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.

Mathematical model of the eroding effect on the surface morphology of Compressed Earth Bricks

2021 ◽  
Vol 9 (3) ◽  
pp. 245-249
Keyword(s):  
Mathematical Model ◽  
Fractal Dimension ◽  
Surface Morphology ◽  
Stochastic Modeling ◽  
Solid Surface ◽  
Fractal Geometry ◽  
Building Materials ◽  
Euclidean Geometry ◽  
Modeling Techniques

A solid-surface morphology has a rough character that prevents it from being described by Euclidean geometry; fractal geometry is plausible. From considering that the deposition of particles and their detachment significantly influences roughness, an expression based on stochastic modeling techniques was obtained to predict the fractal dimension of a surface based on the dynamics of the processes that occur in it. The model obtained was used to characterize whether fiber in building materials made of poured earth influences the surface's morphology and the effects of erosion

scholarly journals Fractal geometry and multifractals in analyzing and processing medical data and images

2002 ◽  
Vol 10 (4) ◽  
pp. 283-293 ◽  
Author(s):  
Irini Reljin ◽  
Branimir Reljin
Keyword(s):  
Fractal Geometry ◽  
Signal Analysis ◽  
Euclidean Geometry ◽  
Medical Data ◽  
Natural Objects ◽  
Complex Shapes ◽  
Medical Signal

A traditional way for describing objects, based on the well-known Euclidean geometry, is not capable to describe different natural objects and phenomena such as clouds, relief shapes, trends in economy, etc. On the contrary fractal geometry and its extension multifractals are new "tools" which can be used for describing, modeling, analyzing and processing different complex shapes and signals. This paper considers fractal geometry and multifractals and their application in signal analyzing and processing particularly in medical signal analysis.

scholarly journals The Fractal Geometry of Growth: Fluctuation–Dissipation Theorem and Hidden Symmetry

2021 ◽  
Vol 9 ◽  
Author(s):  
Petrus H. R. dos Anjos ◽  
Márcio S. Gomes-Filho ◽  
Washington S. Alves ◽  
David L. Azevedo ◽  
Fernando A. Oliveira
Keyword(s):  
Fractal Dimension ◽  
Fractal Geometry ◽  
Euclidean Geometry ◽  
Fractal Structure ◽  
Growth Dynamics ◽  
Kpz Equation ◽  
Field Equations ◽  
Fluctuation Dissipation ◽  
Hidden Symmetry ◽  
Fluctuation Dissipation Theorem

Growth in crystals can be usually described by field equations such as the Kardar-Parisi-Zhang (KPZ) equation. While the crystalline structure can be characterized by Euclidean geometry with its peculiar symmetries, the growth dynamics creates a fractal structure at the interface of a crystal and its growth medium, which in turn determines the growth. Recent work by Gomes-Filho et al. (Results in Physics, 104,435 (2021)) associated the fractal dimension of the interface with the growth exponents for KPZ and provides explicit values for them. In this work, we discuss how the fluctuations and the responses to it are associated with this fractal geometry and the new hidden symmetry associated with the universality of the exponents.

X-Ray Powder Diffraction Patterns as Random Fractals

1995 ◽  
Vol 39 ◽  
pp. 739-746
Author(s):  
Dana T. Griffen ◽  
Kim R. Sullivan
Keyword(s):  
Powder Diffraction ◽  
Fractal Geometry ◽  
Physical World ◽  
Vertical Dimension ◽  
X Ray ◽  
Topographic Profile ◽  
Self Similar ◽  
Diffraction Patterns ◽  
Fractional Dimensions ◽  
East West

