Fractal geometry offers a new approach to describing the morphology of fern leaves. Traditional morphology is based on the Euclidean concept of shape as an area defined by a boundary. This approach has not proven successful with fern leaves because they are so elaborate. Fractal geometry treats forms as relationships between parts rather than as areas. In fern fronds there are often constant relationships between parts. Four fractal methodologies for describing these relations within leaves are explored in (...) this paper. These include recursive line branching algorithms, iterated function systems (IFS), modifications of IFS, and L-systems. The methods are evaluated by comparing their results with measurements and appearances of various ferns. Fractal methods offer objective, quantifiable and succinct descriptions of fern-leaves. We conclude that fractal geometry offers simple descriptions of some elaborate fern shapes and that it will probably have application in investigating different aspects of ferns and other organisms. (shrink)

Different summarized shape indices, like mean shape index (MSI) and area weighted mean shape index (AWMSI) can change over multiple size scales. This variation is important to describe scale heterogeneity of landscapes, but the exact mathematical form of the dependence is rarely known. In this paper, the use of fractal geometry (by the perimeter and area Hausdorff dimensions) made us able to describe the scale dependence of these indices. Moreover, we showed how fractal dimensions can be deducted from (...) existing MSI and AWMSI data. In this way, the equality of a multiscale tabulated MSI and AWMSI dataset and two scale-invariant fractal dimensions has been demonstrated. (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)

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)

"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)

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.

Drinking kills a significant proportion of individuals every year, particularly in low-income communities. An impulsive differential equation system is used to explore the effectiveness of activated charcoal in detoxifying the body after methanol poisoning. Our impression of activated charcoal is shaped by the fractional dynamics of the problem, which leads to speedy and low-cost first aid. The adsorption capacity of activated charcoal is investigated using impulsive differential equations. The ABC fractional operator’s findings paint a more realistic image of first aid (...) in public and primary health centers, which can assist to reduce methanol poisoning deaths. Numerical simulations are provided using generalized Adams–Bashforth–Moulton method. (shrink)

Evolution and geometry generate complexity in similar ways. Evolution drives natural selection while geometry may capture the logic of this selection and express it visually, in terms of specific generic properties representing some kind of advantage. Geometry is ideally suited for expressing the logic of evolutionary selection for symmetry, which is found in the shape curves of vein systems and other natural objects such as leaves, cell membranes, or tunnel systems built by ants. The topology and geometry of symmetry is (...) controlled by numerical parameters, which act in analogy with a biological organism’s DNA. The introductory part of this paper reviews findings from experiments illustrating the critical role of two-dimensional (2D) design parameters, affine geometry and shape symmetry for visual or tactile shape sensation and perception-based decision making in populations of experts and non-experts. It will be shown that 2D fractal symmetry, referred to herein as the “symmetry of things in a thing”, results from principles very similar to those of affine projection. Results from experiments on aesthetic and visual preference judgments in response to 2D fractal trees with varying degrees of asymmetry are presented. In a first experiment (psychophysical scaling procedure), non-expert observers had to rate (on a scale from 0 to 10) the perceived beauty of a random series of 2D fractal trees with varying degrees of fractal symmetry. In a second experiment (two-alternative forced choice procedure), they had to express their preference for one of two shapes from the series. The shape pairs were presented successively in random order. Results show that the smallest possible fractal deviation from “symmetry of things in a thing” significantly reduces the perceived attractiveness of such shapes. The potential of future studies where different levels of complexity of fractal patterns are weighed against different degrees of symmetry is pointed out in the conclusion. (shrink)

This work investigates the ability of the human visual system to discriminate self-similar Gaussian random textures. The power spectra of such textures are similar to themselves when rescaled by some factor h > 1. As such, these textures provide a natural domain for testing the hypothesis that texture perception is based on a set of spatial-frequency channels characterized by filters of similar shape.

