Fractal Geometry as a Bridge between Realms

2013 ◽  
pp. 25-43 ◽  
Author(s):  
Terry Marks-Tarlow
Keyword(s):  
Fractal Geometry ◽  
Open Systems ◽  
Autonomous Systems ◽  
Mathematical Notation ◽  
Deep Space ◽  
The Unconscious ◽  
Self And Other ◽  
Branching Structures ◽  
Self Similar ◽  
Multidimensional Objects

This chapter describes fractal geometry as a bridge between the imaginary and the real, mind and matter, conscious and the unconscious. Fractals are multidimensional objects with self-similar detail across size and/or time scales. Jung conceived of number as the most primitive archetype of order, serving to link observers with the observed. Whereas Jung focused upon natural numbers as the foundation for order that is already conscious in the observer, I offer up the fractal geometry as the underpinnings for a dynamic unconscious destined never to become fully conscious. Throughout nature, fractals model the complex, recursively branching structures of self-organizing systems. When they serve at the edges of open systems, fractal boundaries articulate a paradoxical zone that simultaneously separates as it connects. When modeled by Spencer-Brown’s mathematical notation, full interpenetration between inside and outside edges translates to a distinction that leads to no distinction. By occupying the infinitely deep “space between” dimensions and levels of existence, fractal boundaries contribute to the notion of intersubjectivity, where self and other become most entwined. They also exemplify reentry dynamics of Varela’s autonomous systems, plus Hofstadter’s ever-elusive “tangled hierarchy” between brain and mind.

Application of Fractal Geometry to Fracture Forensics, Research and Production

2006 ◽  
Vol 45 ◽  
pp. 1646-1651 ◽  
Author(s):  
J.J. Mecholsky Jr.
Keyword(s):  
Fractal Analysis ◽  
Fractal Geometry ◽  
Crack Size ◽  
Structural Parameter ◽  
New Materials ◽  
Scale Invariant ◽  
Prepared Material ◽  
Reasonable Approach ◽  
Self Similar ◽  
Production Problems

The fracture surface records past events that occur during the fracture process by leaving characteristic markings. The application of fractal geometry aids in the interpretation and understanding of these events. Quantitative fractographic analysis of brittle fracture surfaces shows that these characteristic markings are self-similar and scale invariant, thus implying that fractal analysis is a reasonable approach to analyzing these surfaces. The fractal dimensional increment, D*, is directly proportional to the fracture energy, γ, during fracture for many brittle materials, i.e., γ = ½ E a0 D* where E is the elastic modulus and a0 is a structural parameter. Also, D* is equal to the crack-size-to-mirror-radius ratio. Using this information can aid in identifying toughening mechanisms in new materials, distinguishing poorly fabricated from well prepared material and identifying stress at fracture for field failures. Examples of the application of fractal analysis in research, fracture forensics and solving production problems are discussed.

Similarity Hashing: A Computer Vision Solution to the Inverse Problem of Linear Fractals

Fractals ◽  
1997 ◽  
Vol 05 (supp01) ◽  
pp. 39-50 ◽  
Author(s):  
John C. Hart ◽  
Wayne O. Cochran ◽  
Patrick J. Flynn
Keyword(s):  
Computer Vision ◽  
Inverse Problem ◽  
Heuristic Search ◽  
Fractal Geometry ◽  
Self Similarity ◽  
Geometric Hashing ◽  
Numerical Minimization ◽  
Self Similar ◽  
Vision Algorithm ◽  
Fractal Representation

The difficult task of finding a fractal representation of an input shape is called the inverse, problem of fractal geometry. Previous attempts at solving this problem have applied techniques from numerical minimization, heuristic search and image compression. The most appropriate domain from which to attack this problem is not numerical analysis nor signal processing, but model-based computer vision. Self-similar objects cause an existing computer vision algorithm called geometric hashing to malfunction. Similarity hashing capitalizes on this observation to not only detect a shape's morphological self-similarity but also find the parameters of its self-transformations.

