fractal dimension

(redirected from Fractal dimensions)
Also found in: Financial.

fractal dimension

(mathematics)
A common type of fractal dimension is the Hausdorff-Besicovich Dimension, but there are several different ways of computing fractal dimension. Fractal dimension can be calculated by taking the limit of the quotient of the log change in object size and the log change in measurement scale, as the measurement scale approaches zero. The differences come in what is exactly meant by "object size" and what is meant by "measurement scale" and how to get an average number out of many different parts of a geometrical object. Fractal dimensions quantify the static *geometry* of an object.

For example, consider a straight line. Now blow up the line by a factor of two. The line is now twice as long as before. Log 2 / Log 2 = 1, corresponding to dimension 1. Consider a square. Now blow up the square by a factor of two. The square is now 4 times as large as before (i.e. 4 original squares can be placed on the original square). Log 4 / log 2 = 2, corresponding to dimension 2 for the square. Consider a snowflake curve formed by repeatedly replacing ___ with _/\_, where each of the 4 new lines is 1/3 the length of the old line. Blowing up the snowflake curve by a factor of 3 results in a snowflake curve 4 times as large (one of the old snowflake curves can be placed on each of the 4 segments _/\_). Log 4 / log 3 = 1.261... Since the dimension 1.261 is larger than the dimension 1 of the lines making up the curve, the snowflake curve is a fractal. [sci.fractals FAQ].
References in periodicals archive ?
The topics include a measure-theoretic result for approximation by Delone sets, self-similar tilings of fractal blow-ups, dimensions of limit sets of Kleinian groups, an overview of complex fractal dimensions: from fractal strings to fractal drums and back, and eigenvalues of the Laplacian on domains with fractal boundary.
The fractal dimensions of the two normal Holter registries between 1.700 and 1.970 occupying 59 and 131 squares in the Kg grids (10 beats/minute) and between 201 and 476 in the Kp grid (5 beats/minute) (see table 2).
trusoft.netmegs.com) for calculation of the fractal dimensions; the STATISTICA software (data analysis software system), version 10 (StatSoft, Inc., 2011).
This paper carried out different-scale gridding and calculated corresponding fractal dimensions to the [T.sup.3x] fault tectonic map (fig.3) of the Fault-Fold Belt of the central Yuanba area using the box covering method.
The instantaneous fractal dimensions of PD signals are given in Fig.5 where limited time segments of given time series signal are analyzed and this process is continuously updated for upcoming signal samples instantaneously.
The calculation of fractal dimensions by means of rheological method is generally applicable to various rheological parameters of a material, including strain at limit of linearity ([[gamma].sub.0]) [20, 32, 34, 38, 60, 68]; storage modulus (G') [8, 21, 32, 34, 84]; elasticity (E) [38, 60, 68]; and shear stress ([sigma]) [8].
From this fact, concepts have been developed for the analysis of fractal dimensions that make it possible to establish diagnostic differences.
where [D.sub.f] and [[??].sub.f] are the fractal dimensions of the damage domain and the damage residual domain, respectively.
A formalism for treating the fractal dimensions and its generalization has been developed and applied to the study of turbulent fluids and other transitions to chaos [41, 47].
In addition, the fractal dimensions of varying grading curves taking different scales into account were calculated to explore the fractal mechanism of particle crushing.
The fractal dimensions of these six samples are varied between 2.19 to 2.62 with correlation coefficient of 0.874-0.997.
In this study, to compare the results of Shannon's entropy and fractal analysis and to evaluate relationships, a graph showing the changes in entropy and fractal dimensions over the years was created (Fig.