Beginning with Erich Auerbach's reflections on the Goethean concept of world literature, Ottmar Ette unfolds the theory and practice of Literatures of the World. Today, only those literary theories which are oriented upon a history of movement are still capable of doing justice to the confusing diversity of highly dynamic, worldwide transformations. This is because they examine transareal pathways in the field of literature. This volume captures literary processes of exchange and transformation between the Mediterranean, Atlantic and Pacific as well (...) as the interplay of different ways of narrating space and time. Thus, this volume speaks from a fractal point of view and unfolds multiple perspectives. Literatures of the World allow the reader to think in different logical frameworks at the same time, therefore shaping our future on the basis of the diversity of humankind. (shrink)

In this paper, chaotic dynamics of a mixed Rayleigh–Liénard oscillator driven by parametric periodic damping and external excitations is investigated analytically and numerically. The equilibrium points and their stability evolutions are analytically analyzed, and the transitions of dynamical behaviors are explored in detail. Furthermore, from the Melnikov method, the analytical criterion for the appearance of the homoclinic chaos is derived. Analytical prediction is tested against numerical simulations based on the basin of attraction of initial conditions. As a result, it is (...) found that for ω = ν, the chaotic region decreases and disappears when the amplitude of the parametric periodic damping excitation increases. Moreover, increasing of F 1 and F 0 provokes an erosion of the basin of attraction and a modification of the geometrical shape of the chaotic attractors. For ω ≠ ν and η = 0.8, the fractality of the basin of attraction increases as the amplitude of the external periodic excitation and constant term increase. Bifurcation structures of our system are performed through the fourth-order Runge–Kutta ode 45 algorithm. It is found that the system displays a remarkable route to chaos. It is also found that the system exhibits monostable and bistable oscillations as well as the phenomenon of coexistence of attractors. (shrink)

For the purposes of analytical clarity it is possible to distinguish two ways in which Nancy's ontology of sense appeals to art. First, he uses 'art' as a metaphorical operator to give features to his ontology (such as surprise and wonder); second, the practice of the contemporary arts instruct the terms of his ontological project because, in his view, this practice catches up with the fragmentation of existence and thus informs ontology about the structure of existence today. These two different (...) roles—in which 'art' is both a general category able to stage the features of sense in general and a particularly striking example of the alteration sense undergoes in our times—make available for Nancy different perspectives on the question of sense. On the one hand, the general category of 'art' allows Nancy to construct a characterology of sense around terms such as surprise and novelty; on the other, the appeal to the fractal practice of the 'contemporary arts' supports the project of giving an account of sense.This paper analyses the effects on Nancy's conception of sense of these different appeals to 'art' and the practice of 'the contemporary arts.' Are the locales from which these different perspectives on sense take shape compatible? In what ways do they inflect each other or, alternatively, undermine the perspectives of the other on the question of sense? Finally, what do these two strands tell us about what Nancy expects of 'art' and what would happen to his ontology of sense without the different appeals he makes to it? (shrink)

Stable adaptive strategy of Homo sapiens (SASH) is a result of the integration in the three-module fractal adaptations based on three independent processes of generation, replication, and the implementation of adaptations — genetic, socio-cultural and symbolic ones. The evolutionary landscape SASH is a topos of several evolutionary multi-dimensional vectors: 1) extraversional projective-activity behavioral intention (adaptive inversion 1), 2) mimesis (socio-cultural inheritance), 3) social (Machiavellian) intelligence, 4) the extension of inter-individual communication beyond their own social groups and their own species (...) in the rest of the world, 5) the symbolic system of communication (symbolic inheritance), 6) spiritualistic trans- formation of emotionally-shaped components of mentality, 7) the dominance of the rationalist thought mentality (enhancer of adaptive inverse 1), 8) a recursive distribution of projective-activity intentions on the man himself his genome, psyche and culture (Adaptive Inversion 2), 9) introversional reorientation of the vector of cognitive activity (adaptive inversion 3). (shrink)

In the course of the twentieth century natural sciences became an integral part not only of the industrial production but also of the semiotics of audio-visual and literary cultures. In the absence of legitimating traditions the techno-scientific models and relations appear to be a readily available habitual source of creative dynamics. Yet this domination contains a paradox. The application of analogy in, for example, visual culture does not work the same way as in mathematics. In the latter when a relation (...) between two points is known it is always possible to assign to a third point its counterpart exactly. In the former it is only the potentiality of the fourth point that analogy can offer. To make the analogy work an onto-poetic step is needed. Repeated applications of such steps detach the model from the original source. The model re-emerges as pata-physics, in its new re-coded art form and territory of application. Models of, for example, atoms, rainbows, meteors, automatons, radiation, entropy, quantum effects, astro-catastrophic singularity, fractals, etc. find their way into literature and arts. This results in new divisions of space and time that in turn determine the shape of archetypal icons through which contemporary meanings are communicated and actualized. (shrink)