Fractal geometry and its uses for characterizing microstructure

1993 ◽  
Vol 51 ◽  
pp. 488-489
Author(s):  
John C. Russ
Keyword(s):  
Fractal Dimension ◽  
Fractal Geometry ◽  
Euclidean Geometry ◽  
Boundary Line ◽  
Natural Objects ◽  
Classical Geometry ◽  
Self Similar ◽  
Fractional Dimensions

Observers of nature at scales from microscopic to global have long recognized that few structures are actually described by Euclidean geometry. Mountains are not cones, clouds are not ellipsoids, and surfaces are not planes. Classical geometry allows dimensions of 0 (point), 1 (line), 2 (surface), and 3 (volume). The advent of a new geometry that allows for fractional dimensions between these integer topological values has stirred much interest because it seems to provide a tool for describing many natural objects. As is the case for many new tools, this fractal geometry is subject to some overuse and abuse.A classic illustration of fractal dimension concerns the length of a boundary line, such as the coast of Britain. Measuring maps with different scales, or striding along the coastline with various measuring rods, produces a result that depends on the resolution. More than this is required for the coastline to be fractal, however: It must also be self-similar.

Fractal Models and the Structure of Materials

MRS Bulletin ◽  
1988 ◽  
Vol 13 (2) ◽  
pp. 22-27 ◽  
Author(s):  
Dale W. Schaefer
Keyword(s):  
Fractal Geometry ◽  
Materials Science ◽  
Rotational Symmetry ◽  
Disordered Materials ◽  
Growth Processes ◽  
Fractal Models ◽  
Kinetic Growth ◽  
Fractal Properties ◽  
New Ideas ◽  
Self Similar

Science often advances through the introdction of new ideas which simplify the understanding of complex problems. Materials science is no exception to this rule. Concepts such as nucleation in crystal growth and spinodal decomposition, for example, have played essential roles in the modern understanding of the structure of materials. More recently, fractal geometry has emerged as an essential idea for understanding the kinetic growth of disordered materials. This review will introduce the concept of fractal geometry and demonstrate its application to the understanding of the structure of materials.Fractal geometry is a natural concept used to describe random or disordered objects ranging from branched polymers to the earth's surface. Disordered materials seldom display translational or rotational symmetry so conventional crystallographic classification is of no value. These materials, however, often display “dilation symmetry,” which means they look geometrically self-similar under transformation of scale such as changing the magnification of a microscope. Surprisingly, most kinetic growth processes produce objects with self-similar fractal properties. It is now becoming clear that the origin of dilation symmetry is found in disorderly kinetic growth processes present in the formation of these materials.

Diffusive transport across irregular and fractal walls

1993 ◽  
Vol 442 (1916) ◽  
pp. 571-583 ◽  
Keyword(s):  
Fractal Geometry ◽  
Numerical Procedure ◽  
Iterative Solution ◽  
Diffusive Transport ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Total Rate ◽  
Nonlinear Algebraic Equations ◽  
Total Transport ◽  
Self Similar

Diffusive transport of heat, mass, and momentum across walls with irregular and fractal geometry is discussed. A configuration is considered in detail, in which the transported variable diffuses across a two-dimensional periodic irregular wall toward an overlying plane wall, while the value of the variable over each wall is maintained at a constant value. Several families of self-similar wall geometries with increasingly finer structure, eventually leading to fractal shapes, are considered in detail using an efficient numerical method that is based on conformal mapping. The numerical procedure involves the iterative solution of a large system of nonlinear algebraic equations. Computed patterns of iso-scalar contours reveal the precise effect of the shape, size, and total length of boundary irregularities on the local and total transport rates, and illustrate the enhancement in transport efficacy with wall refinement. The total rate of transport across walls with self-similar irregularities is shown to be remarkably close to that across walls with random irregularities of same roughness height.

