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)

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)

Ce texte a déjà paru sur Techno-science.net. Des chercheurs découvrent la formule mathématique du rythme et avancent que notre cerveau pourrait être câblé pour y répondre. Une nouvelle étude montre que tout compositeur, de Bach à Brubeck, répète des motifs rythmiques, de sorte que la partie reproduit le tout. Une équipe de recherche dirigée par les neuroscientifiques Daniel Levitin et Vinod Menon, respectivement des universités McGill et Stanford, a analysé les partitions de quelque 2 000 compositions - Mathématiques – Nouvel (...) article. (shrink)

The recent scholarly promotion of Pierre Gassendi to a key position in the formative modern period raises doubts about the portrayal of Descartes as “the father” of the post-Scholastic philosophical conceptualization. I defend the Cartesio-centric account against Thomas M. Lennon’s elliptical alternative. The defense necessitates a reassessment of the root nature of Descartes’s contribution—specifically of the interplay between philosophy and science, the latter being the crucial extraphilosophical component of the new practico-cognitive ensemble. This raises questions about the “philosophically” of Descartes’s (...) activity, and with that questions about the quality of modernity. By way of explaining Gassendis nonoriginative position, his attitude toward the post-Scholastic Humanist thought that Descartes simply repudiated is examined. (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 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)

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)

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 (...) class='Hi'>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)

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)

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)

"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)

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)

Let M be a smooth, compact manifold of dimension n ≥ 5 and sectional curvature | K | ≤ 1. Let Met (M) = Riem(M)/Diff(M) be the space of Riemannian metrics on M modulo isometries. Nabutovsky and Weinberger studied the connected components of sublevel sets (and local minima) for certain functions on Met (M) such as the diameter. They showed that for every Turing machine T e , e ∈ ω, there is a sequence (uniformly effective in e) of homology (...) n-spheres {P k e } k ∈ ω which are also hypersurfaces, such that P k e is diffeomorphic to the standard n-sphere S n (denoted P k e ≈ diff S n ) iff T e halts on input k, and in this case the connected sum N k e =M ♯ P k e ≈ diff M , so N k e ∈ Met(M), and N k e is associated with a local minimum of the diameter function on Met(M) whose depth is roughly equal to the settling time σ e (k) of T e on inputs y i } ∈ ω of c.e. sets so that for all i the settling time of the associated Turing machine for A i dominates that for A i + 1 , even when the latter is composed with an arbitrary computable function. From this, Nabutovsky and Weinberger showed that the basins exhibit a "fractal" like behavior with extremely big basins, and very much smaller basins coming off them, and so on. This reveals what Nabutovsky and Weinberger describe in their paper on fractals as "the astonishing richness of the space of Riemannian metrics on a smooth manifold, up to reparametrization." From the point of view of logic and computability, the Nabutovsky-Weinberger results are especially interesting because: (1) they use c.e. sets to prove structural complexity of the geometry and topology, not merely undecidability results as in the word problem for groups, Hilbert's Tenth Problem, or most other applications; (2) they use nontrivial information about c.e. sets, the Soare sequence {A i } i ∈ ω above, not merely G öodel's c.e. noncomputable set K of the 1930's; and (3) without using computability theory there is no known proof that local minima exist even for simple manifolds like the torus T 5 (see §). (shrink)

Kant's theory of space includes the idea that straight lines and planes can be defined in Euclidean geometry by a concept which nowadays has been revived in the field of fractal geometry: the concept of self-similarity. Absolute self-similarity of straight lines and planes distinguishes Euclidean space from any other geometrical space. Einstein missed this fact in his attempt to refute Kant's theory of space in his article ‘Geometrie und Erfahrung’. Following Hilbert and Schlick he took it for (...) granted, mistakenly, that purely geometrical definitions of the concepts of straight line and plane are not feasible, except as “implicit definitions”. Therefore Einstein's criticism is not persuasive. (shrink)

Field propagation on fractal structures can generate a large scale symmetric space-time geometry. The significance of this fact and the nature of the resulting space-time are discussed.“Each contained all the others, but in this totality each was confused and comingled with all the others without order and system.”—Haïm Vital, 1543–1620 (Kabbalist of Safed); English translation taken from “Sabbatai Sevi” by G. Scholem (Routledge and Kegan Paul, London, 1973), p. 36.

