Showing papers on "Fractal dimension published in 1982"
TL;DR: It is concluded that rain and cloud perimeters are fractals—they have no characteristic horizontal length scale between 1 and 1000 kilometers.
Abstract: Following Mandelbrot9s theory of fractals, the area-perimeter relation is used to investigate the geometry of satellite- and radar-determined cloud and rain areas between 1 and 1.2 x 106square kilometers. The data are well fit by a formula in which the perimeter is given approximately by the square root of the area raised to the power D [See equation in the PDF], where D is interpreted as the fractal dimension of the perimeter. It is concluded that rain and cloud perimeters are fractals—they have no characteristic horizontal length scale between 1 and 1000 kilometers.
653 citations
IBM1
TL;DR: In this article, transmission electron micrographs of thin evaporated gold films were analyzed by computer and the boundary of all clusters is a fractal of dimension $D=2$ while individual boundaries are of fractal dimension ${D}_{C}\ensuremath{\approx}1.9$.
Abstract: Transmission electron micrographs of thin evaporated gold films were analyzed by computer. For length scales above 10 nm, the irregular connected clusters show a perimeter linearly proportional to area. Near the percolation threshold the large-scale power-law correlations and area distributions are consistent with the scaling theory of second-order phase transitions. Geometrically, the boundary of all clusters is a fractal of dimension $D=2$ while individual boundaries are of fractal dimension ${D}_{C}\ensuremath{\approx}1.9$.
166 citations
TL;DR: In this paper, the statistical and coherence properties of the ray-density fluctuations in this regime are calculated as a function of fractal dimension D, and it is shown that in the Brownian case (D = 1.5) the problem can be solved exactly.
Abstract: Consideration of the density of rays emanating from an infinite, corrugated Gaussian surface with fractal slope reveals a hitherto unexplored shortwave scattering regime that is uncomplicated by the presence of caustics in the scattered wave field. The statistical and coherence properties of the ray-density fluctuations in this regime are calculated as a function of fractal dimension D, and it is shown that in the Brownian case (D = 1.5) the problem can be solved exactly. The properties of the intensity pattern in a coherent scattering configuration are also investigated. The contrast of the pattern is computed as a function of propagation distance, and the asymptotic behavior in the strong scattering limit is again found to be exactly solvable when D = 1.5. It is shown that the intensity fluctuations are K distributed in this case. The effects of a finite outer-scale size are evaluated and discussed.
111 citations
TL;DR: This work presents some straightforward algorithms for the generation and display in 3-D of fractal shapes, particularly adapted to shapes which are much more costly to fabricate.
Abstract: We present some straightforward algorithms for the generation and display in 3-D of fractal shapes. These techniques are very general and particularly adapted to shapes which are much more costly t...
80 citations
TL;DR: In this paper, the single-fold statistics of rays emanating from an infinite, corrugated Gaussian surface with fractal slope were investigated, and the low moments of the ray density fluctuation distribution were evaluated as a function of fractal dimension.
Abstract: The single-fold statistics of rays emanating from an infinite, corrugated Gaussian surface with fractal slope are investigated. Low moments of the ray-density fluctuation distribution are evaluated as a function of fractal dimension, D. It is shown that in the Brownian case, D=1.5, the distribution is exactly negative exponential, corresponding to K-distributed intensity fluctuations in a coherent scattering configuration.
63 citations
TL;DR: In this paper, the authors introduced the concept of differential fractal dimension (d.f.d) to characterize the complexity of the path of any randomly walking test particle. And they showed that, as far as the diffusion and the kinetic energy are concerned, they can treat inclusively both microscopic (or thermal) motions and macroscopic (or turbulent) motions by means of this dimension.
Abstract: In order to characterize the complexity of the path of any randomly walking test particle, we introduce the new conception, the differential fractal dimension (in short d.f.d.). This conception seems to have a special importance for the analysis of turbulence, because it clearly represents the complexity of the path observed by a scale r ( r is any given length), while the scales of the observation are particularly important for the theory of turbulence. It is shown that, as far as the diffusion and the kinetic energy are concerned, we can treat inclusively both microscopic (or thermal) motions and macroscopic (or turbulent) motions by means of the d.f.d. As an example, we obtain an analytical expression of the d.f.d. for an one-dimensional random walk with finite mean-free-path.