Fractal Dimensions and Shape Factors of Digested Sludge Particle Aggregates

1994 ◽  
Vol 29 (4) ◽  
pp. 441-456 ◽  
Author(s):  
Juraj Námer ◽  
Jerzy J. Ganczarczyk
Keyword(s):  
Fractal Geometry ◽  
Fractal Dimensions ◽  
Photographic Method ◽  
Image Analysis System ◽  
Digested Sludge ◽  
Shape Factors ◽  
Particle Aggregates ◽  
Self Similar ◽  
Analysis System ◽  
Sludge Particle

Abstract In a previous paper, digested sludge particle aggregates were examined by an image analysis system and by a free-settling multi-exposure photographic method. Based on microscopic observations, it was found that these aggregates exhibited the characteristics of statistically self-similar fractals. The experimentally determined terminal settling velocities and 2-D settling shape factors, which described the irregularity of aggregates during the settling operation, were utilized in this work to estimate the parameters of a settling equation in which the fractal geometry of aggregates was incorporated. Based on these equations, the 2-D and 3-D size-density fractal dimension of digested sludge particles were determined.

Monstrous hermeneutics: Learning from diagrams

Semiotica ◽  
2016 ◽  
Vol 2016 (212) ◽  
pp. 239-258
Author(s):  
Inna Semetsky
Keyword(s):  
Alternative Form ◽  
Crossing Over ◽  
Semiotic System ◽  
The Unconscious ◽  
Self And Other ◽  
Ambiguous Category ◽  
Iconic Signs ◽  
Included Middle

AbstractThis paper addresses a theory/practice nexus represented by a semiotic system of Tarot pictures as iconic signs. Tarot will be analyzed from the perspective of Charles S. Peirce’s and Gilles Deleuze’s philosophies. Tarot functions as a diagram or the included third between “self” and “other,” which are traditionally taken as binary opposites. It thus partakes of the “monster” as a grotesque and ambiguous category that betrays a strict boundary between habitual dualisms, such as mind and world, consciousness and the unconscious, human and divine. While Tarot is usually perceived as irrational and illogical if not altogether “monstrous,” it is the logic of the included middle that enables its functioning. Genuine signs have a triadic structure that includes interpretants crossing over human and non-human natures. The process of reading and interpreting Tarot signs represents specific hermeneutics and constitutes exopedagogy as an alternative form of education partaking of a posthuman dimension. As indices, Tarot pictures refer to the whole gamut of human experiences, and the hermeneutics of Tarot allows us to evaluate experience and to learn from it.

scholarly journals Integration of interdisciplinary directions in the study of fractal geometry elements

2017 ◽  
Vol 6 (4) ◽  
pp. 246-251
Author(s):  
Inna Aleksandrovna Rusanova
Keyword(s):  
Fractal Geometry ◽  
Complex Structure ◽  
Cognitive Activity ◽  
Physical Nature ◽  
Educational Process ◽  
Affine Transformations ◽  
Fractal Sets ◽  
Research Activities ◽  
Self Similar ◽  
And Mathematics

This paper deals with the problem of integrating interdisciplinary areas in research activities that underlie developmental learning. In the conditions of new educational standards introduction deep system transformations of the whole educational process are supposed. The search for solutions to the problems of individualizing the educational route, polar motivation, increasing interest in physics and mathematics lead to the need to design individual methods of pedagogical activity, to implement new approaches and technologies in the natural science cycle of consistent development of holistic research activities, mastering the stages and methods of scientific knowledge. One of the opportunities for the formation of educational and cognitive activity and creative potential in the study of Physics and Mathematics is to study the elements of fractal geometry for analyzing the complex structure of processes of various physical nature, in view of the fact that today there is a large number of problems in Physics, Chemistry, Biology, Geology and Economics, where the fractal structure is the main characteristic of the system. Practical tasks on the construction of fractal sets with the identification of the main signs of self-similarity and the possibility of their computer modeling are considered. Students of grades 9-11 and students of the university are given the task of creating their own images of fractals, investigating the fractality of coastal river lines, constructing self-similar figures according to the algorithm Games in chaos and studying the contracting affine transformations with obtaining various modifications (attractors) of the Serpinsky triangle. The results obtained enable them to conclude that simple mathematical rules can generate self-similar formations with respect to nonlinear transformations, and argue that simple rules can be at the heart of complex structures and processes.

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