Few philosophers have left a legacy like that of Gottfried Wilhelm Leibniz. He has been credited not only with inventing the differential calculus, but also with anticipating the basic ideas of modern logic, information science, and fractal geometry. He made important contributions to such diverse fields as jurisprudence, geology and etymology, while sketching designs for calculating machines, wind pumps, and submarines. But the common presentation of his philosophy as a kind of unworldly idealism is at odds with all (...) this bustling practical activity. In this book Richard. T. W. Arthur offers a fresh reading of Leibniz’s philosophy, clearly situating it in its scientific, political and theological contexts. He argues that Leibniz aimed to provide an improved foundation for the mechanical philosophy based on a new kind of universal language. His contributions to natural philosophy are an integral part of this programme, which his metaphysics, dynamics and organic philosophy were designed to support. Rather than denying that substances really exist in space and time, as the idealist reading proposes, Leibniz sought to provide a deeper understanding of substance and body, and a correct understanding of space as an order of situations and time as an order of successive things. This lively and approachable book will appeal to students of philosophy, as well as anyone seeking a stimulating introduction to Leibniz's thought and its continuing relevance. (shrink)

From one of the most influential scientists of our time, a dazzling exploration of the hidden laws that govern the life cycle of everything from plants and animals to the cities we live in. The former head of the Sante Fe Institute, visionary physicist Geoffrey West is a pioneer in the field of complexity science, the science of emergent systems and networks. The term "complexity" can be misleading, however, because what makes West's discoveries so beautiful is that he has found (...) an underlying simplicity that unites the seemingly complex and diverse phenomena of living systems, including our bodies, our cities and our businesses. Fascinated by issues of aging and mortality, West applied the rigor of a physicist to the biological question of why we live as long as we do and no longer. The result was astonishing, and changed science, creating a new understanding of energy use and metabolism: West found that despite the riotous diversity in the sizes of mammals, they are all, to a large degree, scaled versions of each other. If you know the size of a mammal, you can use scaling laws to learn everything from how much food it eats per day, what its heart-rate is, how long it will take to mature, its lifespan, and so on. Furthermore, the efficiency of the mammal's circulatory systems scales up precisely based on weight: if you compare a mouse, a human and an elephant on a logarithmic graph, you find with every doubling of average weight, a species gets 25% more efficient--and lives 25% longer. This speaks to everything from how long we can expect to live to how many hours of sleep we need. Fundamentally, he has proven, the issue has to do with the fractal geometry of the networks that supply energy and remove waste from the organism's body. (shrink)

Entropy is ubiquitous in physics, and it plays important roles in numerous other disciplines ranging from logic and statistics to biology and economics. However, a closer look reveals a complicated picture: entropy is defined differently in different contexts, and even within the same domain different notions of entropy are at work. Some of these are defined in terms of probabilities, others are not. The aim of this chapter is to arrive at an understanding of some of the most important notions (...) of entropy and to clarify the relations between them, After setting the stage by introducing the thermodynamic entropy, we discuss notions of entropy in information theory, statistical mechanics, dynamical systems theory and fractal geometry. (shrink)

We present a survey of the recent applications of continuous domains for providing simple computational models for classical spaces in mathematics including the real line, countably based locally compact spaces, complete separable metric spaces, separable Banach spaces and spaces of probability distributions. It is shown how these models have a logical and effective presentation and how they are used to give a computational framework in several areas in mathematics and physics. These include fractal geometry, where new results on (...) existence and uniqueness of attractors and invariant distributions have been obtained, measure and integration theory, where a generalization of the Riemann theory of integration has been developed, and real arithmetic, where a feasible setting for exact computer arithmetic has been formulated. We give a number of algorithms for computation in the theory of iterated function systems with applications in statistical physics and in period doubling route to chaos; we also show how efficient algorithms have been obtained for computing elementary functions in exact real arithmetic. (shrink)

