Porosity imaged by a vector projection algorithm correlates with fractal dimension measured on 3D models obtained by microCT

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The unfinished landscape fractal geometry and the aesthetics of ecological design

During the late 20th century it was proposed that a design aesthetic reflecting current ecological concerns was required within the overall domain of the built environment and specifically within landscape design. To address this, some authors suggested various theoretical frameworks upon which such an aesthetic could be based. Within these frameworks there was an underlying theme that the patterns and processes of Nature may have the potential to form this aesthetic — an aesthetic based on fractal rather than Euclidean geometry. In order to understand how fractal geometry, described as the geometry of Nature, could become the referent for a design aesthetic, this research examines the mathematical concepts of fractal Geometry, and the underlying philosophical concepts behind the terms ‘Nature’ and ‘aesthetics’. The findings of this initial research meant that a new definition of Nature was required in order to overcome the barrier presented by the western philosophical Nature¯culture duality. This new definition of Nature is based on the type and use of energy. Similarly, it became clear that current usage of the term aesthetics has more in common with the term ‘style’ than with its correct philosophical meaning. The aesthetic philosophy of both art and the environment recognises different aesthetic criteria related to either the subject or the object, such as: aesthetic experience; aesthetic attitude; aesthetic value; aesthetic object; and aesthetic properties. Given these criteria, and the fact that the concept of aesthetics is still an active and ongoing philosophical discussion, this work focuses on the criteria of aesthetic properties and the aesthetic experience or response they engender. The examination of fractal geometry revealed that it is a geometry based on scale rather than on the location of a point within a three-dimensional space. This enables fractal geometry to describe the complex forms and patterns created through the processes of Wild Nature. Although fractal geometry has been used to analyse the patterns of built environments from a plan perspective, it became clear from the initial review of the literature that there was a total knowledge vacuum about the fractal properties of environments experienced every day by people as they move through them. To overcome this, 21 different landscapes that ranged from highly developed city centres to relatively untouched landscapes of Wild Nature have been analysed. Although this work shows that the fractal dimension can be used to differentiate between overall landscape forms, it also shows that by itself it cannot differentiate between all images analysed. To overcome this two further parameters based on the underlying structural geometry embedded within the landscape are discussed. These parameters are the Power Spectrum Median Amplitude and the Level of Isotropy within the Fourier Power Spectrum. Based on the detailed analysis of these parameters a greater understanding of the structural properties of landscapes has been gained. With this understanding, this research has moved the field of landscape design a step close to being able to articulate a new aesthetic for ecological design.

Fractal and multifractal properties of a family of fractal networks

In this work, we study the fractal and multifractal properties of a family of fractal networks introduced by Gallos et al (2007 Proc. Nat. Acad. Sci. USA 104 7746). In this fractal network model, there is a parameter e which is between 0 and 1, and allows for tuning the level of fractality in the network. Here we examine the multifractal behavior of these networks, the dependence relationship of the fractal dimension and the multifractal parameters on parameter e. First, we find that the empirical fractal dimensions of these networks obtained by our program coincide with the theoretical formula given by Song et al (2006 Nature Phys. 2 275). Then from the shape of the τ(q) and D(q) curves, we find the existence of multifractality in these networks. Last, we find that there exists a linear relationship between the average information dimension 〈D(1)〉 and the parameter e.

Fractal and complex network analyses of protein molecular dynamics

Based on protein molecular dynamics, we investigate the fractal properties of energy, pressure and volume time series using the multifractal detrended fluctuation analysis (MF-DFA) and the topological and fractal properties of their converted horizontal visibility graphs (HVGs). The energy parameters of protein dynamics we considered are bonded potential, angle potential, dihedral potential, improper potential, kinetic energy, Van der Waals potential, electrostatic potential, total energy and potential energy. The shape of the h(q)h(q) curves from MF-DFA indicates that these time series are multifractal. The numerical values of the exponent h(2)h(2) of MF-DFA show that the series of total energy and potential energy are non-stationary and anti-persistent; the other time series are stationary and persistent apart from series of pressure (with H≈0.5H≈0.5 indicating the absence of long-range correlation). The degree distributions of their converted HVGs show that these networks are exponential. The results of fractal analysis show that fractality exists in these converted HVGs. For each energy, pressure or volume parameter, it is found that the values of h(2)h(2) of MF-DFA on the time series, exponent λλ of the exponential degree distribution and fractal dimension dBdB of their converted HVGs do not change much for different proteins (indicating some universality). We also found that after taking average over all proteins, there is a linear relationship between 〈h(2)〉〈h(2)〉 (from MF-DFA on time series) and 〈dB〉〈dB〉 of the converted HVGs for different energy, pressure and volume.