61 citations
TL;DR: Fractal properties of a random pattern formation produced by computer simulation have been analyzed in this article, where the controlling parameter was a tip priority factor $R$ by which a growing tip grows further compared to any site on a branch to produce a new side branch.
Abstract: Fractal properties of a random pattern formation produced by computer simulation have been analyzed. The controlling parameter was a tip priority factor $R$ by which a growing tip grows further compared to any site on a branch to produce a new side branch. The pattern was found to show an approximate self-similarity and associated with two fractal dimensions: One is the inner dimension which measures the fine structure and the other is the outer dimension which measures the framework of the pattern. With increasing tip priority factor, the inner dimension decreases and shows a phase-transition-like behavior at $R=35$. It shows damped oscillations approaching a constant value as the volume is increased. The outer dimension roughly remains 2. A possible mechanism for these behaviors is discussed.
60 citations
01 Jan 1982
TL;DR: An algorithm, which simulates walking a pair of dividers along a curve, used to calculate the fractal dimensions of curves is discussed and results demonstrate the algorithm to be stable and that a curve's fractal dimension can be closely approximated.
Abstract: : This paper discusses an algorithm, which simulates walking a pair of dividers along a curve, used to calculate the fractal dimensions of curves. It also discusses the choice of chord length and the number of solution steps used in computing fracticality. Results demonstrate the algorithm to be stable and that a curve's fractal dimension can be closely approximated. Potential applications for this technique include a new means for curvilinear data compression, description of planimetric feature boundary texture for improved realism in scene generation and possible two-dimensional extension for description of surface feature textures.
48 citations
01 Jan 1982
TL;DR: In this paper, the authors called such objects fractal measures and conjectured that the typical chaotic attractor has a fractal measure, and argued by example that any chaotic object has fractal features.
Abstract: Dimension is an important concept to dynamics because it indicates the number of independent variables inherent in a motion. Assignment of a relevant dimension in chaotic dynamics is nontrivial since, not only do chaotic motions frequently lie on complicated “fractal” surfaces, but in addition, the natural probability measures associated with chaotic motion often possess an intricate microscopic structure that persists down to arbitrarily small length scales. I call such objects fractal measures;1 and arguing by example, conjecture that the typical chaotic attractor has a fractal measure.
40 citations
TL;DR: In this article, the authors investigated the optical turbulence in the limit of Ikeda dispersive bistability with the gaussian noise superimposed on the incident driving field and computed the approximate fractal dimension at various levels of coarse-graining.
12 citations
01 Jan 1982
TL;DR: The Interactive Fractal Analysis System (IFAS) allows the user to measure fractal dimensions of curves and surfaces interactively through the use of virtual maps, character commands and responses, a graphic cursor, and an audible bell.
Abstract: : The Interactive Fractal Analysis System (IFAS) allows the user to measure fractal dimensions of curves and surfaces This is accomplished interactively through the use of virtual maps, character commands and responses, a graphic cursor, and an audible bell With either a curve or surface, the user selects the most appropriate fractal dimension by entering a sampling interval and examining the generated scatterplot, correlation coefficient, and table On a real-time basis, the user also has the capability of determining the fractal dimension for a portion of a curve or surface, editing features, windowing, and creating a perspective view of a surface Several examples demonstrate that IFAS is able to closely approximate the fracticality of curves and surfaces (Author)
01 Jan 1982
TL;DR: In this paper, the response of the chaotic motion under a periodic perturbation is studied using the Lorenz model and the global phase diagram is obtained by computer simulations, which is characterized not only by the subharmonic frequency but also by the symmetry of the orbit in phase space.
Abstract: The response of the chaotic motion under a periodic perturbation is studied using the Lorenz model. The global phase diagram is obtained by computer simulations. The periodic response is characterized not only by the subharmonic frequency but also by the symmetry of the orbit in phase space. On the other hand, the chaotic response is characterized by two chaos pa rameters: fractal dimensions of the strange attract or and of the symbolic time series. The periodic perturbation induces a partially coherent response which is closely connected with the temporal intermittency appearing in the time series of the dynamical variables. Such a coherent behavior is understood in terms of the spectral analysis, fractal dimensions and the information entropy.