Abstract During the last decades it has become widely accepted that scientific observations are ?theory?laden?. Scientists ?see? the world with their theories or theoretical presuppositions. In the present paper it is argued that they ?see? with their scientific instruments as well, as the uses of scientific instruments is an important characteristic of modern natural science. It is further argued that Euclidean geometry is intimately linked to technology, and hence that it plays a fundamental part in the construction and operation (...) of scientific instruments. Finally, Euclidean geometry is compared to fractal geometry, and the question of its a priori status is raised. Although the position that Euclidean geometry is a priori in the original Kantian sense is untenable, the paper concludes that in some restricted sense Euclidean geometry may be said to be a priori. (shrink)

I have argued here for a change in a scientific world-view, from that of the study of forms to that of process. In doing so we need to understand as to how process creates form. In showing this I have at first drawn from the history of Buddhist philosophy; with its concepts of ‘Sunyata’ (Emptiness) and radical interdependency (Huayen). Then showed its parallel with modern Fractal geometries, which thru’ rather simple mathematics, shows as to how process could derive form. (...) I have then gone on to Quantum constructions, which is without doubt the most advanced scientific development in modern times, and attempted to show two vital directions which contradict the classical scientific world view. The usual scientific view retains a Descartian, ‘out there’ deterministic framework. I argue that with the break down of determinism, and the non locality of phenomena suggested by Quantum Mechanics allows for an independent functioning of consciousness; which then is not a mere epiphenomena of neural activity within the brain. Such a view has wide implications on how we live and how we die. (shrink)

The biocognitive theory presented in this paper offers an alternative to the attribution of cause perpetuated by the life sciences in our western culture. Historically, biology has based its epistemology on physics to understand life, whereas cognitive science has grounded its ontology in a convergence of biology, physics, and philosophy to provide models of self that range from a passive acceptance of an outside world to the active creation of an inner world. While Newtonian physics has served us well in (...) the physical sciences, the life sciences continue to embrace the limitations of its reductionism without advancing to the more inclusive concepts offered by complexity and quantum theories. As long as the biological and cognitive sciences remain married to Newtonian physics and Cartesian philosophy, mind will be relegated to an epiphenomenon of biology that will continue to separate cognitive processes from biological functions. Rather than choosing between upward causality that explains cause from the simplest level of the organism and downward causality that explains it from the most complex to the simplest, biocognitive theory offers contextual coemergence where the simultaneous resonance between fields of bioinformation is the genesis of cause. In this model of coemergent causality, cognition, biology, and cultural history are viewed as biocognitions that communicate within a bioinformational field that has both linear processes in Euclidian geometry and non-linear processes in fractal geometry. Because of the simultaneous and reciprocal nature of mind and body communication, it is argued that biology creates thought and thought creates biology. Just as mind and body cannot be separated, to attempt a separation of mind and world would create an artificial split between observer and observation that assumes we can “step out” of the world we are attempting to observe. (shrink)

Both in Force of Imagination: The Sense of the Elemental and in his very recent Logic of Imagination: The Expanse of the Elemental, John Sallis enacts a reconfiguration of the relationship of geometry to elementology, which might be regarded more generally as a rethinking of the relation of mathematics to philosophy. The paper will trace this reconfiguration in two ways: as it lies present but concealed in the history of philosophy, for example, in Descartes’ so-called “dualism” and in Kant’s (...) pure productive imagination, and in its present creative evolution in fractal geometry, as Sallis interprets it. Sallis draws together the mathematical affinity with a fundamental aesthetic drive, likening mathematical patterns to choreographic ones. I conclude by following this strain as it points to specific dance companies, and to my own sense of aesthetic homecoming as presented in my Imagination in Kant’s Critique of Practical Reason. (shrink)

This paper describes a theoretical framework of ecological phase transitions for modeling tree-grass dynamics and analyzing the shifts or phase transitions from one vegetation structure to another in the southern Texas landscape. This framework implements the integration of percolation theory, fractal geometry and phase transition theory as a method for modeling the spatial patterns of tree-grass dynamics, and nonlinear Markov non-equilibrium thermodynamic stability theory as a method for characterizing temporal tree-grass dynamics and phase transition. An historical sequence of (...) aerial photographs at a Prosopis - thornscrub savanna parkland site in southern Texas was used to determine the parameters of the models. The preliminary analytical result accords well with current understanding and field survey of vegetation dynamics in the southern Texas landscape. The potential of such approaches and other relevant theories such as self-organized criticality and synergetics to vegetation dynamics is also discussed. (shrink)

