forth.3 At the time of the paintings' creation even the CIA was in on the act; apparently realizing the ambiguity of the work left it open to political manipulation, they secretly funded the promotion of abstract expressionism as a weapon of the cold war.4 Simultaneously, members of Congress such as George Dondero were actually claiming the opposite—that Pollock was making paintings that were a "means of espionage" (sic) and that "If you know how to read them... [they] will disclose weak spots in U.S. fortifications, and such crucial constructions as Boulder Dam."5Ernst Gombrich's celebrated frustration with abstract art highlights the problems that those following traditional art historical methods had with such art at the time. Writing on Pollock's Number 12 (1952), Gombrich also seemed perplexed in the face of chaos. His response to the lack of explicit representational content or apparently accountable meaning was to claim:It is quite consistent that these [action] painters must counteract all semblance of familiar objects or even of patterns in space. But few of them appear to realise that they can drive into the desired identification only those who know how to apply the various traditional consistency tests and thereby discover the absence of any meaning except the highly ambiguous meaning of traces.6Indeed, if there is any consensus to be communicated to students about the work, it is that a single, specific meaning will always allude them.All of the above reactions to the work makes the claims of three physicists to have found the... (shrink)

"Fractal" is a term coined by mathematician Benoit Mandelbrot to denote the geometry of nature, which traces inherent order in chaotic shapes and processes. Fractal concepts are part of our emerging vocabulary and can be useful in identifying patterns of human behavior, culture, and history, while enhancing our understanding of the nature of consciousness. According to William J. Jackson, the more one studies fractals, the more apparent their connections to the humanities become. In the recursive patterns of religious (...) music, in temple architecture in India, in cathedral structures in Europe and America, in the imagery of religious literature depicting infinity and abundance, and in poetic descriptions of the nature of consciousness, fractal-like configurations are pervasive. Recognition of this structure, which is also found in social organizations and ritual symbolism, requires only that one develop "an eye for fractals" by studying the work of researchers and observing nature. One then begins to see that the separation of humanities and science is convenient oversimplification, not an ultimate fact. Includes a DVD of animated fractals. (shrink)

Habitat fragmentation produces isolated patches characterized by increased edge effects from an originally continuous habitat. The shapes of these patches often show a high degree of irregularity: their shapes deviate significantly from regular geometrical shapes such as rectangular and elliptical ones. In fractal theory, the geometry of patches created by a common landscape transformation process should be statistically similar, i.e. their fractal dimensions and their form factors should be equal. In this paper, we analyze 49 woodlot fragments (Pinus (...) sylvestris L.) in the Belgian Kempen region to study the direct relationship between a transformation process and the concomitant patch geometry. Although the fractal dimension of the woodlots is scattered (i.e. they are not statistically similar), the perimeter-area relation of the fragments is characterized by a single, ‘dimension-like’ exponent. This exponent suggests a certain shape homogeneity among the patches, which is confirmed by the absence of hierarchical levels associated with sharp increases of the fractal dimension at scale transitions. The interaction of different natural (soil factor, vegetation type) and anthropogenic (afforestation, urbanization) processes during patch development is assumed to have generated this feature. Comparison of the area and perimeter fractal dimension with an ecological index for habitat quality, the interior-to-edge ratio, shows that the fractal dimension is suitable for predicting interior habitat presence, which is more likely for patches with smooth perimeters and compact areas. The ratio of the area to the perimeter fractal dimension confirms this observation, with high values for high interior-to-edge ratios, characteristic for regularly shaped patches. (shrink)

This article explores an understanding of organizational management developed from the metaphorical application of complexity science to the field of organizational development. It focuses on the insights that fractality triggers in relation to an alternative way of examining and appreciating organizational hierarchy, and the subsequent implications to liberating creativity, ingenuity and potentiality of individuals working within the organization. Sites where such a fractal-hierarchy mindset appears to be evident are discussed, and the effects on productivity noted.

How a person is on the inside replicates how he is on the outside—so he is a fractal in that sense. And how he is in little matters replicates how he is in big ones—so he is a fractal in that sense as well. And so it is that a person’s identity has a doubly fractal structure.

In an extension of speculations that physical space–time is a fractal which might itself be embedded in a high-dimensional continuum, it is hypothesized to “compensate” for local variations of the fractal dimension by instead varying the metric in such as way that the intrinsic dimensionality remains an integer. Thereby, an extrinsic fractal continuum is intrinsically perceived as a classical continuum. Conversely, it is suggested that any variation of the metric from its Euclidean form can be “shifted” to (...) nontrivial fractal topology. Thereby “holes” or “gaps” in spacetime could give rise to curvature. (shrink)

A system is counterentropic to the extent that it has a fractal structure and entropic to the extent that it doesn't.

