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.

Intermittency and related issues in 16O-Ag/Br collision at 200A GeV/c

2010 ◽  
Vol 88 (8) ◽  
pp. 575-584 ◽  
Author(s):  
M. K. Ghosh ◽  
P. K. Haldar ◽  
S. K. Manna ◽  
A. Mukhopadhyay ◽  
G. Singh
Keyword(s):  
Particle Density ◽  
Simulated Data ◽  
Space Distribution ◽  
Monte Carlo Code ◽  
Final State ◽  
Data Set ◽  
Fractal Properties ◽  
Self Similar ◽  
Singly Charged ◽  
Almost All

In this paper we present some results on the nonstatistical fluctuation in the 1-dimensional (1-d) density distribution of singly charged produced particles in the framework of the intermittency phenomenon. A set of nuclear emulsion data on 16O-Ag/Br interactions at an incident momentum of 200A GeV/c, was analyzed in terms of different statistical methods that are related to the self-similar fractal properties of the particle density function. A comparison of the present experiment with a similar experiment induced by the 32S nuclei and also with a set of results simulated by the Lund Monte Carlo code FRITIOF is presented. A similar comparison between this experiment and a pseudo-random number generated simulated data set is also made. The analysis reveals the presence of a weak intermittency in the 1-d phase space distribution of the produced particles. The results also indicate the occurrence of a nonthermal phase transition during emission of final-state hadrons. Our results on factorial correlators suggests that short-range correlations are present in the angular distribution of charged hadrons, whereas those on oscillatory moments show that such correlations are not restricted only to a few particles. In almost all cases, the simulated results fail to replicate their experimental counterparts.

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.

scholarly journals is regular-closed

2018 ◽  
Vol 40 (1) ◽  
pp. 213-220 ◽  
Author(s):  
YUTARO HIMEKI ◽  
YUTAKA ISHII
Keyword(s):  
Rotational Symmetry ◽  
Iterated Function Systems ◽  
Geometric Proof ◽  
Contraction Ratio ◽  
Iterated Function ◽  
Mandelbrot Sets ◽  
Self Similar ◽  
Regular Closed

For each $n\geq 2$, we investigate a family of iterated function systems which is parameterized by a common contraction ratio $s\in \mathbb{D}^{\times }\equiv \{s\in \mathbb{C}:0<|s|<1\}$ and possesses a rotational symmetry of order $n$. Let ${\mathcal{M}}_{n}$ be the locus of contraction ratio $s$ for which the corresponding self-similar set is connected. The purpose of this paper is to show that ${\mathcal{M}}_{n}$ is regular-closed, that is, $\overline{\text{int}\,{\mathcal{M}}_{n}}={\mathcal{M}}_{n}$ holds for $n\geq 4$. This gives a new result for $n=4$ and a simple geometric proof of the previously known result by Bandt and Hung [Fractal $n$-gons and their Mandelbrot sets. Nonlinearity 21 (2008), 2653–2670] for $n\geq 5$.

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.

The Fractal Properties of Scale Effect on the Density of Rock-Fill Materials

2013 ◽  
Vol 706-708 ◽  
pp. 520-525
Author(s):  
Li Qiang Wu ◽  
Sheng Zhu ◽  
Kuang Min Wei ◽  
Chuan Yin Lu
Keyword(s):  
Relative Density ◽  
Scale Effect ◽  
Fractal Geometry ◽  
Truncation Error ◽  
Rock Fill ◽  
Numerical Tests ◽  
Maximal Diameter ◽  
Fractal Properties ◽  
Physical Tests ◽  
Fill Materials

to study scale effect on the density of rock-fill materials, the relative density tests were carried out by physical tests and numerical tests. Fractal geometry theory was drawn into the gradation of rock-fill materials. Then, the fractal properties of scale effect on the density were investigated. There are close relations between fractal dimension D and densities of rock-fill materials. The densities are maximal when D is the critical value Dc. Furthermore, Dc is independent of the relative density Dr and the maximal diameter dmax. Truncation error is one of the main factors of scale effect of densities of rock-fill materials. The achievements in the paper lay a good foundation for further studying scale effect of rock-fill materials with fractal geometry theory.

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.

The Interpretation of Vernacular Architecture through Fractal Models

2016 ◽  
pp. 55-77
Author(s):  
Ehsan Reza ◽  
Ozgur Dincyurek
Keyword(s):  
Twentieth Century ◽  
Fractal Dimension ◽  
Fractal Geometry ◽  
Vernacular Architecture ◽  
Decision Makers ◽  
Housing Developments ◽  
Mathematical Algorithm ◽  
Fractal Models ◽  
Box Counting ◽  
Box Counting Method

Mathematical algorithm and nonlinear theories were used in order to study the establishment and development of traditional settlements since the second half of twentieth century. In order to interrogate vernacular architecture, fractal geometry is one of the most advanced methodologies in this study. Vernacular architecture is an organic architecture, which is formed in response to environmental, cultural, economical factors. There are plenty of variations in topography; climate and geographical issues among the mountainous areas in Iran. Therefor, there are many useful thought, which can be learnt from the existing vernacular architecture. This study is going to investigate fractal pattern of housing in Masouleh village, Iran. By referring to the fractal dimension calculated with box counting method, different type of information will be collected and this attempt will help decision makers, planners, architects and designers, especially in new housing developments.

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