Topological properties and fractal analysis of a recurrence network constructed from fractional Brownian motions

Many studies have shown that we can gain additional information on time series by investigating their accompanying complex networks. In this work, we investigate the fundamental topological and fractal properties of recurrence networks constructed from fractional Brownian motions (FBMs). First, our results indicate that the constructed recurrence networks have exponential degree distributions; the average degree exponent 〈λ〉 increases first and then decreases with the increase of Hurst index H of the associated FBMs; the relationship between H and 〈λ〉 can be represented by a cubic polynomial function. We next focus on the motif rank distribution of recurrence networks, so that we can better understand networks at the local structure level. We find the interesting superfamily phenomenon, i.e., the recurrence networks with the same motif rank pattern being grouped into two superfamilies. Last, we numerically analyze the fractal and multifractal properties of recurrence networks. We find that the average fractal dimension 〈dB〉 of recurrence networks decreases with the Hurst index H of the associated FBMs, and their dependence approximately satisfies the linear formula 〈dB〉≈2-H, which means that the fractal dimension of the associated recurrence network is close to that of the graph of the FBM. Moreover, our numerical results of multifractal analysis show that the multifractality exists in these recurrence networks, and the multifractality of these networks becomes stronger at first and then weaker when the Hurst index of the associated time series becomes larger from 0.4 to 0.95. In particular, the recurrence network with the Hurst index H=0.5 possesses the strongest multifractality. In addition, the dependence relationships of the average information dimension 〈D(1)〉 and the average correlation dimension 〈D(2)〉 on the Hurst index H can also be fitted well with linear functions. Our results strongly suggest that the recurrence network inherits the basic characteristic and the fractal nature of the associated FBM series.

A note on fractal dimensions of biomedical waveforms

In this paper, we study performance of Katz method of computing fractal dimension of waveforms, and its estimation accuracy is compared with Higuchi's method. The study is performed on four synthetic parametric fractal waveforms for which true fractal dimensions can be calculated, and real sleep electroencephalogram. The dependence of Katz's fractal dimension on amplitude, frequency and sampling frequency of waveforms is noted. Even though the Higuchi's method has given more accurate estimation of fractal dimensions, the study suggests that the results of Katz's based fractal dimension analysis of biomedical waveforms have to be carefully interpreted.

Estimating the Hausdorff-Besicovitch dimension of boundary of basin of attraction in helicopter trim

Helicopter trim involves solution of nonlinear force equilibrium equations. As in many nonlinear dynamic systems, helicopter trim problem can show chaotic behavior. This chaotic behavior is found in the basin of attraction of the nonlinear trim equations which have to be solved to determine the main rotor control inputs given by the pilot. This study focuses on the boundary of the basin of attraction obtained for a set of control inputs. We analyze the boundary by considering it at different magnification levels. The magnified views reveal intricate geometries. It is also found that the basin boundary exhibits the characteristic of statistical self-similarity, which is an essential property of fractal geometries. These results led the authors to investigate the fractal dimension of the basin boundary. It is found that this dimension is indeed greater than the topological dimension. From all the observations, it is evident that the boundary of the basin of attraction for helicopter trim problem is fractal in nature. (C) 2012 Elsevier Inc. All rights reserved.

Search on a fractal lattice using a quantum random walk

The spatial search problem on regular lattice structures in integer number of dimensions d >= 2 has been studied extensively, using both coined and coinless quantum walks. The relativistic Dirac operator has been a crucial ingredient in these studies. Here, we investigate the spatial search problem on fractals of noninteger dimensions. Although the Dirac operator cannot be defined on a fractal, we construct the quantum walk on a fractal using the flip-flop operator that incorporates a Klein-Gordon mode. We find that the scaling behavior of the spatial search is determined by the spectral (and not the fractal) dimension. Our numerical results have been obtained on the well-known Sierpinski gaskets in two and three dimensions.

Fractal Pattern Formation In Anodic Bonding Of Pyrex Glass/Al/Si

Anodic bonding of Pyrex glass/Al/Si is an important bonding technique in micro/nanoelectromechanical systems (MEMS/NEMS) industry. The anodic bonding of Pyrex 7740 glass/Aluminum film/Silicon is completed at the temperature from 300 degrees C to 375 degrees C with a bonding voltage between 150 V and 450 V. The fractal patterns are formed in the intermediate Al thin film. This pattern has the fractal dimension of the typical two-dimensional diffusion-limited aggregation (2D DLA) process, and the fractal dimension is around 1.7. The fractal patterns consist of Al and Si crystalline grains, and their occurrences are due to the limited diffusion, aggregation, and crystallization of Si and Al atoms in the intermediate Al layers. The formation of the fractal pattern is helpful to enhance the bonding strength between the Pyrex 7740 glass and the aluminum thin film coated on the crystal silicon substrates.