Many moral theories hold individuals responsible for their marginal impact on massive patterns (for instance overall value or equality of opportunity) or for following whichever rules would realise that pattern on the whole. But each of these injunctions is problematic. Intuitively, the first gives individuals responsibility for too much, and the second gives them responsibility for too little. I offer the outlines of a new approach to ethics in collective action contexts. I defend a new collaborative principle that assigns recognisably (...) worthwhile responsibilities to agencies wherever possible. This principle supports a fractal model of moral responsibility, one that favours restructuring divisions of labour in ways that enhance the prospects of recognition at every level of social organisation. This essay is programmatic: starting from a conjecture about the moral significance of recognition, it sketches a new way of approaching a range of collective action problems that combines the importance of the perspectives of both moral and political philosophy. (shrink)

Using fractal analyses to study events allows us to capture the scale-independence of those events, that is, no matter at which level we study a phenomenon, we should get roughly the same results because events exhibit similar structure across scales. This is demonstrably true in mathematical fractals but is less assured in behavioral fractals. The current research directly tests the scale-independence hypothesis in the behavioral domain by exploring the fractal structure of aggression, a social phenomenon comprising events that (...) span temporal scales from minutes of face-to-face arguments to centuries of international armed conflicts. Using publicly available data, we examined the temporal fractal structure of four scales of aggression: wars, riots, violent crimes, and body movement during arguments. Our results lend mixed support to the scale-independence hypothesis and provide insight into the self-organization of human interactions. (shrink)

Theterm, fractal, coined by Benoit B. Mandelbrot, describes a shape or pattern within a greater pattern of which it is a scaling piece identical to the greater pattern and in which are reproduced an infinite number of parts or fragments which are also identical to it, thus, identical to the whole at all scales. In this paper, the author describes Hindu cosmology as it is replicated in the elements of the Bharata Natyam, drawing the analogy to fractal patterning.

Fractal dynamics is a unique, systems-based approach of looking at and thinking about organizations. In this view, the organization is viewed as part of a pervasive organizational fractal. Many astounding properties such as fractal completion, fractal influence and fractal universality emerge in fractal space. These offer new insight into the fundamental dynamics animating all organizations. Routine management tasks such as strategy formulation, cost cutting, process redesign, product formulation, management of change and leadership development can (...) be conducted anew with powerful and innovative insights, unavailable when management is approached traditionally through a linear and sequential thinking approach. This article focuses primarily on the rationale for employing fractal dynamics, and on the search for the pervasive organizational fractal that animates all organizations. A brief example of how fractal dynamics offers insights in the case of management of change is also provided. (shrink)

Fractal patterns offer one way to represent the rough complexity of the natural world. Whilst they dominate many of our visual experiences in nature, little large-scale perceptual research has been done to explore how we respond aesthetically to these patterns. Previous research (Taylor et al., 2011) suggests that the fractal patterns with mid-range fractal dimensions have universal aesthetic appeal. Perceptual and aesthetic responses to visual complexity have been more varied with findings suggesting both linear (Forsythe et al., (...) 2011) and curvilinear (Berlyne, 1970) relationships. Individual differences have been found to account for many of the differences we see in aesthetic responses but some, such as culture, have received little attention within the fractal and complexity research fields. This 2-study paper aims to test preference responses to fractal dimension and visual complexity, using a large cohort (N=443) of participants from around the world to allow universality claims to be tested. It explores the extent to which age, culture and gender can predict our preferences for fractally complex patterns. Following exploratory analysis that found strong correlations between fractal dimension and visual complexity, a series of linear mixed-effect models were implemented to explore if each of the individual variables could predict preference. The first tested a linear complexity model (likelihood of selecting the more complex image from the pair of images) and the second a mid-range fractal dimension model (likelihood of selecting an image within mid-range). Results show that individual differences can reliably predict preferences for complexity across culture, gender and age. However, in fitting with current findings the mid-range models show greater consistency in preference not mediated by gender, age or culture. This paper supports the established theory that the mid-range fractal patterns appear to be a universal construct underlying preference but also highlights the fragility of universal claims by demonstrating individual differences in preference for the interrelated concept of visual complexity. This highlights a current stalemate in the field of empirical aesthetics. (shrink)

Fractal structure is a ubiquitous property found in nature and biology, and has been observed in processes at different levels of organization, including rhythmic behavior and musical structure. A temporal process is characterized as fractal when serial long-term correlations and statistical self-similarity (scaling) are present. Previous studies of sensorimotor synchronization using isochronous (non-fractal) stimuli show that participants' errors exhibit persistent structure (positive long-term correlations), while their inter-tap intervals (ITIs) exhibit anti-persistent structure (negative long-term correlations). Auditory-motor synchronization has (...) not been investigated with anti-persistent stimuli. In the current study, we systematically investigated whether the fractal structure of auditory rhythms was reflected in the responses of participants who were asked to coordinate their taps with each event. We asked musicians and non-musicians to tap with 12 different rhythms that ranged from anti-persistent to persistent. The scaling exponents of the ITIs were strongly correlated with the scaling exponents of the stimuli, showing that the long-term structure of the participants' taps scaled with the long-term structure of the stimuli. Surprisingly, the performance of the musicians was not significantly better than that of the non-musicians. Our results imply that humans are able to readily adapt (rather than simply react) to the overall statistical structure of temporally fluctuating stimuli, regardless of musical skill. (shrink)