Val Plumwood's criticism of the ecologically irrational p-centric logic of rationalism, which neglects or denies its dependence on all that is not-p, undercutting its own biological base while denying the illness of the culture it has spawned, is juxtaposed with the clinical picture of the linguistic left hemisphere acting without benefit of input from the more real-time-and-space-centered right. Exploring the metaphor suggests that visual gestalts depicting actual relationships might be effective in drawing our industrial culture's collective attention away from its (...) fascination with endless lines of alphabetic characters and abstract numbers multiplying their way toward infinity, bringing it back to reality and ecological rationality. In illustration, ongoing habitat destruction wrought on the ground as humans follow the detached logic of economic rationalism is seen to reflect, rather than the illusion of eternal "growth" conveyed by Cartesian axes, the relentless finitude of fractal geometry's Sierpinski gasket. (shrink)

This is a companion to another paper. Together they rebut two widespread philosophical doctrines about emergence. The first, and main, doctrine is that emergence is incompatible with reduction. The second is that emergence is supervenience; or more exactly, supervenience without reduction.In the other paper, I develop these rebuttals in general terms, emphasising the second rebuttal. Here I discuss the situation in physics, emphasising the first rebuttal. I focus on limiting relations between theories and illustrate my claims with four examples, each (...) of them a model or a framework for modelling, from well-established mathematics or physics.I take emergence as behaviour that is novel and robust relative to some comparison class. I take reduction as, essentially, deduction. The main idea of my first rebuttal will be to perform the deduction after taking a limit of some parameter. Thus my first main claim will be that in my four examples (and many others), we can deduce a novel and robust behaviour, by taking the limit N→∞ of a parameter N.But on the other hand, this does not show that the N=∞ limit is “physically real”, as some authors have alleged. For my second main claim is that in these same examples, there is a weaker, yet still vivid, novel and robust behaviour that occurs before we get to the limit, i.e. for finite N. And it is this weaker behaviour which is physically real.My examples are: the method of arbitrary functions (in probability theory); fractals (in geometry); superselection for infinite systems (in quantum theory); and phase transitions for infinite systems (in statistical mechanics). (shrink)

In the new physics and in the new field of cosmometry, 1 it is the fundamental pattern that results in the motion from which all is created. Everything starts with the point of infinite potential. The tetrahedron at the point gives birth to the cuboctahedron ; its motion and structure result in the creation of the torus structure. The torus structure is self-referencing on a moment by moment basis since all must pass through the center. But isn't self-referencing the basis (...) for consciousness? It is said that all of creation has awareness, but at different levels. We know plants are aware of threats and of death to other living creatures by the work of Cleve Backster. We know we influence random number generators from the PEAR studies at Princeton. We know baby chicks will influence the movement of robots programed to do a random walk from the work of Rene Peoc'h. Quantum physics has embraced geometry with the work in the equations which explain the Feynman diagrams where particles come in and out of existence; those equations form a structure called the Amplituhedron - which is a quarter of a star tetrahedron. We also know that the tetrahedron can form the star tetrahedron and that each point of the star tetrahedron can form its own star tetrahedron, which can go on infinitely. That is, there is infinity in the finite. As in the fractal and holographic universe, each point or tetrahedron is connected to every other point. If each point has awareness due to formation of the torus, then Hermes teaching of "As above, as below" has meaning in the torus structure. If the torus is the fundamental unit of self-reference, is that the fundamental unit from which consciousness arises? The torus or double torus appears to be fundamental to all of creation - from galaxies to planets to atoms to photons. (shrink)

Contemporary architecture is increasingly grounded in science and mathematics. Architectural discourse has shifted radically from the sometimes disorienting Derridean deconstruction, to engaging scientific terms such as fractals, chaos, complexity, nonlinearity, and evolving systems. That's where the architectural action is -- at least for cutting-edge architects and thinkers -- and every practicing architect and student needs to become conversant with these terms and know what they mean. Unfortunately, the vast majority of architecture faculty are unprepared to explain them to students, not (...) having had a scientific education themselves. Here is an architecture book by an architect/scientist, just in time to help architects in the new millennium. Alexander discusses many of the scientific terms arising in cutting-edge architecture, and explains them to those who don't have scientific training or advanced mathematical knowledge. We find discussions of the evolution of forms; the importance of process in design; iteration; genetic algorithms; sequences of transformations; different levels of scale (i.e. fractals); etc. They are explained here by an architect who is also a scientist, because he wants to change the way architects think and build. Alexander is not merely popularizing other scientists' results and making them accessible to architects: he is in fact presenting new and original scientific work that ties many of these concepts together in a way that will be useful to architects. (shrink)

