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.

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.

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.

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

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.

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.

Fractal geometry of ammonoid sutures

Paleobiology ◽  
1995 ◽  
Vol 21 (3) ◽  
pp. 329-342 ◽  
Author(s):  
Timothy M. Lutz ◽  
George E. Boyajian
Keyword(s):  
Fractal Dimension ◽  
Fractal Geometry ◽  
Two Dimensional ◽  
Evolutionary Time ◽  
Linear Features ◽  
Divider Method ◽  
Richardson Method ◽  
Components Analysis ◽  
Box Method ◽  
Range Of Variation

Interior chamber walls of ammonites range from smoothly undulating surfaces in some taxa to complex surfaces, corrugated on many scales, in others. The ammonite suture, which is the expression of the intersection of these walls on the exterior of the shell, has been used to assess anatomical complexity. We used the fractal dimension to measure sutural complexity and to investigate complexity over evolutionary time and showed that the range of variation in sutural complexity increased through time. In this paper we extend our analyses and consider two new parameters that measure the range of scales over which fractal geometry is a satisfactory metric of a suture. We use a principal components analysis of these parameters and the fractal dimension to establish a two-dimensional morphospace in which the shapes of sutures can be plotted and in which variations and evolution of suture morphology can be investigated. Our results show that morphospace coordinates of ammonitic sutures correspond to visually perceptible differences in suture shape. However, three main classes of sutures (goniatitic, ceratitic, and ammonitic) are not unambiguously discriminated in this morphospace. Interestingly, ammonitic sutures occupy a smaller morphospace than other suture types (roughly one-half of the morphospace of goniatitic and ceratitic sutures combined), and the space they occupied did not change dimensions from the Jurassic to the late Cretaceous.We also compare two methods commonly used to measure the fractal dimension of linear features: the Box method and the Richardson (or divider) method. Both methods yield comparable results for ammonitic sutures but the Richardson method yields more precise results for less complex sutures.

ESTIMATING THE FRACTAL DIMENSION OF PLANTS USING THE TWO-SURFACE METHOD: AN ANALYSIS BASED ON 3D-DIGITIZED TREE FOLIAGE

Fractals ◽  
2006 ◽  
Vol 14 (03) ◽  
pp. 149-163 ◽  
Author(s):  
FRÉDÉRIC BOUDON ◽  
CHRISTOPHE GODIN ◽  
CHRISTOPHE PRADAL ◽  
OLIVIER PUECH ◽  
HERVÉ SINOQUET
Keyword(s):  
Fractal Dimension ◽  
Field Measurements ◽  
Three Dimensions ◽  
Total Leaf Area ◽  
Topological Information ◽  
Surface Method ◽  
Plant Foliage ◽  
Self Similar ◽  
Branching System ◽  
Peach Trees

In this paper, we present a method to estimate the fractal dimension of plant foliage in three dimensions (3D). This method is derived from the two-surface method introduced in the 90s to estimate the fractal dimension of tree species from field measurements on collections of trees. Here we adapted the method to individual plants. The multiscale topology and geometry of the plant must first be digitized in 3D. Then leafy branching systems of different sizes are constructed from the plant database, using the topological information. 3D convex envelops are then computed for each leafy branching system. The fractal dimension of the plant is finally estimated by comparing the total leaf area and the convex envelop area of these leafy modules. The method was assessed on a set of four peach trees entirely digitized at shoot scale. Results show that the peach trees have a marked self-similar foliage with fractal dimension close to 2.4.

scholarly journals A laboratory study of sediment deformation: stress heterogeneity and grain-size evolution

1996 ◽  
Vol 22 ◽  
pp. 167-175 ◽  
Author(s):  
Neal R. Iverson ◽  
Thomas S. Hooyer ◽  
Roger Leb. Hooke
Keyword(s):  
Grain Size ◽  
Fractal Dimension ◽  
Normal Stress ◽  
Size Distribution ◽  
Grain Size Distribution ◽  
Local Stress ◽  
Large Strains ◽  
Self Similar ◽  
Deformation Stress ◽  
Stress Heterogeneity

In shearing sediment beneath glaciers, networks of grains may transiently support shear and normal stresses that are larger than spatial averages. Consistent with studies of fault-gouge genesis, we hypothesize that crushing of grains in such networks is responsible for surrounding larger grains with smaller grains. At sufficiently large strains, this should minimize stress heterogeneity, favor intergranular sliding and abrasion rather than crushing, and result in a self-similar grain-size distribution.This hypothesis is tested with a ring-shear device that slowly shears a large annular sediment sample to high strains. Shearing and comminution of weak equigranular (2.0–3.3 mm) sediment resulted in a self-similar grain-size distribution with a fractal dimension that increased with shear strain toward a steady value of 2.85. This value is significantly larger than that of gouges produced purely by crushing, 2.6, but it is comparable to values for tilts thought to be deforming beneath modern glaciers, 2.8 to nearly 3.0. At low strains, under a steady mean normal stress of 84 kPa, variations in normal stress measured locally ranged in amplitude from 50 to 300 kPa with wavelengths that were 100 times larger than the initial grain diameter. Crushing of grains, observed through the transparent walls of the device, apparently caused the failure of grain networks. At shearing displacements ranging from 0.7 to 1.0 m, the amplitude of local stress fluctuations decreased abruptly. This change is attributed to fine sediment that distributed stresses more uniformly and caused grain networks to fail primarily by intergranular sliding rather than by crushing of grains. Sliding between grains apparently produced silt by abrasion and resulted in a fractal dimension that was higher than if there had been only crushing.A size distribution with a fractal dimension greater than 2.6 is probably a necessary but not sufficient condition for determining whether a basal till has been highly deformed. Stress heterogeneity in subglacial sediment that is shearing through its full thickness should contribute to the erosion of underlying rock.

Fractal Geometry and Properties of Doped BaTiO3 Ceramics

2010 ◽  
Vol 67 ◽  
pp. 42-48 ◽  
Author(s):  
V.V. Mitic ◽  
V.B. Pavlovic ◽  
L. Kocic ◽  
V. Paunovic ◽  
L. Zivkovic
Keyword(s):  
Fractal Dimension ◽  
Solid State ◽  
Fractal Geometry ◽  
Grain Shape ◽  
Grain Structure ◽  
Fractal Method ◽  
Ceramic Structure ◽  
New Approach ◽  
Fractal Modeling ◽  
Dielectrical Properties

Taking into account that the complex grain structure is difficult to describe by using traditional analytical methods, in this study, in order to establish ceramic grain shapes of sintered BaTiO3, new approach on correlation between microstructure and properties of doped BaTiO3 ceramics based on fractal geometry has been developed. BaTiO3 ceramics doped with various dopants (MnCO3, Er2O3, Yb2O3) were prepared using conventional solid state procedure, and were sintered at 1350oC for four hours. The microstructure of sintered specimens was investigated by SEM-5300. Using method of fractal modeling a reconstruction of microstructure configurations, like grains shapes, or intergranular contacts has been successfully done. Furthermore, the area of grains surface was calculated using fractal correction that expresses the irregularity of grains surface through fractal dimension. The presented results, indicate that fractal method for ceramics structure analysis provides a new approach for describing, predicting and modeling the grain shape and relations between the BaTiO3-ceramic structure and dielectrical properties.

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