It is conventionally understood that computers play a rather limited role in theoretical mathematics. While computation is indispensable in applied mathematics and the theory of computing and algorithms is rich and thriving, one does not, even today, expect to find computers in theoretical mathematics settings beyond the theory of computing. Where computers are used, by those studying combinatorics , algebra, number theory, or dynamical systems, the computer most often assumes the role of an automated and speedy theoretician, performing manipulations and (...) checking cases in a way assumed to be possible for human theoreticians, if only they had the time, the memory, and the precision. Automated proofs have become standard tools in mathematical logic, and it is often expected that proofs be published in a computer-checkable format. It is not surprising, then, that most philosophical work on computers in theoretical mathematics has been on computers' roles as supplementary mathematicians. Donald MacKenzie's 2001 book Mechanizing Proof demonstrates the rich potential for social and historical studies to complement the substantial analytic debate in this area of philosophy. But what of computers in theoretical mathematics behaving as computers, and not as mere mechanized mathematicians? Very little role is commonly assumed for computers working as supplements to mathematicians, rather than as supplementary mathematicians themselves. Accordingly, very little philosophy has attempted to grapple with theoretical mathematics in which computers play an essential but essentially non-theoretical role. My presentation will draw on work I conducted as a researcher in harmonic analysis on fractals at Cornell University. I will analyze the explicit and implicit conceptual apparatus employed in my and my fellow researchers' use of computers in the theoretical study of second order differential equations, such as those for sound and heat flow, on various fractal analogues of the Sierpinski gasket. Such gaskets are easy to visualize in very crude approximation in a low number of dimensions. As one increases the complexity of the gasket or the refinement of one's analysis, visualization and precise computation become impossible, and soon computers are unable to produce even approximate data to model differential equations in these situations. We thus had to carefully choose analytic approaches and methods to make our theoretical mathematics amenable to computer simulation. In my case, studying the transformation of the gaskets as they are expanded into increasingly high dimensions, computer simulation eventually required that the problem be reimagined entirely in terms of interlinked systems of parameters. This computer-approximation-driven theoretical orientation shaped my mathematical intuitions toward the problem and guided my fellow researchers and me in both theoretical and computational directions. We discovered both that computer approximation could be incredibly powerful as an aid to intuition, and that it can be incredibly difficult to transfer computer-oriented mathematics back into the purely theoretical standards of our area of specialty. I will address the philosophical implications of computer-driven theoretical mathematics, asking how computer experiments can shape both the content and standards of theoretical sciences. (shrink)

This paper reports on the investigation of the effects of neuronal shape, at both individual cell and network level, on the behavior of neuronal systems. More specifically, two-dimensional biologically realistic neuronal networks are obtained that take explicity into account the position and morphology of neuronal cells, with the respective behavior for associative recall being simulated through a diluted version of Hopfield's model. While a specific probability density function is used for the placement of the cell bodies, images of real neuronal (...) cells (namely alpha and beta ganglion cells from the cat retina) are used to obtain biologically realistic models. Several morphological measures – including fractal dimension, the excluded volume, and integral geometry functionals–are estimated for the considered cells, and their values are correlated with the potential of the network for associative recall, which is quantified in terms of the overlap between distorted version of the trained patterns and their original version. Such an approach allows the quantitative and objective characterization of the relationship between neuronal shape and function, an important issue in neuroscience. The obtained results substantiate interesting relationships between the neural morphology and function as determined by the performance of the network. (shrink)

Introduced in _Psychoanalysis and Motivation _ and further developed in _Self and Motivational Systems_, _The Clinical Exchange _, and _A Spirit of Inquiry _, motivational systems theory aims to identify the components and organization of mental states and the process by which affects, intentions, and goals unfold. Motivation is described as a complex intersubjective process that is cocreated in the developing individual embedded in a matrix of relationships with others. Opening by placing motivational systems theory within a contemporary dynamic systems (...) theory, Lichtenberg, Lachmann, and Fosshage then respond to critics of motivational systems theory. The authors present revisions to their approach to the original five motivational systems, adding two more: an affiliative and a caregiving motivational system. The authors go on to suggest, using ideas garnered from complexity theory and fractals, that motivational systems theory can help us understand how a continuity of self can be maintained despite near-constant fluctuations in interpersonal relations. They then consider how the making of inferences, explicitly and implicitly, is shaped by motivation, before applying their theory to an actual human experience - love - to demonstrate the interplay of multiple shifting motivations within an individual. Last, they present new looks at the clinical applicability of their research. Grounded in observational research of infants but relevant to psychoanalysis at any stage of life, motivational systems theory has evolved via the combined experiences of these three analysts for more than 20 years, and remains an important contribution to our understanding of the driving forces behind human experience. (shrink)