The concept of fractals (or fractional dimensions), although known earlier, was formalized theoretically and given that name in 1967 by Mandelbrot. A fractal is some object, whether mathematically constructed or observed in the physical world, that exhibits scale in variance—that is, it looks essentially the same at all scales, or over some range of scales. Objects that exhibit fractal geometry and that are measured in the same units in both the x and y directions are said to be self-similar; geologic examples of self-similar fractals are a rocky coastline and a topographic contour, for which the east-west and north-south coordinates of any point are expressed in, say, meters or kilometers. Objects that exhibit fractal geometry but which are measured in different units along x and y are said to be selfaffine fractals; an example of a self-affine fractal is a topographic profile, in which the horizontal dimension is measured in kilometers and the vertical dimension is measured in meters.

scholarly journals A survey on fractal dimension for fractal structures

2016 ◽  
Vol 1 (2) ◽  
pp. 437-472 ◽  
Author(s):  
M. Fernández-Martínez
Keyword(s):  
Fractal Dimension ◽  
Fractal Geometry ◽  
Fractal Structures ◽  
Fractal Patterns ◽  
Self Similar

AbstractAlong the years, the foundations of Fractal Geometry have received contributions starting from mathematicians like Cantor, Peano, Hilbert, Hausdorff, Carathéodory, Sierpiński, and Besicovitch, to quote some of them. They were some of the pioneers exploring objects having self-similar patterns or showing anomalous properties with respect to standard analytic attributes. Among the new tools developed to deal with this kind of objects, fractal dimension has become one of the most applied since it constitutes a single quantity which throws useful information concerning fractal patterns on sets. Several years later, fractal structures were introduced from Asymmetric Topology to characterize self-similar symbolic spaces. Our aim in this survey is to collect several results involving distinct definitions of fractal dimension we proved jointly with Prof.M.A. Sánchez-Granero in the context of fractal structures.

scholarly journals Fractal calculus for analysis of wool fiber: Mathematical insight of its biomechanism

2019 ◽  
Vol 14 ◽  
pp. 155892501987220 ◽  
Author(s):  
Jie Fan ◽  
Xue Yang ◽  
Yong Liu
Keyword(s):  
Fractal Dimension ◽  
Temperature Gradient ◽  
Thermal Property ◽  
Fractal Geometry ◽  
Wool Fiber ◽  
Scale Dimension ◽  
Fractal Calculus ◽  
Hierarchic Level ◽  
Self Similar

Wool fiber has a complex hierarchic inner structure. However, like most of the natural things, wool fiber does not have an exactly strict self-similar fractal feature. Here, we calculate the fractal dimension of each hierarchic level of wool fiber using the two-scale dimension method. The obtained fractal dimension of wool fiber in different hierarchic level ranges between 1.37 and 1.47, which is close to that obtained according to the traditional fractal geometry. Thermal property of wool fiber is investigated based on the fractal feature of wool fiber. The result shows that the temperature gradient and the rate of the temperature gradient along the fiber is very slow, suggesting that wool fiber has a good thermal retention property.

Optimal Density Determination of Bouguer Anomaly Using Fractal Analysis (Case of study: Charak area,IRAN)

2005 ◽  
Vol 1 (1) ◽  
pp. 21-24
Author(s):  
Hamid Reza Samadi
Keyword(s):  
Surface Roughness ◽  
Fractal Dimension ◽  
Fractal Analysis ◽  
Fractal Geometry ◽  
Host Rock ◽  
Bouguer Anomaly ◽  
Research Area ◽  
Density Difference ◽  
Optimal Density

In exploration geophysics the main and initial aim is to determine density of under-research goals which have certain density difference with the host rock. Therefore, we state a method in this paper to determine the density of bouguer plate, the so-called variogram method based on fractal geometry. This method is based on minimizing surface roughness of bouguer anomaly. The fractal dimension of surface has been used as surface roughness of bouguer anomaly. Using this method, the optimal density of Charak area insouth of Hormozgan province can be determined which is 2/7 g/cfor the under-research area. This determined density has been used to correct and investigate its results about the isostasy of the studied area and results well-coincided with the geology of the area and dug exploratory holes in the text area