La métaphore est un instrument indispensable pour les analyses scientifiques. Mais elle nécessite un usage réflexif qui en précise le statut car celui-ci peut être divers et n’est pas sans effet sur la théorie comme sur diverses pratiques. Ainsi, les analystes et acteurs des transports ont usé – et probablement abusé – du recours à la physique des fluides pour leurs métaphores. Aujourd’hui, les développements de la physique, par exemple dans les domaines des fractales ou de la percolation, ouvrent des (...) perspectives nouvelles et prometteuses. Cette évolution montre aussi que les chercheurs en sciences sociales auraient intérêt à renouveler les métaphores qu’ils emploient et qui sont souvent assez archaïques. Ainsi, la référence à des espaces à n dimensions et à l’hypertexte permet de faire rebondir les analyses de Georg Simmel, bridées par le recours à une géométrie plane trop sommaire.Metaphors are indispensable tools for scientific investigation. They should be used in such a way that their status as metaphor is clear, however, for their various uses are not without consequence for theory and practice. In the area of transportation, analysts and practitioners have drawn metaphors from the physics of fluids, but recent developments in the theories of fractals and percolation offer promising perspectives. Generally speaking, as these developments suggest, social scientists stand to benefit from renewing their current metaphors, which are often quite archaic. Thus, Georg Simmel’s analyses – which are constrained by the reference to plan – stand to be revitalized by a reference to N-dimensional spaces and to hypertext. (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)

In this revolutionary work, the author sets the stage for the science of the 21st Century, pursuing an unprecedented synthesis of fields previously considered unrelated. Beginning with simple classical concepts, he ends with a complex multidisciplinary theory requiring a high level of abstraction. The work progresses across the sciences in several multidisciplinary directions: Mathematical logic, fundamental physics, computer science and the theory of intelligence. Extraordinarily enough, the author breaks new ground in all these fields. In the field of fundamental physics (...) the author reaches the revolutionary conclusion that physics can be viewed and studied as logic in a fundamental sense, as compared with Einstein's view of physics as space-time geometry. This opens new, exciting prospects for the study of fundamental interactions. A formulation of logic in terms of matrix operators and logic vector spaces allows the author to tackle for the first time the intractable problem of cognition in a scientific manner. In the same way as the findings of Heisenberg and Dirac in the 1930s provided a conceptual and mathematical foundation for quantum physics, matrix operator logic supports an important breakthrough in the study of the physics of the mind, which is interpreted as a fractal of quantum mechanics. Introducing a concept of logic quantum numbers, the author concludes that the problem of logic and the intelligence code in general can be effectively formulated as eigenvalue problems similar to those of theoretical physics. With this important leap forward in the study of the mechanism of mind, the author concludes that the latter cannot be fully understood either within classical or quantum notions. A higher-order covariant theory is required to accommodate the fundamental effect of high-level intelligence. The landmark results obtained by the author will have implications and repercussions for the very foundations of science as a whole. Moreover, Stern's Matrix Logic is suitable for a broad spectrum of practical applications in contemporary technologies. (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)

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)

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)

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)

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.

Die Geometrie ist seit Euklids Elementen nicht nur Vorbild fur Theorieform und Wissenschaftlichkeit. Sie hat auch seit der Antike die erkenntnistheoretische Diskussion uber das Verhaltnis von Wirklichkeit und Erkenntnis beeinflusst. In ihrer formalaxiomatischen Auffassung der physikalischen Geometrie durch A. Einstein wird die Wissenschaftstheorie des 20. Jahrhunderts in ihren Mehrheitspositionen auf einen Logischen Empirismus festgelegt. Vollstandig ausgeklammert bleibt dabei das philosophische Problem der Gegenstandskonstitution. Wovon ist Geometrie eine Wissenschaft, und wodurch erhalt sie ihre Passung auf die Anwendungen in Handwerk, Technik und (...) messender Naturwissenschaft? (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)