This paper reports on the investigation of the effects of neuronal shape, at both individual cell and network level, on the behavior of neuronal systems. More specifically, two-dimensional biologically realistic neuronal networks are obtained that take explicity into account the position and morphology of neuronal cells, with the respective behavior for associative recall being simulated through a diluted version of Hopfield's model. While a specific probability density function is used for the placement of the cell bodies, images of real neuronal (...) cells (namely alpha and beta ganglion cells from the cat retina) are used to obtain biologically realistic models. Several morphological measures – including fractal dimension, the excluded volume, and integral geometry functionals–are estimated for the considered cells, and their values are correlated with the potential of the network for associative recall, which is quantified in terms of the overlap between distorted version of the trained patterns and their original version. Such an approach allows the quantitative and objective characterization of the relationship between neuronal shape and function, an important issue in neuroscience. The obtained results substantiate interesting relationships between the neural morphology and function as determined by the performance of the network. (shrink)

This article uses a Lacanian framework both to map types of posthuman discourse that shape the debates around science, technology and the fate of the human, and to advocate a more psychoanalytic framing of these debates. It identifies three dominant posthumanisms: ‘doomsday’, ‘celebratory’ and ‘critical’. The first adopts an apocalyptic tone in the defence of a supposedly natural human essence; the second unthinkingly embraces the promise of new technologies for augmenting human potential; the third draws on the critique of humanism (...) to balance the first two tendencies. The article then proposes a ‘fractal’ reading of both Freud and Lacan which updates psychoanalysis for the online world today. Finally, the article aligns each of the types of posthumanism with one of Lacan's ‘four discourses’: ‘doomsday’ posthumanism with the discourse of the hysteric, ‘celebratory’ posthumanism with that of the master, and ‘critical’ posthumanism with that of the analyst, thereby putting psychoanalysis at the centre of the posthuman rather than its margins. (shrink)

Diving into the life of the tropical coral reefs and Amadou Hampâté Ba’s reflections on the person conjoin in this work, which is at once philosophical and poetic. The permeable parameters of philosophy, which enable thought to hover between unstable contours rather than to prioritize secure foundations, open to a porous imagination, tracing and retracing panoramic geographies and contemporary tensions of globalization and development. Porous imagination slips, glides, between archipelagos of clay rooftops and refuge dotting the Sudan and the smallest (...) of creatures, source of half the earth’s oxygen and base of the global food chain, the plankton. Concomitantly, ideational codes of nation, continent, humanity, nature, land and water interactively stretch and unwind, shaping a polymetrics of sites and an aesthetics of nourishment. In an epoch beset by the treacherous fractals of overpowering domination and luminous liberation, the morphosis of philosophy through transdisciplinary and transcultural interpretation offers new ways of listening to and retuning songs of sorrow and lament. Pp. 429-435. (shrink)

The concept of geometry may evoke a world of pure platonic shapes, such as spheres and cubes, but a deeper understanding of visual experience demands insight into the perceptual organization of naturalistic form. Japanese gardens excel as designed environments where the complex fractal geometry of nature has been simplified to a structural core that retains the essential properties of the natural landscape, thereby presenting an ideal opportunity for investigating the geometry and perceptual significance of such naturalistic characteristics. Here, (...) fronto-parallel perspective, asymmetrical structuring of the ground plane, spatial arrangement of garden elements, tuning of textural qualities and choice of naturalistic form, are presented as a set of physical features that facilitate a systematic analysis of Japanese garden design per se, as well as the geometry of the particular naturalistic features that it aims to enhance. Comparison with Western landscape design before and after contact between Western and Eastern hemispheres illustrates the degree of naturalness achieved in the Japanese garden, and suggests how classical Western landscape design generally differs in this regard. It further reveals how modern Western gardens culminate in a different naturalistic geometry, thus also a distinctly different vision of the natural landscape, even if these designs were greatly influenced by the gardens of China and Japan. (shrink)

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)

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)

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)

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)

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)

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.

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.

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)

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)

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)

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)

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 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)

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)

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)

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)