Pascal's Prism

2012 ◽  
Vol 96 (536) ◽  
pp. 213-220
Author(s):  
Harlan J. Brothers
Keyword(s):  
Probability Theory ◽  
Fractal Geometry ◽  
Euclidean Geometry ◽  
Fibonacci Series ◽  
Pascal’S Triangle ◽  
Number Sequences ◽  
Definition Of ◽  
Image Position ◽  
Pascal's Triangle

Pascal's triangle is well known for its numerous connections to probability theory [1], combinatorics, Euclidean geometry, fractal geometry, and many number sequences including the Fibonacci series [2,3,4]. It also has a deep connection to the base of natural logarithms, e [5]. This link to e can be used as a springboard for generating a family of related triangles that together create a rich combinatoric object.2. From Pascal to LeibnizIn Brothers [5], the author shows that the growth of Pascal's triangle is related to the limit definition of e.Specifically, we define the sequence sn; as follows [6]:

Fractal and Convolutional Analysis for Deep Atmospheric Turbulence Using Machine Learning

2021 ◽  
Author(s):  
Nicholas Dudu ◽  
Arturo Rodriguez ◽  
Gael Moran ◽  
Jose Terrazas ◽  
Richard Adansi ◽  
...  
Keyword(s):  
Fractal Dimension ◽  
Atmospheric Turbulence ◽  
Isotropic Turbulence ◽  
Passive Scalar ◽  
Counting Method ◽  
Data Set ◽  
Box Counting ◽  
Box Counting Dimension ◽  
Self Similar ◽  
Box Counting Method

Abstract Atmospheric turbulence studies indicate the presence of self-similar scaling structures over a range of scales from the inertial outer scale to the dissipative inner scale. A measure of this self-similar structure has been obtained by computing the fractal dimension of images visualizing the turbulence using the widely used box-counting method. If applied blindly, the box-counting method can lead to misleading results in which the edges of the scaling range, corresponding to the upper and lower length scales referred to above are incorporated in an incorrect way. Furthermore, certain structures arising in turbulent flows that are not self-similar can deliver spurious contributions to the box-counting dimension. An appropriately trained Convolutional Neural Network can take account of both the above features in an appropriate way, using as inputs more detailed information than just the number of boxes covering the putative fractal set. To give a particular example, how the shape of clusters of covering boxes covering the object changes with box size could be analyzed. We will create a data set of decaying isotropic turbulence scenarios for atmospheric turbulence using Large-Eddy Simulations (LES) and analyze characteristic structures arising from these. These could include contours of velocity magnitude, as well as of levels of a passive scalar introduced into the simulated flows. We will then identify features of the structures that can be used to train the networks to obtain the most appropriate fractal dimension describing the scaling range, even when this range is of limited extent, down to a minimum of one order of magnitude.

scholarly journals Thermal and Wet Comfort of Fabrics Based on Fractal Dimension of Silicone Coating

2018 ◽  
Vol 13 (1) ◽  
pp. 155892501801300
Author(s):  
Yunlong Shi ◽  
Liang Wang ◽  
Wenhuan Zhang ◽  
Xiaoming Qian
Keyword(s):  
Fractal Dimension ◽  
Thermal Insulation ◽  
Coating Layer ◽  
Fractal Theory ◽  
Barrier Effect ◽  
Evaporative Resistance ◽  
Silicone Coating ◽  
Self Similar ◽  
Warmth Retention ◽  
Coated Fabrics

In this paper, thermal and wet comforts of silicone coated windbreaker shell jacket fabrics were studied. Both thermal insulation and evaporative resistance of fabric increased with an increase in coating area due to the barrier effect of the silicone coating layer. Moreover, the coated fabrics with self-similar structures showed different thermal insulation and evaporative resistance under the same total coating area. Fractal theory was used to explain this phenomenon. Optimal thermal-wet comfort properties were obtained when the fractal dimension (D=1.599) was close to the Golden Mean (1.618). When the fractal dimension of coating was lower than 1.599, fabric warmth retention was not high enough. In contrast, fabric evaporative resistance was beyond the value at which people would feel comfortable when the fractal dimension was greater than 1.599.

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