Fractal dimensions for unsaturated porous media

The analytical expressions of the fractal dimensions for wetting and non-wetting phases for unsaturated porous media are derived and are found to be a function of porosity, maximum and minimum pore sizes as well as saturation. There is no empirical constant in the proposed fractal dimensions. It is also found that the fractal dimensions increase with porosity of a medium and are meaningful only in a certain range of saturation S-w, i.e. S-w > S-min for wetting phase and S-w < S-max for non-wetting phase at a given porosity, based on real porous media for requirements from both fractal theory and experimental observations. The present analysis of the fractal dimensions is verified to be consistent with the existing experimental observations and it makes possible to analyze the transport properties such as permeability, thermal dispersion in unsaturated porous media by fractal theory and technique.

Orientation preference and fractal character of short fatigue cracks in a weld metal

Short fatigue crack behaviour in a weld metal has been further investigated. The Schmid factor and the fractal dimension of short cracks on iso-stress specimens subjected to reversed bending have been determined and then applied to account for the distribution and orientation characteristics of short fatigue cracks. The result indicates that the orientation preference of short cracks is attributed to the large values of Schmid factor at relevant grains. The Schmid factors of most slip systems, which produced short cracks, are less than or equal to 0.4. Crack length measurements reveal that short crack path, compared to that of long crack, possesses a more stable and relatively larger value of fractal dimension. This is regarded as one of the typical features of short cracks.

Estudo do comportamento caótico e determinação de dimensão fractal em modelos pré-inflacionários não compactos

O caos determinístico é um dos aspectos mais interessantes no que diz respeito à teoria moderna dos sistemas dinâmicos, e está intrinsecamente associado a pequenas variações nas condições iniciais de um dado modelo. Neste trabalho, é feito um estudo acerca do comportamento caótico em dois casos específicos. Primeiramente, estudam-se modelos préinflacionários não-compactos de Friedmann-Robertson-Walker com campo escalar minimamente acoplado e, em seguida, modelos anisotrópicos de Bianchi IX. Em ambos os casos, o componente material é um fluido perfeito. Tais modelos possuem constante cosmológica e podem ser estudados através de uma descrição unificada, a partir de transformações de variáveis convenientes. Estes sistemas possuem estruturas similares no espaço de fases, denominadas centros-sela, que fazem com que as soluções estejam contidas em hipersuperfícies cuja topologia é cilíndrica. Estas estruturas dominam a relação entre colapso e escape para a inflação, que podem ser tratadas como bacias cuja fronteira pode ser fractal, e que podem ser associadas a uma estrutura denominada repulsor estranho. Utilizando o método de contagem de caixas, são calculadas as dimensões características das fronteiras nos modelos, o que envolve técnicas e algoritmos de computação numérica, e tal método permite estudar o escape caótico para a inflação.

Study on the Dissociation Behavior of Gas Hydrate in Porous Sediment Based on Fractal Theory

The dissociation process of gas hydrate was regarded as a gas-solid reaction without solid production layer when the temperature was above the zero centigrade. Based on the shrinking core model and the fractal theory, a fractional dimension dynamical model for gas hydrate dissociation in porous sediment was established. The new approach of evaluating the fractal dimension of the porous media was also presented. The fractional dimension dynamical model for gas hydrate dissociation was examined with the previous experimental data of methane hydrate and carbon dioxide hydrate dissociations, respectively. The calculated results indicate that the fractal dimensions of porous media acquired with this method agree well with the previous study. With the absolute average deviation (AAD) below 10%, the present model provided satisfactory predictions for the dissociation process of methane hydrate and carbon dioxide hydrate.

SAXS measurements of the interface in polyacrylate and epoxy interpenetrating networks with fractal geometry

The interface thickness in two-component interpenetrating polymer networks (IPN) system based on polyacrylate and epoxy were determined using small-angle X-ray scattering (SAXS) in terms of the theory proposed by Ruland. The thickness was found to be nonexistent for the samples at various compositions and synthesized at variable conditions-temperature and initiator concentration. By viewing the system as a two-phase system with a sharp boundary, the roughness of the interface was described by fractal dimension, D, which slightly varies with composition and synthesis condition. Length scales in which surface fractals are proved to be correct exist for each sample and range from 0.02 to 0.4 Angstrom(-1). The interface in the present IPN system was treated as fractal, which reasonably explained the differences between Pored's law and experimental data, and gained an insight into the interaction between different segments on the interface. (C) 1997 Elsevier Science Ltd.